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Data

For demonstration purposes, 20-item dataset GMAT from difNLR R package is used. On this page, you may select one of several toy datasets, mostly offered by ShinyItemAnalysis and difNLR packages or you may upload your own dataset (see below). To return to demonstration dataset, click on Unload data button.


Training datasets



Upload your own datasets

Here you can upload your own dataset. Select all necessary files and use Upload data button on bottom of this page.

Main data file should contain responses of individual respondents (rows) to given items (columns). Data need to be either binary, nominal (e.g. in ABCD format), or ordinal (e.g. in Likert scale). Header may contain item names, no row names should be included. In all data sets header should be either included or excluded. Columns of dataset are by default renamed to Item and number of particular column. If you want to keep your own names, check box Keep item names below. Missing values in scored dataset are by default evaluated as 0. If you want to keep them as missing, check box Keep missing values below.

Data specification
Missing values

For nominal data, it is necessary to upload key of correct answers.

For ordinal data, you are advised to include vector containing cut-score which is used for binarization of uploaded data, i.e., values greater or equal to provided cut-score are set to 1, otherwise to 0. You can either upload dataset of item-specific values, or you can provide one value for whole dataset.

Note: In case that cut-score is not provided, vector of maximal values is used.

For nominal data, it is necessary to upload key of correct answers.

For ordinal data, you are advised to include vector containing cut-score which is used for binarization of uploaded data, i.e., values greater or equal to provided cut-score are set to 1, otherwise to 0. You can either upload dataset of item-specific values, or you can provide one value for whole dataset.

Note: In case that cut-score is not provided, vector of maximal values is used.

For ordinal data, it is optional to upload minimal and maximal values of answers. You can either upload datasets of item-specific values, or you can provide one value for whole dataset.

Note: If no minimal or maximal values are provided, these values are set automatically based on observed values.

Group is binary vector, where 0 represents reference group and 1 represents focal group. Its length needs to be the same as number of individual respondents in the main dataset. If the group is not provided then it won't be possible to run DIF and DDF detection procedures in DIF/Fairness section. Missing values are not supported for group membership vector and such cases/rows of the data should be removed.

Criterion variable is either discrete or continuous vector (e.g. future study success or future GPA in case of admission tests) which should be predicted by the measurement. Its length needs to be the same as number of individual respondents in the main dataset. If the criterion variable is not provided then it wont be possible to run validity analysis in Predictive validity section on Validity page.

DIF matching variable is a vector of the same length as number of observations in your data. If not supplied, total score is automatically computed and utilized by default.








Basic summary

Main dataset


              

Scored test


              

Group


              

Criterion variable


              

DIF matching variable


              

Data exploration

Here you can explore uploaded dataset. Rendering of tables can take some time.


Main dataset


Key (correct answers)


Scored test


Group vector


Criterion variable vector


DIF matching variable vector



Analysis of total scores

Total score, also known as raw score or sum score, is a total number of correct answers.

Summary table

Table below summarizes basic characteristics of total scores including minimum and maximum, mean, median, standard deviation, skewness and kurtosis. The kurtosis here is estimated by sample kurtosis \(\frac{m_4}{s_4}\), where \(m_4\) is the fourth central moment and \(s^2\) is sample variance. The skewness is estimated by sample skewness \(\frac{m_3}{s^3}\), where \(m_3\) is the third central moment. The kurtosis for normally distributed scores is near the value of 3 and the skewness is near the value of 0.

Histogram of total score

For selected cut-score, blue part of histogram shows respondents with total score above the cut-score, grey column shows respondents with total score equal to the cut-score and red part of histogram shows respondents below the cut-score.

Download figure

Selected R code

library(difNLR)
library(ggplot2)
library(moments)

# loading data
data(GMAT)
data 
# total score calculation
score 
# summary of total score 
c(min(score), max(score), mean(score), median(score), sd(score), skewness(score), kurtosis(score))

# colors by cut-score
cut color df 
# histogram
ggplot(df, aes(score)) + 
  geom_histogram(binwidth = 1, fill = color, col = "black") + 
  xlab("Total score") + 
  ylab("Number of respondents") + 
  theme_app()

Standard scores

Total score also known as raw score is a total number of correct answers. It can be used to compare individual score to a norm group, e.g. if the mean is 12, then individual score can be compared to see if it is below or above this average.
Percentile indicates the value below which a percentage of observations falls, e.g. a individual score at the 80th percentile means that the individual score is the same or higher than the scores of 80% of all respondents.
Success rate is the percentage of success, e.g. if the maximum points of test is equal to 20 and individual score is 12 then success rate is 12/20 = 0.6, i.e. 60%.
Z-score or also standardized score is a linear transformation of total score with a mean of 0 and with variance of 1. If X is total score, M its mean and SD its standard deviation then Z-score = (X - M) / SD.
T-score is transformed Z-score with a mean of 50 and standard deviation of 10. If Z is Z-score then T-score = (Z * 10) + 50.

Table by score


Download table

Selected R code

library(difNLR) 

# loading data
data(GMAT) 
data 
# scores calculations
score tosc perc sura zsco tsco 

Reliability

We are typically interested in unobserved true score \(T\), but have available only the observed score \(X\) which is contaminated by some measurement error \(e\), such that \(X = T + e\) and error term is uncorrelated with the true score.

Equation

Reliability is defined as squared correlation of the true and observed score

$$\text{rel}(X) = \text{cor}(T, X)^2$$

Equivalently, reliability can be re-expressed as the ratio of the true score variance to total observed variance

$$\text{rel}(X) = \frac{\sigma^2_T}{\sigma^2_X}$$

Spearman-Brown formula

Equation

For test with \(I\) items total score is calculated as \(X = X_1 + ... + X_I\). Let \(\text{rel}(X)\) be the reliability of the test. For a test consisting of \(I^*\) items (equally precise, measuring the same construct), that is for test which is \(m = \frac{I^*}{I}\) times longer/shorter, the reliability would be

$$\text{rel}(X^*) = \frac{m\cdot \text{rel}(X)}{1 + (m - 1)\cdot\text{rel}(X)}.$$

Spearman-Brown formula can be used to determine reliability of a test with similar items but of different number of items. It can also be used to determine necessary number of items to achieve desired reliability.

In calculations below reliability of original data is by default set to value of Cronbach's \(\alpha\) of the dataset currentli in use. Number of items in original data is by default set to number of items of dataset currently in use.


Estimate of reliability with different number of items

Here you can calculate estimate of reliability of a test consisting of different number of items (equally precise, measuring the same construct).


Necessary number of items for required level of reliability

Here you can calculate necessary number of items (equally precise, measuring the same construct) to gain required level of reliability.


Selected R code

library(psychometrics)
library(ShinyItemAnalysis)

# loading data
data(HCI)
data 
# reliability of original data
rel.original # number of items in original data
items.original 

# number of items in new data
items.new # ratio of tests lengths
m # determining reliability
psychometric::SBrel(Nlength = m, rxx = rel.original)


# desired reliability
rel.new # determining test length
(m.new # number of required items
m.new*items.original


Split-half method

Split-half method uses correlation between two subscores for estimation of reliability. The underlying assumption is that the two halves of the test (or even all items on the test) are equally precise and measure the same underlying construct. Spearman-Brown formula is then used to correct the estimate for the number of items.

Equation

For test with \(I\) items total score is calculated as \(X = X_1 + ... + X_I\). Let \(X^*_1\) and \(X^*_2\) be total scores calculated from items only in the first and second subsets. Then estimate of reliability is given by Spearman-Brown formula (Spearman, 1910; Brown, 1910) with \(m = 2\).

$$\text{rel}(X) = \frac{m\cdot \text{cor}(X^*_1, X^*_2)}{1 + (m - 1)\cdot\text{cor}(X^*_1, X^*_2)} = \frac{2\cdot \text{cor}(X^*_1, X^*_2)}{1 + \text{cor}(X^*_1, X^*_2)}$$

Below you can choose from different split-half approaches. First-last method uses correlation between the first half of items and the second half of items. Even-odd includes even items into the first subset and odd items into the second one. Random method performs random split of items, thus the resulting estimate may be different for each call. Revelle's \(\beta\) is actually the worst split-half (Revelle, 1979). Estimate is here calculated as the lowest split-half reliability of by default 10,000 random splits. Finally, Average considers by default 10,000 split halves and averages the resulting estimates. Number of split halves can be changed below. In case of odd number of items, first subset contains one more item than second one.



Reliability estimate with confidence interval

Estimate of reliability for First-last , Even-odd , Random and Revelle's \(\beta\) is calculated using Spearman-Brown formula. Confidence interval is based on confidence interval of correlation using delta method. Estimate of reliability for Average method is mean value of sampled reliabilities and confidence interval is confidence interval of this mean.


Histogram of reliability estimates

Histogram is based on selected number of split halves estimates (10,000 by default). The current estimate is highlighted by red colour.

Download

Selected R code

library(psych)
library(ShinyItemAnalysis)

# loading data
data(HCI)

# First-last splitting
df1 df2 # total score calculation
ts1 ts2 # correlation
cor.x # apply Spearmann-Brown formula to estimate reliability
(rel.x 
# Even-odd splitting
df1 df2 # total score calculation
ts1 ts2 # correlation
cor.x # apply Spearmann-Brown formula to estimate reliability
(rel.x 
# Random splitting
samp df1 df2 # total score calculation
ts1 ts2 # correlation
cor.x # apply Spearmann-Brown formula to estimate reliability
(rel.x 
# Minimum of 10,000 split-halves (Revelle's beta)
split items1 items2 df1 df2 # total score calculation
ts1 ts2 # correlation
cor.x # apply Spearmann-Brown formula to estimate reliability
(rel.x 
# calculation of CI
z.r n z.low z.upp 
cor.low cor.upp 
rel.x rel.low rel.upp 

# Average 10,000 split-halves
split (rel.x 
# Average all split-halves
split (rel.x 
# calculation of CI
n rel.low rel.upp 

Cronbach's \(\alpha\)

Cronbach's \(\alpha\) is an estimate of internal consistency of a psychometric test. It is a function of the number of items in a test, the average covariance between item-pairs, and the variance of the total score (Cronbach, 1951).

Equation

For test with \(I\) items where \(X = X_1 + ... + X_I\) is a total score, \(\sigma^2_X\) its variance and \(\sigma^2_{X_i}\) variances of items, Cronbach's \(\alpha\) is given by following equation

$$\alpha = \frac{I}{I-1}\left(1 - \frac{\sum_{i = 1}^I \sigma^2_{X_i}}{\sigma^2_X}\right)$$

Estimate with confidence interval

Confidence interval is based on F distribution as proposed by Feldt et al. (1987).

Selected R code

library(psychometric)
library(ShinyItemAnalysis)

# loading data
data(HCI)
data 
# Cronbach's alpha with confidence interval
a psychometric::alpha.CI(a, N = nrow(data), k = ncol(data), level = 0.95)

Correlation structure

Correlation heat map

Correlation heat map displays selected type of correlations between items. The size and shade of circles indicate how much the items are correlated (larger and darker circle mean larger correlations). The color of circles indicates in which way the items are correlated - blue color mean possitive correlation and red color mean negative correlation. Correlation heat map can be reordered using hierarchical clustering method selected below. With number of clusters larger than 1, the rectangles representing clusters are drawn. The values of correlation heatmap may be displayed and also downloaded.

Pearson correlation coefficient describes linear correlation between two random variables \(X\) and \(Y\). It is given by formula

$$\rho = \frac{cov(X,Y)}{\sqrt{var(X)}\sqrt{var(Y)}}.$$

Sample Pearson corelation coefficient may be calculated as

$$ r = \frac{\sum_{i = 1}^{n}(x_{i} - \bar{x})(y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i} - \bar{x})^2}\sqrt{\sum_{i = 1}^{n}(y_{i} - \bar{y})^2}}$$

Pearson correlation coefficient has a value between -1 and +1. Sample correlation of -1 and +1 correspond to all data points lying exactly on a line (decreasing in case of negative linear correlation -1 and increasing for +1). If coefficient is equal to 0 it implies no linear correlation between the variables.

Polychoric/tetrachoric correlation between two ordinal/binary variables is calculated from their contingency table, under the assumption that the ordinal variables dissect continuous latent variables that are bivariate normal.

Spearman's rank correlation coefficient describes strength and direction of monotonic relationship between random variables \(X\) and \(Y\), i.e. dependence between the rankings of two variables. It is given by formula

$$\rho = \frac{cov(rg_{X},rg_{Y})}{\sqrt{var(rg_{X})}\sqrt{var(rg_{Y})}},$$

where \(rg_{X}\) and \(rg_{Y}\) are transformed random variables \(X\) and \(Y\) into ranks, i.e Spearman correlation coefficient is the Pearson correlation coefficient between the ranked variables.

Sample Spearman correlation is calculated by converting \(X\) and \(Y\) to ranks (average ranks are used in case of ties) and by applying Pearson correlation formula. If both \(X\) and \(Y\) have \(n\) unique ranks, i.e. there are no ties, then sample correlation coefficient is given by formula

$$ r = 1 - \frac{6\sum_{i = 1}^{n}d_i^{2}}{n(n-1)}$$

where \(d = rg_{X} - rg_{Y}\) is the difference between two ranks and \(n\) is size of \(X\) and \(Y\). Spearman rank correlation coefficient has value between -1 and 1, where 1 means perfect increasing relationship between variables and -1 means decreasing relationship between the two variables. In case of no repeated values, Spearman correlation of +1 or -1 means all data points lying exactly on some monotone line. If coefficient is equal to 0, it means, there is no tendency for \(Y\) to either increase or decrease with \(X\) increasing.

Clustering methods. Ward's method aims at finding compact clusters based on minimizing the within-cluster sum of squares. Ward's n. 2 method uses squared disimilarities. Single method connects clusters with the nearest neighbours, i.e. the distance between two clusters is calculated as the minimum of distances of observations in one cluster and observations in the other clusters. Complete linkage with farthest neighbours on the other hand uses maximum of distances. Average linkage method uses the distance based on weighted average of the individual distances. McQuitty method uses unweighted average. Median linkage calculates the distance as the median of distances between an observation in one cluster and observation in the other cluster. Centroid method uses distance between centroids of clusters.



Download figure Download matrix

Dendrogram

Download figure

Scree plot

A scree plot displays the eigenvalues associated with an component or a factor in descending order versus the number of the component or factor. Location of a bend (an elbow) suggests a suitable number of factors.

Download figure

Selected R code

library(corrplot) 
library(ggdendro)
library(difNLR) 
library(psych)

# loading data
data(GMAT) 
data 
# calculation of correlation
### Pearson
corP ### Spearman
corP ### Polychoric
corP corP$rho 

# correlation heat map 
corrplot(corP$rho) # correlation plot 
corrplot(corP$rho, order = "hclust", hclust.method = "ward.D", addrect = 3) # correlation plot with 3 clusters using Ward method

# dendrogram
hc ggdendrogram(hc) # dendrogram

library(difNLR) 
library(psych)

# loading data
data(GMAT) 
data 
# scree plot 
ev df 
ggplot(df, aes(x = comp, y = ev)) + 
  geom_point() + 
  geom_line() + 
  ylab("Eigen value") + 
  xlab("Component number") +
  theme_app()

Criterion validity

This section requires criterion variable (e.g. future study success or future GPA in case of admission tests) which should correlate with the measurement. Criterion variable can be uploaded in Data section.

Descriptive plots of criterion variable on total score

Total scores are plotted according to criterion variable. Boxplot or scatterplot is displayed depending on the type of criterion variable - whether it is discrete or continuous. Scatterplot is provided with red linear regression line.

Download figure

Correlation of criterion variable and total score

Test for association between total score and criterion variable is based on Spearman`s \(\rho\). This rank-based measure has been recommended if bivariate normal distribution is not guaranteed. The null hypothesis is that correlation is 0.

Selected R code

library(ShinyItemAnalysis) 
library(difNLR) 

# loading data
data(GMAT) 
data01 # total score calculation
score # criterion variable
criterion # number of respondents in each criterion level
size levels(size) size df 
# descriptive plots 
### boxplot, for discrete criterion
ggplot(df, aes(y = score, x = as.factor(criterion), fill = as.factor(criterion))) +
  geom_boxplot() +
  geom_jitter(shape = 16, position = position_jitter(0.2)) +
  scale_fill_brewer(palette = "Blues") +
  xlab("Criterion group") +
  ylab("Total score") +
  coord_flip() +
  theme_app()

### scatterplot, for continuous criterion
ggplot(df, aes(x = score, y = criterion)) + 
  geom_point() + 
  ylab("Criterion variable") + 
  xlab("Total score") + 
  geom_smooth(method = lm,
              se = FALSE,
              color = "red") + 
  theme_app()

# correlation 
cor.test(criterion, score, method = "spearman", exact = FALSE)

Criterion validity

This section requires criterion variable (e.g. future study success or future GPA in case of admission tests) which should correlate with the measurement. Criterion variable can be uploaded in Data section. Here you can explore how the criterion correlates with individual items.

In distractor analysis based on criterion variable, we are interested in how test takers select the correct answer and how the distractors (wrong answers) with respect to group based on criterion variable.

Distractor plot

With option Combinations all item selection patterns are plotted (e.g. AB, ACD, BC). With option Distractors answers are splitted into distractors (e.g. A, B, C, D).

Download figure

Correlation of criterion variable and scored item

Test for association between total score and criterion variable is based on Spearman`s \(\rho\). This rank-based measure has been recommended if bivariate normal distribution is not guaranteed. The null hypothesis is that correlation is 0.

Selected R code

library(ShinyItemAnalysis) 
library(difNLR) 

# loading data
data("GMAT", "GMATtest", "GMATkey") 
data data01 key criterion 
# distractor plot for item 1 and 3 groups 
plotDistractorAnalysis(data, key, num.groups = 3, item = 1, matching = criterion) 

# correlation for item 1 
cor.test(criterion, data01[, 1], method = "spearman", exact = F)

Traditional item analysis

Traditional item analysis uses proportions of correct answers or correlations to estimate item properties.

Item difficulty/discrimination plot

Displayed is difficulty (red) and discrimination (blue) for all items. Items are ordered by difficulty.
Difficulty of the item is by default estimated as its average scaled score, i.e. average item score divided by its range. Below you can change the estimate of difficulty to average score of the item. For binary items both estimates are equivalent and can be interpreted as percent of respondents who answered item correctly.
Discrimination is by default described by difference of percent correct in upper and lower third of respondents (Upper-Lower Index, ULI). By rule of thumb it should not be lower than 0.2 (borderline in the plot), except for very easy or very difficult items. Discrimination can be customized (see also Martinkova, Stepanek et al., 2017) by changing number of groups and by changing which groups should be compared:


Download figure

Cronbach's alpha

Cronbach's alpha is an estimate of the reliability of a psychometric test. It is a function of the number of items in a test, the average covariance between item-pairs, and the variance of the total score (Cronbach, 1951).

Traditional item analysis table


Download table

Selected R code

library(difNLR) 
library(psych)
library(ShinyItemAnalysis) 

# loading data
data(GMAT) 
data 
# difficulty and discrimination plot 
DDplot(data, discrim = 'ULI', k = 3, l = 1, u = 3) 

# Cronbach alpha 
psych::alpha(data) 

# traditional item analysis table 
ItemAnalysis(data)

Distractor analysis

In distractor analysis, we are interested in how test takers select the correct answer and how the distractors (wrong answers) were able to function effectively by drawing the test takers away from the correct answer.

Distractors plot

With option Combinations all item selection patterns are plotted (e.g. AB, ACD, BC). With option Distractors answers are splitted into distractors (e.g. A, B, C, D).


Download figure

Table with counts

Table with proportions


Barplot of item response patterns

Download figure

Histogram of total scores

Download figure

Table of total scores by groups



Selected R code

library(difNLR)
library(ShinyItemAnalysis) 

# loading data
data(GMATtest) 
data data(GMATkey) 
key 
# combinations - plot for item 1 and 3 groups 
plotDistractorAnalysis(data, key, num.group = 3, item = 1, multiple.answers = TRUE) 

# distractors - plot for item 1 and 3 groups 
plotDistractorAnalysis(data, key, num.group = 3, item = 1, multiple.answers = FALSE) 

# table with counts and margins - item 1 and 3 groups 
DA dcast(as.data.frame(DA), response ~ score.level, sum, margins = TRUE, value.var = "Freq") 

# table with proportions - item 1 and 3 groups 
DistractorAnalysis(data, key, num.groups = 3, p.table = TRUE)[[1]]

Logistic regression on total scores

Various regression models may be fitted to describe item properties in more detail. Logistic regression can model dependency of probability of correct answer on total score by S-shaped logistic curve. Parameter \(b_{0}\) describes horizontal position of the fitted curve, parameter \(b_{1}\) describes its slope.


Plot with estimated logistic curve

Points represent proportion of correct answer with respect to total score. Their size is determined by count of respondents who achieved given level of total score.

Download figure

Equation

$$\mathrm{P}(Y = 1|X, b_0, b_1) = \mathrm{E}(Y|X, b_0, b_1) = \frac{e^{\left( b_{0} + b_1 X\right)}}{1+e^{\left( b_{0} + b_1 X\right) }} $$

Table of parameters


Selected R code

library(difNLR) 
library(ggplot2)

# loading data
data(GMAT) 
data score 
# logistic model for item 1 
fit 
# coefficients 
coef(fit) 

# function for plot 
fun 
# empirical probabilities calculation
df                  y = tapply(data[, 1], score, mean),
                 size = as.numeric(table(score)))

# plot of estimated curve
ggplot(df, aes(x = x, y = y)) +
  geom_point(aes(size = size),
             color = "darkblue",
             fill = "darkblue",
             shape = 21, alpha = 0.5) +
  stat_function(fun = fun, geom = "line",
                args = list(b0 = coef(fit)[1],
                            b1 = coef(fit)[2]),
                size = 1,
                color = "darkblue") +
  xlab("Total score") +
  ylab("Probability of correct answer") +
  ylim(0, 1) +
  ggtitle("Item 1") + 
  theme_app()

Logistic regression on standardized total scores

Various regression models may be fitted to describe item properties in more detail. Logistic regression can model dependency of probability of correct answer on standardized total score (Z-score) by S-shaped logistic curve. Parameter \(b_{0}\) describes horizontal position of the fitted curve (difficulty), parameter \(b_{1}\) describes its slope at inflection point (discrimination).


Plot with estimated logistic curve

Points represent proportion of correct answer with respect to standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score.

Download figure

Equation

$$\mathrm{P}(Y = 1|Z, b_0, b_1) = \mathrm{E}(Y|Z, b_0, b_1) = \frac{e^{\left( b_{0} + b_1 Z\right) }}{1+e^{\left( b_{0} + b_1 Z\right) }} $$

Table of parameters


Selected R code

library(difNLR) 
library(ggplot2)

# loading data
data(GMAT) 
data zscore 
# logistic model for item 1 
fit 
# coefficients 
coef(fit) 

# function for plot 
fun 
# empirical probabilities calculation
df                  y = tapply(data[, 1], zscore, mean),
                 size = as.numeric(table(zscore)))

# plot of estimated curve
ggplot(df, aes(x = x, y = y)) +
  geom_point(aes(size = size),
             color = "darkblue",
             fill = "darkblue",
             shape = 21, alpha = 0.5) +
  stat_function(fun = fun, geom = "line",
                args = list(b0 = coef(fit)[1],
                            b1 = coef(fit)[2]),
                size = 1,
                color = "darkblue") +
  xlab("Standardized total score") +
  ylab("Probability of correct answer") +
  ylim(0, 1) +
  ggtitle("Item 1") + 
  theme_app()

Logistic regression on standardized total scores with IRT parameterization

Various regression models may be fitted to describe item properties in more detail. Logistic regression can model dependency of probability of correct answer on standardized total score (Z-score) by s-shaped logistic curve. Note change in parametrization - the IRT parametrization used here corresponds to the parametrization used in IRT models. Parameter \(b\) describes horizontal position of the fitted curve (difficulty), parameter \(a\) describes its slope at inflection point (discrimination).


Plot with estimated logistic curve

Points represent proportion of correct answer with respect to standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score.

Download figure

Equation

$$\mathrm{P}(Y = 1|Z, a, b) = \mathrm{E}(Y|Z, a, b) = \frac{e^{ a\left(Z - b\right) }}{1+e^{a\left(Z - b\right)}} $$

Table of parameters


Selected R code

library(difNLR) 
library(ggplot2)

# loading data
data(GMAT) 
data zscore 
# logistic model for item 1 
fit 
# coefficients
coef coef  

# function for plot 
fun 
# empirical probabilities calculation
df                  y = tapply(data[, 1], zscore, mean),
                 size = as.numeric(table(zscore)))

# plot of estimated curve
ggplot(df, aes(x = x, y = y)) +
  geom_point(aes(size = size),
             color = "darkblue",
             fill = "darkblue",
             shape = 21, alpha = 0.5) +
  stat_function(fun = fun, geom = "line",
                args = list(a = coef[1],
                            b = coef[2]),
                size = 1,
                color = "darkblue") +
  xlab("Standardized total score") +
  ylab("Probability of correct answer") +
  ylim(0, 1) +
  ggtitle("Item 1") + 
  theme_app()

Nonlinear three parameter regression on standardized total scores with IRT parameterization

Various regression models may be fitted to describe item properties in more detail. Nonlinear regression can model dependency of probability of correct answer on standardized total score (Z-score) by s-shaped logistic curve. The IRT parametrization used here corresponds to the parametrization used in IRT models. Parameter \(b\) describes horizontal position of the fitted curve (difficulty), parameter \(a\) describes its slope at inflection point (discrimination). This model allows for nonzero lower left asymptote \(c\) (pseudo-guessing parameter).


Plot with estimated nonlinear curve

Points represent proportion of correct answer with respect to standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score.

Download figure

Equation

$$\mathrm{P}(Y = 1|Z, a, b, c) = \mathrm{E}(Y|Z, a, b, c) = c + \left( 1-c \right) \cdot \frac{e^{a\left(Z-b\right) }}{1+e^{a\left(Z-b\right) }} $$

Table of parameters


Selected R code

library(difNLR) 
library(ggplot2)

# loading data
data(GMAT) 
data zscore 
# NLR 3P model for item 1 
fun 
fit            algorithm = "port", 
           start = startNLR(data, GMAT[, "group"], model = "3PLcg", parameterization = "classic")[[1]][1:3],
           lower = c(-Inf, -Inf, 0,),
           upper = c(Inf, Inf, 1)) 
# coefficients 
coef(fit) 

# empirical probabilities calculation
df                  y = tapply(data[, 1], zscore, mean),
                 size = as.numeric(table(zscore)))

# plot of estimated curve
ggplot(df, aes(x = x, y = y)) +
  geom_point(aes(size = size),
             color = "darkblue",
             fill = "darkblue",
             shape = 21, alpha = 0.5) +
  stat_function(fun = fun, geom = "line",
                args = list(a = coef(fit)[1],
                            b = coef(fit)[2],
                            c = coef(fit)[3]),
                size = 1,
                color = "darkblue") +
  xlab("Standardized total score") +
  ylab("Probability of correct answer") +
  ylim(0, 1) +
  ggtitle("Item 1") + 
  theme_app()

Nonlinear four parameter regression on standardized total scores with IRT parameterization

Various regression models may be fitted to describe item properties in more detail. Nonlinear four parameter regression can model dependency of probability of correct answer on standardized total score (Z-score) by s-shaped logistic curve. The IRT parametrization used here corresponds to the parametrization used in IRT models. Parameter \(b\) describes horizontal position of the fitted curve (difficulty), parameter \(a\) describes its slope at inflection point (discrimination), pseudo-guessing parameter \(c\) is describes lower asymptote and inattention parameter \(d\) describes upper asymptote.


Plot with estimated nonlinear curve

Points represent proportion of correct answer with respect to standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score.

Download figure

Equation

$$\mathrm{P}(Y = 1|Z, a, b, c,d) = \mathrm{E}(Y|Z, a, b, c, d) = c + \left( d-c \right) \cdot \frac{e^{a\left(Z-b\right) }}{1+e^{a\left(Z-b\right) }} $$

Table of parameters


Selected R code

library(difNLR) 
library(ggplot2)

# loading data
data(GMAT) 
data zscore 
# NLR 4P model for item 1 
fun 
fit            algorithm = "port", 
           start = startNLR(data, GMAT[, "group"], model = "4PLcgdg", parameterization = "classic")[[1]][1:4],
           lower = c(-Inf, -Inf, 0, 0),
           upper = c(Inf, Inf, 1, 1)) 
# coefficients 
coef(fit) 

# empirical probabilities calculation
df                  y = tapply(data[, 1], zscore, mean),
                 size = as.numeric(table(zscore)))

# plot of estimated curve
ggplot(df, aes(x = x, y = y)) +
  geom_point(aes(size = size),
             color = "darkblue",
             fill = "darkblue",
             shape = 21, alpha = 0.5) +
  stat_function(fun = fun, geom = "line",
                args = list(a = coef(fit)[1],
                            b = coef(fit)[2],
                            c = coef(fit)[3],
                            d = coef(fit)[4]),
                size = 1,
                color = "darkblue") +
  xlab("Standardized total score") +
  ylab("Probability of correct answer") +
  ylim(0, 1) +
  ggtitle("Item 1") + 
  theme_app()

Logistic regression model selection

Here you can compare classic 2PL logistic regression model to non-linear model item by item using some information criteria:

  • AIC is the Akaike information criterion (Akaike, 1974),
  • BIC is the Bayesian information criterion (Schwarz, 1978)

Another approach to nested models can be likelihood ratio chi-squared test. Significance level is set to 0.05. As tests are performed item by item, it is possible to use multiple comparison correction method.

Table of comparison statistics

Rows BEST indicate which model has the lowest value of criterion, or is the largest significant model by likelihood ratio test.


Selected R code

library(difNLR) 

# loading data
data(GMAT) 
Data zscore 
# function for fitting models
fun 
# starting values for item 1
start 
# 2PL model for item 1 
fit2PL               algorithm = "port", 
              start = start[1:2]) 
# NLR 3P model for item 1 
fit3PL               algorithm = "port", 
              start = start[1:3],
              lower = c(-Inf, -Inf, 0), 
              upper = c(Inf, Inf, 1)) 
# NLR 4P model for item 1 
fit3PL               algorithm = "port", 
              start = start,
              lower = c(-Inf, -Inf, 0, 0), 
              upper = c(Inf, Inf, 1, 1)) 

# comparison 
### AIC
AIC(fit2PL); AIC(fit3PL); AIC(fit4PL) 
### BIC
BIC(fit2PL); BIC(fit3PL); BIC(fit4PL) 
### LR test, using Benjamini-Hochberg correction
###### 2PL vs NLR 3P
LRstat LRdf LRpval LRpval ###### NLR 3P vs NLR 4P
LRstat LRdf LRpval LRpval 

Cumulative logit regression

Various regression models may be fitted to describe item properties in more detail. Cumulative logit regression can model cumulative probabilities, i.e., probabilities to obtain item score higher than or equal to 1, 2, 3, etc.

Cumulative logit model can be fitted on selected matching criterion - total scores or standardized scores, using classical (slope/intercept) or IRT parametrization.


Plot of cumulative probabilities

Lines determine the cumulative probabilities \(P(Y \geq k)\). Circles represent proportion of answers with at least \(k\) points with respect to the matching criterion, i.e., the empirical cumulative probabilities. The size of the points is determined by the count of respondents who achieved given level of the matching criterion.

Download figure

Plot of category probabilities

Lines determine the category probabilities \(P(Y = k)\). Circles represent proportion of answers with \(k\) points with respect to the matching criterion, i.e., the empirical category probabilities. The size of the points is determined by the count of respondents who achieved given level of the matching criterion.

Download figure

Equation

Table of parameters

Selected R code

library(ShinyItemAnalysis)
library(VGAM)

# loading data
data 
# total score calculation
score key maxval data[, 1] 
# cummulative logit model for item 1
fit.cum # coefficients for item 1
coefs 
# plotting cumulative probabilities
plotCumulative(fit.cum, type = "cumulative", matching.name = "Total score")
# plotting category probabilities
plotCumulative(fit.cum, type = "category", matching.name = "Total score")


Adjacent category logit regression

Models for ordinal responses need not use cumulative probabilities. Adjacent categories model assumes linear form of logarithm of ratio of probabilities of two successive scores (e.g. 1 vs. 2, 2 vs. 3, etc.), i.e., of the adjacent category logits.

Adjacent category logit model can be fitted on selected matching criterion - total scores or standardized scores, using classical (slope/intercept) or IRT parametrization.


Plot with category probabilities

Lines determine the category probabilities \(P(Y = k)\). Circles represent the proportion of answers with k points with respect to the total score, i. e., the empirical category probabilities. The size of the circles is determined by the count of respondents who achieved given level of the total score.

Download figure

Equation

Table of parameters

Selected R code

library(ShinyItemAnalysis)
library(VGAM)

# loading data
data 
# total score calculation
score key maxval data[, 1] 
# adjacent category logit model for item 1
fit.adj # coefficients for item 1
coefs 
# plotting category probabilities
plotAdjacent(fit.adj, matching.name = "Total score")


Multinomial regression on standardized total scores

Various regression models may be fitted to describe item properties in more detail. Multinomial regression allows for simultaneous modelling of probability of choosing given distractors on standardized total score (Z-score).


Plot with estimated curves of multinomial regression

Points represent proportion of selected option with respect to standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score and who selected given option.

Download figure

Equation

Table of parameters

Interpretation:

Selected R code

library(difNLR) 
library(nnet) 
library(ShinyItemAnalysis)

# loading data
data(GMAT, GMATtest, GMATkey) 
zscore data key 
# multinomial model for item 1 
fit 
# coefficients 
coef(fit)

# plot for item 1
plotMultinomial(fit, zscore, matching.name = "Z-score")

Rasch model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

In Rasch model (Rasch, 1960), all items are assumed to have the same slope in inflection point, i.e., the same discrimination parameter \(a\) which is fixed to value of 1. Items may differ in location of their inflection point, i.e. they may differ in difficulty parameter \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow normal distribution with freely estimated variance.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, b_{j} \right) = \frac{e^{\left(\theta_{i}-b_{j}\right) }}{1+e^{\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Test information function

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.


Download table

Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.


Download abilities

Scatter plot of factor scores and standardized total scores

Download figure

Wright map

Wright map (Wilson, 2005; Wright & Stone, 1979), also called item-person map, is a graphical tool to display person ability estimates and item parameters. The person side (left) represents histogram of estimated abilities of respondents. The item side (right) displays estimates of difficulty parameters of individual items.

Download figure

Selected R code

library(difNLR)
library(mirt) 
library(ShinyItemAnalysis)

# loading data
data(GMAT) 
data 
# fitting Rasch model
fit 
# Item Characteristic Curves 
plot(fit, type = 'trace', facet_items = F) 
# Item Information Curves 
plot(fit, type = 'infotrace', facet_items = F) 
# Test Information Function 
plot(fit, type = 'infoSE') 

# Coefficients 
coef(fit, simplify = TRUE) 
coef(fit, IRTpars = TRUE, simplify = TRUE) 

# Item fit statistics 
itemfit(fit) 

# Factor scores vs Standardized total scores 
fs sts plot(fs ~ sts) 

# Wright Map 
b ggWrightMap(fs, b)

Rasch model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

In Rasch model (Rasch, 1960), all items are assumed to have the same slope in inflection point, i.e., the same discrimination parameter \(a\) which is fixed to value of 1. Items may differ in location of their inflection point, i.e. they may differ in difficulty parameter \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow normal distribution with freely estimated variance.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, b_{j} \right) = \frac{e^{\left(\theta_{i}-b_{j}\right) }}{1+e^{\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando & Thissen, 2000). SX2 is computed only when no missing data are present. In such a case consider using imputed dataset!

One parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

In 1PL IRT model, all items are assumed to have the same slope in inflection point, i.e., the same discrimination \(a\). Its value corresponds to standard deviation of ability estimates in Rasch model. Items can differ in location of their inflection point, i.e., in item difficulty parameters \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a, b_{j} \right) = \frac{e^{a\left(\theta_{i}-b_{j}\right) }}{1+e^{a\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Test information function

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.


Download table

Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.


Download abilities

Scatter plot of factor scores and standardized total scores

Download figure

Wright map

Wright map (Wilson, 2005; Wright & Stone, 1979), also called item-person map, is a graphical tool to display person ability estimates and item parameters. The person side (left) represents histogram of estimated abilities of respondents. The item side (right) displays estimates of difficulty parameters of individual items.

Download figure

Selected R code

library(difNLR)
library(mirt) 
library(ShinyItemAnalysis)

# loading data
data(GMAT) 
data 
# fitting 1PL model
fit 
# Item Characteristic Curves 
plot(fit, type = 'trace', facet_items = F) 
# Item Information Curves 
plot(fit, type = 'infotrace', facet_items = F) 
# Test Information Function 
plot(fit, type = 'infoSE') 

# Coefficients 
coef(fit, simplify = TRUE) 
coef(fit, IRTpars = TRUE, simplify = TRUE) 

# Item fit statistics 
itemfit(fit) 

# Factor scores vs Standardized total scores 
fs sts plot(fs ~ sts) 

# Wright Map 
b ggWrightMap(fs, b)



# You can also use ltm library for IRT models 
#  fitting 1PL model
fit # for Rasch model use 
# fit 
# Item Characteristic Curves 
plot(fit) 
# Item Information Curves 
plot(fit, type = 'IIC') 
# Test Information Function 
plot(fit, items = 0, type = 'IIC') 

# Coefficients 
coef(fit) 

# Factor scores vs Standardized total scores 
df1 FS df2 df2$Obs STS df plot(FS ~ STS, data = df, xlab = 'Standardized total score', ylab = 'Factor score')

1PL model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

In 1PL IRT model, all items are assumed to have the same slope in inflection point, i.e., the same discrimination \(a\). Its value corresponds to standard deviation of ability estimates in Rasch model. Items can differ in location of their inflection point, i.e., in item difficulty parameters \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a, b_{j} \right) = \frac{e^{a\left(\theta_{i}-b_{j}\right) }}{1+e^{a\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.

Two parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

2PL IRT model allows for different slopes in inflection point, i.e., different discrimination parameters \(a\). Items can also differ in location of their inflection point, i.e., in item difficulty parameters \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}\right) = \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Test information function

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.


Download table

Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.


Download abilities

Scatter plot of factor scores and standardized total scores

Download figure

Selected R code

library(difNLR)
library(mirt)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- mirt(data, model = 1, itemtype = "2PL", SE = T)
# Item Characteristic Curves
plot(fit, type = "trace", facet_items = F)
# Item Information Curves
plot(fit, type = "infotrace", facet_items = F)
# Test Information Function
plot(fit, type = "infoSE")
# Coefficients
coef(fit, simplify = TRUE)
coef(fit, IRTpars = TRUE, simplify = TRUE)
# Item fit statistics
itemfit(fit)
# Factor scores vs Standardized total scores
fs <- as.vector(fscores(fit))
sts <- as.vector(scale(apply(data, 1, sum)))
plot(fs ~ sts)


# You can also use ltm library for IRT models
library(difNLR)
library(ltm)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- ltm(data ~ z1, IRT.param = TRUE)
# Item Characteristic Curves
plot(fit)
# Item Information Curves
plot(fit, type = "IIC")
# Test Information Function
plot(fit, items = 0, type = "IIC")
# Coefficients
coef(fit)
# Factor scores vs Standardized total scores
df1 <- ltm::factor.scores(fit, return.MIvalues = T)$score.dat
FS <- as.vector(df1[, "z1"])
df2 <- df1
df2$Obs <- df2$Exp <- df2$z1 <- df2$se.z1 <- NULL
STS <- as.vector(scale(apply(df2, 1, sum)))
df <- data.frame(FS, STS)
plot(FS ~ STS, data = df, xlab = "Standardized total score", ylab = "Factor score")

Two parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

2PL IRT model allows for different slopes in inflection point, i.e., different discrimination parameters \(a\). Items can also differ in location of their inflection point, i.e., in item difficulty parameters \(b\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}\right) = \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.

Three parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

3PL IRT model allows for different discriminations of items \(a\), different item difficulties \(b\) and allows also for nonzero left asymptote, pseudo-guessing \(c\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}, c_{j} \right) = c_{j} + \left(1 - c_{j}\right) \cdot \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Test information function

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.


Download table

Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.


Download abilities

Scatter plot of factor scores and standardized total scores

Download figure

Selected R code

library(difNLR)
library(mirt)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- mirt(data, model = 1, itemtype = "3PL", SE = T)
# Item Characteristic Curves
plot(fit, type = "trace", facet_items = F)
# Item Information Curves
plot(fit, type = "infotrace", facet_items = F)
# Test Information Function
plot(fit, type = "infoSE")
# Coefficients
coef(fit, simplify = TRUE)
coef(fit, IRTpars = TRUE, simplify = TRUE)
# Item fit statistics
itemfit(fit)
# Factor scores vs Standardized total scores
fs <- as.vector(fscores(fit))
sts <- as.vector(scale(apply(data, 1, sum)))
plot(fs ~ sts)
# You can also use ltm library for IRT models


library(difNLR)
library(ltm)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- tpm(data, IRT.param = TRUE)
# Item Characteristic Curves
plot(fit)
# Item Information Curves
plot(fit, type = "IIC")
# Test Information Function
plot(fit, items = 0, type = "IIC")
# Coefficients
coef(fit)
# Factor scores vs Standardized total scores
df1 <- ltm::factor.scores(fit, return.MIvalues = T)$score.dat
FS <- as.vector(df1[, "z1"])
df2 <- df1
df2$Obs <- df2$Exp <- df2$z1 <- df2$se.z1 <- NULL
STS <- as.vector(scale(apply(df2, 1, sum)))
df <- data.frame(FS, STS)
plot(FS ~ STS, data = df, xlab = "Standardized total score", ylab = "Factor score")

Three parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

3PL IRT model allows for different discriminations of items \(a\), different item difficulties \(b\) and allows also for nonzero left asymptote, pseudo-guessing \(c\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}, c_{j} \right) = c_{j} + \left(1 - c_{j}\right) \cdot \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.

Four parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

4PL IRT model allows for different discriminations of items \(a\), different item difficulties \(b\), nonzero left asymptote, pseudo-guessing \(c\) and also for upper asymptote lower than one, i.e, inattention parameter \(d\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}, c_{j}, d_{j} \right) = c_{j} + \left(d_{j} - c_{j}\right) \cdot \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

Download figure

Item information curves

Download figure

Test information function

Download figure

Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.


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Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.


Download abilities

Scatter plot of factor scores and standardized total scores

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Selected R code

library(difNLR)
library(mirt)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- mirt(data, model = 1, itemtype = "4PL", SE = T)
# Item Characteristic Curves
plot(fit, type = "trace", facet_items = F)
# Item Information Curves
plot(fit, type = "infotrace", facet_items = F)
# Test Information Function
plot(fit, type = "infoSE")
# Coefficients
coef(fit, simplify = TRUE)
coef(fit, IRTpars = TRUE, simplify = TRUE)
# Item fit statistics
itemfit(fit)
# Factor scores vs Standardized total scores
fs <- as.vector(fscores(fit))
sts <- as.vector(scale(apply(data, 1, sum)))
plot(fs ~ sts)

Four parameter Item Response Theory model

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters.

4PL IRT model allows for different discriminations of items \(a\), different item difficulties \(b\), nonzero left asymptote, pseudo-guessing \(c\) and also for upper asymptote lower than one, i.e, inattention parameter \(d\). Model parameters are estimated using marginal maximum likelihood (MML) method. Ability \(\theta\) is assumed to follow standard normal distribution.

Equation

$$\mathrm{P}\left(Y_{ij} = 1\vert \theta_{i}, a_{j}, b_{j}, c_{j}, d_{j} \right) = c_{j} + \left(d_{j} - c_{j}\right) \cdot \frac{e^{a_{j}\left(\theta_{i}-b_{j}\right) }}{1+e^{a_{j}\left(\theta_{i}-b_{j}\right) }} $$

Item characteristic curves

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Item information curves

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Table of estimated parameters

Estimates of parameters are completed by SX2 item fit statistics (Orlando and Thissen, 2000). SX2 statistics are computed only when no missing data are present.

Item Response Theory model selection

Item Response Theory (IRT) models are mixed-effect regression models in which respondent ability \(\theta\) is assumed to be latent and is estimated together with item paramters. Model parameters are estimated using marginal maximum likelihood (MML) method, in 1PL, 2PL, 3PL and 4PL IRT models, ability \(\theta\) is assumed to follow standard normal distribution.

IRT models can be compared by several information criteria:

  • AIC is the Akaike information criterion (Akaike, 1974),
  • AICc is AIC with a correction for finite sample size,
  • BIC is the Bayesian information criterion (Schwarz, 1978).
  • SABIC is the Sample-sized adjusted BIC criterion,

Another approach to compare IRT models can be likelihood ratio chi-squared test. Significance level is set to 0.05.

Table of comparison statistics

Row BEST indicates which model has the lowest value of criterion, or is the largest significant model by likelihood ratio test.


Selected R code

library(difNLR) 
library(mirt)

# loading data
data(GMAT) 
data 
# 1PL IRT model 
s            "CONSTRAIN = (1-", ncol(data), ", a1)")
model fit1PL # 2PL IRT model 
fit2PL # 3PL IRT model 
fit3PL # 4PL IRT model 
fit4PL 
# comparison 
anova(fit1PL, fit2PL) 
anova(fit2PL, fit3PL) 
anova(fit3PL, fit4PL)

Bock's nominal Item Response Theory model

The nominal response model (NRM) was introduced by Bock (1972) as a way to model responses to items with two or more nominal categories. This model is suitable for multiple-choice items with no particular ordering of distractors. It is also generalization of some models for ordinal data, e.g. generalized partial credit model (GPCM) or its restricted versions partial credit model (PCM) and rating scale model (RSM).

Equation

For \(K\) possible test choices is the probability of the choice \(k\) for person \(i\) with latent trait \(\theta\) in item \(j\) given by the following equation: $$\mathrm{P}(Y_{ij} = k|\theta_i, a_{j1}, al_{j(l-1)}, d_{j(l-1)}, l = 1, \dots, K) = \frac{e^{(ak_{j(k-1)} * a_{j1} * \theta_i + d_{j(k-1)})}}{\sum_l e^{(al_{j(l-1)} * a_{j1} * \theta_i + d_{j(l-1)})}}$$

Item characteristic curves

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Item information curves

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Test information function

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Table of parameters

Ability estimates

This table shows the response score of only six respondents. If you want to see scores for all respondents, click on Download abilities button.

Download abilities

Scatter plot of factor scores and standardized total scores

Download figure

Selected R code

library(difNLR) 
library(mirt)
library(ShinyItemAnalysis)

# loading data
data("dataMedicalgraded") 
data 
# model 
fit 
# item characteristic curves 
plot(fit, type = "trace", facet_items = F) 
# item information curves 
plot(fit, type = "infotrace", facet_items = F) 
# test information function 
plot(fit, type = "infoSE") 

# coefficients 
coef(fit, simplify = TRUE) 
coef(fit, IRTpars = TRUE, simplify = TRUE) 

# factor scores vs standardized total scores 
fs sts plot(fs ~ sts)


Bock's nominal Item Response Theory model

The nominal response model (NRM) was introduced by Bock (1972) as a way to model responses to items with two or more nominal categories. This model is suitable for multiple-choice items with no particular ordering of distractors. It is also generalization of some models for ordinal data, e.g. generalized partial credit model (GPCM) or its restricted versions partial credit model (PCM) and rating scale model (RSM).

Equation

For \(K\) possible test choices is the probability of the choice \(k\) for person \(i\) with latent trait \(\theta\) in item \(j\) given by the following equation: $$\mathrm{P}(Y_{ij} = k|\theta_i, a_{j1}, al_{j(l-1)}, d_{j(l-1)}, l = 1, \dots, K) = \frac{e^{(ak_{j(k-1)} * a_{j1} * \theta_i + d_{j(k-1)})}}{\sum_l e^{(al_{j(l-1)} * a_{j1} * \theta_i + d_{j(l-1)})}}$$

Item characteristic curves

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Item information curves

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Table of parameters



Dichotomous models

Dichotomous models are used for modelling items producing a simple binary response (i.e., true/false). Most complex unidimensional dichotomous IRT model described here is 4PL IRT model. Rasch model (Rasch, 1960) assumes discrimination fixed to \(a = 1\) guessing fixed to \(c = 0\) and innatention to \(d = 1\). Similarly, other restricted models (1PL, 2PL and 3PL models) can be obtained by fixing appropriate parameters in 4PL model.

In this section, you can explore behavior of two item characteristic curves \(\mathrm{P}\left(\theta\right)\) and their item information functions \(\mathrm{I}\left(\theta\right)\) in 4PL IRT model.

Parameters

Select parameters \(a\) (discrimination), \(b\) (difficulty), \(c\) (guessing) and \(d\) (inattention). By constraining \(a = 1\), \(c = 0\), \(d = 1\) you get Rasch model. With option \(c = 0\) and \(d = 1\) you get 2PL model and with option \(d = 1\) 3PL model.

When different curve parameters describe properties of the same item but for different groups of respondents, this phenomenon is called Differential Item Functioning (DIF). See further section for more information.

Select also the value of latent ability \(\theta\) to see the interpretation of the item characteristic curves.

Equations

$$\mathrm{P}\left(\theta \vert a, b, c, d \right) = c + \left(d - c\right) \cdot \frac{e^{a\left(\theta-b\right) }}{1+e^{a\left(\theta-b\right) }} $$ $$\mathrm{I}\left(\theta \vert a, b, c, d \right) = a \cdot \left(d - c\right) \cdot \frac{e^{a\left(\theta-b\right) }}{\left[1+e^{a\left(\theta-b\right)}\right]^2} $$


Exercise 1

Consider the following 2PL items with parameters
Item 1: \(a = 2.5, b = -0.5\)
Item 2: \(a = 1.5, b = 0\)
For these items fill the following exercises with an accuracy of up to 0.05. Then click on Submit answers button. If you need a hint, click on blue button with question mark.

  • Sketch item characteristic and information curves.
  • Calculate probability of correct answer for latent abilities \(\theta = -2, -1, 0, 1, 2\).
    Item 1:
    Item 2:
  • For what level of ability \(\theta\) are the probabilities equal?
  • Which item provides more information for weak (\(\theta = -2\)), average (\(\theta = 0\)) and strong (\(\theta = 2\)) students?
    \(\theta = -2\)
    \(\theta = 0\)
    \(\theta = 2\)

Exercise 2

Consider now 2 items with following parameters
Item 1: \(a = 1.5, b = 0, c = 0, d = 1\)
Item 2: \(a = 1.5, b = 0, c = 0.2, d = 1\)
For these items fill the following exercises with an accuracy of up to 0.05. Then click on Submit answers button.

  • What is the lower asymptote for items?
    Item 1:
    Item 2:
  • What is the probability of correct answer for latent ability \(\theta = b\)?
    Item 1:
    Item 2:
  • Which item is more informative?

Exercise 3

Consider now 2 items with following parameters
Item 1: \(a = 1.5, b = 0, c = 0, d = 0.9\)
Item 2: \(a = 1.5, b = 0, c = 0, d = 1\)
For these items fill the following exercises with an accuracy of up to 0.05. Then click on Submit answers button.

  • What is the upper asymptote for items?
    Item 1:
    Item 2:
  • What is the probability of correct answer for latent ability \(\theta = b\)?
    Item 1:
    Item 2:
  • Which item is more informative?


Selected R code

library(ggplot2)
library(data.table)

# parameters 
a1 a2 
# latent ability 
theta # latent ability level
theta0 
# function for IRT characteristic curve 
icc_irt 
# calculation of characteristic curves
df                  "icc1" = icc_irt(theta, a1, b1, c1, d1),
                 "icc2" = icc_irt(theta, a2, b2, c2, d2))
df 
# plot for characteristic curves 
ggplot(df, aes(x = theta, y = value, color = variable)) + 
  geom_line() + 
  geom_segment(aes(y = icc_irt(theta0, a = a1, b = b1, c = c1, d = d1), 
                   yend = icc_irt(theta0, a = a1, b = b1, c = c1, d = d1), 
                   x = -4, xend = theta0), 
               color = "gray", linetype = "dashed") + 
  geom_segment(aes(y = icc_irt(theta0, a = a2, b = b2, c = c2, d = d2), 
                   yend = icc_irt(theta0, a = a2, b = b2, c = c2, d = d2), 
                   x = -4, xend = theta0), 
               color = "gray", linetype = "dashed") + 
  geom_segment(aes(y = 0, 
                   yend = max(icc_irt(theta0, a = a1, b = b1, c = c1, d = d1), 
                              icc_irt(theta0, a = a2, b = b2, c = c2, d = d2)), 
                   x = theta0, xend = theta0),
               color = "gray", linetype = "dashed") + 
  xlim(-4, 4) + 
  xlab("Ability") + 
  ylab("Probability of correct answer") + 
  theme_bw() + 
  ylim(0, 1) + 
  theme(axis.line = element_line(colour = "black"), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Item characteristic curve") 

# function for IRT information function 
iic_irt 
# calculation of information curves
df                  "iic1" = iic_irt(theta, a1, b1, c1, d1),
                 "iic2" = iic_irt(theta, a2, b2, c2, d2))
df 
# plot for information curves 
ggplot(df, aes(x = theta, y = value, color = variable)) + 
  geom_line() + 
  xlim(-4, 4) + 
  xlab("Ability") + 
  ylab("Information") + 
  theme_bw() + 
  ylim(0, 4) + 
  theme(axis.line = element_line(colour = "black"), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Item information curve") 

Polytomous models

Polytomous models are used when partial score is possible, or when items are graded on Likert scale (e.g. from Totally disagree to Totally agree); some polytomous models can also be used when analyzing multiple-choice items. In this section you can explore item response functions of some polytomous models.


Two main classes of polytomous IRT models are considered:

Difference models are defined by setting mathematical form to cumulative probabilities, while category probabilities are calculated as their difference. These models are also sometimes called cumulative logit models as they set linear form to cumulative logits.

As an example, Graded Response Model (GRM; Samejima, 1970) uses 2PL IRT model to describe cumulative probabilities (probabilities to obtain score higher than 1, 2, 3, etc.). Category probabilities are then described as differences of two subsequent cumulative probabilities.


For divide-by-total models response category probabilities are defined as the ratio between category-related functions and their sum.

In Generalized Partial Credit Model (GPCM; Muraki, 1992), probability of the successful transition from one category score to the next category score is modelled by 2PL IRT model, while Partial Credit Model (PCM; Masters, 1982) uses 1PL IRT model to describe this probability. Even more restricted version, the Rating Scale Model (RSM; Andrich, 1978) assumes exactly the same K response categories for each item and threshold parameters which can be split into a response-threshold parameter and an item-specific location parameter. These models are also sometimes called adjacent-category logit models as they set linear form to adjacent logits.

To model distractor properties in multiple-choice items, Nominal Response Model (NRM; Bock, 1972) can be used. NRM is an IRT analogy of multinomial regression model. This model is also generalization of GPCM/PCM/RSM ordinal models. NRM is also sometimes called baseline-category logit model as it sets linear form to log of odds of selecting given category to selecting a baseline category. Baseline can be chosen arbitrary, although usually the correct answer or the first answer is chosen.

Graded response model

Graded response model (GRM; Samejima, 1970) uses 2PL IRT model to describe cumulative probabilities (probabilities to obtain score higher than 1, 2, 3, etc.). Category probabilities are then described as differences of two subsequent cumulative probabilities.

It belongs to class of difference models, which are defined by setting mathematical form to cumulative probabilities, while category probabilities are calculated as their difference. These models are also sometimes called cumulative logit models, as they set linear form to cumulative logits.

Parameters

Select number of responses and difficulty for cummulative probabilities \(b\) and common discrimination parameter \(a\). Cummulative probability \(P(Y \geq 0)\) is always equal to 1 and it is not displayed, corresponding category probability \(P(Y = 0)\) is displayed with black color.




Equations

$$\pi_k* = \mathrm{P}\left(Y \geq k \vert \theta, a, b_k\right) = \frac{e^{a\left(\theta-b\right) }}{1+e^{a\left(\theta-b\right) }} $$ $$\pi_k =\mathrm{P}\left(Y = k \vert \theta, a, b_k, b_{k+1}\right) = \pi_k* - \pi_{k+1}* $$ $$\mathrm{E}\left(Y \vert \theta, a, b_1, \dots, b_K\right) = \sum_{k = 0}^K k\pi_k$$

Plots


Exercise

Consider item following graded response model rated \(0-1-2-3\), with discrimination \(a = 1\) and difficulties \(b_{1} = − 0.5\), \(b_{2} = 1\) and \(b_{3} = 1.5\).

  • Calculate probabilities of obtaining \(k\) and more points for specific level of ability \(\theta\)
    \(k \geq 0 \)
    \(k \geq 1 \)
    \(k \geq 2 \)
    \(k \geq 3 \)
  • Calculate probabilities of obtaining exactly \(k\) points for specific level of ability \(\theta\)
    \(k = 0\)
    \(k = 1\)
    \(k = 2\)
    \(k = 3\)
  • What is the expected item score for specific level of ability \(\theta\)?
    \(\theta = -2\)
    \(\theta = -1\)
    \(\theta = 0\)
    \(\theta = 1\)
    \(\theta = 2\)

Selected R code

library(ggplot2) 
library(data.table) 

# setting parameters 
a b theta 
# calculating cummulative probabilities 
ccirt df1 df1 
# plotting cummulative probabilities 
ggplot(data = df1, aes(x = theta, y = value, col = variable)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Cummulative probability") + 
  xlim(-4, 4) + 
  ylim(0, 1) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Cummulative probabilities") + 
  scale_color_manual("", values = c("red", "yellow", "green", "blue"), labels = paste0("P(Y >= ", 1:4, ")")) 

# calculating category probabilities 
df2 df2 df2 
# plotting category probabilities 
ggplot(data = df2, aes(x = theta, y = value, col = variable)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Category probability") + 
  xlim(-4, 4) + 
  ylim(0, 1) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Category probabilities") + 
  scale_color_manual("", values = c("black", "red", "yellow", "green", "blue"), labels = paste0("P(Y >= ", 0:4, ")"))

# calculating expected item score
df3 df3 df3 
# plotting category probabilities 
ggplot(data = df3, aes(x = theta, y = exp)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Expected item score") + 
  xlim(-4, 4) + 
  ylim(0, 4) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Expected item score")


Generalized partial credit model

In Generalized Partial Credit Model (GPCM; Muraki, 1992), probability of the successful transition from one category score to the next category score is modelled by 2PL IRT model. The response category probabilities are then ratios between category-related functions (cumulative sums of exponentials) and their sum.

Two simpler models can be derived from GPCM by restricting some parameters: Partial Credit Model (PCM; Masters, 1982) uses 1PL IRT model to describe this probability, thus parameters \(\alpha = 1\). Even more restricted version, the Rating Scale Model (RSM; Andrich, 1978) assumes exactly the same K response categories for each item and threshold parameters which can be split into a response-threshold parameter \(\lambda_t\) and an item-specific location parameter \(\delta_i\). These models are also sometimes called adjacent logit models, as they set linear form to adjacent logits.

Parameters

Select number of responses and their threshold parameters \(\delta\) and common discrimination parameter \(\alpha\). With \(\alpha = 1\) you get PCM. Numerator of \(\pi_0 = P(Y = 0)\) is set to 1 and \(\pi_0\) is displayed with black color.




Equations

$$\pi_k =\mathrm{P}\left(Y = k \vert \theta, \alpha, \delta_0, \dots, \delta_K\right) = \frac{\exp\sum_{t = 0}^k \alpha(\theta - \delta_t)}{\sum_{r = 0}^K\exp\sum_{t = 0}^r \alpha(\theta - \delta_t)} $$ $$\mathrm{E}\left(Y \vert \theta, \alpha, \delta_0, \dots, \delta_K\right) = \sum_{k = 0}^K k\pi_k$$

Plots


Exercise

Consider item following generalized partial credit model rated \(0-1-2\), with discrimination \(a = 1\) andthreshold parameters \(d_{1} = − 1\) and \(d_{2} = 1\).

  • For what ability levels do the category probability curves cross?
  • What is the expected item score for these ability levels?
    \(\theta = -1.5\)
    \(\theta = 0\)
    \(\theta = 1.5\)
  • Change discrimination to \(a = 2\). Do the category probability curves cross for the same ability levels?
  • How did the expected item score change for these ability levels?
    \(\theta = -1.5\)
    \(\theta = 0\)
    \(\theta = 1.5\)

Selected R code

library(ggplot2) 
library(data.table) 

# setting parameters 
a d theta 
# calculating category probabilities 
ccgpcm df pk pk pk denom df df1 
# plotting category probabilities 
ggplot(data = df1, aes(x = theta, y = value, col = variable)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Category probability") + 
  xlim(-4, 4) + 
  ylim(0, 1) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Category probabilities") + 
  scale_color_manual("", values = c("black", "red", "yellow", "green", "blue"), labels = paste0("P(Y = ", 0:4, ")"))

# calculating expected item score
df2 # plotting expected item score 
ggplot(data = df2, aes(x = theta, y = exp)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Expected item score") + 
  xlim(-4, 4) + 
  ylim(0, 4) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Expected item score")


Nominal response model

In Nominal Response Model (NRM; Bock, 1972), probability of selecting given category over baseline category is modelled by 2PL IRT model. This model is also sometimes called baseline-category logit model, as it sets linear form to log of odds of selecting given category to selecting a baseline category. Baseline can be chosen arbitrary, although usually the correct answer or the first answer is chosen. NRM model is generalization of GPCM model by setting item-specific and category-specific intercept and slope parameters.

Parameters

Select number of distractors and their threshold parameters \(\delta\) and discrimination parameters \(\alpha\). Parameters of \(\pi_0 = P(Y = 0)\) are set to zeros and \(\pi_0\) is displayed with black color.



Equations

$$\pi_k =\mathrm{P}\left(Y = k \vert \theta, \alpha_0, \dots, \alpha_K, \delta_0, \dots, \delta_K\right) = \frac{\exp(\alpha_k\theta + \delta_k)}{\sum_{r = 0}^K\exp(\alpha_r\theta + \delta_r)} $$

Plots

Download figure

Selected R code

library(ggplot2) 
library(data.table) 

# setting parameters 
a d theta 
# calculating category probabilities 
ccnrm df df denom df df1 
# plotting category probabilities 
ggplot(data = df1, aes(x = theta, y = value, col = variable)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Category probability") + 
  xlim(-4, 4) + 
  ylim(0, 1) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Category probabilities") + 
  scale_color_manual("", values = c("black", "red", "yellow", "green", "blue"), labels = paste0("P(Y = ", 0:4, ")"))

# calculating expected item score
df2 
# plotting expected item score
ggplot(data = df2, aes(x = theta, y = exp)) + 
  geom_line() + 
  xlab("Ability") + 
  ylab("Expected item score") + 
  xlim(-4, 4) + 
  ylim(0, 4) + 
  theme_bw() + 
  theme(text = element_text(size = 14), 
        panel.grid.major = element_blank(), 
        panel.grid.minor = element_blank()) + 
  ggtitle("Expected item score")


Differential Item/Distractor Functioning

Differential item functioning (DIF) occurs when respondents from different social groups (such as defined by gender or ethnicity) with the same underlying ability have a different probability of answering the item correctly or endorsing the item. If some item functions differently for two groups, it is potentially unfair and should be checked for wording. In general, two types of DIF can be distinguished: The uniform DIF describes a situation when the item advantages one of the groups at all levels of the latent ability (left figure). In such a case, the item has different difficulty (location parameters) for given two groups, while the item discrimination is the same. Contrary, the non-uniform DIF (right figure) means that the item advantages one of the groups at lower ability levels, and the other group at higher ability levels. In this case, the item has different discrimination (slope) parameters and possibly also different difficulty parameters for the given two groups.



Differential distractor functioning (DDF) occurs when respondents from different groups but with the same latent ability have different probability of selecting at least one distractor choice. Again, two types of DDF can be distinguished - uniform (left figure below) and non-uniform DDF (right figure below).


Total scores and other matching variables

DIF analysis may come to a different conclusion than test of group differences in total scores. Two groups may have the same distribution of total scores, yet, some items may function differently for the two groups. Also, one of the groups may have significantly lower total score, yet, it may happen that there is no DIF item (Martinkova et al., 2017). This section examines the differences in total scores only. Explore further DIF sections to analyze differential item functioning.

DIF can also be explored with respect to matching criteria other than the total score of analyzed items. For example, to analyze instructional sensitivity, Martinkova et al. (2020) analyzed differential item functioning in change (DIF-C) by analyzing DIF on Grade 9 item answers while matching on Grade 6 total scores of the same respondents in a longitudinal setting (see toy data Learning to Learn 9 in Data section).

Summary of for groups

Histograms of for groups

Download figure

Comparison of

Notes: Test for difference in between the reference and the focal group is based on Welch two sample t-test.
Diff. (CI) - difference in means of with 95% confidence interval, \(t\)-value - test statistic, df - degrees of freedom, \(p\)-value - value lower than 0.05 means significant difference in between the reference and the focal group.

Selected R code

library(ggplot2)
library(moments)
library(ShinyItemAnalysis)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Total score calculation wrt group
score score0 score1 
# Summary of total score
rbind(
  c(length(score0), min(score0), max(score0), mean(score0), median(score0), sd(score0), skewness(score0), kurtosis(score0)),
  c(length(score1), min(score1), max(score1), mean(score1), median(score1), sd(score1), skewness(score1), kurtosis(score1))
)

df 
# Histogram of total scores wrt group
ggplot(data = df, aes(x = score, fill = group, col = group)) +
  geom_histogram(binwidth = 1, position = "dodge2", alpha = 0.75) +
  xlab("Total score") +
  ylab("Number of respondents") +
  scale_fill_manual(values = c("dodgerblue2", "goldenrod2"), labels = c("Reference", "Focal")) +
  scale_colour_manual(values = c("dodgerblue2", "goldenrod2"), labels = c("Reference", "Focal")) +
  theme_app() +
  theme(legend.position = "left"))

# t-test to compare total scores
t.test(score0, score1)

Delta plot

Delta plot (Angoff & Ford, 1973) compares the proportions of correct answers per item in the two groups. It displays non-linear transformation of these proportions using quantiles of standard normal distributions (so-called delta scores) for each item for the two genders in a scatterplot called diagonal plot or delta plot (see Figure below). Item is under suspicion of DIF if the delta point considerably departs from the main axis of the ellipsoid formed by delta scores.

Method specification

The detection threshold is either fixed to the value of 1.5 or it is based on bivariate normal approximation (Magis & Facon, 2012). The item purification algorithms offered when using the threshold based on normal approximationare are as follows: IPP1 uses the threshold obtained after the first run in all following runs, IPP2 updates only the slope parameter of the threshold formula and thus lessens the impact of DIF items, IPP3 adjusts every single parameter and completely discards the effect of items flagged as DIF from the computation of the threshold (for further details see Magis & Facon, 2013). When using the fixed threshold and item purification, this threshold (1.5) stays the same henceforward during the purification algorithm.


Delta plot

Download figure

Summary table

Summary table contains information about proportions of correct answers in the reference and the focal group together with their transformations into delta scores. It also includes distances of delta scores from the main axis of the ellipsoid formed by delta scores.




Purification process


Download table

Selected R code

library(deltaPlotR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Delta scores with fixed threshold
(DS_fixed # Delta plot
diagPlot(DS_fixed, thr.draw = TRUE)

# Delta scores with normal threshold
(DS_normal # Delta plot
diagPlot(DS_normal, thr.draw = TRUE)

Mantel-Haenszel test

Mantel-Haenszel test is a DIF detection method based on contingency tables which are calculated for each level of the total score (Mantel & Haenszel, 1959).

Method specification

Here you can select correction method for multiple comparison, and/or item purification.


Summary table

Summary table contains information about Mantel-Haenszel \(\chi^2\) statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Moreover, table offers values of Mantel-Haenszel estimates of odds ratio \(\alpha_{\mathrm{MH}}\), which incorporate all levels of total score, and their transformations into D-DIF indices \(\Delta_{\mathrm{MH}} = -2.35 \log(\alpha_{\mathrm{MH}})\) to evaluate DIF effect size.




Purification process



Selected R code

library(difR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Mantel-Haenszel test
(fit 

Mantel-Haenszel test

Mantel-Haenszel test is a DIF detection method based on contingency tables which are calculated for each level of total score (Mantel & Haenszel, 1959).

Contingency tables and odds ratio calculation

For selected item and for selected level of total score you can display contingency table and calculates odds ratio of answering item correctly. This can be compared to Mantel-Haenszel estimate of odds ratio \(\alpha_{\mathrm{MH}}\), which incorporates all levels of total score. Further, \(\alpha_{\mathrm{MH}}\) can be transformed into Mantel-Haenszel D-DIF index \(\Delta_{\mathrm{MH}}\) to evaluate DIF effect size.


Selected R code

library(difR)
library(reshape2)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# contingency table for item 1 and score 12
item cut 
df colnames(df) df$Answer df$Group score df dcast(data.frame(xtabs(~ Group + Answer, data = df)),
  Group ~ Answer,
  value.var = "Freq", margins = TRUE, fun = sum
)

# Mantel-Haenszel estimate of OR
(fit fit$alphaMH

# D-DIF index calculation
-2.35 * log(fit$alphaMH)

Logistic regression

Logistic regression method allows for detection of uniform and non-uniform DIF (Swaminathan & Rogers, 1990) by including a group specific intercept \(b_{2}\) (uniform DIF) and group specific interaction \(b_{3}\) (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can choose what type of DIF to be tested. You can also select correction method for multiple comparison or item purification. Finally, you may change the DIF matching variable. While matching on the standardized total score is typical, upload of other DIF matching variable is possible in Section Data. Using a pre-test (standardized) total score as DIF matching variable allows for testing differential item functioning in change (DIF-C) to provide proofs of instructional sensitivity (Martinkova et al., 2020), also see Learning To Learn 9 toy dataset.

Equation

$$\mathrm{P}\left(Y_{ij} = 1 | X_i, G_i, b_0, b_1, b_2, b_3\right) = \frac{e^{b_0 + b_1 X_i + b_2 G_i + b_3 X_i G_i}}{1+e^{b_0 + b_1 X_i + b_2 G_i + b_3 X_i G_i}} $$

Summary table

Summary table contains information about DIF test statistics \(LR(\chi^2)\), corresponding \(p\)-values considering selected adjustement, and significance codes. Moreover, it offers values of Nagelkerke's \(R^2\) with DIF effect size classifications. Table also provides estimated parameters for the best fitted model for each item.




Purification process


Selected R code

library(difR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Logistic regression DIF detection method
(fit 
# Loading data
data(LearningToLearn, package = "ShinyItemAnalysis")
Data group match 
# Detecting differential item functioning in change (DIF-C) using
# logistic regression DIF detection method
# and standardized total score from Grade 6 as matching criterion
(fit 

Logistic regression

Logistic regression method allows for detection of uniform and non-uniform DIF (Swaminathan & Rogers, 1990) by including a group specific intercept \(b_{2}\) (uniform DIF) and group specific interaction \(b_{3}\) (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can choose what type of DIF to be tested. You can also select correction method for multiple comparison or item purification. Finally, you may change the DIF matching variable. While matching on the standardized total score is typical, upload of other DIF matching variable is possible in Section Data. Using a pre-test (standardized) total score as DIF matching criterion allows for testing differential item functioning in change (DIF-C) to provide proofs of instructional sensitivity (Martinkova et al., 2020), also see Learning To Learn 9 toy dataset. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF logistic curve

Points represent proportion of correct answer (empirical probabilities) with respect to the DIF matching variable. Their size is determined by count of respondents who achieved given level of DIF matching variable with respect to the group membership.

Download figure

Equation

$$\mathrm{P}\left(Y_{ij} = 1 | X_i, G_i, b_0, b_1, b_2, b_3\right) = \frac{e^{b_0 + b_1 X_i + b_2 G_i + b_3 X_i G_i}}{1+e^{b_0 + b_1 X_i + b_2 G_i + b_3 X_i G_i}} $$

Table of parameters

Table summarizes estimated item parameters together with standard errors.


Selected R code

library(difR)
library(ShinyItemAnalysis)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Logistic regression DIF detection method
(fit 
# Plot of characteristic curve for item 1
plotDIFLogistic(fit, item = 1, Data = Data, group = group)

# Estimated coefficients for item 1
fit$logitPar[1, ]

Generalized logistic regression

Generalized logistic regression models are extensions of logistic regression method which account for possibility of guessing by allowing for nonzero lower asymptote - pseudo-guessing \(c\) (Drabinova & Martinkova, 2017) or upper asymptote lower than one - inattention \(d\). Similarly to logistic regression, its extensions also provide detection of uniform and non-uniform DIF by letting the difficulty parameter \(b\) (uniform) and the discrimination parameter \(a\) (non-uniform) differ for groups and by testing for difference in their values. Moreover, these extensions allow for testing differences in pseudo-guessing and inattention parameters and they can be seen as proxies of 3PL and 4PL IRT models for DIF detection.

Method specification

Here you can specify the assumed model. In 3PL and 4PL models, the abbreviations \(c_{g}\) or \(d_{g}\) mean that parameters \(c\) or \(d\) are assumed to be the same for both groups, otherwise they are allowed to differ. With type you can specify the type of DIF to be tested by choosing the parameters in which difference between groups should be tested. You can also select correction method for multiple comparison or item purification.

Finally, you may change the DIF matching variable. While matching on standardized total score is typical, upload of other DIF matching variable is possible in section Data. Using a pre-test (standardized) total score allows for testing differential item functioning in change (DIF-C) to provide proofs of instructional sensitivity (Martinkova et al., 2020), also see Learning To Learn 9 toy dataset.

Equation

Displayed equation is based on model selected below

Summary table

Summary table contains information about DIF test statistic \(LR(\chi^2)\), corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for the best fitted model for each item. Note that \(a_{jG_i}\) (and also other parameters) from the equation above consists of parameter for the reference group and parameter for the difference between focal and reference groups, i.e., \(a_{jG_i} = a_{j} + a_{jDif}G_{i}\), where \(G_{i} = 0\) for the reference group and \(G_{i} = 1\) for the focal group, as stated in the table below.




Purification process



Selected R code

library(difNLR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Generalized logistic regression DIF method
# using 3PL model with the same guessing parameter for both groups
(fit 
# Loading data
data(LearningToLearn, package = "ShinyItemAnalysis")
Data group match 
# Detecting differential item functioning in change (DIF-C) using
# generalized logistic regression DIF method with 3PL model
# with the same guessing parameter for both groups
# and standardized total score from Grade 6 as matching criterion
(fit 

Generalized logistic regression

Generalized logistic regression models are extensions of logistic regression method which account for possibility of guessing by allowing for nonzero lower asymptote - pseudo-guessing \(c\) (Drabinova & Martinkova, 2017) or upper asymptote lower than one - inattention \(d\). Similarly to logistic regression, its extensions also provide detection of uniform and non-uniform DIF by letting the difficulty parameter \(b\) (uniform) and the discrimination parameter \(a\) (non-uniform) differ for groups and by testing for difference in their values. Moreover, these extensions allow for testing differences in pseudo-guessing and inattention parameters and they can be seen as proxies of 3PL and 4PL IRT models for DIF detection.

Method specification

Here you can specify the assumed model. In 3PL and 4PL models, the abbreviations \(c_{g}\) or \(d_{g}\) mean that parameters \(c\) or \(d\) are assumed to be the same for both groups, otherwise they are allowed to differ. With type you can specify the type of DIF to be tested by choosing the parameters in which difference between groups should be tested. You can also select correction method for multiple comparison or item purification.

Finally, you may change the DIF matching variable. While matching on standardized total score is typical, upload of other DIF matching variable is possible in section Data. Using a pre-test (standardized) total score allows for testing differential item functioning in change (DIF-C) to provide proofs of instructional sensitivity (Martinkova et al., 2020), also see Learning To Learn 9 toy dataset. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF generalized logistic curve

Points represent proportion of correct answer (empirical probabilities) with respect to the DIF matching variable. Their size is determined by count of respondents who achieved given level of DIF matching variable with respect to the group membership.

Download figure

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors. Note that \(a_{jG_i}\) (and also other parameters) from the equation above consists of parameter for the reference group and parameter for the difference between focal and reference groups, i.e., \(a_{jG_i} = a_{j} + a_{jDif}G_{i}\), where \(G_{i} = 0\) for the reference group and \(G_{i} = 1\) for the focal group, as stated in the table below.


Selected R code

library(difNLR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# Generalized logistic regression DIF method
# using 3PL model with the same guessing parameter for both groups
(fit 
# Plot of characteristic curve of item 1
plot(fit, item = 1)

# Estimated coefficients for item 1 with standard errors
coef(fit, SE = TRUE)

Lord test for IRT models

To detect DIF, Lord test (Lord, 1980) compares item parameters of selected IRT model, fitted separately on data of the two groups. Model is either 1PL, 2PL, or 3PL with guessing which is the same for the two groups. In case of 3PL model, the guessing parameter is estimated based on the whole dataset and is subsequently considered fixed. In statistical terms, Lord statistic is equal to Wald statistic.

Method specification

Here you can choose underlying IRT model used to test DIF. You can also select correction method for multiple comparison, and/or item purification.

Equation

Summary table

Summary table contains information about Lord's \(\chi^2\)-statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for both groups. Note that item parameters might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant. Also note that under the 3PL model, the guessing parameter \(c\) is estimated from the whole dataset, and is considered fixed in the final models, thus no standard error is displayed.




Purification process



Selected R code

library(difR)
library(ltm)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# 1PL IRT MODEL
(fit1PL 
# 2PL IRT MODEL
(fit2PL 
# 3PL IRT MODEL with the same guessing for groups
guess (fit3PL 

Lord test for IRT models

To detect DIF, Lord test (Lord, 1980) compares item parameters of selected IRT model, fitted separately on data of the two groups. Model is either 1PL, 2PL, or 3PL with guessing which is the same for the two groups. In case of 3PL model, the guessing parameter is estimated based on the whole dataset and is subsequently considered fixed. In statistical terms, Lord statistic is equal to Wald statistic.

Method specification

Here you can choose underlying IRT model used to test DIF. You can also select correction method for multiple comparison, and/or item purification. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF characteristic curve

Note that plots might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant.

Download figure

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors. Note that item parameters might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant. Also note that under the 3PL model, the guessing parameter \(c\) is estimated from the whole dataset, and is considered fixed in the final models, thus no standard error is displayed.


Selected R code

library(difR)
library(ltm)
library(ShinyItemAnalysis)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# 1PL IRT MODEL
(fit1PL # Estimated coefficients for all items
(coef1PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef1PL, item = 1, test = "Lord")

# 2PL IRT MODEL
(fit2PL # Estimated coefficients for all items
(coef2PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef2PL, item = 1, test = "Lord")

# 3PL IRT MODEL with the same guessing for groups
guess (fit3PL # Estimated coefficients for all items
(coef3PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef3PL, item = 1, test = "Lord")

Raju test for IRT models

To detect DIF, Raju test (Raju, 1988, 1990) uses area between the item charateristic curves of selected IRT model, fitted separately on data of the two groups. Model is either 1PL, 2PL, or 3PL with guessing which is the same for the two groups. In case of 3PL model, the guessing parameter is estimated based on the whole dataset and is subsequently considered fixed.

Method specification

Here you can choose underlying IRT model used to test DIF. You can also select correction method for multiple comparison, and/or item purification.

Equation

Summary table

Summary table contains information about Raju's \(Z\)-statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for both groups. Note that item parameters might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant. Also note that under the 3PL model, the guessing parameter \(c\) is estimated from the whole dataset, and is considered fixed in the final models, thus no standard error is displayed.




Purification process



Selected R code

library(difR)
library(ltm)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# 1PL IRT MODEL
(fit1PL 
# 2PL IRT MODEL
(fit2PL 
# 3PL IRT MODEL with the same guessing for groups
guess (fit3PL 

Raju test for IRT models

To detect DIF, Raju test (Raju, 1988, 1990) uses area between the item charateristic curves of selected IRT model, fitted separately on data of the two groups. Model is either 1PL, 2PL, or 3PL with guessing which is the same for the two groups. In case of 3PL model, the guessing parameter is estimated based on the whole dataset and is subsequently considered fixed.

Method specification

Here you can choose underlying IRT model used to test DIF. You can also select correction method for multiple comparison, and/or item purification. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF characteristic curve

Note that plots might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant.

Download figure

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors. Note that item parameters might slightly differ even for non-DIF items as two seperate models are fitted, however this difference is non-significant. Also note that under the 3PL model, the guessing parameter \(c\) is estimated from the whole dataset, and is considered fixed in the final models, thus no standard error is available.


Selected R code

library(difR)
library(ltm)
library(ShinyItemAnalysis)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# 1PL IRT MODEL
(fit1PL # Estimated coefficients for all items
(coef1PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef1PL, item = 1, test = "Raju")

# 2PL IRT MODEL
(fit2PL # Estimated coefficients for all items
(coef2PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef2PL, item = 1, test = "Raju")

# 3PL IRT MODEL with the same guessing for groups
guess (fit3PL # Estimated coefficients for all items
(coef3PL # Plot of characteristic curve of item 1
plotDIFirt(parameters = coef3PL, item = 1, test = "Raju")

SIBTEST

The SIBTEST method (Shealy & Stout, 1993) allows for detection of uniform DIF without requiring an item response model. Its modified version, the Crossing-SIBTEST (Chalmers, 2018; Li & Stout, 1996), focuses on detection of non-uniform DIF.

Method specification

Here you can choose type of DIF to test. With uniform DIF, SIBTEST is applied, while with non-uniform DIF, the Crossing-SIBTEST method is used instead. You can also select correction method for multiple comparison or item purification.


Summary table

Summary table contains estimates of \(\beta\) together with standard errors (only available when testing uniform DIF), corresponding \(\chi^2\)-statistics with \(p\)-values considering selected adjustement, and significance codes.




Purification process


Download table


Selected code

library(difR)

# Loading data
data(GMAT, package = "difNLR")
Data group 
# SIBTEST (uniform DIF)
(fit_udif 
# Crossing-SIBTEST (non-uniform DIF)
(fit_nudif 

Method comparison

Here you can compare all offered DIF detection methods. In the table below, columns represent DIF detection methods, and rows represent item number. If the method detects item as DIF, value 1 is assigned to that item, otherwise 0 is assigned. In case that any method fail to converge or cannot be fitted, NA is displayed instead of 0/1 values. Available methods:

  • Delta is delta plot method (Angoff & Ford, 1973; Magis & Facon, 2012),
  • MH is Mantel-Haenszel test (Mantel & Haenszel, 1959),
  • LR is logistic regression (Swaminathan & Rogers, 1990),
  • NLR is generalized (non-linear) logistic regression (Drabinova & Martinkova, 2017),
  • LORD is Lord chi-square test (Lord, 1980),
  • RAJU is Raju area method (Raju, 1990),
  • SIBTEST is SIBTEST (Shealy & Stout, 1993) and crossing-SIBTEST method (Chalmers, 2018; Li & Stout, 1996).

Table with method comparison

Settings for individual methods (DIF matching criterion, type of DIF to be tested, correction method, item purification) are taken from subsection pages of given methods. In case your settings are not unified, you can set some of them below. Note that changing the options globaly can be computationaly demanding. This especially applies for purification request. To see the complete setting of all analyses, please refer to the note below the table. The last column shows how many methods detect certain item as DIF. The last row shows how many items are detected as DIF by a certain method.





Cumulative logit regression model for DIF detection

Cumulative logit regression allows for detection of uniform and non-uniform DIF among ordinal data by adding a group specific intercept \(b_2\) (uniform DIF) and interaction \(b_3\) between group and DIF matching variable (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can change DIF matching variable and choose type of DIF to be tested. You can also select correction method for multiple comparison or item purification.

Equation

The probability that person \(p\) with DIF matching variable (e.g., standardized total score) \(Z_p\) and group membership \(G_p\) obtained at least \(k\) points in item \(i\) is given by the following equation:

The probability that person \(p\) with DIF matching variable (e.g., standardized total score) \(Z_p\) and group membership \(G_p\) obtained exactly \(k\) points in item \(i\) is then given as differnce between probabilities of obtaining at least \(k\) and \(k + 1\) points:

Summary table

Summary table contains information about \(\chi^2\)-statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for the best fitted model for each item.


              

Selected R code

library(difNLR)

# Loading data
data(dataMedicalgraded, package = "ShinyItemAnalysis")
Data group 
# DIF with cumulative logit regression model
(fit 


Cumulative logit regression model for DIF detection

Cumulative logit regression allows for detection of uniform and non-uniform DIF among ordinal data by adding a group specific intercept \(b_2\) (uniform DIF) and interaction \(b_3\) between group and DIF matching variable (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can change DIF matching variable and choose type of DIF to be tested. You can also select correction method for multiple comparison or item purification. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF curves

Points represent proportion of obtained score with respect to DIF matching variable. Their size is determined by count of respondents who achieved given level of DIF matching variable and who selected given option with respect to the group membership.

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors.


Selected R code

library(difNLR)

# Loading data
data(dataMedicalgraded, package = "ShinyItemAnalysis")
Data group 
# DIF with cumulative logit regression model
(fit 
# Plot of characteristic curves for item X2003, cumulative probabilities
plot(fit, item = "X2003", plot.type = "cumulative")

# Plot of characteristic curves for item X2003, category probabilities
plot(fit, item = "X2003", plot.type = "category")

# Estimated coefficients for all items with standard errors
coef(fit, SE = TRUE)


Adjacent category logit regression model for DIF detection

Adjacent category logit regression model allows for detection of uniform and non-uniform DIF among ordinal data by adding a group specific intercept \(b_2\) (uniform DIF) and interaction \(b_3\) between group and DIF matching variable (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can change DIF matching variable and choose type of DIF to be tested. You can also select correction method for multiple comparison or item purification.

Equation

The probability that person \(p\) with DIF matching variable (e.g., standardized total score) \(Z_p\) and group membership \(G_p\) obtained \(k\) points in item \(i\) is given by the following equation:

Summary table

Summary table contains information about \(\chi^2\)-statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for the best fitted model for each item.


              

Selected R code

library(difNLR)

# Loading data
data(dataMedicalgraded, package = "ShinyItemAnalysis")
Data group 
# DIF with cumulative logit regression model
(fit 


Adjacent category logit regression model for DIF detection

Adjacent category logit regression model allows for detection of uniform and non-uniform DIF among ordinal data by adding a group specific intercept \(b_2\) (uniform DIF) and interaction \(b_3\) between group and DIF matching variable (non-uniform DIF) into model and by testing for their significance.

Method specification

Here you can change DIF matching variable and choose type of DIF to be tested. You can also select correction method for multiple comparison or item purification. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DIF curves

Points represent proportion of obtained score with respect to DIF matching variable. Their size is determined by count of respondents who achieved given level of DIF matching variable and who selected given option with respect to the group membership.

Download figure

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors.


Selected R code

library(difNLR)

# Loading data
data(dataMedicalgraded, package = "ShinyItemAnalysis")
Data group 
# DIF with cumulative logit regression model
(fit 
# Plot of characteristic curves for item X2003
plot(fit, item = "X2003")

# Estimated coefficients for all items with standard errors
coef(fit, SE = TRUE)


Multinomial regression model for DDF detection

Differential Distractor Functioning (DDF) occurs when people from different groups but with the same knowledge have different probability of selecting at least one distractor choice. DDF is here examined by multinomial log-linear regression model with Z-score and group membership as covariates.

Method specification

Here you can change DIF matching variable and choose type of DDF to be tested. You can also select correction method for multiple comparison or item purification.

Equation

For \(K\) possible test choices is the probability of the correct answer for person \(p\) with DIF matching variable (e.g., standardized total score) \(Z_p\) and group membership \(G_p\) in item \(i\) given by the following equation:

$$\mathrm{P}(Y_{ip} = K|Z_p, G_p, b_{il0}, b_{il1}, b_{il2}, b_{il3}, l = 1, \dots, K-1) = \frac{1}{1 + \sum_l e^{\left( b_{il0} + b_{il1} Z_p + b_{il2} G_p + b_{il3} Z_p:G_p\right)}}$$

The probability of choosing distractor \(k\) is then given by:

$$\mathrm{P}(Y_{ip} = k|Z_p, G_p, b_{il0}, b_{il1}, b_{il2}, b_{il3}, l = 1, \dots, K-1) = \frac{e^{\left( b_{ik0} + b_{ik1} Z_p + b_{ik2} G_p + b_{ik3} Z_p:G_p\right)}} {1 + \sum_l e^{\left( b_{il0} + b_{il1} Z_p + b_{il2} G_p + b_{il3} Z_p:G_p\right)}}$$

Summary table

Summary table contains information about \(\chi^2\)-statistics, corresponding \(p\)-values considering selected adjustement, and significance codes. Table also provides estimated parameters for the best fitted model for each item.


              

Selected R code

library(difNLR)

# Loading data
data(GMATtest, GMATkey, package = "difNLR")
Data group key 
# DDF with multinomial  regression model
(fit 


Multinomial regression model for DDF detection

Differential Distractor Functioning (DDF) occurs when people from different groups but with the same knowledge have different probability of selecting at least one distractor choice. DDF is here examined by Multinomial Log-linear Regression model with Z-score and group membership as covariates.

Method specification

Here you can change DIF matching variable and choose type of DDF to be tested. You can also select correction method for multiple comparison or item purification. For selected item you can display plot of its characteristic curves and table of its estimated parameters with standard errors.

Plot with estimated DDF curves

Points represent proportion of selected answer with respect to DIF matching variable. Their size is determined by count of respondents who achieved given level of DIF matching variable and who selected given option with respect to the group membership.

Download figure

Equation

Table of parameters

Table summarizes estimated item parameters together with standard errors.


Selected R code

library(difNLR)

# Loading data
data(GMATtest, GMATkey, package = "difNLR")
Data group key 
# DDF with multinomial  regression model
(fit 
# Plot of characteristic curves for item 1
plot(fit, item = 1)

# Estimated coefficients for all items with standard errors
coef(fit, SE = TRUE)


Download report

Settings of report

ShinyItemAnalysis offers an option to download a report in HTML or PDF format. PDF report creation requires latest version of MiKTeX (or other TeX distribution). If you don't have the latest installation, please, use the HTML report.

There is an option to use customized settings. When checking the Customize settings local settings will be offered and used for each selected section of the report. Otherwise, the settings will be taken from sections of the application. You may also include your name into the report, as well as the name of analyzed dataset.

Content of report

Reports by default contain summary of total scores, table of standard scores, item analysis, distractor plots for each item and multinomial regression plots for each item. Other analyses can be selected below.


Validity


Difficulty/discrimination plot


Distractors plots



DIF method selection

Delta plot settings

Mantel-Haenszel test settings

Logistic regression settings

Multinomial regression settings



Recommendation: Report generation can be faster and more reliable when you first check sections of intended contents. For example, if you wish to include a 3PL IRT model, you can first visit IRT models section and 3PL subsection.





Welcome

Welcome to ShinyItemAnalysis!

ShinyItemAnalysis is an interactive online application for psychometric analysis of educational and other psychological tests and their items, built on R and shiny. You can simply start using the application by choosing toy dataset (or upload your own one) in section Data and run analysis including:

  • Exploration of total and standard scores in Summary section
  • Analysis of measurement error in Reliability section
  • Correlation structure and criterion validity analysis in Validity section
  • Item and distractor analysis in Item analysis section
  • Item analysis with regression models in Regression section
  • Item analysis by item response theory models in IRT models section
  • Differential item functioning (DIF) and differential distractor functioning (DDF) methods in DIF/Fairness section

All graphical outputs and selected tables can be downloaded via download button. Moreover, you can automatically generate HTML or PDF report in Reports section. All offered analyses are complemented by selected R code which is ready to be copy-pasted into your R console, hence a similar analysis can be run and modified in R.

Visit www.ShinyItemAnalysis.org webpage to learn more about ShinyItemAnalysis!


Availability

Application can be downloaded as an R package from CRAN.
It is also available online at Czech Academy of Sciences and shinyapps.io .

Versions

Current CRAN version is 1.3.3.
Version available online is 1.3.3.
The newest development version available on GitHub is 1.3.3.


Feedback

If you discover a problem with this application please contact the project maintainer at martinkova(at)cs.cas.cz or use GitHub. We also encourage you to provide your feedback using Google form.


License

This program is free software and you can redistribute it and or modify it under the terms of the GNU GPL 3 as published by the Free Software Foundation. This program is distributed in the hope that it will be useful, but without any warranty; without even the implied warranty of merchantability of fitness for a particular purpose.

To cite ShinyItemAnalysis in publications, please use:

Martinkova P., & Drabinova A. (2018).
ShinyItemAnalysis for teaching psychometrics and to enforce routine analysis of educational tests.
The R Journal, 10(2), 503-515. doi: 10.32614/RJ-2018-074

Acknowledgments

Project was supported by Czech Science Foundation grant GJ15-15856Y 'Estimation of psychometric measures as part of admission test development' and by Charles University under project PRIMUS/17/HUM/11 'Center for Educational Measurement and Psychometrics (CEMP)'.



R packages

  • corrplot Wei, T. & Simko, V. (2017). R package "corrplot": Visualization of a Correlation Matrix. R package version 0.84. See online.
  • cowplot Claus O. Wilke (2018). cowplot: Streamlined Plot Theme and Plot Annotations for "ggplot2". R package version 0.9.3. See online.
  • CTT Willse, J. & Willse, T. (2018). CTT: Classical Test Theory Functions. R package version 2.3.3. See online.
  • data.table Dowle, M. & Srinivasan, A. (2019). data.table: Extension of "data.frame". R package version 1.12.8. See online.
  • deltaPlotR Magis, D. & Facon, B. (2014). deltaPlotR: An R Package for Differential Item Functioning Analysis with Angoff`s Delta Plot. Journal of Statistical Software, Code Snippets, 59(1), 1-19. See online.
  • difNLR Drabinova, A., Martinkova, P. (2020). difNLR: DIF and DDF Detection by Non-Linear Regression Models. R package version 1.3.2. See online.
  • difR Magis, D., Beland, S., Tuerlinckx, F. & De Boeck, P. (2010). A general framework and an R package for the detection of dichotomous differential item functioning. Behavior Research Methods, 42847-862.
  • DT Xie, Y., Cheng, J. & Tan, X. (2019). DT: A Wrapper of the JavaScript Library "DataTables". R package version 0.10. See online.
  • ggdendro de Vries, A. & Ripley, B.D. (2016). ggdendro: Create Dendrograms and Tree Diagrams Using "ggplot2". R package version 0.1-20. See online.
  • ggplot2 Wickham, H. (2016). ggplot2: Elegant Graphics for Data Analysis. See online.
  • gridExtra Auguie, B. (2017). gridExtra: Miscellaneous Functions for "Grid" Graphics. R package version 2.3. See online.
  • knitr Xie, Y. (2019). knitr: A General-Purpose Package for Dynamic Report Generation in R. R package version 1.26. See online.
  • latticeExtra Sarkar, D. & Andrews, F. (2016). latticeExtra: Extra Graphical Utilities Based on Lattice. R package version 0.6-28. See online.
  • ltm Rizopoulos, D. (2006). ltm: An R package for Latent Variable Modelling and Item Response Theory Analyses. Journal of Statistical Software, 17(5), 1-25. See online.
  • mirt Chalmers, R. & Chalmers, P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29.
  • moments Komsta, L. & Novomestky, F. (2015). moments: Moments, cumulants, skewness, kurtosis and related tests. R package version 0.14. See online.
  • msm Jackson, C. & Jackson, H. (2011). Multi-State Models for Panel Data: The msm Package for R. Journal of Statistical Software, 38(8), 1-29. See online.
  • nnet Venables, C. & Ripley, C. (2002). Modern Applied Statistics with S. See online.
  • plotly Sievert, C., Parmer, C., Hocking, T., Chamberlain, S., Ram, K., Corvellec, M. & Despouy, P. (2017). plotly: Create Interactive Web Graphics via "plotly.js". R package version 4.9.1. See online.
  • psych Revelle, W. (2018). psych: Procedures for Psychological, Psychometric, and Personality Research. R package version 1.8.12. See online.
  • psychometric Fletcher, T. & Fletcher, D. (2010). psychometric: Applied Psychometric Theory. R package version 2.2. See online.
  • reshape2 Wickham, H. (2007). Reshaping Data with the reshape Package. Journal of Statistical Software, 21(12), 1-20. See online.
  • rmarkdown Xie, Y., Allaire, J.J. & Grolemund G. (2018). R Markdown: The Definitive Guide. Chapman and Hall/CRC. ISBN 9781138359338. See online.
  • shiny Chang, W., Cheng, J., Allaire, J., Xie, Y. & McPherson, J. (2019). shiny: Web Application Framework for R. R package version 1.4.0. See online.
  • shinyBS Bailey, E. (2015). shinyBS: Twitter Bootstrap Components for Shiny. R package version 0.61. See online.
  • shinydashboard Chang, W. & Borges Ribeiro, B. (2018). shinydashboard: Create Dashboards with "Shiny". R package version 0.7.1 See online.
  • ShinyItemAnalysis Martinkova, P., & Drabinova, A. (2018). ShinyItemAnalysis for teaching psychometrics and to enforce routine analysis of educational tests. The R Journal, 10(2), 503-515. See online.
  • shinyjs Attali, D. (2018). shinyjs: Easily Improve the User Experience of Your Shiny Apps in Seconds. R package version 1.0. See online.
  • stringr Wickham, H. (2019). stringr: Simple, Consistent Wrappers for Common String Operations. R package version 1.4.0. See online.
  • xtable Dahl, D., Scott, D., Roosen, C., Magnusson, A.& Swinton, J. (2019). xtable: Export Tables to LaTeX or HTML. R package version 1.8-4. See online.
  • VGAM Yee, T. W. (2019). VGAM: Vector Generalized Linear and Additive Models. R package version 1.1-2. See online.

References

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  • Angoff, W. H., & Ford, S. F. (1973). Item-Race Interaction on a Test of Scholastic Aptitude. Journal of Educational Measurement, 10(2), 95-105. See online.
  • Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002). The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.
  • Bock, R. D. (1972). Estimating Item Parameters and Latent Ability when Responses Are Scored in Two or More Nominal Categories. Psychometrika, 37(1), 29-51. See online.
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  • Chalmers, R. P. (2018). Improving the Crossing-SIBTEST Statistic for Detecting Non-uniform DIF. Psychometrika, 83(2), 376-386. See online.
  • Cronbach, L. J. (1951). Coefficient Alpha and the Internal Structure of Tests. Psychometrika, 16(3), 297-334. See online.
  • Drabinova, A., & Martinkova, P. (2017). Detection of Differential Item Functioning with Non-Linear Regression: Non-IRT Approach Accounting for Guessing. Journal of Educational Measurement, 54(4), 498-517 See online.
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  • Magis, D., & Facon, B. (2012). Angoffs Delta Method Revisited: Improving DIF Detection under Small Samples. British Journal of Mathematical and Statistical Psychology, 65(2), 302-321. See online.
  • Magis, D., & Facon, B. (2013). Item purification does not always improve DIF detection: a counter-example with Angoffs Delta plot. Educational and Psychological Measurement, 73(2), 293-311. See online.
  • Mantel, N., & Haenszel, W. (1959). Statistical Aspects of the Analysis of Data from Retrospective Studies. Journal of the National Cancer Institute, 22(4), 719-748. See online.
  • Martinkova, P., Drabinova, A., & Houdek, J. (2017). ShinyItemAnalysis: Analyza Prijimacich a Jinych Znalostnich ci Psychologickych Testu. [ShinyItemAnalysis: Analyzing Admission and Other Educational and Psychological Tests] TESTFORUM, 6(9), 16-35. See online.
  • Martinkova, P., Drabinova, A., Liaw, Y. L., Sanders, E. A., McFarland, J. L., & Price, R. M. (2017). Checking Equity: Why Differential Item Functioning Analysis Should Be a Routine Part of Developing Conceptual Assessments. CBE-Life Sciences Education, 16(2), rm2. See online
  • Martinkova, P., Stepanek, L., Drabinova, A., Houdek, J., Vejrazka, M., & Stuka, C. (2017). Semi-real-time Analyses of Item Characteristics for Medical School Admission Tests. In Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, 189-194. See online.
  • Martinkova, P., Drabinova, A., & Potuznikova, E. (2020). Is academic tracking related to gains in learning competence? Using propensity score matching and differential item change functioning analysis for better understanding of tracking implications. Learning and Instruction 66(April). See online
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  • Orlando, M., & Thissen, D. (2000). Likelihood-based item-fit indices for dichotomous item response theory models. Applied Psychological Measurement, 24(1), 50-64. See online.
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Settings

IRT models setting

Set the number of cycles for IRT 1PL, 2PL, 3PL and 4PL models.

Figure downloads

Here you can change setting for download of figures.