### Description

ShinyItemAnalysis provides analysis of educational tests (such as admission tests) and their items including:

• Exploration of total and standard scores on Summary page.
• Correlation structure and predictive validity analysis on Validity page.
• Item and distractor analysis on Item analysis page.
• Item analysis by logistic models on Regression page.
• Item analysis by item response theory models on IRT models page.
• Differential item functioning (DIF) and differential distractor functioning (DDF) methods on DIF/Fairness page.

This application is based on the free statistical software R and its shiny package.

For all graphical outputs a download button is provided. Moreover, on Reports page HTML or PDF report can be created. Additionaly, all application outputs are complemented by selected R code hence the similar analysis can be run and modified in R.

#### Data

For demonstration purposes, by default, 20-item dataset GMAT from R difNLR package is used. Other four datasets are available: GMAT2 and MSAT-B from difNLR package and Medical 100 and HCI from ShinyItemAnalysis package. You can change the dataset (and try your own one) on page Data.

#### Availability

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

#### Version

Current version of ShinyItemAnalysis available on CRAN is 1.2.7. Version available online is 1.2.7. The newest development version available on GitHub is 1.2.7.

#### List of packages used

library(corrplot)
library(CTT)
library(data.table)
library(deltaPlotR)
library(DT)
library(difNLR)
library(difR)
library(ggplot2)
library(grid)
library(gridExtra)
library(knitr)
library(latticeExtra)
library(ltm)
library(mirt)
library(moments)
library(msm)
library(nnet)
library(plotly)
library(psych)
library(psychometric)
library(reshape2)
library(rmarkdown)
library(shiny)
library(shinyBS)
library(shinyjs)
library(stringr)
library(WrightMap)
library(xtable)

#### References

To cite package ShinyItemAnalysis in publications please use:

Martinkova P., Drabinova A., Leder O., & Houdek J. (2018). ShinyItemAnalysis: Test and item analysis via shiny. R package version 1.2.6. https://CRAN.R-project.org/package=ShinyItemAnalysis

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. doi:10.5817/TF2017-9-129

#### Bug reports

If you discover a problem with this application please contact the project maintainer at martinkova(at)cs.cas.cz or use GitHub.

#### Acknowledgments

Project was supported by grant funded by Czech Science Foundation under number GJ15-15856Y.

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.

### 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

Used dataset GMAT (Martinkova, et al., 2017) is generated dataset based on parameters of real Graduate Management Admission Test (GMAT) (Kingston et al., 1985). However, first two items were simulated to function differently in uniform and non-uniform way respectively. The dataset represents responses of 2,000 subjects (1,000 males, 1,000 females) to multiple-choice test of 20 items. The distribution of total scores is the same for both groups. See Martinkova, et al. (2017) for further discussion. GMAT containts simulated continuous criterion variable.

GMAT2 (Drabinova & Martinkova, 2017) is also simulated dataset based on parameters of GMAT (Kingston et al., 1985) from difNLR R package . Again, first two items were generated to function differently in uniform and non-uniform way respectively. The dataset represents responses of 1,000 subjects (500 males, 500 females) to multiple-choice test of 20 items.

MSAT-B (Drabinova & Martinkova, 2017) is a subset of real Medical School Admission Test in Biology in Czech Republic. The dataset represents responses of 1,407 subjects (484 males, 923 females) to multiple-choice test of 20 items. First item was previously detected as functioning differently. For more details of item selection see Drabinova and Martinkova (2017). Dataset can be found in difNLR R package.

Medical 100 is a real dataset of admission test to medical school from ShinyItemAnalysis R package. The data set represents responses of 2,392 subjects (750 males, 1,633 females and 9 subjects without gender specification) to multiple-choice test of 100 items. Medical 100 contains criterion variable - indicator whether student studies standardly or not.

HCI (McFarland et al., 2017) is a real dataset of Homeostasis Concept Inventory from ShinyItemAnalysis R package. The dataset represents responses of 651 subjects (405 males, 246 females) to multiple-choice test of 20 items. HCI contains criterion variable - indicator whether student plans to major in the life sciences.

Main data file should contain responses of individual respondents (rows) to given items (columns). Header may contain item names, no row names should be included. If responses are in unscored ABCD format, the key provides correct response for each item. If responses are scored 0-1, key is vector of 1s.

Group is 0-1 vector, where 0 represents reference group and 1 represents focal group. Its length need to be the same as number of individual respondents in main dataset. If the group is not provided then it wont be possible to run DIF and DDF detection procedures on DIF/Fairness page.

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. Again, 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.

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 items 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 exploration

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

### Analysis of total scores

#### 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 $$\frac{m_4}{m_2^2}$$, where $$m_4$$ is the fourth central moment and $$m_2$$ is the second central moment. The skewness is estimated by sample skewness $$\frac{m_3}{m_2^{3/2}}$$, 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_bw() +   theme(legend.title = element_blank(),         axis.line  = element_line(colour = "black"),         panel.grid.major = element_blank(),         panel.grid.minor = element_blank(),         text = element_text(size = 14))

### 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

### Correlation structure

#### Polychoric correlation heat map

Polychoric correlation heat map is a correlation plot which displays a polychoric correlations of items. The size and shade of circles indicate how much the items are correlated (larger and darker circle means larger correlation). The color of circles indicates in which way the items are correlated - blue color shows possitive correlation and red color shows negative correlation.

Polychoric correlation heat map can be reordered using hierarchical clustering method below. Ward's method aims at finding compact clusters based on minimizing the within-cluster sum of squares. Ward's n. 2 method used 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, i.e. maximum of distances. Average linkage method used the distance based on weighted average of the individual distances. With McQuitty method used 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 used distance between centroids of clusters.

With number of clusters larger than 1, the rectangles representing clusters are drawn.

#### 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.

### 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 = T) # distractors - plot for item 1 and 3 groups plotDistractorAnalysis(data, key, num.group = 3, item = 1, multiple.answers = F) # 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 = T, value.var = "Freq") # table with proportions - item 1 and 3 groups DistractorAnalysis(data, key, num.groups = 3, p.table = T)[[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")

### 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")

### 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")

### 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")

### 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")

### 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 a random effect and is estimated together with item paramters. Ability (theta) is often assumed to follow normal distibution.

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.

#### 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 parameters with item fit statistics

Estimates of parameters are completed by SX2 item fit statistics (Ames & Penfield, 2015). SX2 is computed only when no missing data are present. In such a case consider using imputed dataset!

#### 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(WrightMap)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- 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"])
wrightMap(fs, b, item.side = itemClassic)

### 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 a random effect and is estimated together with item paramters. Ability (theta) is often assumed to follow normal distibution.

In 1PL IRT model, all items are assumed to have the same slope in inflection point, i.e., the same discrimination a. Items can differ in location of their inflection point, i.e., in item difficulty parameters b.

#### 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 parameters with item fit statistics

Estimates of parameters are completed by SX2 item fit statistics (Ames & Penfield, 2015). SX2 is computed only when no missing data are present. In such a case consider using imputed dataset!

#### 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(WrightMap)
data(GMAT)
data <- GMAT[, 1:20]

# Model
fit <- 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"])
wrightMap(fs, b, item.side = itemClassic)

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

# Model
fit <- 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") ### 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 a random effect and is estimated together with item paramters. Ability (theta) is often assumed to follow normal distibution. 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. #### 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 parameters with item fit statistics Estimates of parameters are completed by SX2 item fit statistics (Ames & Penfield, 2015). SX2 is computed only when no missing data are present. In such a case consider using imputed dataset! #### 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")

### 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 a random effect and is estimated together with item paramters. Ability (theta) is often assumed to follow normal distibution.

3PL IRT model allows for different discriminations of items a, different item difficulties b, and allows also for nonzero left asymptote, pseudo-guessing c.

#### 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 parameters with item fit statistics

Estimates of parameters are completed by SX2 item fit statistics (Ames & Penfield, 2015). SX2 is computed only when no missing data are present. In such a case consider using imputed dataset!

#### 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
fit$logitPar ### Logistic regression on standardized total scores with IRT parameterization Logistic regression allows for detection of uniform and non-uniform DIF (Swaminathan & Rogers, 1990) by adding a group specific intercept bDIF (uniform DIF) and group specific interaction aDIF (non-uniform DIF) into model and by testing for their significance. #### Equation $$\mathrm{P}\left(Y_{ij} = 1 | Z_i, G_i, a_j, b_j, a_{\text{DIF}j}, b_{\text{DIF}j}\right) = \frac{e^{\left(a_j + a_{\text{DIF}j} G_i\right) \left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}}{1+e^{\left(a_j + a_{\text{DIF}j} G_i\right) \left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}}$$ #### Summary table Here you can choose what type of DIF to test. You can also select correction method for multiple comparison. #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] scaled.score <- scale(score) # Logistic regression DIF detection method fit <- difLogistic(Data = data, group = group, focal.name = 1, type = "both", match = scaled.score, p.adjust.method = "none", purify = F) fit ### Logistic regression on standardized total scores with IRT parameterization Logistic regression allows for detection of uniform and non-uniform DIF by adding a group specific intercept bDIF (uniform DIF) and group specific interaction aDIF (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. 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 with respect to the group membership. NOTE: Plots and tables are based on DIF logistic procedure without any correction method. #### Equation $$\mathrm{P}\left(Y_{ij} = 1 | Z_i, G_i, a_j, b_j, a_{\text{DIF}j}, b_{\text{DIF}j}\right) = \frac{e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}} {1+e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}}$$ #### Table of parameters #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] scaled.score <- scale(score) # Logistic regression DIF detection method fit <- difLogistic(Data = data, group = group, focal.name = 1, type = "both", match = scaled.score, p.adjust.method = "none", purify = F) fit # Plot of characteristic curve for item 1 plotDIFLogistic(data, group, type = "both", item = 1, IRT = T, p.adjust.method = "BH") # Coefficients for item 1 - recalculation coef_old <- fit$logitPar[1, ]
coef <- c()
# a = b1, b = -b0/b1, adif = b3, bdif = -(b1b2-b0b3)/(b1(b1+b3))
coef[1] <- coef_old[2]
coef[2] <- -(coef_old[1] / coef_old[2])
coef[3] <- coef_old[4]
coef[4] <- -(coef_old[2] * coef_old[3] + coef_old[1] * coef_old[4] ) / (coef_old[2] * (coef_old[2] + coef_old[4]))

### Nonlinear regression on standardized total scores with IRT parameterization

Nonlinear regression model allows for nonzero lower asymptote - pseudoguessing c (Drabinova & Martinkova, 2017). Similarly to logistic regression, also nonlinear regression allows for detection of uniform and non-uniform DIF by adding a group specific intercept bDIF (uniform DIF) and group specific interaction aDIF (non-uniform DIF) into the model and by testing for their significance.

#### Equation

$$\mathrm{P}\left(Y_{ij} = 1 | Z_i, G_i, a_j, b_j, c_j, a_{\text{DIF}j}, b_{\text{DIF}j}\right) = c_j + \left(1 - c_j\right) \cdot \frac{e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}} {1+e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}}$$

#### 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(GMAT)
Data <- GMAT[, 1:20]
group <- GMAT[, "group"]

# Nonlinear regression DIF method
fit <- difNLR(Data = Data, group = group, focal.name = 1, model = "3PLcg", type = "both", p.adjust.method = "none")
fit

### Nonlinear regression on standardized total scores with IRT parameterization

Nonlinear regression model allows for nonzero lower asymptote - pseudoguessing c (Drabinova & Martinkova, 2017). Similarly to logistic regression, also nonlinear regression allows for detection of uniform and non-uniform DIF by adding a group specific intercept bDIF (uniform DIF) and group specific interaction aDIF (non-uniform DIF) into the model and by testing for their significance.

#### Plot with estimated DIF nonlinear 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 standardized total score. Their size is determined by count of respondents who achieved given level of standardized total score with respect to the group membership.

#### Equation

$$\mathrm{P}\left(Y_{ij} = 1 | Z_i, G_i, a_j, b_j, c_j, a_{\text{DIF}j}, b_{\text{DIF}j}\right) = c_j + \left(1 - c_j\right) \cdot \frac{e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}} {1+e^{\left(a_j + a_{\text{DIF}j} G_i\right)\left(Z_i -\left(b_j + b_{\text{DIF}j} G_i\right)\right)}}$$

#### Selected R code

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

# Nonlinear regression DIF method
fit <- difNLR(Data = Data, group = group, focal.name = 1, model = "3PLcg", type = "both", p.adjust.method = "none")
# Plot of characteristic curve of item 1
plot(fit, item = 1)
# Coefficients
fit$nlrPAR ### Lord test for IRT models Lord test (Lord, 1980) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the difference between item parameters for the two groups to detect DIF. In statistical terms, Lord statistic is equal to Wald statistic. #### Summary table Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification. #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] # 1PL IRT MODEL fit1PL <- difLord(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F) fit1PL # 2PL IRT MODEL fit2PL <- difLord(Data = data, group = group, focal.name = 1, model = "2PL", p.adjust.method = "none", purify = F) fit2PL # 3PL IRT MODEL with the same guessing for groups guess <- itemParEst(data, model = "3PL")[, 3] fit3PL <- difLord(Data = data, group = group, focal.name = 1, model = "3PL", c = guess, p.adjust.method = "none", purify = F) fit3PL ### Lord test for IRT models Lord test (Lord, 1980) is based on IRT model (1PL, 2PL, or 3PL with the same guessing). It uses the difference between item parameters for the two groups to detect DIF. In statistical terms, Lord statistic is equal to Wald statistic. #### Plot with estimated DIF characteristic curve Here you can choose model to test DIF. You can also select correction method for multiple comparison or item purification. NOTE: Plots and tables are based on larger DIF IRT model. #### Equation #### Table of parameters #### Selected R code library(difNLR) library(difR) data(GMAT) data <- GMAT[, 1:20] group <- GMAT[, "group"] # 1PL IRT MODEL fit1PL <- difLord(Data = data, group = group, focal.name = 1, model = "1PL", p.adjust.method = "none", purify = F) fit1PL # Coefficients for all items tab_coef1PL <- fit1PL$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.

#### 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

#### 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 whether to use customize settings. By checking the Customize settings local settings will be offered and use for each selected section of report. Otherwise the settings will be taken from pages of application. You can also include your name into report as well as the name of dataset which was used.

#### Content of report

Reports by default contain summary of total scores, table of standard scores, item analysis, distractors 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.

### References

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