### Data

For demonstration purposes, 20-item dataset GMAT from difNLR R package is used. On this page, you may select one of five datasets offered by difNLR and ShinyItemAnalysis packages or you may upload your own dataset (see below). To return to demonstration dataset, refresh this page in your browser (F5) .

#### Training datasets

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 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.

### Data exploration

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

### 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.

#### Selected R code

library(difNLR)library(ggplot2)library(moments)# loading datadata(GMAT)data <- GMAT[, 1:20]# total score calculationscore <- apply(data, 1, sum)# summary of total score c(min(score), max(score), mean(score), median(score), sd(score), skewness(score), kurtosis(score))# colors by cut-scorecut <- median(score) # cut-score color <- c(rep("red", cut - min(score)), "gray", rep("blue", max(score) - cut))df <- data.frame(score)# histogramggplot(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.

#### Selected R code

library(difNLR) # loading datadata(GMAT) data <- GMAT[, 1:20] # scores calculationsscore <- apply(data, 1, sum)             # Total score tosc <- sort(unique(score))              # Levels of total score perc <- cumsum(prop.table(table(score))) # Percentiles sura <- 100 * (tosc / max(score))        # Success rate zsco <- sort(unique(scale(score)))       # Z-score tsco <- 50 + 10 * zsco                   # T-score

### 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 datadata(HCI)data <- HCI[, 1:20]# reliability of original datarel.original <- psychometric::alpha(data)# number of items in original dataitems.original <- ncol(data)# number of items in new dataitems.new <- 30# ratio of tests lengthsm <- items.new/items.original# determining reliabilitypsychometric::SBrel(Nlength = m, rxx = rel.original)# desired reliabilityrel.new <- 0.8# determining test length(m.new <- psychometric::SBlength(rxxp = rel.new, rxx = rel.original))# number of required itemsm.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.

### 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).

#### Selected R code

library(difNLR)library(ShinyItemAnalysis) # loading datadata(GMATtest) data <- GMATtest[, 1:20] data(GMATkey) key <- GMATkey # 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 <- DistractorAnalysis(data, key, num.groups = 3)[[1]] 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 b0 describes horizontal position of the fitted curve, parameter b1 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.

#### Equation

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

#### Selected R code

library(difNLR) library(ggplot2)# loading datadata(GMAT) data <- GMAT[, 1:20] score <- apply(data, 1, sum) # total score# logistic model for item 1 fit <- glm(data[, 1] ~ score, family = binomial) # coefficients coef(fit) # function for plot fun <- function(x, b0, b1){exp(b0 + b1 * x) / (1 + exp(b0 + b1 * x))} # empirical probabilities calculationdf <- data.frame(x = sort(unique(score)),                 y = tapply(data[, 1], score, mean),                 size = as.numeric(table(score)))# plot of estimated curveggplot(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 b0 describes horizontal position of the fitted curve (difficulty), parameter b1 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.

#### Equation

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

#### Selected R code

library(difNLR) library(ggplot2)# loading datadata(GMAT) data <- GMAT[, 1:20] zscore <- scale(apply(data, 1, sum)) # standardized total score# logistic model for item 1 fit <- glm(data[, 1] ~ zscore, family = binomial) # coefficients coef(fit) # function for plot fun <- function(x, b0, b1){exp(b0 + b1 * x) / (1 + exp(b0 + b1 * x))} # empirical probabilities calculationdf <- data.frame(x = sort(unique(zscore)),                 y = tapply(data[, 1], zscore, mean),                 size = as.numeric(table(zscore)))# plot of estimated curveggplot(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.

#### Equation

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

#### Selected R code

library(difNLR) library(ggplot2)# loading datadata(GMAT) data <- GMAT[, 1:20] zscore <- scale(apply(data, 1, sum)) # standardized total score# logistic model for item 1 fit <- glm(data[, 1] ~ zscore, family = binomial) # coefficientscoef <- c(a = coef(fit)[2], b = - coef(fit)[1] / coef(fit)[2]) coef  # function for plot fun <- function(x, a, b){exp(a * (x - b)) / (1 + exp(a * (x - b)))} # empirical probabilities calculationdf <- data.frame(x = sort(unique(zscore)),                 y = tapply(data[, 1], zscore, mean),                 size = as.numeric(table(zscore)))# plot of estimated curveggplot(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.

#### Equation

$$\mathrm{P}(Y = 1|Z, b_0, b_1, c) = \mathrm{E}(Y|Z, b_0, b_1, c) = c + \left( 1-c \right) \cdot \frac{e^{a\left(Z-b\right) }}{1+e^{a\left(Z-b\right) }}$$

#### Selected R code

library(difNLR) library(ggplot2)# loading datadata(GMAT) data <- GMAT[, 1:20] zscore <- scale(apply(data, 1, sum)) # standardized total score# NLR 3P model for item 1 fun <- function(x, a, b, c){c + (1 - c) * exp(a * (x - b)) / (1 + exp(a * (x - b)))} fit <- nls(data[, 1] ~ fun(zscore, a, b, c),            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 calculationdf <- data.frame(x = sort(unique(zscore)),                 y = tapply(data[, 1], zscore, mean),                 size = as.numeric(table(zscore)))# plot of estimated curveggplot(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.

#### Equation

$$\mathrm{P}(Y = 1|Z, b_0, b_1, c) = \mathrm{E}(Y|Z, b_0, b_1, c) = c + \left( d-c \right) \cdot \frac{e^{a\left(Z-b\right) }}{1+e^{a\left(Z-b\right) }}$$

#### Selected R code

library(difNLR) library(ggplot2)# loading datadata(GMAT) data <- GMAT[, 1:20] zscore <- scale(apply(data, 1, sum)) # standardized total score# NLR 4P model for item 1 fun <- function(x, a, b, c, d){c + (d - c) * exp(a * (x - b)) / (1 + exp(a * (x - b)))} fit <- nls(data[, 1] ~ fun(zscore, a, b, c, d),            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 calculationdf <- data.frame(x = sort(unique(zscore)),                 y = tapply(data[, 1], zscore, mean),                 size = as.numeric(table(zscore)))# plot of estimated curveggplot(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 datadata(GMAT) Data <- GMAT[, 1:20] zscore <- scale(apply(Data, 1, sum)) # standardized total score# function for fitting modelsfun <- function(x, a, b, c, d){c + (d - c) * exp(a * (x - b)) / (1 + exp(a * (x - b)))} # starting values for item 1start <- startNLR(Data, GMAT[, "group"], model = "4PLcgdg", parameterization = "classic")[[1]][, 1:4]# 2PL model for item 1 fit2PL <- nls(Data[, 1] ~ fun(zscore, a, b, c = 0, d = 1),               algorithm = "port",               start = start[1:2]) # NLR 3P model for item 1 fit3PL <- nls(Data[, 1] ~ fun(zscore, a, b, c, d = 1),               algorithm = "port",               start = start[1:3],              lower = c(-Inf, -Inf, 0),               upper = c(Inf, Inf, 1)) # NLR 4P model for item 1 fit3PL <- nls(Data[, 1] ~ fun(zscore, a, b, c, d),               algorithm = "port",               start = start,              lower = c(-Inf, -Inf, 0, 0),               upper = c(Inf, Inf, 1, 1)) # comparison ### AICAIC(fit2PL); AIC(fit3PL); AIC(fit4PL) ### BICBIC(fit2PL); BIC(fit3PL); BIC(fit4PL) ### LR test, using Benjamini-Hochberg correction###### 2PL vs NLR 3PLRstat <- -2 * (sapply(fit2PL, logLik) - sapply(fit3PL, logLik)) LRdf <- 1 LRpval <- 1 - pchisq(LRstat, LRdf) LRpval <- p.adjust(LRpval, method = "BH") ###### NLR 3P vs NLR 4PLRstat <- -2 * (sapply(fit3PL, logLik) - sapply(fit4PL, logLik)) LRdf <- 1 LRpval <- 1 - pchisq(LRstat, LRdf) LRpval <- p.adjust(LRpval, method = "BH")

### 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.

Interpretation:

#### Selected R code

library(difNLR) library(nnet) # loading datadata(GMAT, GMATtest, GMATkey) zscore <- scale(apply(GMAT[, 1:20] , 1, sum)) # standardized total scoredata <- GMATtest[, 1:20] key <-GMATkey# multinomial model for item 1 fit <- multinom(relevel(data[, 1], ref = paste(key[1])) ~ zscore) # coefficients coef(fit)

### 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

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

#### 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.

#### 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.

#### 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.

#### Selected R code

library(difNLR)library(mirt) library(ShinyItemAnalysis)# loading datadata(GMAT) data <- GMAT[, 1:20] # fitting Rasch modelfit <- mirt(data, model = 1, itemtype = 'Rasch', 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) # Wright Map b <- sapply(1:ncol(data), function(i) coef(fit)[[i]][, 'd']) 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

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

#### 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

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

#### 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.

#### 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.

#### 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.

library(difNLR)library(mirt) library(ShinyItemAnalysis)# loading datadata(GMAT) data <- GMAT[, 1:20] # fitting 1PL modelfit <- mirt(data, model = 1, itemtype = '2PL', constrain = list((1:ncol(data)) + seq(0, (ncol(data) - 1)*3, 3)), 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) # Wright Map b <- sapply(1:ncol(data), function(i) coef(fit)[[i]][, 'd']) ggWrightMap(fs, b)# You can also use ltm library for IRT models #  fitting 1PL modelfit <- rasch(data) # for Rasch model use # fit <- rasch(data, constraint = cbind(ncol(data) + 1, 1)) # 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') ### 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 #### Item information curves #### 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 #### Item information curves #### Test information function #### 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. #### 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. #### Scatter plot of factor scores and standardized total scores #### 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

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

#### 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

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

#### 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.

#### 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.

#### 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

### Logistic regression on total scores

Logistic regression allows for detection of uniform and non-uniform DIF (Swaminathan & Rogers, 1990) by adding a group specific intercept b2 (uniform DIF) and group specific interaction b3 (non-uniform DIF) into model and by testing for their significance.

#### 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

Here you can choose what type of DIF to test. You can also select correction method for multiple comparison or item purification.

#### Selected R code

library(difNLR)
library(difR)
data(GMAT)
data <- GMAT[, 1:20]
group <- GMAT[, "group"]

# Logistic regression DIF detection method
fit <- difLogistic(Data = data, group = group, focal.name = 1, type = "both", p.adjust.method = "none", purify = F)
fit

### Logistic regression on total scores

Logistic regression allows for detection of uniform and non-uniform DIF (Swaminathan & Rogers, 1990) by adding a group specific intercept b2 (uniform DIF) and group specific interaction b3 (non-uniform DIF) into model and by testing for their significance.

#### Plot with estimated DIF logistic curve

Here you can choose what type of DIF to test. You can also select correction method for multiple comparison or item purification.

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 with respect to the group membership.

NOTE: Plots and tables are based on DIF logistic procedure without any correction method.

#### Equation

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

#### Selected R code

library(difNLR)
library(difR)
data(GMAT)
data <- GMAT[, 1:20]
group <- GMAT[, "group"]

# Logistic regression DIF detection method
fit <- difLogistic(Data = data, group = group, focal.name = 1, type = "both", p.adjust.method = "none", purify = F)
fit
# Plot of characteristic curve for item 1
plotDIFLogistic(data, group, type = "both", item = 1, IRT = F, p.adjust.method = "none", purify = F)
# Coefficients

### Lord test for IRT models

Lord test (Lord, 1980) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the difference between item parameters for the two groups to detect DIF. In statistical terms, Lord statistic is equal to Wald statistic.

#### Summary table

Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification.

#### Selected R code

library(difNLR)
library(difR)
data(GMAT)
data <- GMAT[, 1:20]
group <- GMAT[, "group"]

# 1PL IRT MODEL
fit1PL <- difLord(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F)
fit1PL

# 2PL IRT MODEL
fit2PL <- difLord(Data = data, group = group, focal.name = 1, model = "2PL", p.adjust.method = "none", purify = F)
fit2PL

# 3PL IRT MODEL with the same guessing for groups
guess <- itemParEst(data, model = "3PL")[, 3]
fit3PL <- difLord(Data = data, group = group, focal.name = 1, model = "3PL", c = guess, p.adjust.method = "none", purify = F)
fit3PL

### Lord test for IRT models

Lord test (Lord, 1980) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the difference between item parameters for the two groups to detect DIF. In statistical terms, Lord statistic is equal to Wald statistic.

#### Plot with estimated DIF characteristic curve

Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification.

NOTE: Plots and tables are based on larger DIF IRT model.

#### Selected R code

library(difNLR)
library(difR)
data(GMAT)
data <- GMAT[, 1:20]
group <- GMAT[, "group"]

# 1PL IRT MODEL
fit1PL <- difLord(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F)
fit1PL
# Coefficients for all items
tab_coef1PL <- fit1PL$itemParInit # Plot of characteristic curve of item 1 plotDIFirt(parameters = tab_coef1PL, item = 1, test = "Lord") # 2PL IRT MODEL fit2PL <- difLord(Data = data, group = group, focal.name = 1, model = "2PL", p.adjust.method = "none", purify = F) fit2PL # Coefficients for all items tab_coef2PL <- fit2PL$itemParInit
# Plot of characteristic curve of item 1
plotDIFirt(parameters = tab_coef2PL, item = 1, test = "Lord")

# 3PL IRT MODEL with the same guessing for groups
guess <- itemParEst(data, model = "3PL")[, 3]
fit3PL <- difLord(Data = data, group = group, focal.name = 1, model = "3PL", c = guess, p.adjust.method = "none", purify = F)
fit3PL
# Coefficients for all items
tab_coef3PL <- fit3PL$itemParInit # Plot of characteristic curve of item 1 plotDIFirt(parameters = tab_coef3PL, item = 1, test = "Lord") ### Raju test for IRT models Raju test (Raju, 1988, 1990) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the area between the item charateristic curves for the two groups to detect DIF. #### Summary table Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification. #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] # 1PL IRT MODEL fit1PL <- difRaju(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F) fit1PL # 2PL IRT MODEL fit2PL <- difRaju(Data = data, group = group, focal.name = 1, model = "2PL", p.adjust.method = "none", purify = F) fit2PL # 3PL IRT MODEL with the same guessing for groups guess <- itemParEst(data, model = "3PL")[, 3] fit3PL <- difRaju(Data = data, group = group, focal.name = 1, model = "3PL", c = guess, p.adjust.method = "none", purify = F) fit3PL ### Raju test for IRT models Raju test (Raju, 1988, 1990) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the area between the item charateristic curves for the two groups to detect DIF. #### Plot with estimated DIF characteristic curve Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification. NOTE: Plots and tables are based on larger DIF IRT model. #### Equation #### Table of parameters #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] # 1PL IRT MODEL fit1PL <- difRaju(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F) fit1PL # Coefficients for all items tab_coef1PL <- fit1PL$itemParInit
# Plot of characteristic curve of item 1
plotDIFirt(parameters = tab_coef1PL, item = 1, test = "Raju")

# 2PL IRT MODEL
fit2PL <- difRaju(Data = data, group = group, focal.name = 1, model = "2PL", p.adjust.method = "none", purify = F)
fit2PL
# Coefficients for all items
tab_coef2PL <- fit2PL$itemParInit # Plot of characteristic curve of item 1 plotDIFirt(parameters = tab_coef2PL, item = 1, test = "Raju") # 3PL IRT MODEL with the same guessing for groups guess <- itemParEst(data, model = "3PL")[, 3] fit3PL <- difRaju(Data = data, group = group, focal.name = 1, model = "3PL", c = guess, p.adjust.method = "none", purify = F) fit3PL # Coefficients for all items tab_coef3PL <- fit3PL$itemParInit
# Plot of characteristic curve of item 1
plotDIFirt(parameters = tab_coef3PL, item = 1, test = "Raju")

### SIBTEST

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

#### Summary table

Here you can choose type of DIF to be tested. 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.

#### Selected code

library(difNLR)library(difR)# loading datadata(GMAT)data <- GMAT[, 1:20]group <- GMAT[, "group"]# SIBTEST (uniform DIF)fit <- difMH(Data = data, group = group, focal.name = 1, type = "udif", p.adjust.method = "none", purify = F)fit# Crossing-SIBTEST (non-uniform DIF)fit <- difMH(Data = data, group = group, focal.name = 1, type = "nudif", p.adjust.method = "none", purify = F)fit

### Differential Distractor Functioning with multinomial log-linear regression model

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.

#### Equation

For K possible test choices is the probability of the correct answer for person i with standardized total score Z and group membership G in item j given by the following equation:

$$\mathrm{P}(Y_{ij} = K|Z_i, G_i, b_{jl0}, b_{jl1}, b_{jl2}, b_{jl3}, l = 1, \dots, K-1) = \frac{1}{1 + \sum_l e^{\left( b_{il0} + b_{il1} Z + b_{il2} G + b_{il3} Z:G\right)}}$$

The probability of choosing distractor k is then given by:

$$\mathrm{P}(Y_{ij} = k|Z_i, G_i, b_{jl0}, b_{jl1}, b_{jl2}, b_{jl3}, l = 1, \dots, K-1) = \frac{e^{\left( b_{jk0} + b_{jk1} Z_i + b_{jk2} G_i + b_{jk3} Z_i:G_i\right)}} {1 + \sum_l e^{\left( b_{jl0} + b_{jl1} Z_i + b_{jl2} G_i + b_{jl3} Z_i:G_i\right)}}$$

#### Summary table

Here you can choose what type of DIF to test. You can also select correction method for multiple comparison or item purification.

#### Selected R code

library(difNLR)
data(GMATtest, GMATkey)
Data <- GMATtest[, 1:20]
group <- GMATtest[, "group"]
key <- GMATkey

# DDF with difNLR package
fit <- ddfMLR(Data, group, focal.name = 1, key, type = "both", p.adjust.method = "none")
fit

### Differential Distractor Functioning with multinomial log-linear regression model

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.

#### Plot with estimated DDF curves

Here you can choose what type of DIF to test. You can also select correction method for multiple comparison or item purification.

Points represent proportion of selected answer 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 with respect to the group membership.

#### Selected R code

library(difNLR)
data(GMATtest, GMATkey)
Data <- GMATtest[, 1:20]
group <- GMATtest[, "group"]
key <- GMATkey

# DDF with difNLR package
fit <- ddfMLR(Data, group, focal.name = 1, key, type = "both", p.adjust.method = "none")
# Estimated coefficients of item 1
fit\$mlrPAR[[1]]

#### 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

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, built on R and shiny, for psychometric analysis of educational and other psychological tests and their items. 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.

#### Availability

It is also available online at Czech Academy of Sciences and shinyapps.io .

#### Versions

Current CRAN version is 1.3.0.
Version available online is 1.3.0.
The newest development version available on GitHub is 1.3.0.

#### 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.

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

In Czech written papers you can also use

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. In Czech].
TESTFORUM, 6(9), 16-35. doi: 10.5817/TF2017-9-129

#### 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 grant 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 2.3.2. See online.
• data.table Dowle, M. & Srinivasan, A. (2018). data.table: Extension of data.frame. R package version 1.11.4. See online.
• deltaPlotR Magis, D. & Facon, B. (2014). deltaPlotR: An R Package for Differential Item Functioning Analysis with Angoffs Delta Plot. Journal of Statistical Software, Code Snippets, 59(1), 1--19. See online.
• difNLR Drabinova, A., Martinkova, P. & Zvara, K. (2018). difNLR: DIF and DDF Detection by Non-Linear Regression Models. R package version 1.2.2. 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. (2018). DT: A Wrapper of the JavaScript Library DataTables. R package version 0.4. See online.
• ggdendro Andrie de Vries & Brian D. Ripley (2018). 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. (2018). knitr: A General-Purpose Package for Dynamic Report Generation in R. R package version 1.20. See online.
• lattice Sarkar, D. (2008). Lattice: Multivariate Data Visualization with R. 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.
• MASS Venables, C. & Ripley, C. (2002). Modern Applied Statistics with S. 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.
• multilevel Bliese, P. (2016). multilevel: Multilevel Functions. R package version 2.6. See online.
• nlme Pinheiro, J., Bates, D., DebRoy, S., Sarkar, D. & NULL, R. (2018). nlme: Linear and Nonlinear Mixed Effects Models. R package version 3.1-137. 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 4.7.1. See online.
• polycor Fox, J. (2016). polycor: Polychoric and Polyserial Correlations. R package version 0.7-9. See online.
• psych Revelle, W. (2018). psych: Procedures for Psychological, Psychometric, and Personality Research. R package version 1.8.4. See online.
• psychometric Fletcher, T. & Fletcher, D. (2010). psychometric: Applied Psychometric Theory. R package version 2.2. See online.
• RColorBrewer Neuwirth, E. (2014). RColorBrewer: ColorBrewer Palettes. R package version 1.1-2. See online.
• reshape2 Wickham, H. (2007). Reshaping Data with the reshape Package. Journal of Statistical Software, 21(12), 1--20. See online.
• rmarkdown Allaire, J., Xie, Y., McPherson, J., Luraschi, J., Ushey, K., Atkins, A., Wickham, H., Cheng, J. & Chang, W. (2018). rmarkdown: Dynamic Documents for R. R package version 1.10. See online.
• shiny Chang, W., Cheng, J., Allaire, J., Xie, Y. & McPherson, J. (2018). shiny: Web Application Framework for R. R package version 1.1.0. See online.
• shinyBS Bailey, E. (2015). shinyBS: Twitter Bootstrap Components for Shiny. R package version 0.61. 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. (2018). stringr: Simple, Consistent Wrappers for Common String Operations. R package version 1.3.1. See online.
• xtable` Dahl, D. & Dahl, B. (2016). xtable: Export Tables to LaTeX or HTML. R package version 1.8-2. See online.

### References

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• Ames, A. J., & Penfield, R. D. (2015). An NCME Instructional Module on Item-Fit Statistics for Item Response Theory Models. Educational Measurement: Issues and Practice, 34(3), 39-48. See online.
• Andrich, D. (1978). A Rating Formulation for Ordered Response Categories. Psychometrika, 43(4), 561-573. See online.
• 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.
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• 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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• 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 Psychologických 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.
• Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174. See online.
• McFarland, J. L., Price, R. M., Wenderoth, M. P., Martinkova, P., Cliff, W., Michael, J., ... & Wright, A. (2017). Development and Validation of the Homeostasis Concept Inventory. CBE-Life Sciences Education, 16(2), ar35. See online.
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• Swaminathan, H., & Rogers, H. J. (1990). Detecting Differential Item Functioning Using Logistic Regression Procedures. Journal of Educational Measurement, 27(4), 361-370. See online.
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• Revelle, W. (1979). Hierarchical cluster analysis and the internal structure of tests. Multivariate Behavioral Research, 14(1), 57-74. See online.
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• Wilson, M. (2005). Constructing Measures: An Item Response Modeling Approach.
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#### IRT models setting

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