Item Response Theory

Advanced Psychometrics using Item Response Theory, the Rasch Model, and related concepts.

We previously examined psychometrics using measures such as alpha, GLB, and related measures, to help us look whether questions are representative and might be worthwhile using. These measures are generally applicable to a number of measures, especially questionnaires and scales in which there is no correct answer. When we have a set of questions that can be thought of as a knowledge test, or at least a directional scale we are trying to use to discriminate, there are additional psychometric properties related to distinguishing people that we might be interested in. For example, a question might be easier or more difficult, and we might want to include both easy and difficult questions. Easy questions might help distinguish between the least knowledgeable people and everybody else. Similarly, a difficult question might distinguish between the most knowledgeable person and every one else. In addition, we might want to know whether an individual question is good at separating your tested population into two groups, regardless of difficulty. Looking at the discriminability of items and tests can be helpful by allowing us to design tests appropriate to their use.

For example, the quantitative GRE discriminates those with low math skills from those with better math skills, but it might not be appropriate for predicting success in a mathematics graduate program, because everyone who gets admitted will be in the 90th percentile or above. Similarly, a higher-level calculus test might be good at this, but would be useful as a measure of math achievement for elementary school children.
A systematic approach for doing this can simply use regression modeling to try to capture different aspects of test items and the people taking the test. If we think about modeling an entire test by considering the chance of getting any item correct, we can frame a set of scored test questions as a single DV (correctness) and a single IV (question ID). Then, the intercept or additive constant for each question essentially identifies how difficult or easy the question is. Similarly, we might also include student as a second predictor. This is similar to how we do repeated measures models so that all observations of an individual can be linked, but this essentially captures how some people are better on average than others. This model helps provide understanding of whether individual questions are predictive of the whole. As an example, we will begin by fitting a logistic regression to two parallel tests–an easy and a difficult one–given to a single group of people.

Fitting subject parameters in logistic regression

In a regression model, it is common to include participant as a predictor variable to account for overall individual variability, and this is essentially how mixed-effects models work when you give subject a random intercept. Suppose that you have a test with ten questions, and with individual variability across 50 individuals. Also, let’s suppose that each question has a different difficulty.

For participant j and question i, we can think about the log-odds of successfully answering a question as being related to both the difficulty of the question and the ability of the person. The simplest version of this would be to take a factor \(\theta_j\) related to the ability of the person and add to it a value related to the easiness of each question. This is the same as subtracting \(b_i\)–something related to the difficulty of a question. So, a linear prediction in log-odds space would be (\(\theta_j -b_i\))

logodds <- function(p) {
    log(p/1 - p)
}  ##The probability of a 'yes' for a given set of predictor values. 
logit <- function(lo) {
    1/(1 + exp(-lo))
}  ##This is the inverse of the logodds 

set.seed(1009)
numsubs <- 50
numqs <- 20
skilllevel <- rnorm(numsubs)
questiondiff <- rnorm(numqs)
combined <- outer(skilllevel, questiondiff, function(x, y) {
    x - y
})
pcorrect <- logit(combined)
pcorrect.2 <- logit(combined + 2)  ## An easier test with the same subjects and problems.

Now, the matrix pcorrect indicates the probability of each person answering each question correctly. We can simulate a given experiment by comparing each probability value to a randomly chosen number uniformly between 0 and 1

sim1 <- pcorrect > runif(numsubs * numqs)
sim2 <- pcorrect.2 > runif(numsubs * numqs)

Now, because this is all framed in terms of a log-odds an logistic transforms, we should be able to take the data in sim1 and estimate the parameters used to create them using logistic regression. To do so, we need to put the matrix in long format:

simdat <- data.frame(sub = factor(rep(1:numsubs, numqs)), question = factor(rep(1:numqs,
    each = numsubs)), corr = as.vector(sim1) + 0)

simdat.2 <- data.frame(sub = factor(rep(1:numsubs, numqs)), question = factor(rep(1:numqs,
    each = numsubs)), corr = as.vector(sim2) + 0)

Now, we just fit a logistic regression model. Because the baseline data had no intercept, we can re-estimate the parameters using a no-intercept model (specify +0 in the predictors)

model <- glm(corr ~ 0 + sub + question, family = binomial(), data = simdat)
summary(model)


Call:
glm(formula = corr ~ 0 + sub + question, family = binomial(), 
    data = simdat)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.7131  -0.8781   0.2908   0.8382   2.3009  

Coefficients:
           Estimate Std. Error z value Pr(>|z|)    
sub1         3.5041     0.7518   4.661 3.14e-06 ***
sub2         1.1419     0.7397   1.544 0.122642    
sub3         4.7411     0.9359   5.066 4.06e-07 ***
sub4         0.4008     0.8128   0.493 0.621959    
sub5         4.7411     0.9359   5.066 4.06e-07 ***
sub6         5.5506     1.1788   4.709 2.49e-06 ***
sub7         4.2264     0.8377   5.045 4.53e-07 ***
sub8         3.8329     0.7844   4.886 1.03e-06 ***
sub9         2.2103     0.7030   3.144 0.001666 ** 
sub10        1.7097     0.7117   2.402 0.016290 *  
sub11        3.2141     0.7306   4.399 1.09e-05 ***
sub12        4.2264     0.8377   5.045 4.53e-07 ***
sub13        2.4525     0.7039   3.484 0.000494 ***
sub14        3.2141     0.7306   4.399 1.09e-05 ***
sub15        3.2141     0.7306   4.399 1.09e-05 ***
 [ reached getOption("max.print") -- omitted 54 rows ]
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1386.3  on 1000  degrees of freedom
Residual deviance: 1009.8  on  931  degrees of freedom
AIC: 1147.8

Number of Fisher Scoring iterations: 16

anova(model, test = "Chisq")

Analysis of Deviance Table

Model: binomial, link: logit

Response: corr

Terms added sequentially (first to last)

         Df Deviance Resid. Df Resid. Dev  Pr(>Chi)    
NULL                      1000     1386.3              
sub      50   195.73       950     1190.6 < 2.2e-16 ***
question 19   180.72       931     1009.8 < 2.2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

model2 <- glm(corr ~ 0 + sub + question, family = binomial(), data = simdat.2)
summary(model2)


Call:
glm(formula = corr ~ 0 + sub + question, family = binomial(), 
    data = simdat.2)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.75946   0.00006   0.24361   0.49496   1.64680  

Coefficients:
           Estimate Std. Error z value Pr(>|z|)    
sub1          4.560      1.287   3.544 0.000394 ***
sub2          2.038      1.126   1.810 0.070318 .  
sub3          5.346      1.468   3.641 0.000272 ***
sub4          4.059      1.220   3.327 0.000878 ***
sub5         21.895   2272.318   0.010 0.992312    
sub6         21.895   2272.318   0.010 0.992312    
sub7         21.895   2272.318   0.010 0.992312    
sub8         21.895   2272.318   0.010 0.992312    
sub9          5.346      1.468   3.641 0.000272 ***
sub10         2.788      1.141   2.442 0.014599 *  
sub11         4.560      1.287   3.544 0.000394 ***
sub12         5.346      1.468   3.641 0.000272 ***
sub13         3.672      1.186   3.098 0.001951 ** 
sub14         4.059      1.220   3.327 0.000878 ***
sub15         4.560      1.287   3.544 0.000394 ***
 [ reached getOption("max.print") -- omitted 54 rows ]
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1386.29  on 1000  degrees of freedom
Residual deviance:  555.27  on  931  degrees of freedom
AIC: 693.27

Number of Fisher Scoring iterations: 18

We have a lack of identifiability here, because for any set of parameters, I can always add a constant to all subject parameters while subtracting it from all question parameters and obtain the same values. This can be seen in the model coefficients, which don’t have a question1. The performance on question1 is taken as a baseline, and all subject and question parameters are scaled to match it. We can use sum-to-zero contrasts and play with the intercept, and now our question coefficients will sum to 0, meaning the average difficulty of the questions is 0, easy questions have a positive coefficient, and difficult questions have a negative coefficient. Because we are fitting coefficients for each person, this does not mean the average accuracy of a question is 50%. Note that the subject coefficients have a similar interpretation: M

library(ggplot2)
library(tidyverse)
contrasts(simdat$question) <- contr.sum(levels(simdat$question))
contrasts(simdat.2$question) <- contr.sum(levels(simdat.2$question))

model <- glm(corr ~ 0 + sub + question, family = binomial(), data = simdat)
model2 <- glm(corr ~ 0 + sub + question, family = binomial(), data = simdat.2)



items <- data.frame(names = names(model$coefficients[51:69]), model1 = model$coefficients[51:69],
    model2 = model2$coefficients[51:69])



knitr::kable(items)

	names	model1	model2
question1	question1	1.4387805	1.2641483
question2	question2	-1.2205912	-1.7364275
question3	question3	-0.7084914	-1.5774645
question4	question4	16.7194570	16.7384285
question5	question5	-0.8126159	-1.5774645
question6	question6	-1.2205912	-1.8882096
question7	question7	-0.8126159	0.0491793
question8	question8	-1.5257122	-0.2971166
question9	question9	-0.0378902	-0.2971166
question10	question10	-1.5257122	-1.5774645
question11	question11	-1.1192838	0.0491793
question12	question12	-0.1574964	-0.5792980
question13	question13	0.3526097	-0.8211821
question14	question14	-0.7084914	-1.2299573
question15	question15	-3.0993342	-2.5814248
question16	question16	-1.2205912	-1.2299573
question17	question17	-1.3219110	0.0491793
question18	question18	-1.2205912	-1.2299573
question19	question19	-1.5257122	-1.2299573

items %>%
    ggplot(aes(x = model1, y = model2)) + geom_point() + geom_abline(intercept = 0,
    slope = 1) + theme_bw() + ggtitle("Item parameters")

## Person coefficients

person <- data.frame(names = names(model$coefficients[1:50]), model1 = model$coefficients[1:50],
    model2 = model2$coefficients[1:50])


knitr::kable(person)

	names	model1	model2
sub1	sub1	2.0653239	3.2956455
sub2	sub2	-0.2968900	0.7736769
sub3	sub3	3.3023419	4.0819608
sub4	sub4	-1.0379936	2.7946231
sub5	sub5	3.3023419	20.6306860
sub6	sub6	4.1118608	20.6306860
sub7	sub7	2.7876581	20.6306860
sub8	sub8	2.3940758	20.6306860
sub9	sub9	0.7715201	4.0819608
sub10	sub10	0.2709648	1.5234510
sub11	sub11	1.7752804	3.2956455
sub12	sub12	2.7876581	4.0819608
sub13	sub13	1.0136767	2.4081400
sub14	sub14	1.7752804	2.7946231
sub15	sub15	1.7752804	3.2956455
sub16	sub16	1.2576773	3.2956455
sub17	sub17	0.2709648	1.7917401
sub18	sub18	-0.0001059	1.5234510
sub19	sub19	0.5258563	2.4081400
sub20	sub20	1.5092340	20.6306860
sub21	sub21	1.0136767	2.0820668
sub22	sub22	0.2709648	4.0819608
sub23	sub23	2.3940758	2.0820668
sub24	sub24	-0.0001059	1.5234510
sub25	sub25	-0.0001059	1.5234510
sub26	sub26	2.3940758	20.6306860
sub27	sub27	1.7752804	4.0819608
sub28	sub28	2.0653239	20.6306860
sub29	sub29	-0.0001059	1.7917401
sub30	sub30	2.3940758	20.6306860
sub31	sub31	1.7752804	20.6306860
sub32	sub32	1.2576773	20.6306860
sub33	sub33	2.0653239	3.2956455
sub34	sub34	1.0136767	4.0819608
sub35	sub35	1.7752804	3.2956455
sub36	sub36	1.5092340	1.7917401
sub37	sub37	-0.0001059	4.0819608
sub38	sub38	0.7715201	4.0819608
sub39	sub39	1.2576773	3.2956455
sub40	sub40	2.3940758	20.6306860
sub41	sub41	1.2576773	20.6306860
sub42	sub42	1.0136767	4.0819608
sub43	sub43	1.2576773	3.2956455
sub44	sub44	0.5258563	2.7946231
sub45	sub45	-1.5647423	0.7736769
sub46	sub46	1.5092340	4.0819608
sub47	sub47	-0.2968900	1.7917401
sub48	sub48	-0.2968900	2.4081400
sub49	sub49	0.5258563	2.7946231
sub50	sub50	0.7715201	2.7946231

person %>%
    ggplot(aes(x = model1, y = model2)) + geom_point() + geom_abline(intercept = 0,
    slope = 1) + theme_bw() + ggtitle("Person parameters")

Notice that when framed this way, the item parameters estimate values about the same, but the person parameters are different for the two tests. The two regressions do not know about one another, so there is no reason they should try to constrain the ability to be the same for each person across the two tests, but this does raise the question of how this should be done ideally. We might try to force the person parameters to have an average of 0 and the items parameters to scale to fit the difficulty of the test, which could make more sense.

So, how good is it? Let’s compare our estimated parameters to our actual parameters:

library(ggplot2)
qplot(x = model$coef[1:numsubs], y = skilllevel) + geom_point(size = 4) + xlab("Estimated Model coefficient") +
    ylab("Person Ability") + geom_label(label = 1:numsubs) + theme_bw()

cor(model$coef[1:numsubs], skilllevel)

[1] 0.8558655

This is not bad–we predict fairly well the skill level of each person based on 10 yes/no answers. How about assessing the question difficulty. In our sum-to-zero coding, question 20 is the negative average of the other items.

itempars <- c((model$coef[-(1:numsubs)]), -mean(model$coef[-(1:numsubs)]))
itempars2 <- c((model2$coef[-(1:numsubs)]), -mean(model2$coef[-(1:numsubs)]))

qplot(x = itempars, y = questiondiff) + geom_label(label = 1:length(questiondiff)) +
    xlab("Estimated Model coefficient") + ylab("Question difficulty") + theme_bw()

cor(itempars, questiondiff)

[1] -0.8104594

We could have scored each person and each question according to accuracy:

library(gridExtra)
rowMeans(sim1)

 [1] 0.75 0.30 0.90 0.20 0.90 0.95 0.85 0.80 0.50 0.40 0.70 0.85 0.55 0.70 0.70
[16] 0.60 0.40 0.35 0.45 0.65 0.55 0.40 0.80 0.35 0.35 0.80 0.70 0.75 0.35 0.80
[31] 0.70 0.60 0.75 0.55 0.70 0.65 0.35 0.50 0.60 0.80 0.60 0.55 0.60 0.45 0.15
[46] 0.65 0.30 0.30 0.45 0.50

colMeans(sim1)

 [1] 0.90 0.50 0.60 1.00 0.58 0.50 0.58 0.44 0.72 0.44 0.52 0.70 0.78 0.60 0.18
[16] 0.50 0.48 0.50 0.44 0.68

grid.arrange(qplot(rowMeans(sim1), model$coef[1:numsubs], xlab = "Person accuracy",
    ylab = "Person ability parameter") + theme_bw(), qplot(colMeans(sim1), itempars,
    xlab = "Question accuracy", ylab = "Item difficulty parameter") + theme_bw(),
    ncol = 2)

grid.arrange(qplot(rowMeans(sim2), model2$coef[1:numsubs], xlab = "Person accuracy",
    ylab = "Person ability parameter") + theme_bw(), qplot(colMeans(sim2), itempars2,
    xlab = "Question accuracy", ylab = "Item difficulty parameter") + theme_bw(),
    ncol = 2)

Notice that there is a fairly close mapping between the question accuracy and the difficulty. What if we look at the two different tests and compare parameter estimates:

abilities <- data.frame(set1 = model$coef[1:50], set2 = model2$coef[1:50])
cor(abilities)

          set1      set2
set1 1.0000000 0.5772977
set2 0.5772977 1.0000000

ggplot(abilities, aes(x = set1, y = set2)) + geom_point() + ggtitle("Person abilities") +
    theme_bw()

probdifficulty <- data.frame(set1 = itempars, set2 = itempars2)

cor(probdifficulty)

          set1      set2
set1 1.0000000 0.9841231
set2 0.9841231 1.0000000

ggplot(probdifficulty, aes(x = set1, y = set2)) + geom_point() + ggtitle("Question difficulty") +
    theme_bw()

We are able to extract ‘person’ related coefficients across the two tests that are reasonably well related. Furthermore, we get high correlations for the test parameters

This analysis is essentially equivalent to what is known as the Rasch model of Item Response Theory (IRT). The ltm package estimates these directly from a wide data format. We might look especially at how the Rasch model chooses to anchor difficulty parameters.

library(ltm)

p1 <- sim1 + 0
p2 <- sim2 + 0
irt1 <- rasch(p1)
irt2 <- rasch(p2)
summary(irt1)


Call:
rasch(data = p1)

Model Summary:
  log.Lik      AIC      BIC
 -565.682 1173.364 1213.517

Coefficients:
                  value    std.err  z.vals
Dffclt.Item 1   -2.6972     0.6433 -4.1927
Dffclt.Item 2    0.0009     0.3591  0.0026
Dffclt.Item 3   -0.5142     0.3703 -1.3888
Dffclt.Item 4  -28.3941 71439.4339 -0.0004
Dffclt.Item 5   -0.4094     0.3662 -1.1180
Dffclt.Item 6    0.0008     0.3591  0.0024
Dffclt.Item 7   -0.4095     0.3662 -1.1181
Dffclt.Item 8    0.3074     0.3630  0.8467
Dffclt.Item 9   -1.1912     0.4173 -2.8544
Dffclt.Item 10   0.3074     0.3630  0.8469
Dffclt.Item 11  -0.1009     0.3595 -0.2806
Dffclt.Item 12  -1.0699     0.4063 -2.6331
Dffclt.Item 13  -1.5870     0.4606 -3.4453
Dffclt.Item 14  -0.5143     0.3703 -1.3890
Dffclt.Item 15   1.8911     0.5009  3.7752
Dffclt.Item 16   0.0010     0.3591  0.0027
Dffclt.Item 17   0.1027     0.3595  0.2857
Dffclt.Item 18   0.0009     0.3591  0.0026
Dffclt.Item 19   0.3074     0.3630  0.8469
Dffclt.Item 20  -0.9533     0.3968 -2.4027
Dscrmn           0.9356     0.1343  6.9684

Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0019 
quasi-Newton: BFGS

summary(irt2)


Call:
rasch(data = p2)

Model Summary:
   log.Lik      AIC      BIC
 -335.3924 712.7847 752.9372

Coefficients:
                  value    std.err  z.vals
Dffclt.Item 1   -3.7372     1.0006 -3.7349
Dffclt.Item 2   -1.3398     0.3768 -3.5559
Dffclt.Item 3   -1.4609     0.3917 -3.7292
Dffclt.Item 4  -21.8561 48471.2855 -0.0005
Dffclt.Item 5   -1.4608     0.3917 -3.7291
Dffclt.Item 6   -1.2254     0.3639 -3.3676
Dffclt.Item 7   -2.7467     0.6423 -4.2763
Dffclt.Item 8   -2.4679     0.5719 -4.3151
Dffclt.Item 9   -2.4674     0.5718 -4.3151
Dffclt.Item 10  -1.4609     0.3917 -3.7291
Dffclt.Item 11  -2.7455     0.6420 -4.2766
Dffclt.Item 12  -2.2411     0.5219 -4.2938
Dffclt.Item 13  -2.0490     0.4842 -4.2320
Dffclt.Item 14  -1.7281     0.4296 -4.0226
Dffclt.Item 15  -0.7187     0.3212 -2.2373
Dffclt.Item 16  -1.7277     0.4295 -4.0222
Dffclt.Item 17  -2.7448     0.6418 -4.2767
Dffclt.Item 18  -1.7277     0.4295 -4.0223
Dffclt.Item 19  -1.7277     0.4295 -4.0223
Dffclt.Item 20  -2.4680     0.5720 -4.3151
Dscrmn           1.2155     0.2000  6.0774

Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0053 
quasi-Newton: BFGS

## this is an alternative to alpha in psych package
descript(p1)


Descriptive statistics for the 'p1' data-set

Sample:
 20 items and 50 sample units; 0 missing values

Proportions for each level of response:
[[1]]
[1] 0.1 0.9

[[2]]
[1] 0.5 0.5

[[3]]
[1] 0.4 0.6

[[4]]
[1] 1

[[5]]
[1] 0.42 0.58

[[6]]
[1] 0.5 0.5

[[7]]
[1] 0.42 0.58

[[8]]
[1] 0.56 0.44

[[9]]
[1] 0.28 0.72

[[10]]
[1] 0.56 0.44

[[11]]
[1] 0.48 0.52

[[12]]
[1] 0.3 0.7

[[13]]
[1] 0.22 0.78

[[14]]
[1] 0.4 0.6

[[15]]
[1] 0.82 0.18

[[16]]
[1] 0.5 0.5

[[17]]
[1] 0.52 0.48

[[18]]
[1] 0.5 0.5

[[19]]
[1] 0.56 0.44

[[20]]
[1] 0.32 0.68



Frequencies of total scores:
     0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Freq 0 0 0 1 1 0 3 5 3 3  3  4  5  3  6  3  5  2  2  1  0


Cronbach's alpha:
                   value
All Items         0.7616
Excluding Item 1  0.7542
Excluding Item 2  0.7527
Excluding Item 3  0.7553
Excluding Item 4  0.7637
Excluding Item 5  0.7543
Excluding Item 6  0.7371
Excluding Item 7  0.7618
Excluding Item 8  0.7559
Excluding Item 9  0.7351
Excluding Item 10 0.7507
Excluding Item 11 0.7338
Excluding Item 12 0.7602
Excluding Item 13 0.7735
Excluding Item 14 0.7457
Excluding Item 15 0.7504
Excluding Item 16 0.7561
Excluding Item 17 0.7468
Excluding Item 18 0.7509
Excluding Item 19 0.7498
Excluding Item 20 0.7484


Pairwise Associations:
   Item i Item j p.value
1       1      8   1.000
2       1     12   1.000
3       1     13   1.000
4       1     15   1.000
5       1     18   1.000
6       1     19   1.000
7       2      4   1.000
8       2      7   1.000
9       2     13   1.000
10      2     18   1.000

Compare to our logistic regression:

plot(itempars, irt1$coef[, 1], main = "Comparison of model Item coefficients", xlab = "Logistic coefficients",
    ylab = "IRT coefficients")
abline(0, 1)

plot(itempars2, irt2$coef[, 1], main = "Comparison of model Item coefficients", xlab = "Logistic coefficients",
    ylab = "IRT coefficients")
abline(0, 1)

The item coefficients are not centered the same so their absolute values are different. These models use person ability parameters, but do not return those coefficients so it is not easy to make a direct comparison to the logistic, or do a person-type analysis. Instead, the focus of IRT is on the items or questions

Visualizing the Rasch Model

If you plot() the model, it will display the inferred logistic curves for all the questions

plot(irt1)

plot(irt2)

Notice that each curve is identical but shifted. The slope of the model is fit as a common value for all items, with different constant offsets (i.e., intercepts) for each question.

Boundary conditions of the Rasch Model

The data/questions in this example were all created as if they obeyed IRT. Thus, the model worked fairly well. If we have any violations of the model, the estimates can get less precise, and the small number of respondents (50) for the questions we chose (20) would not be enough. Typically you would want more, and the more complicated the model, the more participants.

What happens if they don’t–if we have ‘bad’ questions. One way to do this is to recode a few questions in the opposite direction, so that the people with high ability are more likely to get it wrong

set.seed(10010)
irt2 <- rasch(sim2 + 0)
sim3 <- sim2
sim3[, 1:5] <- (runif(5 * numsubs) < 0.5) + 0
irt3 <- rasch(sim3)
summary(irt2)


Call:
rasch(data = sim2 + 0)

Model Summary:
   log.Lik      AIC      BIC
 -335.3924 712.7847 752.9372

Coefficients:
                  value    std.err  z.vals
Dffclt.Item 1   -3.7372     1.0006 -3.7349
Dffclt.Item 2   -1.3398     0.3768 -3.5559
Dffclt.Item 3   -1.4609     0.3917 -3.7292
Dffclt.Item 4  -21.8561 48471.2855 -0.0005
Dffclt.Item 5   -1.4608     0.3917 -3.7291
Dffclt.Item 6   -1.2254     0.3639 -3.3676
Dffclt.Item 7   -2.7467     0.6423 -4.2763
Dffclt.Item 8   -2.4679     0.5719 -4.3151
Dffclt.Item 9   -2.4674     0.5718 -4.3151
Dffclt.Item 10  -1.4609     0.3917 -3.7291
Dffclt.Item 11  -2.7455     0.6420 -4.2766
Dffclt.Item 12  -2.2411     0.5219 -4.2938
Dffclt.Item 13  -2.0490     0.4842 -4.2320
Dffclt.Item 14  -1.7281     0.4296 -4.0226
Dffclt.Item 15  -0.7187     0.3212 -2.2373
Dffclt.Item 16  -1.7277     0.4295 -4.0222
Dffclt.Item 17  -2.7448     0.6418 -4.2767
Dffclt.Item 18  -1.7277     0.4295 -4.0223
Dffclt.Item 19  -1.7277     0.4295 -4.0223
Dffclt.Item 20  -2.4680     0.5720 -4.3151
Dscrmn           1.2155     0.2000  6.0774

Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0053 
quasi-Newton: BFGS

summary(irt3)


Call:
rasch(data = sim3)

Model Summary:
   log.Lik      AIC      BIC
 -444.2396 930.4791 970.6316

Coefficients:
                 value std.err  z.vals
Dffclt.Item 1  -0.2581  0.4654 -0.5546
Dffclt.Item 2   0.3966  0.4693  0.8451
Dffclt.Item 3  -0.5225  0.4755 -1.0989
Dffclt.Item 4   0.2659  0.4652  0.5716
Dffclt.Item 5   0.3968  0.4693  0.8455
Dffclt.Item 6  -1.8679  0.6167 -3.0287
Dffclt.Item 7  -4.3637  1.1820 -3.6919
Dffclt.Item 8  -3.8917  1.0450 -3.7240
Dffclt.Item 9  -3.8921  1.0451 -3.7240
Dffclt.Item 10 -2.2409  0.6778 -3.3061
Dffclt.Item 11 -4.3638  1.1820 -3.6919
Dffclt.Item 12 -3.5111  0.9459 -3.7121
Dffclt.Item 13 -3.1954  0.8705 -3.6707
Dffclt.Item 14 -2.6720  0.7584 -3.5231
Dffclt.Item 15 -1.0778  0.5170 -2.0847
Dffclt.Item 16 -2.6712  0.7583 -3.5228
Dffclt.Item 17 -4.3641  1.1821 -3.6918
Dffclt.Item 18 -2.6713  0.7583 -3.5228
Dffclt.Item 19 -2.6720  0.7584 -3.5231
Dffclt.Item 20 -3.8914  1.0450 -3.7240
Dscrmn          0.6761  0.1298  5.2080

Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0072 
quasi-Newton: BFGS

plot((cbind(irt1$coef[, 1], irt3$coef[, 1])), main = "Item coefficients with bad questions",
    xlab = "test 2", ylab = "test 3")

plot((cbind(irt1$coef[, 1], irt3$coef[, 1])), main = "Item coefficients with bad questions (zoomed)",
    xlab = "test 2", ylab = "test 3", ylim = c(-5, 5))

In this case, the ‘bad’ questions all ended up with negative difficulty coefficients. If we examine the questions using item.fit, it will test whether each question fits into the basic model. When everything was from the model, none of the items were selected as bad. Once we made 5 items (25%) bad, in this case a bunch of items get flagged. This includes all the items 1..5, but also 6, 7, and 9 and maybe 18. Strangely, a few bad questions might make other questions look bad as well.

item.fit(irt2)


Item-Fit Statistics and P-values

Call:
rasch(data = sim2 + 0)

Alternative: Items do not fit the model
Ability Categories: 10

          X^2 Pr(>X^2)
It 1   2.4798   0.9286
It 2  13.5244   0.0603
It 3   5.2193   0.6332
It 4   0.0000        1
It 5  13.3812   0.0633
It 6   6.6327   0.4681
It 7   3.6856   0.8152
It 8   4.6883   0.6979
It 9  14.4979    0.043
It 10  8.9407   0.2569
It 11  6.8692   0.4426
It 12  5.4804   0.6016
It 13 10.4973   0.1621
It 14  4.3378   0.7401
It 15  5.0282   0.6565
It 16  8.0741   0.3261
It 17  7.3144   0.3969
It 18  7.9045   0.3411
It 19  9.9131   0.1936
It 20  4.0405   0.7751

item.fit(irt3)


Item-Fit Statistics and P-values

Call:
rasch(data = sim3)

Alternative: Items do not fit the model
Ability Categories: 10

          X^2 Pr(>X^2)
It 1   3.5064   0.7431
It 2   3.8323   0.6994
It 3   2.5409   0.8639
It 4   5.8604    0.439
It 5   8.6429   0.1947
It 6   3.9761   0.6799
It 7   4.3075   0.6351
It 8   7.5153   0.2758
It 9   3.8459   0.6975
It 10  5.2996    0.506
It 11  3.0357   0.8044
It 12  7.8566   0.2488
It 13  3.3830   0.7595
It 14  6.1092   0.4111
It 15 10.6329   0.1004
It 16  3.8659   0.6948
It 17 10.7470   0.0965
It 18  2.6317   0.8534
It 19  3.6177   0.7283
It 20  2.8663   0.8254

Some of the other things we can look at to examine the fit of the model:

person.fit(irt2)


Person-Fit Statistics and P-values

Call:
rasch(data = sim2 + 0)

Alternative: Inconsistent response pattern under the estimated model

  It 1 It 2 It 3 It 4 It 5 It 6 It 7 It 8 It 9 It 10 It 11 It 12 It 13 It 14
1    0    0    0    1    0    1    1    0    1     1     1     1     1     1
2    1    0    0    1    0    0    0    1    1     1     1     0     0     0
3    1    0    0    1    0    0    1    1    1     0     0     0     1     0
  It 15 It 16 It 17 It 18 It 19 It 20       L0      Lz Pr(<Lz)
1     0     1     1     0     1     1 -12.6539 -1.0533  0.1461
2     0     1     1     0     1     1 -10.8261  0.6039  0.7271
3     0     0     1     1     1     1 -10.1098  1.0793  0.8598
 [ reached 'max' / getOption("max.print") -- omitted 33 rows ]

person.fit(irt3)


Person-Fit Statistics and P-values

Call:
rasch(data = sim3)

Alternative: Inconsistent response pattern under the estimated model

  It 1 It 2 It 3 It 4 It 5 It 6 It 7 It 8 It 9 It 10 It 11 It 12 It 13 It 14
1    0    0    0    0    1    1    1    1    1     0     1     0     1     0
2    0    0    0    0    1    1    1    1    1     0     1     1     1     1
3    0    0    0    1    0    0    0    1    1     1     1     0     0     0
  It 15 It 16 It 17 It 18 It 19 It 20       L0      Lz Pr(<Lz)
1     1     0     1     1     0     1 -12.1349 -0.6812  0.2479
2     0     0     0     1     1     1 -10.9041 -0.2671  0.3947
3     0     1     1     0     1     1 -12.9750 -0.8758  0.1906
 [ reached 'max' / getOption("max.print") -- omitted 43 rows ]

Extending and Constraining IRT

Slope of the item characteristic function

In the Rasch model, all items are of the same family, and have the same slope, or steepness. A very steep function means that there is a sharp cut-off between who can answer it correctly and who cannot. This is often called ‘discriminability’. A good test item typically has high discriminabilty, and a good test has a set of highly-discriminable items that have different difficulty. Typically, high discriminability will correspond to good part-whole item correlations. As a sort of ideal situation, the easiest item will be answered correctly by everyone but the lowest-ability person, the hardest item will only be answered correctly by the highest-ability person, and the person’s ability will directly control how many of the items they can answer. As a rule of thumb, higher discriminability values (greater than 1.0, or better yet greater than 2.0) are good. By default, the rasch model estimates a slope. However, the default logistic model will have a slope of 1.0, and so this is sometimes considered a simpler model. You might do this if you have limited data–maybe a test from a class with relatively few students, because it will hopefully make estimation more stable.

For example, The following is are the results of a psychology test:

dat <- read.csv("testscores.csv")

## descript(dat) ##doesn't work. Thus compute Cronbach's alahp on the data
descript(dat, chi.squared = F)


Descriptive statistics for the 'dat' data-set

Sample:
 47 items and 21 sample units; 0 missing values

Proportions for each level of response:
$q1
1 
1 

$q2
     0      1 
0.3333 0.6667 

$q3
1 
1 

$q4
     0      1 
0.8095 0.1905 

$q5
     0      1 
0.1429 0.8571 

$q6
     0      1 
0.3333 0.6667 

$q7
     0      1 
0.2381 0.7619 

$q8
     0      1 
0.1429 0.8571 

$q9
1 
1 

$q10
     0      1 
0.0476 0.9524 

$q11
    0     1 
0.619 0.381 

$q12
    0     1 
0.381 0.619 

$q13
     0      1 
0.0476 0.9524 

$q14
1 
1 

$q15
     0      1 
0.7619 0.2381 

$q16
     0      1 
0.0476 0.9524 

$q17
1 
1 

$q18
     0      1 
0.0476 0.9524 

$q19
     0      1 
0.3333 0.6667 

$q20
     0      1 
0.2381 0.7619 

$q21
1 
1 

$q22
     0      1 
0.2857 0.7143 

$q23
     0      1 
0.2381 0.7619 

$q24
1 
1 

$q25
     0      1 
0.6667 0.3333 

$q26
1 
1 

$q27
     0      1 
0.7143 0.2857 

$q28
    0     1 
0.381 0.619 

$q29
    0     1 
0.381 0.619 

$q31
     0      1 
0.6667 0.3333 

$q32
     0      1 
0.6667 0.3333 

$q33
     0      1 
0.4762 0.5238 

$q34
     0      1 
0.0952 0.9048 

$q35
     0      1 
0.3333 0.6667 

$q36
    0     1 
0.619 0.381 

$q37
     0      1 
0.3333 0.6667 

$q38
     0      1 
0.1905 0.8095 

$q39
     0      1 
0.7619 0.2381 

$q40
     0      1 
0.4762 0.5238 

$q41
     0      1 
0.0476 0.9524 

$q42
     0      1 
0.1429 0.8571 

$q43
     0      1 
0.0952 0.9048 

$q44
    0     1 
0.381 0.619 

$q45
    0     1 
0.381 0.619 

$q47
     0      1 
0.2381 0.7619 

$q48
    0     1 
0.619 0.381 

$q49
     0      1 
0.7143 0.2857 



Frequencies of total scores:
     0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Freq 0 0 0 0 0 0 0 0 0 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  3
     28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Freq  0  2  3  3  2  1  1  1  0  0  1  3  0  1  0  0  0  0  0  0


Cronbach's alpha:
               value
All Items     0.6291
Excluding q1  0.6294
Excluding q2  0.6504
Excluding q3  0.6294
Excluding q4  0.6317
Excluding q5  0.6138
Excluding q6  0.5851
Excluding q7  0.6021
Excluding q8  0.6313
Excluding q9  0.6294
Excluding q10 0.6211
Excluding q11 0.6217
Excluding q12 0.6122
Excluding q13 0.6297
Excluding q14 0.6294
Excluding q15 0.6309
Excluding q16 0.6254
Excluding q17 0.6294
Excluding q18 0.6297
Excluding q19 0.5979
Excluding q20 0.6139
Excluding q21 0.6294
Excluding q22 0.6342
Excluding q23 0.6251
Excluding q24 0.6294
Excluding q25 0.6426
Excluding q26 0.6294
Excluding q27 0.6118
Excluding q28 0.5952
Excluding q29 0.6168
Excluding q31 0.6124
Excluding q32 0.6147
Excluding q33 0.6177
Excluding q34 0.6232
Excluding q35 0.6302
Excluding q36 0.6526
Excluding q37 0.6406
Excluding q38 0.6174
Excluding q39 0.6200
Excluding q40 0.6107
Excluding q41 0.6317
Excluding q42 0.6334
Excluding q43 0.6337
Excluding q44 0.6075
Excluding q45 0.6425
Excluding q47 0.6116
Excluding q48 0.6428
Excluding q49 0.6047

# force the discrimination parameter to be 1
model1 <- rasch(dat, constraint = cbind(length(dat) + 1, 1))
model1


Call:
rasch(data = dat, constraint = cbind(length(dat) + 1, 1))

Coefficients:
 Dffclt.q1   Dffclt.q2   Dffclt.q3   Dffclt.q4   Dffclt.q5   Dffclt.q6  
   -25.566      -0.773     -25.566       1.591      -1.947      -0.773  
 Dffclt.q7   Dffclt.q8   Dffclt.q9  Dffclt.q10  Dffclt.q11  Dffclt.q12  
    -1.281      -1.946     -25.566      -3.188       0.531      -0.546  
Dffclt.q13  Dffclt.q14  Dffclt.q15  Dffclt.q16  Dffclt.q17  Dffclt.q18  
    -3.188     -25.566       1.281      -3.188     -25.566      -3.188  
Dffclt.q19  Dffclt.q20  Dffclt.q21  Dffclt.q22  Dffclt.q23  Dffclt.q24  
    -0.773      -1.281     -25.566      -1.016      -1.281     -25.566  
Dffclt.q25  Dffclt.q26  Dffclt.q27  Dffclt.q28  Dffclt.q29  Dffclt.q31  
     0.762     -25.566       1.009      -0.545      -0.546       0.762  
Dffclt.q32  Dffclt.q33  Dffclt.q34  Dffclt.q35  Dffclt.q36  Dffclt.q37  
     0.762      -0.114      -2.425      -0.773       0.531      -0.773  
Dffclt.q38  Dffclt.q39  Dffclt.q40  Dffclt.q41  Dffclt.q42  Dffclt.q43  
    -1.583       1.281      -0.115      -3.188      -1.946      -2.424  
Dffclt.q44  Dffclt.q45  Dffclt.q47  Dffclt.q48  Dffclt.q49      Dscrmn  
    -0.546      -0.545      -1.281       0.531       1.009       1.000  

Log.Lik: -430.349

# summary(model1) allow discrimination parameter to be estimated
model2 <- rasch(dat)
model2


Call:
rasch(data = dat)

Coefficients:
 Dffclt.q1   Dffclt.q2   Dffclt.q3   Dffclt.q4   Dffclt.q5   Dffclt.q6  
   -49.028      -1.416     -49.028       2.930      -3.615      -1.418  
 Dffclt.q7   Dffclt.q8   Dffclt.q9  Dffclt.q10  Dffclt.q11  Dffclt.q12  
    -2.362      -3.612     -49.028      -5.965       0.984      -0.996  
Dffclt.q13  Dffclt.q14  Dffclt.q15  Dffclt.q16  Dffclt.q17  Dffclt.q18  
    -5.966     -49.028       2.361      -5.967     -49.028      -5.966  
Dffclt.q19  Dffclt.q20  Dffclt.q21  Dffclt.q22  Dffclt.q23  Dffclt.q24  
    -1.417      -2.364     -49.028      -1.869      -2.362     -49.028  
Dffclt.q25  Dffclt.q26  Dffclt.q27  Dffclt.q28  Dffclt.q29  Dffclt.q31  
     1.407     -49.028       1.862      -0.996      -0.996       1.408  
Dffclt.q32  Dffclt.q33  Dffclt.q34  Dffclt.q35  Dffclt.q36  Dffclt.q37  
     1.408      -0.206      -4.517      -1.416       0.984      -1.416  
Dffclt.q38  Dffclt.q39  Dffclt.q40  Dffclt.q41  Dffclt.q42  Dffclt.q43  
    -2.928       2.361      -0.201      -5.966      -3.612      -4.518  
Dffclt.q44  Dffclt.q45  Dffclt.q47  Dffclt.q48  Dffclt.q49      Dscrmn  
    -0.996      -0.996      -2.364       0.984       1.862       0.521  

Log.Lik: -426.767

# summary(model2)
par(mfrow = c(1, 2))
plot(model1)
plot(model2)

Notice that several of the questions have difficulty parameters of -49.02. These are the problems that everybody got correct. This also likely led to the error messages returned by the models. It is difficult to estimate the difficulty of such questions, because they must be really easy. We fit two different models; one in which has a discrimination parameter. Is it worthwhile using this extra parameter?

anova(model1, model2)


 Likelihood Ratio Table
          AIC     BIC log.Lik  LRT df p.value
model1 954.70 1003.79 -430.35                
model2 949.53  999.67 -426.77 7.16  1   0.007

Results show that there is no difference between the two, despite the fact that the mean discriminability when estimated was .5 instead of 1.

Fitting individual difficulty parameters

In other cases, it is likely that different items have different discriminabilities, and you might want to use this to help create a better simpler test. You might be able to choose 5 highly discriminable items to replace 50 low-discriminable items, for example. The two-parameter IRT model can estimate a difficulty and discriminibility for each item. It is fit with the ltm() function in ltm.

The ltm function has more bells and whistles that we won’t deal with. For example, it lets you estimate a set of latent predictors–sort of a factor analysis. We will use a single factor, which ends up being the two-parameter model. The syntax is a bit different. You need to write a formula, and tell it how many latent factors to estimate. We will specify a single factor by doing data~z1, but you can use two by doing data~z1 + z2. This model is sometimes referred to as the two-parameter logistic (TPL) model.

model3 <- ltm(dat ~ z1)
model3


Call:
ltm(formula = dat ~ z1)

Coefficients:
            Dffclt   Dscrmn
q1   -2.417364e+08    0.000
q2   -7.760000e-01    0.976
q3   -2.417364e+08    0.000
q4   -2.119000e+00   -0.734
q5    1.382000e+00  -22.068
q6    6.560000e-01  -31.827
q7    1.803000e+00   -0.800
q8    1.352400e+01   -0.134
q9   -2.417364e+08    0.000
q10   3.708000e+00   -0.976
q11  -2.900000e-01   -4.347
q12  -3.407000e+00    0.139
q13  -3.773000e+00    0.874
q14  -2.417364e+08    0.000
q15   2.212000e+00    0.608
q16   2.059000e+00  -30.676
q17  -2.417364e+08    0.000
q18  -1.668000e+00   21.067
q19   2.454000e+00   -0.304
q20   2.715000e+00   -0.474
q21  -2.417364e+08    0.000
q22  -1.178080e+02    0.008
q23  -3.029000e+00    0.385
q24  -2.417364e+08    0.000
q25  -5.009000e+00   -0.136
q26  -2.417364e+08    0.000
q27  -1.051000e+00   -0.980
q28  -8.553000e+00    0.056
q29  -2.193000e+00    0.214
q31  -1.614000e+00   -0.423
q32  -2.699000e+00   -0.251
q33   3.370000e-01   -0.445
q34   1.691000e+00  -42.311
q35   4.430000e+00   -0.162
q36   1.202000e+00    0.474
q37  -2.966000e+00    0.229
q38   2.450000e+00   -0.693
 [ reached getOption("max.print") -- omitted 10 rows ]

Log.Lik: -400.606

plot(model3)

# summary(model3)

Notice that items vary in their difficulty and discriminibility, and that some are negatively discrimination. It is sort of a mess. This is not unexpected because we have so few participants in this test–there just isn’t enough information to reliably and stably estimate anything. Before we go on, we can look at a few things about how well the model fits:

item.fit(model3)


Item-Fit Statistics and P-values

Call:
ltm(formula = dat ~ z1)

Alternative: Items do not fit the model
Ability Categories: 10

         X^2 Pr(>X^2)
q1    0.0000        1
q2    8.1511   0.4189
q3    0.0000        1
q4    7.6834    0.465
q5    0.1379        1
q6    0.2621        1
q7    9.2744   0.3197
q8    8.7169   0.3667
q9    0.0000        1
q10   4.8496   0.7735
q11   1.5435    0.992
q12   3.4025   0.9066
q13   6.8417   0.5538
q14   0.0000        1
q15   6.0480   0.6419
q16   9.9338   0.2697
q17   0.0000        1
q18 133.4230  <0.0001
q19   7.9250   0.4408
q20   6.6433   0.5756
q21   0.0000        1
q22   7.9145   0.4419
q23  11.6092   0.1695
q24   0.0000        1
q25  14.2920   0.0745
q26   0.0000        1
q27   5.5064   0.7023
q28   9.7243   0.2849
q29   7.9233    0.441
q31   6.2791    0.616
q32   6.0579   0.6407
q33   6.5497   0.5859
q34   0.0000        1
q35   9.1083   0.3332
q36   2.4719    0.963
q37   8.9905   0.3431
q38   7.7650   0.4568
 [ reached 'max' / getOption("max.print") -- omitted 10 rows ]

This gives a ‘fit’ parameter for each question. A few items, like Q18, have bad fit parameters. Looking at the psych::alpha function, it has very low item-whole correlation.

psych::alpha(dat)

Some items ( q2 q7 q8 q12 q15 q18 q23 q28 q29 q36 q37 q42 q44 q45 ) were negatively correlated with the total scale and 
probably should be reversed.  
To do this, run the function again with the 'check.keys=TRUE' option


Reliability analysis   
Call: psych::alpha(x = dat)

  raw_alpha std.alpha G6(smc) average_r S/N  ase mean    sd median_r
      0.63      0.62     0.6      0.04 1.6 0.11 0.69 0.093    0.038

    95% confidence boundaries 
         lower alpha upper
Feldt     0.36  0.63  0.82
Duhachek  0.41  0.63  0.85

 Reliability if an item is dropped:
    raw_alpha std.alpha G6(smc) average_r S/N var.r med.r
q2       0.65      0.64    0.63     0.045 1.8 0.050 0.040
q4       0.63      0.62    0.61     0.042 1.7 0.050 0.037
q5       0.62      0.60    0.58     0.038 1.5 0.049 0.038
q6       0.59      0.58    0.56     0.035 1.4 0.049 0.037
q7       0.60      0.60    0.58     0.037 1.5 0.049 0.037
q8       0.63      0.63    0.61     0.042 1.7 0.050 0.040
q10      0.62      0.61    0.60     0.039 1.6 0.050 0.038
q11      0.62      0.61    0.59     0.040 1.6 0.049 0.038
q12      0.62      0.60    0.60     0.039 1.5 0.050 0.038
q13      0.63      0.62    0.61     0.042 1.6 0.049 0.040
 [ reached 'max' / getOption("max.print") -- omitted 29 rows ]

 Item statistics 
     n  raw.r  std.r   r.cor r.drop mean   sd
q2  21 -0.048 -0.093 -0.2498 -0.157 0.67 0.48
q4  21  0.135  0.119  0.0231  0.042 0.19 0.40
q5  21  0.375  0.410  0.3985  0.301 0.86 0.36
q6  21  0.642  0.636  0.6890  0.569 0.67 0.48
q7  21  0.496  0.450  0.4491  0.415 0.76 0.44
q8  21  0.119  0.099 -0.0027  0.037 0.86 0.36
q10 21  0.293  0.303  0.2607  0.246 0.95 0.22
q11 21  0.287  0.280  0.2300  0.177 0.38 0.50
q12 21  0.382  0.346  0.3159  0.279 0.62 0.50
q13 21  0.083  0.147  0.0588  0.033 0.95 0.22
 [ reached 'max' / getOption("max.print") -- omitted 29 rows ]

Non missing response frequency for each item
       0    1 miss
q2  0.33 0.67    0
q4  0.81 0.19    0
q5  0.14 0.86    0
q6  0.33 0.67    0
q7  0.24 0.76    0
q8  0.14 0.86    0
q10 0.05 0.95    0
q11 0.62 0.38    0
q12 0.38 0.62    0
q13 0.05 0.95    0
q15 0.76 0.24    0
q16 0.05 0.95    0
q18 0.05 0.95    0
q19 0.33 0.67    0
q20 0.24 0.76    0
q22 0.29 0.71    0
q23 0.24 0.76    0
q25 0.67 0.33    0
q27 0.71 0.29    0
q28 0.38 0.62    0
q29 0.38 0.62    0
q31 0.67 0.33    0
q32 0.67 0.33    0
q33 0.48 0.52    0
q34 0.10 0.90    0
 [ reached getOption("max.print") -- omitted 14 rows ]

We can look at the person-parameters. These tell us how well each person is described by the model. The tests are like a chi-squared test. We assume that a person of a certain ability should be getting problems easier than they are right, and more difficult than they are wrong. If this assumption is violated for a person, that indicates they violate this model, and are somehow different from the rest of the class which determines the model. Possibly they were cheating so they didn’t get many right but the ones they got right were the most difficult.

person.fit(model3)


Person-Fit Statistics and P-values

Call:
ltm(formula = dat ~ z1)

Alternative: Inconsistent response pattern under the estimated model

  q1 q2 q3 q4 q5 q6 q7 q8 q9 q10 q11 q12 q13 q14 q15 q16 q17 q18 q19 q20 q21
1  1  0  1  0  1  0  0  1  1   1   0   0   1   1   0   1   1   1   1   1   1
  q22 q23 q24 q25 q26 q27 q28 q29 q31 q32 q33 q34 q35 q36 q37 q38 q39 q40 q41
1   1   0   1   1   1   1   0   0   0   1   0   1   0   1   1   0   0   0   1
  q42 q43 q44 q45 q47 q48 q49       L0      Lz Pr(<Lz)
1   1   1   0   0   1   0   0 -22.3816 -1.7917  0.0366
 [ reached 'max' / getOption("max.print") -- omitted 20 rows ]

These are not bad–most people are reasonably-well fit in the model. The margins() function looks at whether there are particular parings of items that happen more often than by chance.

margins(model3)


Call:
ltm(formula = dat ~ z1)

Fit on the Two-Way Margins

Response: (0,0)
  Item i Item j Obs  Exp (O-E)^2/E    
1     13     37   1 0.11      6.86 ***
2      7     28   5 1.72      6.28 ***
3     13     42   1 0.14      5.32 ***

Response: (1,0)
  Item i Item j Obs  Exp (O-E)^2/E    
1      7     33   2 0.37      7.24 ***
2     30     33   1 0.15      5.04 ***
3      4     41   2 0.51      4.31 ***

Response: (0,1)
  Item i Item j Obs  Exp (O-E)^2/E    
1     16     30   1 0.07     11.81 ***
2      5      7   3 0.88      5.15 ***
3     33     43   2 0.49      4.65 ***

Response: (1,1)
  Item i Item j Obs  Exp (O-E)^2/E    
1     30     47   5 2.20      3.55 ***
2      4     15   2 0.71      2.32    
3     39     46   7 4.02      2.21    

'***' denotes a chi-squared residual greater than 3.5

For example, consider the first line. According to the model, we’d expect 0.11 people to get both 13 and 37 wrong. But the margins show 1 person got them both wrong, which would be unlikely to happen by chance. We can check the table here:

table(dat[, 13], dat[, 37])

These might indicate that there are questions that are not independent and so may violate the model assumptions. For 30 and 47, we’d expect only 2.06 people to get them both correct, but 5 did. In these cases, the two questions might be redundant. In other cases, this could capture things like ‘leakage’, where you learn the answer to one question based on another; or exclusive-or choices, where there are two parallel questions with one attractive answer so that people who are guessing are likely to either get them both right or wrong, or are likely to get only one right and the other wrong.

Multiple latent traits

The simple ltm model is essentially logistic regression, but at its core assumes your test measures a single ability dimension. What if your test meaured multiple dimensions that differed systematically and indepedently across people? Usually, you might do a PCA or factor analysis to examine this, but the ltm model will let you test up to two latent traits directly. This should also remind you a bit of how MANOVA works.

As a brief example, here is how we’d do multiple latent traits.

model4a <- ltm(dat[, 1:15] ~ z1)
plot(model4a)

model4a


Call:
ltm(formula = dat[, 1:15] ~ z1)

Coefficients:
            Dffclt  Dscrmn
q1   -1.802629e+08   0.000
q2    9.910000e-01  -0.864
q3   -1.802629e+08   0.000
q4    5.842000e+00   0.254
q5   -1.494000e+00   1.581
q6   -6.030000e-01  28.245
q7   -8.320000e-01   2.301
q8   -9.028000e+00   0.198
q9   -1.802629e+08   0.000
q10  -1.449000e+00  12.044
q11   3.520000e-01  27.475
q12   1.140900e+01  -0.043
q13   2.567000e+00  -1.604
q14  -1.802629e+08   0.000
q15  -1.656000e+00  -0.750

Log.Lik: -99.018

# summary(model4)
item.fit(model4a)


Item-Fit Statistics and P-values

Call:
ltm(formula = dat[, 1:15] ~ z1)

Alternative: Items do not fit the model
Ability Categories: 10

        X^2 Pr(>X^2)
q1   0.0000        1
q2  15.4790   0.0505
q3   0.0000        1
q4  13.1159   0.1079
q5   9.8377   0.2766
q6   0.0956        1
q7   5.3522   0.7194
q8   8.1935   0.4148
q9   0.0000        1
q10  1.9348   0.9829
q11  0.0621        1
q12 13.6124   0.0924
q13  5.8759   0.6611
q14  0.0000        1
q15  6.9981   0.5368

model4b <- ltm(dat[, 1:15] ~ z1 + z2)
model4b


Call:
ltm(formula = dat[, 1:15] ~ z1 + z2)

Coefficients:
     (Intercept)       z1        z2
q1        65.566    0.000     0.000
q2         1.236   -0.537     1.942
q3        65.566    0.000     0.000
q4      -121.843  105.913    59.674
q5       185.353  107.015  -142.073
q6        47.280   98.106   -39.255
q7       103.551  147.976    37.214
q8         2.165    0.775     0.896
q9        65.566    0.000     0.000
q10      143.812   93.717   -18.639
q11      -74.365  111.343  -148.643
q12        0.523    0.274     0.552
q13        3.472   -0.503     0.906
q14       65.566    0.000     0.000
q15       -1.578   -0.019     1.505

Log.Lik: -85.347

anova(model4a, model4b)


 Likelihood Ratio Table
           AIC    BIC log.Lik   LRT df p.value
model4a 258.04 289.37  -99.02                 
model4b 260.69 307.70  -85.35 27.34 15   0.026

# item.fit(model4b)
fs <- factor.scores(model4b)
barplot(t(fs$coef), beside = T)

plot(fs$coef[, 2], fs$coef[, 3])

For the z1 only model, we get a difficulty and discriminability parameter, which is like a mean and variance. For the z1 + z2 model, we get intercept which is going to be akin to the difficulty parameter, and z1/z2 represent variance along two principle components. Here, we don’t see too much similarity across models, but with a larger data set it is likely we would. It is also likely that z1 or z2 is going to be similar to the discriminability parameter, but not guaranteed. Here, we see z2 is somewhat correlated with the discriminability

plot(rank(coef(model4a)[, 1]), rank(coef(model4b)[, 1]))

plot(coef(model4a)[, 2], coef(model4b)[, 2])

plot(coef(model4a)[, 2], coef(model4b)[, 3])

Guessing parameters: the three-parameter model

If you have questions that differ in the ability to get the question right by chance, you might want to incorporate a guessing parameter. These are just the normal ltm model, but bottom out at a lower level that you either estimate or specify. This might be useful if you have a true/false test, where accuracy should be at least 50%, especially if this is mixed with other questions like short answer or multiple choice where guessing accuracy would be lower. In this case, you could fix the parameters based on question type. Otherwise, you might want to estimate them directly–but you would need to be sure you had enough data to get good estimates.

This model is called the three-parameter model (TPM). It incorporates a guessing value, if the chance of getting an answer right is non-zero by guessing.

model9 <- tpm(dat[, 1:15], type = "latent.trait", max.guessing = 0.5)
model9


Call:
tpm(data = dat[, 1:15], type = "latent.trait", max.guessing = 0.5)

Coefficients:
     Gussng         Dffclt   Dscrmn
q1    0.025  -4.775907e+08    0.000
q2    0.054  -9.160000e-01    0.806
q3    0.029  -4.775662e+08    0.000
q4    0.174  -1.884500e+01   -0.200
q5    0.113   1.031000e+00   -3.043
q6    0.014   4.510000e-01  -18.446
q7    0.463   1.230000e-01   -6.073
q8    0.051   2.353800e+01   -0.074
q9    0.032  -4.775418e+08    0.000
q10   0.048   1.513000e+00   -6.885
q11   0.000  -3.800000e-01  -22.118
q12   0.072  -2.603000e+00    0.144
q13   0.048  -1.916000e+00    3.428
q14   0.036  -4.775173e+08    0.000
q15   0.003   1.715000e+00    0.722

Log.Lik: -98.513

plot(model9)

Notice how different items bottom out at different levels.

With a small class, there are a lot of items with negative discriminability. Let’s look at how they work out, by comparing average test score to particular answers:

par(mfrow = c(2, 2))
boxplot(dat$q5, rowMeans(dat), main = "Correct on q5", names = c("Incorrect (3)",
    "correct (18)"))

boxplot(dat$q4, rowMeans(dat), main = "Correct on q4", names = c("Incorrect (1)",
    "correct (20)"))

boxplot(dat$q10, rowMeans(dat), main = "Correct on q10", names = c("Incorrect (3)",
    "correct (18)"))

boxplot(dat$q11, rowMeans(dat), main = "Correct on q11", names = c("Incorrect (13)",
    "correct (8)"))

We can see that for some of these, accuracy on the question is negatively correlated with accuracy on the test. For others, there are other strange things, like very small numbers of errors that might make estimation difficult.

Information curves

Each questions can be transformed into an information score, which is the distribution of information implied by the cumulative score. Also, you can plot the characteristic of the entire test:

plot(model4a)

plot(model4a, legend = T, type = "IIC", items = 1:5)

plot(model4a, type = "IIC", legend = T, item = c(1:15)[c(-11, -6, -10)])

The height of the curve indicates where the most informative ability level for each question is. A very discriminative question will have a sharp rise at a specific point, and you would be good at separating those below from those above.

model5 <- ltm(dat[, c(1, 3, 5, 9, 14)] ~ z1)
model5


Call:
ltm(formula = dat[, c(1, 3, 5, 9, 14)] ~ z1)

Coefficients:
            Dffclt  Dscrmn
q1   -1.321958e+11   0.000
q3   -1.321958e+11   0.000
q5    1.188000e+00  -3.667
q9   -1.321958e+11   0.000
q14  -1.321958e+11   0.000

Log.Lik: -8.612

plot(model5, legend = T, type = "IIC")

You can specify different items, or items=0 tells you the entire test. This tells you the range of abilities that the test or items will be good at. You can also specify a range to integrate over, to see which range the test is best at discriminating. This can be used to understand whether the test is good at discrimating low-performers (maybe a test for remidial instruction) on high-performers (a test for entrance into a competitive class or program).

plot(model5, legend = T, type = "IIC", items = 0)

info <- information(model5, c(-4, 4))
info


Call:
ltm(formula = dat[, c(1, 3, 5, 9, 14)] ~ z1)

Total Information = 3.67
Information in (-4, 4) = 3.67 (100%)
Based on all the items

Graded response model and partial credit model.

The ltm library provides five models, rasch, ltm, tpm, grm (graded response model-polytomous) and gpcm (general partial credit-polytomous). These last two are appropriate for rank-order and categorical responses, and might be useful for evaluating personality scales.

The basic assumptions of IRT is that you have a binary outcome (correct or incorrect). But it could be interesting to do an IRT-like analysis for non-binary responses. If you have a set of likert-scale responses, where they are all coded in the same direction, and they each independently give support for some construct, you can use a graded response model. This might be useful for personality data, for example. Let’s consider measures from the big five personality questionnaire we have examined in the past.

A related model in the ltm package is the graded partial credit model (gpcm). This would allow you to place an ordinal scale on correctness, and do an IRT analysis. Maybe in a short answer response, you score full credit for one response, and partial credit for another. We won’t cover this model here, but it has some similarity to the GRM.

To examine the GRM, Let’s obtain just the introversion/extraversion values, and reverse code so they are all in the same direction. for convenience, I’ll also remove any values that are NA.

big5 <- read.csv("bigfive.csv")
qtype <- c("E", "A", "C", "N", "O", "E", "A", "C", "N", "O", "E", "A", "C", "N",
    "O", "E", "A", "C", "N", "O", "E", "A", "C", "N", "O", "E", "A", "C", "N", "O",
    "E", "A", "C", "N", "O", "E", "A", "C", "N", "O", "O", "A", "C", "O")
valence <- c(1, -1, 1, 1, 1, -1, 1, -1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1,
    1, -1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1)
## reverse code
for (i in 2:ncol(big5)) {
    if (valence[i - 1] == -1) {
        big5[, i] <- 6 - big5[, i]
    }
}
ei <- big5[, c(T, qtype == "E")]
ei <- ei[!is.na(rowSums(ei)), ]

Now, the graded response model in ltm (grm) will do a irt-like analysis, treating these as ordinal values. You can use a constrained or unconstrained model–the constrained model fits an equal discriminability across all questions. Because we have a lot of data, this model takes a while to fit.

g1 <- grm(ei[, -1], constrained = TRUE)
g1


Call:
grm(data = ei[, -1], constrained = TRUE)

Coefficients:
     Extrmt1  Extrmt2  Extrmt3  Extrmt4  Dscrmn
Q1    -2.210   -0.924   -0.150    1.233   1.684
Q6    -1.531    0.123    0.838    2.007   1.684
Q11   -2.729   -1.194   -0.359    1.068   1.684
Q16   -2.853   -1.410   -0.370    1.061   1.684
Q21   -1.356    0.100    0.702    1.863   1.684
Q26   -2.003   -0.861   -0.116    1.360   1.684
Q31   -1.389    0.368    0.865    1.971   1.684
Q36   -2.579   -1.099   -0.468    0.984   1.684

Log.Lik: -10583.71

summary(g1)


Call:
grm(data = ei[, -1], constrained = TRUE)

Model Summary:
   log.Lik      AIC     BIC
 -10583.71 21233.41 21395.7

Coefficients:
$Q1
         value
Extrmt1 -2.210
Extrmt2 -0.924
Extrmt3 -0.150
Extrmt4  1.233
Dscrmn   1.684

$Q6
         value
Extrmt1 -1.531
Extrmt2  0.123
Extrmt3  0.838
Extrmt4  2.007
Dscrmn   1.684

$Q11
         value
Extrmt1 -2.729
Extrmt2 -1.194
Extrmt3 -0.359
Extrmt4  1.068
Dscrmn   1.684

$Q16
         value
Extrmt1 -2.853
Extrmt2 -1.410
Extrmt3 -0.370
Extrmt4  1.061
Dscrmn   1.684

$Q21
         value
Extrmt1 -1.356
Extrmt2  0.100
Extrmt3  0.702
Extrmt4  1.863
Dscrmn   1.684

$Q26
         value
Extrmt1 -2.003
Extrmt2 -0.861
Extrmt3 -0.116
Extrmt4  1.360
Dscrmn   1.684

$Q31
         value
Extrmt1 -1.389
Extrmt2  0.368
Extrmt3  0.865
Extrmt4  1.971
Dscrmn   1.684

$Q36
         value
Extrmt1 -2.579
Extrmt2 -1.099
Extrmt3 -0.468
Extrmt4  0.984
Dscrmn   1.684


Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0094 
quasi-Newton: BFGS

We can see that each question is modeled with its own IRT-like model. There are five levels here, and four transitions between levels, which are modeled as sort of difficulty parameters for each transition between items.

Plotting each question gives us another look

par(mfrow = c(4, 2))
plot(g1, items = 1)
plot(g1, items = 2)
plot(g1, items = 3)
plot(g1, items = 4)
plot(g1, items = 5)
plot(g1, items = 6)
plot(g1, items = 7)
plot(g1, items = 8)

The margins() function works here as well. We can see that there are a couple that violate the two-way independence (q1-q21; q6-q21, etc.)

margins(g1)


Call:
grm(data = ei[, -1], constrained = TRUE)

Fit on the Two-Way Margins

    Q1  Q6     Q11    Q16    Q21    Q26    Q31    Q36   
Q1  -    25.82  50.38  37.85 102.57  67.00  82.91  72.89
Q6      -       44.89  50.86 124.08  73.31  44.03  37.03
Q11            -       96.73  55.00  76.17  84.33  45.93
Q16            ***    -       55.38  51.31  65.46  52.74
Q21 *** ***                  -       53.24  74.63  29.73
Q26                                 -       62.84  37.69
Q31                                        -       34.28
Q36                                               -     

'***' denotes pairs of items with lack-of-fit

Let’s fit this unconstrained:

g2 <- grm(ei[, -1], constrained = FALSE)
g2


Call:
grm(data = ei[, -1], constrained = FALSE)

Coefficients:
     Extrmt1  Extrmt2  Extrmt3  Extrmt4  Dscrmn
Q1    -1.927   -0.814   -0.137    1.076   2.215
Q6    -1.531    0.121    0.838    2.010   1.682
Q11   -2.916   -1.276   -0.384    1.140   1.509
Q16   -2.766   -1.370   -0.363    1.029   1.773
Q21   -1.269    0.090    0.655    1.742   1.907
Q26   -2.549   -1.088   -0.137    1.732   1.164
Q31   -1.511    0.395    0.939    2.150   1.459
Q36   -2.302   -0.988   -0.426    0.877   2.090

Log.Lik: -10539.91

summary(g2)


Call:
grm(data = ei[, -1], constrained = FALSE)

Model Summary:
   log.Lik      AIC      BIC
 -10539.91 21159.82 21356.53

Coefficients:
$Q1
         value
Extrmt1 -1.927
Extrmt2 -0.814
Extrmt3 -0.137
Extrmt4  1.076
Dscrmn   2.215

$Q6
         value
Extrmt1 -1.531
Extrmt2  0.121
Extrmt3  0.838
Extrmt4  2.010
Dscrmn   1.682

$Q11
         value
Extrmt1 -2.916
Extrmt2 -1.276
Extrmt3 -0.384
Extrmt4  1.140
Dscrmn   1.509

$Q16
         value
Extrmt1 -2.766
Extrmt2 -1.370
Extrmt3 -0.363
Extrmt4  1.029
Dscrmn   1.773

$Q21
         value
Extrmt1 -1.269
Extrmt2  0.090
Extrmt3  0.655
Extrmt4  1.742
Dscrmn   1.907

$Q26
         value
Extrmt1 -2.549
Extrmt2 -1.088
Extrmt3 -0.137
Extrmt4  1.732
Dscrmn   1.164

$Q31
         value
Extrmt1 -1.511
Extrmt2  0.395
Extrmt3  0.939
Extrmt4  2.150
Dscrmn   1.459

$Q36
         value
Extrmt1 -2.302
Extrmt2 -0.988
Extrmt3 -0.426
Extrmt4  0.877
Dscrmn   2.090


Integration:
method: Gauss-Hermite
quadrature points: 21 

Optimization:
Convergence: 0 
max(|grad|): 0.0097 
quasi-Newton: BFGS

par(mfrow = c(4, 2))
plot(g2, items = 1)
plot(g2, items = 2)
plot(g2, items = 3)
plot(g2, items = 4)
plot(g2, items = 5)
plot(g2, items = 6)
plot(g2, items = 7)
plot(g2, items = 8)

For this model, we might consider the midpoint transition (extrm2) is the ‘center’ of the question. We can see that Q36 and Q26 are low, while Q21 and Q31 are high. We might also use this to infer that a 4 on Q36 is about equivalent to a 3 on Q21.

Discussion and Conclusions about the IRT model

The basic IRT/Rasch model is ubiquitous in areas of testing and measurement. It is helpful for designing tests that are maximally predictive using the smallest number of questions. It does not completely replace more traditional psychometric approaches, but augments them so that you can design smaller more effective tests, and is an important tool that both educators and researchers are typically not aware of, even if it can be helpful.

Additional resources

There are special-purpose standalon software packages many people use for IRT style models. SPSS recommends that you use an R package via their R connector.
The mirt library estimates IRT models. It also has a companion mirtCat and ggmirt library for using it in other contexts. (ggmirt is not in the CRAN and needs to be downloaded). A number of people have shown how to use ggplot with mirt on various blogs. MIRT looks like it can use syntax from some commonly IRT system, and you need to use R to program that other system via a text string, but blogs appear to show simple ways to apply it as well.
The irt library has objects for representing testbeds and items in a computerized adaptive testing framework, but I’m not sure if it can estimate parameters directly from data. It looks more useful for driving an adaptive test or simulating tests.
Other packages noted by Chalmers, the developer of mirt, include eRm, MCMCpack, and he notes that mirt was the only one to include confirmatory item factor analysis methods, at least in 2012.

Item Response Theory