Latent Trait Models for Item Response Theory Analyses

Description

This package provides a flexible framework for Item Response Theory analyses for dichotomous and polytomous data under a Marginal Maximum Likelihood approach. The fitting algorithms provide valid inferences under Missing At Random missing data mechanisms.

Details

The following options are available:

Descriptives:

samples proportions, missing values information, biserial correlation of items with total score, pairwise associations between items, Cronbach's \alpha, unidimensionality check using modified parallel analysis, nonparametric correlation coefficient, plotting of sample proportions versus total score.

Dichotomous data:

Rasch Model, Two Parameter Logistic Model, Birnbaum's Three Parameter Model, and Latent Trait Model up to two latent variables (allowing also for nonlinear terms between the latent traits).

Polytomous data:

Samejima's Graded Response Model and the Generalized Partial Credit Model.

Goodness-of-Fit:

Bootstrapped Pearson \chi^2 for Rasch and Generalized Partial Credit models, fit on the two- and three-way margins for all models, likelihood ratio tests between nested models (including AIC and BIC criteria values), and item- and person-fit statistics.

Factor Scoring - Ability Estimates:

Empirical Bayes (i.e., posterior modes), Expected a posteriori (i.e., posterior means), Multiple Imputed Empirical Bayes, and Component Scores for dichotomous data.

Test Equating:

Alternate Form Equating (where common and unique items are analyzed simultaneously) and Across Sample Equating (where different sets of unique items are analyzed separately based on previously calibrated anchor items).

Plotting:

Item Characteristic Curves, Item Information Curves, Test Information Functions, Standard Error of Measurement, Standardized Loadings Scatterplot (for the two-factor latent trait model), Item Operation Characteristic Curves (for ordinal polytomous data), Item Person Maps.

Author(s)

Dimitris Rizopoulos

Maintainer: Dimitris Rizopoulos <d.rizopoulos@erasmusmc.nl>

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

Attitude Towards Abortion

Description

The data contain responses given by 410 individuals to four out of seven items concerning attitude to abortion. A small number of individual did not answer to some of the questions and this data set contains only the complete cases.

Format

379 individuals answered to the following questions after being asked if the law should allow abortion under the circumstances presented under each item,

Item 1

The woman decides on her own that she does not.

Item 2

The couple agree that they do not wish to have a child.

Item 3

The woman is not married and does not wish to marry the man.

Item 4

The couple cannot afford any more children.

Source

1986 British Social Attitudes Survey (McGrath and Waterton, 1986).

References

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Knott, M., Albanese, M. and Galbraith, J. (1990) Scoring attitudes to abortion. The Statistician, 40, 217–223.

McGrath, K. and Waterton, J. (1986) British social attitudes, 1983-86 panel survey. London: SCPR.

Examples

## Descriptive statistics for Abortion data
dsc <- descript(Abortion)
dsc
plot(dsc)

Attitude to the Environment

Description

This data set comes from the Environment section of the 1990 British Social Attitudes Survey (Brook et al., 1991). A sample of 291 responded to the questions below:

Format

All of the below items were measured on a three-group scale with response categories "very concerned", "slightly concerned" and "not very concerned":

LeadPetrol

Lead from petrol.

RiverSea

River and sea pollution.

RadioWaste

Transport and storage of radioactive waste.

AirPollution

Air pollution.

Chemicals

Transport and disposal of poisonous chemicals.

Nuclear

Risks from nuclear power station.

References

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Brook, L., Taylor, B. and Prior, G. (1991) British Social Attitudes, 1990, Survey. London: SCPR.

Examples

## Descriptive statistics for Environment data
descript(Environment)

Goodness of Fit for Rasch Models

Description

Performs a parametric Bootstrap test for Rasch and Generalized Partial Credit models.

Usage

GoF.gpcm(object, simulate.p.value = TRUE, B = 99, seed = NULL, ...)

GoF.rasch(object, B = 49, ...)

Arguments

Details

GoF.gpcm and GoF.rasch perform a parametric Bootstrap test based on Pearson's chi-squared statistic defined as

\sum\limits_{r = 1}^{2^p} \frac{\{O(r) - E(r)\}^2}{E(r)},

where r represents a response pattern, O(r) and E(r) represent the observed and expected frequencies, respectively and p denotes the number of items. The Bootstrap approximation to the reference distribution is preferable compared with the ordinary Chi-squared approximation since the latter is not valid especially for large number of items (=> many response patterns with expected frequencies smaller than 1).

In particular, the Bootstrap test is implemented as follows:

Step 0:

Based on object compute the observed value of the statistic T_{obs}.

Step 1:

Simulate new parameter values, say \theta^*, from N(\hat{\theta}, C(\hat{\theta})), where \hat{\theta} are the MLEs and C(\hat{\theta}) their large sample covariance matrix.

Step 2:

Using \theta^* simulate new data (with the same dimensions as the observed ones), fit the generalized partial credit or the Rasch model and based on this fit calculate the value of the statistic T_i.

Step 3:

Repeat steps 1-2 B times and estimate the p-value using [1 + \sum\limits_{i=1}^B I(T_i > T_{obs})] / (B + 1).

Furthermore, in GoF.gpcm when simulate.p.value = FALSE, then the p-value is based on the asymptotic chi-squared distribution.

Value

An object of class GoF.gpcm or GoF.rasch with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

person.fit, item.fit, margins, gpcm, rasch

Examples

## GoF for the Rasch model for the LSAT data:
fit <- rasch(LSAT)
GoF.rasch(fit)

The Law School Admission Test (LSAT), Section VI

Description

The LSAT is a classical example in educational testing for measuring ability traits. This test was designed to measure a single latent ability scale.

Format

A data frame with the responses of 1000 individuals to 5 questions.

Source

This LSAT example is a part of a data set given in Bock and Lieberman (1970).

References

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Bock, R. and Lieberman, M. (1970) Fitting a response model for n dichotomously scored items. Psychometrika, 35, 179–197.

Examples

## Descriptive statistics for LSAT data
dsc <- descript(LSAT)
dsc
plot(dsc)

Women's Mobility

Description

A rural subsample of 8445 women from the Bangladesh Fertility Survey of 1989.

Format

The dimension of interest is women's mobility of social freedom. Women were asked whether they could engage in the following activities alone (1 = yes, 0 = no):

Item 1

Go to any part of the village/town/city.

Item 2

Go outside the village/town/city.

Item 3

Talk to a man you do not know.

Item 4

Go to a cinema/cultural show.

Item 5

Go shopping.

Item 6

Go to a cooperative/mothers' club/other club.

Item 7

Attend a political meeting.

Item 8

Go to a health centre/hospital.

Source

Bangladesh Fertility Survey of 1989 (Huq and Cleland, 1990).

References

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Huq, N. and Cleland, J. (1990) Bangladesh Fertility Survey, 1989. Dhaka: National Institute of Population Research and Training (NIPORT).

Examples

## Descriptive statistics for Mobility data
descript(Mobility)

Attitude to Science and Technology

Description

This data set comes from the Consumer Protection and Perceptions of Science and Technology section of the 1992 Euro-Barometer Survey (Karlheinz and Melich, 1992) based on a sample from Great Britain. The questions asked are given below:

Format

All of the below items were measured on a four-group scale with response categories "strongly disagree", "disagree to some extent", "agree to some extent" and "strongly agree":

Comfort

Science and technology are making our lives healthier, easier and more comfortable.

Environment

Scientific and technological research cannot play an important role in protecting the environment and repairing it.

Work

The application of science and new technology will make work more interesting.

Future

Thanks to science and technology, there will be more opportunities for the future generations.

Technology

New technology does not depend on basic scientific research.

Industry

Scientific and technological research do not play an important role in industrial development.

Benefit

The benefits of science are greater than any harmful effect it may have.

References

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Karlheinz, R. and Melich, A. (1992) Euro-Barometer 38.1: Consumer Protection and Perceptions of Science and Technology. INRA (Europe), Brussels. [computer file]

Examples

## Descriptive statistics for Science data
descript(Science)

Workplace Industrial Relation Survey Data

Description

These data were taken from a section of the 1990 Workplace Industrial Relation Survey (WIRS) dealing with management/worker consultation in firms. The questions asked are given below:

Format

Please consider the most recent change involving the introduction of the new plant, machinery and equipment. Were discussions or consultations of any of the type on this card held either about the introduction of the change or about the way it was to be implemented.

Item 1

Informal discussion with individual workers.

Item 2

Meeting with groups of workers.

Item 3

Discussions in established joint consultative committee.

Item 4

Discussions in specially constituted committee to consider the change.

Item 5

Discussions with the union representatives at the establishment.

Item 6

Discussions with paid union officials from outside.

References

Bartholomew, D. (1998) Scaling unobservable constructs in social science. Applied Statistics, 47, 1–13.

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Examples

## Descriptive statistics for Wirs data
descript(WIRS)

Anova method for fitted IRT models

Description

Performs a Likelihood Ratio Test between two nested IRT models.

Usage

## S3 method for class 'gpcm'
anova(object, object2, simulate.p.value = FALSE, 
    B = 200, verbose = getOption("verbose"), seed = NULL, ...)

## S3 method for class 'grm'
anova(object, object2, ...)

## S3 method for class 'ltm'
anova(object, object2, ...)

## S3 method for class 'rasch'
anova(object, object2, ...)

## S3 method for class 'tpm'
anova(object, object2, ...)

Arguments

Details

anova.gpcm() also includes the option to estimate the p-value of the LRT using a parametric Bootstrap approach. In particular, B data sets are simulated under the null hypothesis (i.e., under the generalized partial credit model object), and both the null and alternative models are fitted and the value of LRT is computed. Then the p-value is approximate using [1 + \sum\limits_{i=1}^B I(T_i > T_{obs})] / (B + 1), where T_{obs} is the value of the likelihood ratio statistic in the original data set, and T_i the value of the statistic in the ith Bootstrap sample.

In addition, when simulate.p.value = TRUE objects of class aov.gpcm have a method for the plot() generic function that produces a QQ plot comparing the Bootstrap sample of likelihood ration statistic with the asymptotic chi-squared distribution. For instance, you can use something like the following: lrt <- anova(obj1, obj2, simulate.p.value = TRUE); plot(lrt).

Value

An object of either class aov.gpcm, aov.grm, class aov.ltm or class aov.rasch with components,

Warning

The code does not check if the models are nested! The user is responsible to supply nested models in order the LRT to be valid.

When object2 represents a three parameter model, note that the null hypothesis in on the boundary of the parameter space for the guessing parameters. Thus, the Chi-squared reference distribution used by these function might not be totally appropriate.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

GoF.gpcm, GoF.rasch, gpcm, grm, ltm, rasch, tpm

Examples

## LRT between the constrained and unconstrained GRMs 
## for the Science data:
fit0 <- grm(Science[c(1,3,4,7)], constrained = TRUE)
fit1 <- grm(Science[c(1,3,4,7)])
anova(fit0, fit1)


## LRT between the one- and two-factor models 
## for the WIRS data:
anova(ltm(WIRS ~ z1), ltm(WIRS ~ z1 + z2))


## An LRT between the Rasch and a constrained 
## two-parameter logistic model for the WIRS data: 
fit0 <- rasch(WIRS)
fit1 <- ltm(WIRS ~ z1, constraint = cbind(c(1, 3, 5), 2, 1))
anova(fit0, fit1)


## An LRT between the constrained (discrimination 
## parameter equals 1) and the unconstrained Rasch
## model for the LSAT data: 
fit0 <- rasch(LSAT, constraint = rbind(c(6, 1)))
fit1 <- rasch(LSAT)
anova(fit0, fit1)


## An LRT between the Rasch and the two-parameter 
## logistic model for the LSAT data: 
anova(rasch(LSAT), ltm(LSAT ~ z1))

Point-Biserial Correlation

Description

Computes the point-biserial correlation between a dichotomous and a continuous variable.

Usage

biserial.cor(x, y, use = c("all.obs", "complete.obs"), level = 1)

Arguments

Details

The point biserial correlation computed by biserial.cor() is defined as follows

r = \frac{(\overline{X}_1 - \overline{X}_0)\sqrt{\pi (1 - \pi)}}{S_x},

where \overline{X}_1 and \overline{X}_0 denote the sample means of the X-values corresponding to the first and second level of Y, respectively, S_x is the sample standard deviation of X, and \pi is the sample proportion for Y = 1. The first level of Y is defined by the level argument; see Examples.

Value

the (numeric) value of the point-biserial correlation.

Note

Changing the order of the levels for y will produce a different result. By default, the first level is used as a reference level

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

Examples

# the point-biserial correlation between
# the total score and the first item, using
# '0' as the reference level
biserial.cor(rowSums(LSAT), LSAT[[1]])

# and using '1' as the reference level
biserial.cor(rowSums(LSAT), LSAT[[1]], level = 2)

Extract Estimated Loadings

Description

Extracts the estimated parameters from either grm, ltm, rasch or tpm objects.

Usage

## S3 method for class 'gpcm'
coef(object, ...)

## S3 method for class 'grm'
coef(object, ...)

## S3 method for class 'ltm'
coef(object, standardized = FALSE, prob = FALSE, order = FALSE, ...)

## S3 method for class 'rasch'
coef(object, prob = FALSE, order = FALSE, ...)

## S3 method for class 'tpm'
coef(object, prob = FALSE, order = FALSE, ...)

Arguments

Details

The standardization of the factor loadings is useful in order to form a link to the Underlying Variable approach. In particular, the standardized form of the factor loadings represents the correlation coefficient between the latent variables and the underlying continuous variables based on which the dichotomous outcomes arise (see Bartholomew and Knott, 1999, p.87-88 or Bartholomew et al., 2002, p.191).

The standardized factor loadings are computed only for the linear one- and two-factor models, fitted by ltm().

Value

A list or a matrix of the estimated parameters for the fitted model.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Bartholomew, D. and Knott, M. (1999) Latent Variable Models and Factor Analysis, 2nd ed. London: Arnold.

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

See Also

gpcm, grm, ltm, rasch, tpm

Examples

fit <- grm(Science[c(1,3,4,7)])
coef(fit)

fit <- ltm(LSAT ~ z1)
coef(fit, TRUE, TRUE)

m <- rasch(LSAT)
coef(fit, TRUE, TRUE)

Cronbach's alpha

Description

Computes Cronbach's alpha for a given data-set.

Usage

cronbach.alpha(data, standardized = FALSE, CI = FALSE, 
    probs = c(0.025, 0.975), B = 1000, na.rm = FALSE)

Arguments

Details

The Cronbach's alpha computed by cronbach.alpha() is defined as follows

\alpha = \frac{p}{p - 1}\left(1 - \frac{\sum_{i=1}^p \sigma_{y_i}^2}{\sigma_x^2}\right),

where p is the number of items \sigma_x^2 is the variance of the observed total test scores, and \sigma_{y_i}^2 is the variance of the ith item.

The standardized Cronbach's alpha computed by cronbach.alpha() is defined as follows

\alpha_s = \frac{p \cdot \bar{r}}{1 + (p - 1) \cdot \bar{r}},

where p is the number of items, and \bar{r} is the average of all (Pearson) correlation coefficients between the items. In this case if na.rm = TRUE, then the complete observations (i.e., rows) are used.

The Bootstrap confidence interval is calculated by simply taking B samples with replacement from data, calculating for each \alpha or \alpha_s, and computing the quantiles according to probs.

Value

cronbach.alpha() returns an object of class cronbachAlpha with components

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Cronbach, L. J. (1951) Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297–334.

Examples

# Cronbach's alpha for the LSAT data-set
# with a Bootstrap 95% CI
cronbach.alpha(LSAT, CI = TRUE, B = 500)

Descriptive Statistics

Description

Computes descriptive statistics for dichotomous and polytomous response matrices.

Usage

descript(data, n.print = 10, chi.squared = TRUE, B = 1000)

Arguments

Details

The following descriptive statistics are returned by descript():

(i)

the proportions for all the possible response categories for each item. In case all items are dichotomous, the logit of the proportion for the positive responses is also included.

(ii)

the frequencies of all possible total scores. The total score of a response pattern is simply its sum. For dichotomous items this is the number of positive responses, whereas for polytomous items this is the sum of the levels represented as numeric values (e.g., the response categories "very concerned", "slightly concerned", and "not very concerned" in Environment are represented as 1, 2, and 3).

(iii)

Cronbach's alpha, for all items and excluding each time one of the items.

(iv)

for dichotomous response matrices two versions of the point biserial correlation of each item with the total score are returned. In the first one the item is included in the computation of the total score, and in the second one is excluded.

(v)

pairwise associations between items. Before an analysis with latent variable models, it is useful to inspect the data for evidence of positive correlations. In the case of binary or polytomous data, this ad hoc check is performed by constructing the 2 \times 2 contingency tables for all possible pairs of items and examine the Chi-squared p-values. In case any expected frequencies are smaller than 5, simulate.p.value is turned to TRUE in chisq.test(), using B resamples.

Value

descript() returns an object of class descript with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

plot.descript, unidimTest

Examples

## Descriptives for LSAT data:
dsc <- descript(LSAT, 3)
dsc
plot(dsc, type = "b", lty = 1, pch = 1:5)
legend("topleft", names(LSAT), pch = 1:5, col = 1:5, lty = 1, bty = "n")

Factor Scores - Ability Estimates

Description

Computation of factor scores for grm, ltm, rasch and tpm models.

Usage

factor.scores(object, ...)

## S3 method for class 'gpcm'
factor.scores(object, resp.patterns = NULL, 
        method = c("EB", "EAP", "MI"), B = 5, robust.se = FALSE, 
        prior = TRUE, return.MIvalues = FALSE, ...)

## S3 method for class 'grm'
factor.scores(object, resp.patterns = NULL, 
        method = c("EB", "EAP", "MI"), B = 5, prior = TRUE, 
        return.MIvalues = FALSE, ...)

## S3 method for class 'ltm'
factor.scores(object, resp.patterns = NULL, 
        method = c("EB", "EAP", "MI", "Component"), B = 5, 
        robust.se = FALSE, prior = TRUE, return.MIvalues = FALSE, 
        ...)

## S3 method for class 'rasch'
factor.scores(object, resp.patterns = NULL, 
        method = c("EB", "EAP", "MI"), B = 5, robust.se = FALSE,
	    prior = TRUE, return.MIvalues = FALSE, ...)

## S3 method for class 'tpm'
factor.scores(object, resp.patterns = NULL, 
        method = c("EB", "EAP", "MI"), B = 5, prior = TRUE, 
        return.MIvalues = FALSE, ...)

Arguments

Details

Factor scores or ability estimates are summary measures of the posterior distribution p(z|x), where z denotes the vector of latent variables and x the vector of manifest variables.

Usually as factor scores we assign the modes of the above posterior distribution evaluated at the MLEs. These Empirical Bayes estimates (use method = "EB") and their associated variance are good measures of the posterior distribution while p \rightarrow \infty, where p is the number of items. This is based on the result

p(z|x)=p(z|x; \hat{\theta})(1+O(1/p)),

where \hat{\theta} are the MLEs. However, in cases where p and/or n (the sample size) is small we ignore the variability of plugging-in estimates but not the true parameter values. A solution to this problem can be given using Multiple Imputation (MI; use method = "MI"). In particular, MI is used the other way around, i.e.,

Step 1:

Simulate new parameter values, say \theta^*, from N(\hat{\theta}, C(\hat{\theta})), where C(\hat{\theta}) is the large sample covariance matrix of \hat{\theta} (if robust.se = TRUE, C(\hat{\theta}) is based on the sandwich estimator).

Step 2:

Maximize p(z|x; \theta^*) wrt z and also compute the associated variance to this mode.

Step 3:

Repeat steps 1-2 B times and combine the estimates using the known formulas of MI.

This scheme explicitly acknowledges the ignorance of the true parameter values by drawing from their large sample posterior distribution while taking into account the sampling error. The modes of the posterior distribution p(z|x; \theta) are numerically approximated using the BFGS algorithm in optim().

The Expected a posteriori scores (use method = "EAP") computed by factor.scores() are defined as follows:

\int z p(z | x; \hat{\theta}) dz.

The Component scores (use method = "Component") proposed by Bartholomew (1984) is an alternative method to scale the sample units in the latent dimensions identified by the model that avoids the calculation of the posterior mode. However, this method is not valid in the general case where nonlinear latent terms are assumed.

Value

An object of class fscores is a list with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Bartholomew, D. (1984) Scaling binary data using a factor model. Journal of the Royal Statistical Society, Series B, 46, 120–123.

Bartholomew, D. and Knott, M. (1999) Latent Variable Models and Factor Analysis, 2nd ed. London: Arnold.

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

Rizopoulos, D. and Moustaki, I. (2008) Generalized latent variable models with nonlinear effects. British Journal of Mathematical and Statistical Psychology, 61, 415–438.

See Also

plot.fscores, gpcm, grm, ltm, rasch, tpm

Examples

## Factor Scores for the Rasch model
fit <- rasch(LSAT)
factor.scores(fit) # Empirical Bayes


## Factor scores for all subjects in the
## original dataset LSAT
factor.scores(fit, resp.patterns = LSAT)


## Factor scores for specific patterns,
## including NA's, can be obtained by 
factor.scores(fit, resp.patterns = rbind(c(1,0,1,0,1), c(NA,1,0,NA,1)))


## Not run: 
## Factor Scores for the two-parameter logistic model
fit <- ltm(Abortion ~ z1)
factor.scores(fit, method = "MI", B = 20) # Multiple Imputation

## Factor Scores for the graded response model
fit <- grm(Science[c(1,3,4,7)])
factor.scores(fit, resp.patterns = rbind(1:4, c(NA,1,2,3)))

## End(Not run)

Fitted Values for IRT model

Description

Computes the expected frequencies for vectors of response patterns.

Usage

## S3 method for class 'gpcm'
fitted(object, resp.patterns = NULL, 
    type = c("expected", "marginal-probabilities",
    "conditional-probabilities"), ...)

## S3 method for class 'grm'
fitted(object, resp.patterns = NULL, 
    type = c("expected", "marginal-probabilities",
    "conditional-probabilities"), ...)

## S3 method for class 'ltm'
fitted(object, resp.patterns = NULL, 
    type = c("expected", "marginal-probabilities", 
    "conditional-probabilities"), ...)

## S3 method for class 'rasch'
fitted(object, resp.patterns = NULL, 
    type = c("expected", "marginal-probabilities", 
    "conditional-probabilities"), ...)

## S3 method for class 'tpm'
fitted(object, resp.patterns = NULL, 
    type = c("expected", "marginal-probabilities", 
    "conditional-probabilities"), ...)

Arguments

Value

a numeric matrix or a list containing either the response patterns of interest with their expected frequencies or marginal probabilities, if type == "expected" || "marginal-probabilities" or the conditional probabilities for each response pattern and item, if type == "conditional-probabilities".

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

residuals.gpcm, residuals.grm, residuals.ltm, residuals.rasch, residuals.tpm

Examples

fit <- grm(Science[c(1,3,4,7)])
fitted(fit, resp.patterns = matrix(1:4, nr = 4, nc = 4))

fit <- rasch(LSAT)
fitted(fit, type = "conditional-probabilities")

Gauss-Hermite Quadrature Points

Description

Table with Gauss-Hermite Quadrature Points

Generalized Partial Credit Model - Polytomous IRT

Description

Fits the Generalized Partial Credit model for ordinal polytomous data, under the Item Response Theory approach.

Usage

gpcm(data, constraint = c("gpcm", "1PL", "rasch"), IRT.param = TRUE, 
    start.val = NULL, na.action = NULL, control = list())

Arguments

Details

The Generalized Partial Credit Model is an IRT model, that can handle ordinal manifest variables. This model was discussed by Masters (1982) and it was extended by Muraki (1992).

The model is defined as follows

P_{ik}(z) = \frac{\exp \sum \limits_{c = 0}^k \beta_i (z - \beta_{ic}^*)}{ \sum \limits_{r = 0}^{m_i} \exp \sum \limits_{c = 0}^r \beta_i (z - \beta_{ic}^*)},

where P_{ik}(z) denotes the probability of responding in category k for item i, given the latent ability z, \beta_{ic}^* are the item-category parameters, \beta_i is the discrimination parameter, m_i is the number of categories for item i, and

\sum \limits_{c = 0}^0 \beta_i (z - \beta_{ic}^*) \equiv 0.

If constraint = "rasch", then the discrimination parameter \beta_i is assumed equal for all items and fixed at one. If constraint = "1PL", then the discrimination parameter \beta_i is assumed equal for all items but is estimated. If constraint = "gpcm", then each item has its one discrimination parameter \beta_i that is estimated. See Examples for more info.

If IRT.param = FALSE, then the linear predictor is of the form \beta_i z + \beta_{ic}.

The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.

Value

An object of class gpcm with components,

Warning

In case the Hessian matrix at convergence is not positive definite try to re-fit the model by specifying the starting values or using start.val = "random".

Note

gpcm() can also handle binary items and can be used instead of rasch and ltm though it is less efficient. However, gpcm() can handle a mix of dichotomous and polytomous items that neither rasch nor ltm can.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Masters, G. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.

Muraki, E. (1992). A generalized partial credit model: application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.

See Also

coef.gpcm, fitted.gpcm, summary.gpcm, anova.gpcm, plot.gpcm, vcov.gpcm, GoF.gpcm, margins, factor.scores

Examples

## The Generalized Partial Credit Model for the Science data:
gpcm(Science[c(1,3,4,7)])

## The Generalized Partial Credit Model for the Science data,
## assuming equal discrimination parameters across items:
gpcm(Science[c(1,3,4,7)], constraint = "1PL")

## The Generalized Partial Credit Model for the Science data,
## assuming equal discrimination parameters across items
## fixed at 1:
gpcm(Science[c(1,3,4,7)], constraint = "rasch")

## more examples can be found at:
## http://wiki.r-project.org/rwiki/doku.php?id=packages:cran:ltm#sample_analyses

Graded Response Model - Polytomous IRT

Description

Fits the Graded Response model for ordinal polytomous data, under the Item Response Theory approach.

Usage

grm(data, constrained = FALSE, IRT.param = TRUE, Hessian = FALSE, 
    start.val = NULL, na.action = NULL, control = list())

Arguments

Details

The Graded Response Model is a type of polytomous IRT model, specifically designed for ordinal manifest variables. This model was first discussed by Samejima (1969) and it is mainly used in cases where the assumption of ordinal levels of response options is plausible.

The model is defined as follows

\log\left(\frac{\gamma_{ik}}{1-\gamma_{ik}}\right) = \beta_i z - \beta_{ik},

where \gamma_{ik} denotes the cumulative probability of a response in category kth or lower to the ith item, given the latent ability z. If constrained = TRUE it is assumed that \beta_i = \beta for all i.

If IRT.param = TRUE, then the parameters estimates are reported under the usual IRT parameterization, i.e.,

\log\left(\frac{\gamma_{ik}}{1-\gamma_{ik}}\right) = \beta_i (z - \beta_{ik}^*),

where \beta_{ik}^* = \beta_{ik} / \beta_i.

The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.

Value

An object of class grm with components,

Warning

In case the Hessian matrix at convergence is not positive definite try to re-fit the model, using start.val = "random".

Note

grm() returns the parameter estimates such that the discrimination parameter for the first item \beta_1 is positive.

When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE), their standard errors are calculated using the Delta method.

grm() can also handle binary items, which should be coded as ‘1, 2’ instead of ‘0, 1’.

Some parts of the code used for the calculation of the log-likelihood and the score vector have been based on polr() from package MASS.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement, 34, 100–114.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

See Also

coef.grm, fitted.grm, summary.grm, anova.grm, plot.grm, vcov.grm, margins, factor.scores

Examples

## The Graded Response model for the Science data:
grm(Science[c(1,3,4,7)])

## The Graded Response model for the Science data,
## assuming equal discrimination parameters across items:
grm(Science[c(1,3,4,7)], constrained = TRUE)

## The Graded Response model for the Environment data
grm(Environment)

Area under the Test or Item Information Curves

Description

Computes the amount of test or item information for a fitted IRT model, in a specified range.

Usage

information(object, range, items = NULL, ...)

Arguments

Details

The amount of information is computed as the area under the Item or Test Information Curve in the specified interval, using integrate().

Value

A list of class information with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

plot.gpcm, plot.grm, plot.ltm, plot.rasch

Examples

fit <- rasch(LSAT)
information(fit, c(-2, 0))
information(fit, c(0, 2), items = c(3, 5))

Item-Fit Statistics and P-values

Description

Computation of item fit statistics for ltm, rasch and tpm models.

Usage

item.fit(object, G = 10, FUN = median, 
         simulate.p.value = FALSE, B = 100)

Arguments

Details

The item-fit statistic computed by item.fit() has the form:

\sum \limits_{j = 1}^G \frac{N_j (O_{ij} - E_{ij})^2}{E_{ij} (1 - E_{ij})},

where i is the item, j is the interval created by grouping sample units on the basis of their ability estimates, G is the number of sample units groupings (i.e., G argument), N_j is the number of sample units with ability estimates falling in a given interval j, O_{ij} is the observed proportion of keyed responses on item i for interval j, and E_{ij} is the expected proportion of keyed responses on item i for interval j based on the IRT model (i.e., object) evaluated at the ability estimate z^* within the interval, with z^* denoting the result of FUN applied to the ability estimates in group j.

If simulate.p.value = FALSE, then the p-values are computed assuming a chi-squared distribution with degrees of freedom equal to the number of groups G minus the number of estimated parameters. If simulate.p.value = TRUE, a Monte Carlo procedure is used to approximate the distribution of the item-fit statistic under the null hypothesis. In particular, the following steps are replicated B times:

Step 1:

Simulate a new data-set of dichotomous responses under the assumed IRT model, using the maximum likelihood estimates \hat{\theta} in the original data-set, extracted from object.

Step 2:

Fit the model to the simulated data-set, extract the maximum likelihood estimates \theta^* and compute the ability estimates z^* for each response pattern.

Step 3:

For the new data-set, and using z^* and \theta^*, compute the value of the item-fit statistic.

Denote by T_{obs} the value of the item-fit statistic for the original data-set. Then the p-value is approximated according to the formula

\left(1 + \sum_{b = 1}^B I(T_b \geq T_{obs})\right) / (1 + B),

where I(.) denotes the indicator function, and T_b denotes the value of the item-fit statistic in the bth simulated data-set.

Value

An object of class itemFit is a list with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Reise, S. (1990) A comparison of item- and person-fit methods of assessing model-data fit in IRT. Applied Psychological Measurement, 14, 127–137.

Yen, W. (1981) Using simulation results to choose a latent trait model. Applied Psychological Measurement, 5, 245–262.

See Also

person.fit, margins, GoF.gpcm, GoF.rasch

Examples

# item-fit statistics for the Rasch model
# for the Abortion data-set
item.fit(rasch(Abortion))

# Yen's Q1 item-fit statistic (i.e., 10 latent ability groups; the
# mean ability in each group is used to compute fitted proportions) 
# for the two-parameter logistic model for the LSAT data-set
item.fit(ltm(LSAT ~ z1), FUN = mean)

Latent Trait Model - Latent Variable Model for Binary Data

Description

Fit a latent trait model under the Item Response Theory (IRT) approach.

Usage

ltm(formula, constraint = NULL, IRT.param, start.val, 
    na.action = NULL, control = list())

Arguments

.

Details

The latent trait model is the analogue of the factor analysis model for binary observed data. The model assumes that the dependencies between the observed response variables (known as items) can be interpreted by a small number of latent variables. The model formulation is under the IRT approach; in particular,

\log\left(\frac{\pi_{i}}{1-\pi_{i}}\right)=\beta_{0i} + \beta_{1i}z_1 + \beta_{2i}z_2,

where \pi_i is the the probability of a positive response in the ith item, \beta_{i0} is the easiness parameter, \beta_{ij} (j=1,2) are the discrimination parameters and z_1, z_2 denote the two latent variables.

The usual form of the latent trait model assumes linear latent variable effects (Bartholomew and Knott, 1999; Moustaki and Knott, 2000). ltm() fits the linear one- and two-factor models but also provides extensions described by Rizopoulos and Moustaki (2006) to include nonlinear latent variable effects. These are incorporated in the linear predictor of the model, i.e.,

\log\left (\frac{\pi_{i}}{1-\pi_{i}}\right)=\beta_{0i} + \beta_{1i}z_1 + \beta_{2i}z_2 + \beta_{nl}^tf(z_1, z_2),

where f(z_1, z_2) is a function of z_1 and z_2 (e.g., f(z_1, z_2) = z_1z_2, f(z_1, z_2) = z_1^2, etc.) and \beta_{nl} is a matrix of nonlinear terms parameters (look also at the Examples).

If IRT.param = TRUE, then the parameters estimates for the two-parameter logistic model (i.e., the model with one factor) are reported under the usual IRT parameterization, i.e.,

\log\left(\frac{\pi_i}{1-\pi_i}\right) = \beta_{1i} (z - \beta_{0i}^*).

The linear two-factor model is unidentified under orthogonal rotations on the factors' space. To achieve identifiability you can fix the value of one loading using the constraint argument.

The parameters are estimated by maximizing the approximate marginal log-likelihood under the conditional independence assumption, i.e., conditionally on the latent structure the items are independent Bernoulli variates under the logit link. The required integrals are approximated using the Gauss-Hermite rule. The optimization procedure used is a hybrid algorithm. The procedure initially uses a moderate number of EM iterations (see control argument iter.em) and then switches to quasi-Newton (see control arguments method and iter.qN) iterations until convergence.

Value

An object of class ltm with components,

Warning

In case the Hessian matrix at convergence is not positive definite, try to re-fit the model; ltm() will use new random starting values.

The inclusion of nonlinear latent variable effects produces more complex likelihood surfaces which might possess a number of local maxima. To ensure that the maximum likelihood value has been reached re-fit the model a number of times (simulations showed that usually 10 times are adequate to ensure global convergence).

Conversion of the parameter estimates to the usual IRT parameterization works only for the two-parameter logistic model.

Note

In the case of the one-factor model, the optimization algorithm works under the constraint that the discrimination parameter of the first item \beta_{11} is always positive. If you wish to change its sign, then in the fitted model, say m, use m$coef[, 2] <- -m$coef[, 2].

When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE), their standard errors are calculated using the Delta method.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Bartholomew, D. and Knott, M. (1999) Latent Variable Models and Factor Analysis, 2nd ed. London: Arnold.

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Moustaki, I. and Knott, M. (2000) Generalized latent trait models. Psychometrika, 65, 391–411.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

Rizopoulos, D. and Moustaki, I. (2008) Generalized latent variable models with nonlinear effects. British Journal of Mathematical and Statistical Psychology, 61, 415–438.

See Also

coef.ltm, fitted.ltm, summary.ltm, anova.ltm, plot.ltm, vcov.ltm, item.fit, person.fit, margins, factor.scores

Examples

## The two-parameter logistic model for the WIRS data
## with the constraint that (i) the easiness parameter 
## for the 1st item equals 1 and (ii) the discrimination
## parameter for the 6th item equals -0.5

ltm(WIRS ~ z1, constr = rbind(c(1, 1, 1), c(6, 2, -0.5)))


## One-factor and a quadratic term
## using the Mobility data
ltm(Mobility ~ z1 + I(z1^2))

## Two-factor model with an interaction term
## using the WIRS data
ltm(WIRS ~ z1 * z2)


## The two-parameter logistic model for the Abortion data 
## with 20 quadrature points and 20 EM iterations;
## report results under the usual IRT parameterization
ltm(Abortion ~ z1, control = list(GHk = 20, iter.em = 20))

Fit of the model on the margins

Description

Checks the fit on the two- and three-way margins for grm, ltm, rasch and tpm objects.

Usage

margins(object, ...)

## S3 method for class 'gpcm'
margins(object, type = c("two-way", "three-way"), rule = 3.5, ...)

## S3 method for class 'grm'
margins(object, type = c("two-way", "three-way"), rule = 3.5, ...)

## S3 method for class 'ltm'
margins(object, type = c("two-way", "three-way"), rule = 3.5, 
        nprint = 3, ...)

## S3 method for class 'rasch'
margins(object, type = c("two-way", "three-way"), rule = 3.5, 
        nprint = 3, ...)

## S3 method for class 'tpm'
margins(object, type = c("two-way", "three-way"), rule = 3.5, 
        nprint = 3, ...)

Arguments

Details

Rather than looking at the whole set of response patterns, we can look at the two- and three-way margins. For the former, we construct the 2 \times 2 contingency tables obtained by taking the variables two at a time. Comparing the observed and expected two-way margins is analogous to comparing the observed and expected correlations when judging the fit of a factor analysis model. For Bernoulli and Ordinal variates, the comparison is made using the so called Chi-squared residuals. As a rule of thumb residuals greater than 3.5 are indicative of poor fit. For a more strict rule of thumb use the rule argument. The analogous procedure is followed for the three-way margins.

Value

An object of either class margins.ltm if object inherits from class ltm, class rasch or class tpm, or an object of class margins.grm if object inherits from class grm, with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Bartholomew, D. (1998) Scaling unobservable constructs in social science. Applied Statistics, 47, 1–13.

Bartholomew, D. and Knott, M. (1999) Latent Variable Models and Factor Analysis, 2nd ed. London: Arnold.

Bartholomew, D., Steel, F., Moustaki, I. and Galbraith, J. (2002) The Analysis and Interpretation of Multivariate Data for Social Scientists. London: Chapman and Hall.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

See Also

person.fit, item.fit, GoF.rasch,

Examples

## Two- and Three-way residuals for the Rasch model
fit <- rasch(LSAT)
margins(fit)
margins(fit, "three")


## Two- and Three-way residuals for the one-factor model
fit <- ltm(WIRS ~ z1)
margins(fit)
margins(fit, "three")


## Two- and Three-way residuals for the graded response model
fit <- grm(Science[c(1,3,4,7)])
margins(fit)
margins(fit, "three")

Multiple Choice Items to Binary Responses

Description

It converts multiple choice items to a matrix of binary responses.

Usage

mult.choice(data, correct)

Arguments

Value

a matrix of 0/1 values indicating wrong/correct answers.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

Examples

dat <- data.frame(It1 = sample(4, 100, TRUE),
                  It2 = sample(4, 100, TRUE),
                  It3 = sample(5, 100, TRUE),
                  It4 = sample(5, 100, TRUE),
                  It5 = sample(4, 100, TRUE),
                  It6 = sample(5, 100, TRUE))
dat[] <- lapply(dat, function (x) { x[sample(100, 4)] <- NA; x })
crct <- c(3, 2, 5, 3, 4, 5)
####################
mult.choice(dat, crct)

Person-Fit Statistics and P-values

Description

Computation of person fit statistics for ltm, rasch and tpm models.

Usage

person.fit(object, alternative = c("less", "greater", "two.sided"), 
           resp.patterns = NULL, FUN = NULL, simulate.p.value = FALSE, 
           B = 1000)

Arguments

Details

The statistics calculated by default (i.e., if FUN = NULL) by person.fit() are the L_0 statistic of Levine and Rubin (1979) and its standardized version L_z proposed by Drasgow et al. (1985). If simulate.p.value = FALSE, the p-values are calculated for the L_z assuming a standard normal distribution for the statistic under the null. If simulate.p.value = TRUE, a Monte Carlo procedure is used to approximate the distribution of the person-fit statistic(s) under the null hypothesis. In particular, the following steps are replicated B times for each response pattern:

Step 1:

Simulate a new ability estimate, say z^*, from a normal distribution with mean the ability estimate of the response pattern under the fitted model (i.e., object), and standard deviation the standard error of the ability estimate, as returned by the factor.scores function.

Step 2:

Simulate a new response pattern of dichotomous items under the assumed IRT model, using z^* and the maximum likelihood estimates under object.

Step 4:

For the new response pattern and using z^* and the MLEs, compute the values of the person-fit statistic.

Denote by T_{obs} the value of the person-fit statistic for the original data-set. Then the p-value is approximated according to the formula

\left(1 + \sum_{b = 1}^B I(T_b \leq T_{obs})\right) / (1 + B),

if alternative = "less",

\left(1 + \sum_{b = 1}^B I(T_b \geq T_{obs})\right) / (1 + B),

if alternative = "greater", or

\left(1 + \sum_{b = 1}^B I(|T_b| \geq |T_{obs}|)\right) / (1 + B),

if alternative = "two.sided", where T_b denotes the value of the person-fit statistic in the bth simulated data-set, I(.) denotes the indicator function, and |.| denotes the absolute value. For the L_z statistic, negative values (i.e., alternative = "less") indicate response patterns that are unlikely, given the measurement model and the ability estimate. Positive values (i.e., alternative = "greater") indicate that the examinee's response pattern is more consistent than the probabilistic IRT model expected. Finally, when alternative = "two.sided" both the above settings are captured.

This simulation scheme explicitly accounts for the fact that ability values are estimated, by drawing from their large sample distribution. Strictly speaking, drawing z^* from a normal distribution is not theoretically appropriate, since the posterior distribution for the latent abilities is not normal. However, the normality assumption will work reasonably well, especially when a large number of items is considered.

Value

An object of class persFit is a list with components,

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Drasgow, F., Levine, M. and Williams, E. (1985) Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38, 67–86.

Levine, M. and Rubin, D. (1979) Measuring the appropriateness of multiple-choice test scores. Journal of Educational Statistics, 4, 269–290.

Meijer, R. and Sijtsma, K. (2001) Methodology review: Evaluating person fit. Applied Psychological Measurement, 25, 107–135.

Reise, S. (1990) A comparison of item- and person-fit methods of assessing model-data fit in IRT. Applied Psychological Measurement, 14, 127–137.

See Also

item.fit, margins, GoF.gpcm, GoF.rasch

Examples

# person-fit statistics for the Rasch model
# for the Abortion data-set
person.fit(rasch(Abortion))

# person-fit statistics for the two-parameter logistic model
# for the LSAT data-set
person.fit(ltm(LSAT ~ z1), simulate.p.value = TRUE, B = 100)

Plot method for fitted IRT models

Description

Produces the Item Characteristic or Item Information Curves for fitted IRT models.

Usage

## S3 method for class 'gpcm'
plot(x, type = c("ICC", "IIC", "OCCu", "OCCl"), 
    items = NULL, category = NULL, zrange = c(-3.8, 3.8), 
    z = seq(zrange[1], zrange[2], length = 100), annot,
    labels = NULL, legend = FALSE, cx = "top", cy = NULL, ncol = 1, 
    bty = "n", col = palette(), lty = 1, pch, xlab, ylab, main, sub = NULL, 
    cex = par("cex"), cex.lab = par("cex.lab"), cex.main = par("cex.main"), 
    cex.sub = par("cex.sub"), cex.axis = par("cex.axis"), ask = TRUE,
    plot = TRUE, ...)

## S3 method for class 'grm'
plot(x, type = c("ICC", "IIC", "OCCu", "OCCl"), 
    items = NULL, category = NULL, zrange = c(-3.8, 3.8), 
    z = seq(zrange[1], zrange[2], length = 100), annot,
    labels = NULL, legend = FALSE,
    cx = "top", cy = NULL, ncol = 1, bty = "n", col = palette(), lty = 1, pch,
    xlab, ylab, main, sub = NULL, cex = par("cex"), cex.lab = par("cex.lab"),
    cex.main = par("cex.main"), cex.sub = par("cex.sub"), 
    cex.axis = par("cex.axis"), ask = TRUE, plot = TRUE, ...)

## S3 method for class 'ltm'
plot(x, type = c("ICC", "IIC", "loadings"), 
    items = NULL, zrange = c(-3.8, 3.8),
    z = seq(zrange[1], zrange[2], length = 100), annot,
    labels = NULL, legend = FALSE, cx = "topleft", cy = NULL, 
    ncol = 1, bty = "n", col = palette(), lty = 1, pch, xlab, 
    ylab, zlab, main, sub = NULL, cex = par("cex"),
    cex.lab = par("cex.lab"), cex.main = par("cex.main"), 
    cex.sub = par("cex.sub"), ask = TRUE,
    cex.axis = par("cex.axis"), plot = TRUE, ...)

## S3 method for class 'rasch'
plot(x, type = c("ICC", "IIC"), items = NULL, 
    zrange = c(-3.8, 3.8), z = seq(zrange[1], zrange[2], length = 100), 
    annot, labels = NULL, legend = FALSE, cx = "topleft", cy = NULL, 
    ncol = 1, bty = "n", col = palette(), lty = 1, pch, xlab, ylab, 
    main, sub = NULL, cex = par("cex"), cex.lab = par("cex.lab"),
    cex.main = par("cex.main"), cex.sub = par("cex.sub"), 
    cex.axis = par("cex.axis"), plot = TRUE, ...)

## S3 method for class 'tpm'
plot(x, type = c("ICC", "IIC"), items = NULL, 
    zrange = c(-3.8, 3.8), z = seq(zrange[1], zrange[2], length = 100), 
    annot, labels = NULL, legend = FALSE, cx = "topleft", cy = NULL, 
    ncol = 1, bty = "n", col = palette(), lty = 1, 
    pch, xlab, ylab, main, sub = NULL, cex = par("cex"), 
    cex.lab = par("cex.lab"), cex.main = par("cex.main"), 
    cex.sub = par("cex.sub"), cex.axis = par("cex.axis"), 
    plot = TRUE, ...)

Arguments

Details

Item response category characteristic curves show how the probability of responding in the kth category, in each item, changes with the values of the latent variable (ability).

The item information curves indicate the relative ability of an item to discriminate among contiguous trait scores at various locations along the trait continuum. The test information curve, which is the sum of item information curves, provides a visual depiction of where along the trait continuum a test is most discriminating (Reise and Waller, 2002).

Value

The values used to create the plot, i.e., the x-, y-coordinates. This is either a matrix or a list in which the first column or element provides the latent variable values used, and the remaining columns or elements correspond to either probabilities or information or loadings, depending on the value of the type argument.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Reise, S. and Waller, N. (2002) Item response theory for dichotomous assessment data. In Drasgow, F. and Schmitt, N., editors, Measuring and Analyzing Behavior in Organizations. San Francisco: Jossey-Bass.

See Also

information, gpcm, grm, ltm, rasch, tpm

Examples

# Examples for plot.grm()

fit <- grm(Science[c(1,3,4,7)])

## Item Response Category Characteristic Curves for 
## the Science data
op <- par(mfrow = c(2, 2))
plot(fit, lwd = 2, legend = TRUE, ncol = 2)
# re-set par()
par(op)

## Item Characteristic Curves for the 2nd category,
## and items 1 and 3
plot(fit, category = 2, items = c(1, 3), lwd = 2, legend = TRUE, cx = "right")

## Item Information Curves for the Science data;
plot(fit, type = "IIC", legend = TRUE, cx = "topright", lwd = 2, cex = 1.4)

## Test Information Function for the Science data;
plot(fit, type = "IIC", items = 0, lwd = 2)


###################################################


# Examples for plot.ltm()

## Item Characteristic Curves for the two-parameter logistic
## model; plot only items 1, 2, 4 and 6; take the range of the
## latent ability to be (-2.5, 2.5):
fit <- ltm(WIRS ~ z1)
plot(fit, items = c(1, 2, 4, 6), zrange = c(-2.5, 2.5), lwd = 3, cex = 1.4)

## Test Information Function under the two-parameter logistic
## model for the Lsat data
fit <- ltm(LSAT ~ z1)
plot(fit, type = "IIC", items = 0, lwd = 2, cex.lab = 1.2, cex.main = 1.3)
info <- information(fit, c(-3, 0))
text(x = 2, y = 0.5, labels = paste("Total Information:", round(info$InfoTotal, 3), 
    "\n\nInformation in (-3, 0):", round(info$InfoRange, 3), 
    paste("(", round(100 * info$PropRange, 2), "%)", sep = "")), cex = 1.2)

## Item Characteristic Surfaces for the interaction model:
fit <- ltm(WIRS ~ z1 * z2)
plot(fit, ticktype = "detailed", theta = 30, phi = 30, expand = 0.5, d = 2, 
     cex = 0.7, col = "lightblue")

###################################################


# Examples for plot.rasch()

## Item Characteristic Curves for the WIRS data;
## plot only items 1, 3 and 5:
fit <- rasch(WIRS)
plot(fit, items = c(1, 3, 5), lwd = 3, cex = 1.4)
abline(v = -4:4, h = seq(0, 1, 0.2), col = "lightgray", lty = "dotted")

fit <- rasch(LSAT)

## Item Characteristic Curves for the LSAT data;
## plot all items plus a legend and use only black:
plot(fit, legend = TRUE, cx = "right", lwd = 3, cex = 1.4, 
     cex.lab = 1.6, cex.main = 2, col = 1, lty = c(1, 1, 1, 2, 2),
     pch = c(16, 15, 17, 0, 1))
abline(v = -4:4, h = seq(0, 1, 0.2), col = "lightgray", lty = "dotted")

## Item Information Curves, for the first 3 items; include a legend
plot(fit, type = "IIC", items = 1:3, legend = TRUE, lwd = 2, cx = "topright")

## Test Information Function
plot(fit, type = "IIC", items = 0, lwd = 2, cex.lab = 1.1, 
     sub = paste("Call: ", deparse(fit$call)))

## Total information in (-2, 0) based on all the items
info.Tot <- information(fit, c(-2, 0))$InfoRange
## Information in (-2, 0) based on items 2 and 4
info.24 <- information(fit, c(-2, 0), items = c(2, 4))$InfoRange
text(x = 2, y = 0.5, labels = paste("Total Information in (-2, 0):", 
    round(info.Tot, 3), 
    "\n\nInformation in (-2, 0) based on\n Items 2 and 4:", round(info.24, 3), 
    paste("(", round(100 * info.24 / info.Tot, 2), "%)", sep = "")), 
    cex = 1.2)

## The Standard Error of Measurement can be plotted by
vals <- plot(fit, type = "IIC", items = 0, plot = FALSE)
plot(vals[, "z"], 1 / sqrt(vals[, "info"]), type = "l", lwd = 2,
     xlab = "Ability", ylab = "Standard Error", 
     main = "Standard Error of Measurement")

###################################################


# Examples for plot.tpm()

## Compare the Item Characteristic Curves for the LSAT data,
## under the constraint Rasch model, the unconstraint Rasch model,
## and the three parameter model assuming equal discrimination
## across items
par(mfrow = c(2, 2))
pl1 <- plot(rasch(LSAT, constr = cbind(length(LSAT) + 1, 1)))
text(2, 0.35, "Rasch model\nDiscrimination = 1")
pl2 <- plot(rasch(LSAT))
text(2, 0.35, "Rasch model")
pl3 <- plot(tpm(LSAT, type = "rasch", max.guessing = 1))
text(2, 0.35, "Rasch model\nwith Guessing parameter")

## Compare the Item Characteristic Curves for Item 4
## (you have to run the above first)
plot(range(pl1[, "z"]), c(0, 1), type = "n", xlab = "Ability", 
     ylab = "Probability", main = "Item Characteristic Curves - Item 4")
lines(pl1[, c("z", "Item 4")], lwd = 2, col = "black")
lines(pl2[, c("z", "Item 4")], lwd = 2, col = "red")
lines(pl3[, c("z", "Item 4")], lwd = 2, col = "blue")
legend("right", c("Rasch model Discrimination = 1", "Rasch model", 
       "Rasch model with\nGuessing parameter"), lwd = 2, col = c("black", 
       "red", "blue"), bty = "n")

Descriptive Statistics Plot method

Description

The plot method for descript objects currently works for dichotomous response patterns, and produces the xy-plot of the total score versus the proportion of correct responses for each item.

Usage

## S3 method for class 'descript'
plot(x, items = NULL, includeFirstLast = FALSE, xlab, ylab, ...)

Arguments

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

descript

Examples

## Descriptives for WIRS data:
dsc <- descript(WIRS)
dsc
plot(dsc, includeFirstLast = TRUE, type = "b", lty = 1, pch = 1:6)
legend("topleft", names(WIRS), pch = 1:6, col = 1:6, lty = 1, bty = "n")

Factor Scores - Ability Estimates Plot method

Description

Plots a Kernel Density Estimation of the distribution of the factor scores (i.e., person parameters). Provides also the option to include in the plot the item difficulty parameters (similar to the Item Person Maps).

Usage

## S3 method for class 'fscores'
plot(x, bw = "nrd0", adjust = 2, kernel = "gaussian", 
    include.items = FALSE, tol = 0.2, xlab = "Ability", ylab = "Density", 
    main = "Kernel Density Estimation for Ability Estimates", 
    pch = 16, cex = 1.5, ...)

Arguments

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

factor.scores

Examples

## Factor Scores for LSAT data:
fsc <- factor.scores(rasch(LSAT))
plot(fsc, include.items = TRUE, main = "KDE for Person Parameters")
legend("left", "item parameters", pch = 16, cex = 1.5, bty = "n")

Rasch Model

Description

Fit the Rasch model under the Item Response Theory approach.

Usage

rasch(data, constraint = NULL, IRT.param = TRUE, start.val = NULL, 
    na.action = NULL, control = list(), Hessian = TRUE)

Arguments

Details

The Rasch model is a special case of the unidimensional latent trait model when all the discrimination parameters are equal. This model was first discussed by Rasch (1960) and it is mainly used in educational testing where the aim is to study the abilities of a particular set of individuals.

The model is defined as follows

\log\left(\frac{\pi_i}{1-\pi_i}\right) = \beta_{i} + \beta z,

where \pi_i denotes the conditional probability of responding correctly to the ith item given z, \beta_{i} is the easiness parameter for the ith item, \beta is the discrimination parameter (the same for all the items) and z denotes the latent ability.

If IRT.param = TRUE, then the parameters estimates are reported under the usual IRT parameterization, i.e.,

\log\left(\frac{\pi_i}{1-\pi_i}\right) = \beta (z - \beta_i^*).

The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.

Value

An object of class rasch with components,

Warning

In case the Hessian matrix at convergence is not positive definite, try to re-fit the model using rasch(..., start.val = "random").

Note

Although the common formulation of the Rasch model assumes that the discrimination parameter is fixed to 1, rasch() estimates it. If you wish to fit the constrained version of the model, use the constraint argument accordingly. See Examples for more info.

The optimization algorithm works under the constraint that the discrimination parameter \beta is always positive.

When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE), their standard errors are calculated using the Delta method.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Rasch, G. (1960) Probabilistic Models for Some Intelligence and Attainment Tests. Copenhagen: Paedagogiske Institute.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

See Also

coef.rasch, fitted.rasch, summary.rasch, anova.rasch, plot.rasch, vcov.rasch, GoF.rasch, item.fit, person.fit, margins, factor.scores

Examples

## The common form of the Rasch model for the 
## LSAT data, assuming that the discrimination
## parameter equals 1
rasch(LSAT, constraint = cbind(ncol(LSAT) + 1, 1))


## The Rasch model for the LSAT data under the 
## normal ogive; to do that fix the discrimination
## parameter to 1.702
rasch(LSAT, constraint = cbind(ncol(LSAT) + 1, 1.702))

## The Rasch model for the LSAT data with
## unconstraint discrimination parameter
rasch(LSAT)

## The Rasch model with (artificially created) 
## missing data
data <- LSAT
data[] <- lapply(data, function(x){
    x[sample(1:length(x), sample(15, 1))] <- NA
    x
})
rasch(data)

Pairwise Associations between Items using a Correlation Coefficient

Description

Computes and tests the pairwise associations between items using a correlation coefficient

Usage

rcor.test(mat, p.adjust = FALSE, p.adjust.method = "holm", ...)

Arguments

Value

An object of class rcor.test with components,

The print method for class rcor.test returns a square matrix in which the upper diagonal part contains the estimates of the correlation coefficients, and the lower diagonal part contains the corresponding p-values.

Note

rcor.test() is more appropriate for informal testing of association between polytomous items.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

Examples

## pairwise associations for Environment data:
rcor.test(data.matrix(Environment), method = "kendall")

## pairwise associations for independent normal random variates:
mat <- matrix(rnorm(1000), 100, 10, dimnames = list(NULL, LETTERS[1:10]))
rcor.test(mat)
rcor.test(mat, method = "kendall")
rcor.test(mat, method = "spearman")

Residuals for IRT models

Description

Computes the residuals for vectors of response patterns.

Usage

## S3 method for class 'gpcm'
residuals(object, resp.patterns = NULL, order = TRUE, ...)

## S3 method for class 'grm'
residuals(object, resp.patterns = NULL, order = TRUE, ...)

## S3 method for class 'ltm'
residuals(object, resp.patterns = NULL, order = TRUE, ...)

## S3 method for class 'rasch'
residuals(object, resp.patterns = NULL, order = TRUE, ...)

## S3 method for class 'tpm'
residuals(object, resp.patterns = NULL, order = TRUE, ...)

Arguments

Details

The following residuals are computed:

\frac{O_i - E_i}{\sqrt{E_i}},

where O_i and E_i denote the observed and expected frequencies for the ith response pattern.

Value

A numeric matrix containing the observed and expected frequencies as well as the residual value for each response pattern.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

fitted.gpcm, fitted.grm, fitted.ltm, fitted.rasch, fitted.tpm

Examples

fit <- ltm(LSAT ~ z1)
residuals(fit)
residuals(fit, order = FALSE)

Generate Random Responses Patterns under Dichotomous and Polytomous IRT models

Description

Produces Bernoulli or Multinomial random variates under the Rasch, the two-parameter logistic, the three parameter, the graded response, and the generalized partial credit models.

Usage

rmvlogis(n, thetas, IRT = TRUE, link = c("logit", "probit"), 
         distr = c("normal", "logistic", "log-normal", "uniform"), 
         z.vals = NULL)

rmvordlogis(n, thetas, IRT = TRUE, model = c("gpcm", "grm"), 
    link = c("logit", "probit"), 
    distr = c("normal", "logistic", "log-normal", "uniform"), 
    z.vals = NULL)

Arguments

Details

The binary variates can be simulated under the following parameterizations for the probability of correctly responding in the ith item. If IRT = TRUE

\pi_i = c_i + (1 - c_i) g(\beta_{2i} (z - \beta_{1i})),

whereas if IRT = FALSE

\pi_i = c_i + (1 - c_i) g(\beta_{1i} + \beta_{2i} z),

z denotes the latent variable, \beta_{1i} and \beta_{2i} are the first and second columns of thetas, respectively, and g() is the link function. If thetas is a three-column matrix then the third column should contain the guessing parameters c_i's.

The ordinal variates are simulated according to the generalized partial credit model or the graded response model depending on the value of the model argument. Check gpcm and grm to see how these models are defined, under both parameterizations.

Value

a numeric matrix with n rows and columns the number of items, containing the simulated binary or ordinal variates.

Note

For options distr = "logistic", distr = "log-normal" and distr = "uniform" the simulated random variates for z simulated under the Logistic distribution with location = 0 and scale = 1, the log-Normal distribution with meanlog = 0 and sdlog = 1 and the Uniform distribution with min = -3.5 and max = 3.5, respectively. Then, the simulated z variates are standardized, using the theoretical mean and variance of the Logistic, log-Normal and Uniform distribution, respectively.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

gpcm, grm, ltm, rasch, tpm

Examples

# 10 response patterns under a Rasch model
# with 5 items
rmvlogis(10, cbind(seq(-2, 2, 1), 1))

# 10 response patterns under a GPCM model
# with 5 items, with 3 categories each
thetas <- lapply(1:5, function(u) c(seq(-1, 1, len = 2), 1.2))
rmvordlogis(10, thetas)

Summary method for fitted IRT models

Description

Summarizes the fit of either grm, ltm, rasch or tpm objects.

Usage

## S3 method for class 'gpcm'
summary(object, robust.se = FALSE, ...)

## S3 method for class 'grm'
summary(object, ...)

## S3 method for class 'ltm'
summary(object, robust.se = FALSE, ...)

## S3 method for class 'rasch'
summary(object, robust.se = FALSE, ...)

## S3 method for class 'tpm'
summary(object, ...)

Arguments

Value

An object of either class summ.gpcm, class summ.grm, class summ.ltm or class summ.rasch with components,

Note

For the parameters that have been constrained, the standard errors and z-values are printed as NA.

When the coefficients' estimates are reported under the usual IRT parameterization (i.e., IRT.param = TRUE in the call of either grm, ltm or rasch), their standard errors are calculated using the Delta method.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

gpcm, grm, ltm, rasch, tpm

Examples

# use Hessian = TRUE if you want standard errors
fit <- grm(Science[c(1,3,4,7)], Hessian = TRUE)
summary(fit)

## One factor model using the WIRS data;
## results are reported under the IRT
## parameterization
fit <- ltm(WIRS ~ z1)
summary(fit)

Prepares Data for Test Equating

Description

Test equating by common items.

Usage

testEquatingData(DataList, AnchoringItems = NULL)

Arguments

Details

The purpose of this function is to combine items from different forms. Two cases are considered. Alternate Form Equating (where common and unique items are analyzed simultaneously) and Across Sample Equating (where different sets of unique items are analyzed separately based on previously calibrated anchor items).

Value

A matrix containing the common and unique items.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Yu, C.-H. and Osborn Popp, S. (2005) Test equating by common items and common subjects: concepts and applications. Practical Assessment Research and Evaluation, 10(4), 1–19.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

Examples

# Let two data-sets with common and unique items
dat1 <- as.data.frame(rmvlogis(20, cbind(c(-2, 1, 2, 1), 1)))
names(dat1) <- c("CIt2", "CIt3", "CIt4", "W")

dat2 <- as.data.frame(rmvlogis(10, cbind(c(-2, -1, 1, 2, 0.95), 1)))
names(dat2) <- c("CIt1", "CIt2", "CIt3", "CIt4", "K")

# combine in one data-set by
lisForms <- list(dat1, dat2)
testEquatingData(lisForms)

Birnbaum's Three Parameter Model

Description

Fit Birnbaum's three parameter model under the Item Response Theory approach.

Usage

tpm(data, type = c("latent.trait", "rasch"), constraint = NULL, 
    max.guessing = 1, IRT.param = TRUE, start.val = NULL, 
    na.action = NULL, control = list())

Arguments

Details

Birnbaum's three parameter model is usually employed to handle the phenomenon of non-random guessing in the case of difficult items.

The model is defined as follows

\pi_i = c_i + (1 - c_i) \frac{\exp(\beta_{1i} + \beta_{2i} z)}{1 + \exp(\beta_{1i} + \beta_{2i} z)},

where \pi_i denotes the conditional probability of responding correctly to the ith item given z, c_i denotes the guessing parameter, \beta_{1i} is the easiness parameter, \beta_{2i} is the discrimination parameter, and z denotes the latent ability. In case type = "rasch", \beta_{2i} is assumed equal for all items.

If IRT.param = TRUE, then the parameters estimates are reported under the usual IRT parameterization, i.e.,

\pi_i = c_i + (1 - c_i) \frac{\exp[\beta_{2i} (z - \beta_{1i}^*)]}{1 + \exp[\beta_{2i} (z - \beta_{1i}^*)]}.

The fit of the model is based on approximate marginal Maximum Likelihood, using the Gauss-Hermite quadrature rule for the approximation of the required integrals.

Value

An object of class tpm with components,

Warning

The three parameter model is known to have numerical problems like non-convergence or convergence on the boundary, especially for the guessing parameters. These problems usually result in a zero estimate for some guessing parameters and/or in a non positive definite Hessian matrix or in a high absolute value for the score vector (returned by the summary method) at convergence. In case of estimates on the boundary, the constraint argument can be used to set the guessing parameter(s) for the problematic item(s) to zero. In addition, tpm() has a number of control parameters that can be tuned in order to obtain successful convergence; the most important of these are the starting values, the parameter scaling vector and the optimizer.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Baker, F. and Kim, S-H. (2004) Item Response Theory, 2nd ed. New York: Marcel Dekker.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In F. M. Lord and M. R. Novick (Eds.), Statistical Theories of Mental Test Scores, 397–479. Reading, MA: Addison-Wesley.

Rizopoulos, D. (2006) ltm: An R package for latent variable modelling and item response theory analyses. Journal of Statistical Software, 17(5), 1–25. URL doi: 10.18637/jss.v017.i05

See Also

coef.tpm, fitted.tpm, summary.tpm, anova.tpm, plot.tpm, vcov.tpm, item.fit, person.fit, margins, factor.scores

Examples

# the three parameter model
tpm(LSAT)

# use 'nlminb' as optimizer
tpm(LSAT, control = list(optimizer = "nlminb"))


# the three parameter model with equal 
# discrimination parameter across items
# fix the guessing parameter for the third item to zero
tpm(LSAT, type = "rasch", constraint = cbind(3, 1, 0))


# the three parameter model for the Abortion data
fit <- tpm(Abortion)
fit

# the guessing parameter estimates for items 1, 3, and 4 seem to be on
# the boundary; update the fit by fixing them to zero
update(fit, constraint = cbind(c(1, 3, 4), 1, 0))

Unidimensionality Check using Modified Parallel Analysis

Description

An empirical check for the unidimensionality assumption for ltm, rasch and tpm models.

Usage

unidimTest(object, data, thetas, IRT = TRUE, z.vals = NULL, 
           B = 100, ...)

Arguments

Details

This function implements the procedure proposed by Drasgow and Lissak (1983) for examining the latent dimensionality of dichotomously scored item responses. The statistic used for testing unidimensionality is the second eigenvalue of the tetrachoric correlations matrix of the dichotomous items. The tetrachoric correlations between are computed using function polycor() from package ‘polycor’, and the largest one is taken as communality estimate.

A Monte Carlo procedure is used to approximate the distribution of this statistic under the null hypothesis. In particular, the following steps are replicated B times:

Step 1:

If object is supplied, then simulate new ability estimates, say z^*, from a normal distribution with mean the ability estimates \hat{z} in the original data-set, and standard deviation the standard error of \hat{z} (in this case the z.vals argument is ignored). If object is not supplied and the z.vals argument has been specified, then set z^* = z.vals. Finally, if object is not supplied and the z.vals argument has not been specified, then simulate z^* from a standard normal distribution.

Step 2:

Simulate a new data-set of dichotomous responses, using z^*, and parameters the estimated parameters extracted from object (if it is supplied) or the parameters given in the thetas argument.

Step 3:

For the new data-set simulated in Step 2, compute the tetrachoric correlations matrix and take the largest correlations as communalities. For this matrix compute the eigenvalues.

Denote by T_{obs} the value of the statistic (i.e., the second eigenvalue) for the original data-set. Then the p-value is approximated according to the formula \left(1 + \sum_{b = 1}^B I(T_b \geq T_{obs})\right) / (1 + B), where I(.) denotes the indicator function, and T_b denotes the value of the statistic in the bth data-set.

Value

An object of class unidimTest is a list with components,

Note

For ltm objects you can also use a likelihood ratio test to check unidimensionality. In particular, fit0 <- ltm(data ~ z1); fit1 <- ltm(data ~ z1 + z2); anova(fit0, fit1).

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

References

Drasgow, F. and Lissak, R. (1983) Modified parallel analysis: a procedure for examining the latent dimensionality of dichotomously scored item responses. Journal of Applied Psychology, 68, 363–373.

See Also

descript

Examples

## Not run: 
# Unidimensionality Check for the LSAT data-set
# under a Rasch model:
out <- unidimTest(rasch(LSAT))
out
plot(out, type = "b", pch = 1:2)
legend("topright", c("Real Data", "Average Simulated Data"), lty = 1, 
    pch = 1:2, col = 1:2, bty = "n")

## End(Not run)

vcov method for fitted IRT models

Description

Extracts the asymptotic variance-covariance matrix of the MLEs from either gpcm, grm, ltm, rasch or tpm objects.

Usage

## S3 method for class 'gpcm'
vcov(object, robust = FALSE, ...)

## S3 method for class 'grm'
vcov(object, ...)

## S3 method for class 'ltm'
vcov(object, robust = FALSE, ...)

## S3 method for class 'rasch'
vcov(object, robust = FALSE, ...)

## S3 method for class 'tpm'
vcov(object, ...)

Arguments

Value

a numeric matrix representing the estimated covariance matrix of the maximum likelihood estimates. Note that this covariance matrix is for the parameter estimates under the additive parameterization and not under the usual IRT parameterization; for more info check the Details section of grm, ltm, rasch, and tpm.

Author(s)

Dimitris Rizopoulos d.rizopoulos@erasmusmc.nl

See Also

gpcm, grm, ltm, rasch, tpm

Examples

fit <- rasch(WIRS)
vcov(fit)
sqrt(diag(vcov(fit))) # standard errors under additive parameterization