Estimate and plot IRT models for binary responses

Description

Details

The irtoys package contains a bunch of functions potentially useful to those teaching or learning Item Response Theory (IRT). R being particularly good at graphics, there is a rich array of plotting functions to visualize IRT models or assess their fit. Ability estimates can be estimated by MLE, BME, EAP, WLE. Various IRT scaling methods are supported: MM, MS, Stocking-Lord, and Hebaera. Last not least, irtoys may be the only package to estimate Haberman's interaction model, although a new and more powerful package is in the making.

Currently, there are several R packages that can estimate the item parameters in various uni- and multidimensional IRT models, but only several years ago one had to use stand-alone programs that had wildly different and often unwieldy user interfaces. Besides, no single program does everything one needs. One of the original purposes of irtoys was to provide a simple and unified interface to some of the most basic functions in ICL, BILOG, and R's own ltm, such that beginners could concentrate on learning IRT rather than syntaxes. Once that these steps have been made, those wishing to take advantage of the full functionality of ICL, BILOG & Co. must still master the syntax of their program of choice.

To take full advantage of irtoys, some IRT software is needed. Package ltm is automatically loaded. ICL by Brad Hanson can be downloaded from his site, http://www.openirt.com/b-a-h/software/. Executables and a very detailed manual (not needed to use ICL with irtoys are provided for Windows, Linux, and Macintosh. The Mac version probably does not work with the current OS versions but I still use the Windows and Linux versions without any problem.

BILOG is commercial software sold by SSI — see https://ssicentral.com/ for further detail.

On Windows, make sure that the executable files (icl.exe for ICL, BLM1.EXE, BLM2.EXE, and BLM3.EXE for BILOG) are located in a directory that is included in the PATH variable. On Linux, BILOG, being a Windows program, is run with wine, and should also be on a path where wine can find it. On my machine, I have simply put the three files in ~/.wine/drive_c/windows/. It seems that new versions of wine expect them to be explicitly tagged as executable. On Macintosh, at least ltm should work in all cases.

NOTE: Starting with version 0.2.0, function est returns a list of three elements: est contains the parameter estimates and is thus identical to the output in earlier versions, se contains the standard errors, in a similar format, and vcm contains the variance-covariance matrices (NULL when using ICL). When passing item parameters to another function that only needs the estimates, irf(ip) and irf(ip$est) can be used interchangeably. This facilitates using simulated item parameters. A function that does require the complete object is .

Also, function itf now returns item fit statistics as a vector rather than a list. Finally, since most of the functions in irtoys have been written with the "logistic" metric in mind (i.e., a_j(\theta_i-b_j) rather than 1.7a^*_j(\theta_i-b_j), function est now estimates item parameters only in the logistic metric.

Author(s)

Ivailo Partchev <partchev@gmail.com>

References

S. E. Embretson and S. P. Reise (2000), Item Response Theory for Psychologists, Lawrence Erlbaum Associates, Mahwah, NJ

Binary (true/false) responses to a test

Description

Real-life data set containing the responses to a test, scored as true or false.

Format

A data set with 472 persons and 18 items.

Example item parameters

Description

Item parameter estimates for the 2PL model, estimated with ltm from the example data set Scored. These are provided as a check, and to speed up the examples for the various functions in the package.

Format

The object returned by est

Original, unscored multiple-choice responses to a test

Description

Real-life data set containing the responses to a test, before they have been recoded as true or false. Can be used with only two functions in the package: scoreResponses and tgf. All other functions expect binary data, which can be produced with scoreResponses.

Format

A data set with 472 persons and 18 items. Each item has 4 possible answers, of which only one is true. There are many NA, which can be treated as wrong responses.

Estimate ability

Description

A wrapper around all functions for ability estimation

Usage

ability(resp, ip, method = "WLE", mu = 0, sigma = 1, n = 5)

Arguments

Value

Depends on the method.

Author(s)

Ivailo Partchev

See Also

mlebme, eap, wle, dpv, qrs

Examples

theta <- ability(resp=Scored, method="WLE", ip=Scored2pl)

The api appropriateness index

Description

Computes the api appropriateness index, a measure of person fit in IRT models

Usage

api(resp, ip, theta)

Arguments

Value

A vector of length equal to the number of rows in resp, containing the appropriateness indices

Author(s)

Ivailo Partchev

References

Drasgow, F., Levine, M. V., & Williams, E. A. (1985). Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38, 67–80

Examples

api(Scored, Scored2pl)

Example item parameters 1PL (Bilog)

Description

Item parameter estimates for the 1PL model, estimated with bilog from the example data set Scored. These are provided because not users will have BILOG-MG available.

Format

The object returned by est

Example item parameters 2PL (Bilog)

Description

Item parameter estimates for the 2PL model, estimated with bilog from the example data set Scored. These are provided because not users will have BILOG-MG available.

Format

The object returned by est

Example item parameters 3PL (Bilog)

Description

Item parameter estimates for the 3PL model, estimated with bilog from the example data set Scored. These are provided because not users will have BILOG-MG available.

Format

The object returned by est

Draw plausible values

Description

Draws (by rejection sampling) plausible values from the posterior distribution of ability

Usage

dpv(resp, ip, mu = 0, sigma = 1, n = 5)

Arguments

Value

A matrix with n columns

Author(s)

Ivailo Partchev

See Also

mlebme, eap

Examples

plval <- dpv(resp=Scored, ip=Scored2pl)

EAP estimation of ability

Description

Estimates the expectation of the posterior distribution of the latent variable ("ability") for each person.

Usage

eap(resp, ip, qu)

Arguments

Value

A matrix with the ability estimates in column 1, and their standard errors of measurement (SEM) in column 2, and the number of non-missing reponses in column 3

Author(s)

Ivailo Partchev

See Also

mlebme, normal.qu, read.qu.icl

Examples

th.eap <- eap(resp=Scored, ip=Scored2pl, qu=normal.qu())

Plot empirical response function

Description

Produces a plot of the empirical response function (proportion of correct responses at each possible sum score).

Usage

erf(resp, item)

Arguments

Value

None

Author(s)

Ivailo Partchev

Examples

erf(Scored, 3)

Estimate item parameters

Description

Estimate IRT item parameters using either ICL, BILOG, or ltm. Provides access to the most widely used options in these programs.

Usage

est(
  resp,
  model = "2PL",
  engine = "icl",
  omitsWrong = FALSE,
  nqp = 20,
  est.distr = FALSE,
  nch = 5,
  a.prior = TRUE,
  b.prior = FALSE,
  c.prior = TRUE,
  bilog.defaults = TRUE,
  rasch = FALSE,
  run.name = "mymodel"
)

Arguments

Details

Estimate the parameters of an IRT model defined in the most general case ("3PL") as

P(U_{ij}=1|\theta_i,a_j,b_j,c_j)=c_j+(1-c_j)\frac{\displaystyle\exp(a_j(\theta_i-b_j))}{1+\displaystyle\exp(a_j(\theta_i-b_j))}

where U_{ij} is a binary response given by person i to item j, \theta_i is the value of the latent variable ("ability") for person i, a_j is the discrimination parameter for item j, b_j is the difficulty parameter for item j, c_j is the asymptote for item j.

Some authors prefer to represent the model with a logit 1.7a^*_j(\theta_i-b_j) rather than a_j(\theta_i-b_j). This option has been removed from irtoys as it is not supported by the remaining functions of the package.

In the 2PL model (model="2PL"), all asymptotes c_j are 0. In the 1PL model (model="1PL"), all asymptotes c_j are 0 and the discriminations a_j are equal for all items (and sometimes to 1).

Package irtoys provides a simple common interface to the estimation of item parameters with three different programs. It only accesses the most basic and widely used options in these programs. Each of the three programs has a much wider choice of options and cababilities, and serious users must still learn the corresponding syntax in order to access the advanced features. Even when models are fit "by hand", irtoys may be useful in plotting results, doing comparisons across programs etc.

Estimation of the more complex IRT models (2PL and 3PL) for some "difficult" data sets often has to use prior distributions for the item parameters. irtoys adopts the default behaviour of BILOG: no priors for b in any model, priors for a in the 2PL and 3PL models, priors for c in the 3PL model. This can be overriden by changing the values of a.prior, b.prior, and c.prior.

If priors are used at all, they will be the same for all items. Note that both ICL and BILOG can, at some additional effort, set different priors for any individual item. At default, the common priors are the BILOG defaults: normal(0,2) for b, lognormal (0, 0.5) for a, and beta(20*p+1, 20(1-p)+1) for c; p is 1 over the number of choices in the original item formulations, which can be set with the parameter nch, and is again assumed the same for all items.

When engine="icl" and bilog.defaults=F, any priors used will be the ICL default ones, and based on the 4-parameter beta distribution: beta(1.01, 1.01, -6, 6) for b, beta(1.75, 3, 0, 3) for a, and beta(3.5, 4, 0, 0.5) for c. When engine="ltm", all commands involving priors are ignored.

est only works when some IRT software is installed. Package ltm is automatically loaded. ICL can be downloaded from http://www.openirt.com/b-a-h/software/. BILOG is commercial software sold by SSI — see https://ssicentral.com/ for further detail. On Windows, make sure that the executable files (icl.exe for ICL, blm1.exe, blm2.exe, and blm3.exe, for BILOG) are located in directories that are included in the PATH variable.

Value

A list with three elements: est and se are matrices containing the parameter estimates and their standard errors, correspondingly. Each matrix has three columns: [,1] item discrimination a, [,2] item difficulty b, and [,3] asymptote c. For the 1PL and 2PL models, all asymptotes are equal to 0; for the 1PL, the discriminations are all equal but not necessarily equal to 1. The third element, vcm, is a list of variance-covariance matrices for the item parameters. Currently, vcm is only used in function irfPlot. When ICL is used as estimation engine, se and vcm are NULL as ICL does not compute standard errors for the item parameter estimates.

Author(s)

Ivailo Partchev

Examples

p.1pl <- est(Scored, model="1PL", engine="ltm")
p.2pl <- est(Scored, model="2PL", engine="ltm")

Item information function

Description

The item information function (IIF) for the 3PL model can be computed as

I(\theta) = a^2\frac{Q(\theta)}{P(\theta)}\left[\frac{P(\theta)-c}{1-c}\right]^2,

where \theta is the value of the latent variable for a person, a is the discrimination parameter for the item, P is the IRF for the person and item, and Q=1-P. For the 1PL and 2PL models, the expression reduces to a^2PQ.

Usage

iif(ip, items = NULL, x = NULL)

Arguments

Details

A common use of this function would be to obtain a plot of the IIF.

Value

A list of:

Author(s)

Ivailo Partchev

See Also

plot.iif, irf

Examples

plot(iif(Scored2pl, items=1:3))

Interaction model

Description

Estimates the parameters of the interaction model. The plots can be produced with the appropriate plotting method.

Usage

interactionModel(x, eps = 0.1, rasch = FALSE)

Arguments

Value

A matrix

Author(s)

Ivailo Partchev, using C code and theory by Gunter Maris

References

Shelby Haberman. The interaction model. Chapter in: Mathias von Davier & Claus H. Carstensen (Eds.) (2007). Multivariate and Mixture Distribution Rasch Models. New York:Springer

Item response function

Description

Returns the item response function of the 3PL (1PL, 2PL) model, the i.e. the probabilities defined by

P(U_{ij}=1|\theta_i,a_j,b_j,c_j)=c_j+(1-c_j)\frac{\displaystyle\exp(a_j(\theta_i-b_j))}{1+\displaystyle\exp(a_j(\theta_i-b_j))}

where U_{ij} is a binary response given by person i to item j, \theta_i is the value of the latent variable ("ability") for person i, a_j is the discrimination parameter for item j, b_j is the difficulty parameter for item j, c_j is the asymptote for item j. Some authors call the IRF "the item characteristic curve".

Usage

irf(ip, items = NULL, x = NULL)

Arguments

Details

In the 2PL model (model="2PL"), all asymptotes c_j are 0. In the 1PL model (model="1PL"), all asymptotes c_j are 0 and the discriminations a_j are equal for all items (and sometimes to 1).

A common use of this function would be to obtain a plot of the IRF.

Value

A list of:

Author(s)

Ivailo Partchev

See Also

plot.irf

Examples

plot(irf(Scored2pl, item=1))

Item fit plot

Description

Produces a plot that compares the estimated trace line (item response function) for an item with the data. The data is represented with a non-parametric trace line computed as in tgf (not npp). Approximate confidence intervals for the parametric line are also shown.

Usage

irfPlot(
  resp,
  ip,
  x = NULL,
  item,
  main = "Parametric vs non-parametric trace line",
  use.sm = FALSE
)

Arguments

Details

Comparing the outputs of npp and irf has at least two drawbacks: (i) package sm is used as a black box, and (ii) the confidence intervals are drawn around the data (even though represented by the nonparametric regression) rather than the parametric curve. In this function, the parametric curve is shown in red, with a 68 The non-parametric curve representing the data is shown in black. It is computed by applying Bayes theorem to the density of ability. The standard error of the parametric curve is computed by applying the delta theorem on the standard error of the item parameters, as computed by the underlying software (currently only available with ICL).

Author(s)

Ivailo Partchev

See Also

irf, npp, tgf

Examples

# a plot for item 5 
irfPlot(Scored, ip=b3, item=4)

Test item fit

Description

Returns a statistic of item fit together with its degrees of freedom and p-value. Optionally produces a plot.

Usage

itf(
  resp,
  ip,
  item,
  stat = "lr",
  theta,
  standardize = TRUE,
  mu = 0,
  sigma = 1,
  bins = 9,
  breaks = NULL,
  equal = "count",
  type = "means",
  do.plot = TRUE,
  main = "Item fit"
)

Arguments

Details

Given a long test, say 20 items or more, a large-test statistic of item fit could be constructed by dividing examinees into groups of similar ability, and comparing the observed proportion of correct answers in each group with the expected proportion under the proposed model. Different statistics have been proposed for this purpose.

The chi-squared statistic

X^2=\sum_g(N_g\frac{(p_g-\pi_g)^2}{\pi_g(1-\pi_g)},

where N_g is the number of examinees in group g, p_g=r_g/N_g, r_g is the number of correct responses to the item in group g, and \pi_g is the IRF of the proposed model for the median ability in group g, is attributed by Embretson & Reise to R. D. Bock, although the article they cite does not actually mention it. The statistic is the sum of the squares of quantities that are often called "Pearson residuals" in the literature on categorical data analysis.

BILOG uses the likelihood-ratio statistic

X^2=2\sum_g\left[r_g\log\frac{p_g}{\pi_g} + (N_g-r_g)\log\frac{(1-p_g)}{(1-\pi_g)}\right],

where \pi_g is now the IRF for the mean ability in group g, and all other symbols are as above.

Both statistics are assumed to follow the chi-squared distribution with degrees of freedom equal to the number of groups minus the number of parameters of the model (eg 2 in the case of the 2PL model). The first statistic is obtained in itf with stat="chi", and the second with stat="lr" (or not specifying stat at all).

In the real world we can only work with estimates of ability, not with ability itself. irtoys allows use of any suitable ability measure via the argument theta. If theta is not specified, itf will compute EAP estimates of ability, group them in 9 groups having approximately the same number of cases, and use the means of the ability eatimates in each group. This is the approximate behaviour of BILOG.

If the test has less than 20 items, itf will issue a warning. For tests of 10 items or less, BILOG has a special statistic of fit, which can be found in the BILOG output. Also of interest is the fit in 2- and 3-way marginal tables in package ltm.

Value

A vector of three numbers:

Author(s)

Ivailo Partchev

References

S. E. Embretson and S. P. Reise (2000), Item Response Theory for Psychologists, Lawrence Erlbaum Associates, Mahwah, NJ

M. F. Zimowski, E. Muraki, R. J. Mislevy and R. D. Bock (1996), BILOG–MG. Multiple-Group IRT Analysis and Test Maintenance for Binary Items, SSI Scientific Software International, Chicago, IL

See Also

eap, qrs

Examples

fit   <- itf(resp=Scored, ip=Scored2pl, item=7)

Maximum likelihood and Bayes Modal estimation of ability

Description

Estimates the value of the latent variable ("ability") for each person by direct optimization

Usage

mlebme(resp, ip, mu = 0, sigma = 1, method = "ML")

Arguments

Value

A matrix with the ability estimates in column 1 and their standard errors of measurement (SEM) in column 2, and the number of non-missing responses in column 3

Author(s)

Ivailo Partchev

See Also

eap

Examples

th.mle <- mlebme(resp=Scored, ip=Scored2pl)

Normal quadrature points and weights

Description

Quadrature points and weights based on the Normal distribution. Quadrature objects are used when estimating abilities with eap and for some of the scaling methods in sca.

Usage

normal.qu(n = 15, lower = -4, upper = 4, mu = 0, sigma = 1, scaling = "points")

Arguments

Value

A list of:

Author(s)

Ivailo Partchev

See Also

read.qu.icl, eap, sca

Examples

quad <- normal.qu(n=20)

Non-parametric characteristic curves

Description

A plotting routine producing non-parametric analogues of the IRF not unlike those in Jim Ramsay's TestGraf program. The curves are produced by a kernel binomial regression of the actual responses to an item on some estimates of the latent variable, by courtesy of package sm.

Usage

npp(
  resp,
  x,
  items,
  from = -4,
  to = 4,
  co = 1,
  main = "Non-parametric response function",
  add = FALSE,
  bands = FALSE,
  label = FALSE
)

Arguments

Author(s)

Ivailo Partchev

References

James O. Ramsay (2000). TestGraf: A program for the graphical analysis of multiple choice test and questionnaire data. McGill University, Montreal, Canada

See Also

qrs, irf, plot.irf

Examples

# plot items 1:5 in different colours, label
npp(Scored, items=1:5, co=NA, label=TRUE)

# For item 7, compare npp with the 2PL parametric IRF 
npp(Scored, items=7, bands=TRUE)
plot(irf(ip=Scored2pl, items=7), co=3, add=TRUE)

A plot method for item information functions

Description

Useful for plotting item information functions. The x argument of iif should better be left out unless something special is required.

Usage

## S3 method for class 'iif'
plot(
  x,
  add = FALSE,
  main = "Item information function",
  co = 1,
  label = FALSE,
  ...
)

Arguments

Author(s)

Ivailo Partchev

See Also

iif

Examples

# plot IIF for all items in red, label with item number
plot(iif(Scored2pl), co="red", label=TRUE)
# plot IIF for items 2, 3, and 7 in different colours
plot(iif(Scored2pl, items=c(2,3,7)), co=NA)

A plot method for the interaction model

Description

Plot the item-total regressions fit by the interaction (or Rasch) model

Usage

## S3 method for class 'imp'
plot(
  x,
  items = NULL,
  shade = 10,
  highlight = -Inf,
  add = FALSE,
  main = "Item-total regression",
  label = FALSE,
  ...
)

Arguments

Author(s)

Ivailo Partchev, using theory and C code by Gunter Maris

See Also

interactionModel

Examples

plot(interactionModel(Scored), highlight=-.3)

A plot method for item response functions

Description

Useful for plotting item response functions. The x argument of irf should better be left out unless something special is required.

Usage

## S3 method for class 'irf'
plot(
  x,
  add = FALSE,
  main = "Item response function",
  co = 1,
  label = FALSE,
  ...
)

Arguments

Author(s)

Ivailo Partchev

See Also

irf

Examples

# plot IRF for all items in red, label with item number
plot(irf(Scored2pl), co="red", label=TRUE)
# plot IRF for items 2, 3, and 7 in different colours
plot(irf(Scored2pl, items=c(2,3,7)), co=NA)

A plot method for test information functions

Description

Useful for plotting test information functions. The x argument of tif should better be left out unless something special is required.

Usage

## S3 method for class 'tif'
plot(x, add = FALSE, main = "Test information function", co = 1, ...)

Arguments

Author(s)

Ivailo Partchev

See Also

tif

Examples

plot(tif(Scored2pl))

A plot method for test response functions

Description

Useful for plotting test response functions. The x argument of trf should better be left out unless something special is required.

Usage

## S3 method for class 'trf'
plot(x, add = FALSE, main = "Test response function", co = 1, ...)

Arguments

Author(s)

Ivailo Partchev

See Also

trf

Examples

plot(trf(Scored2pl))

Quantiles of the ranked sum scores

Description

A rough estimate of the values of the latent variable ("ability"). The sum scores (number of correct responses) are ranked, breaking ties at random. The ranks are divided by the sample size + 1, and the corresponding quantiles of the standard Normal distribution are returned. Used as default in the non-parametric IRF plots produced by npp in analogy to Jim Ramsay's TestGraf. Another possible use is in itf.

Usage

qrs(resp)

Arguments

Value

A one-column matrix of values

Author(s)

Ivailo Partchev

See Also

npp, itf

Examples

sc <- qrs(Scored)

Read in parameter estimates

Description

From BILOG output, read in estimates of item parameters. Invoked automatically when the model is estimated via irtoys. If that is not the case, file must be a file produced with the >SAVE PARm file; command in BILOG.

Usage

read.ip.bilog(file)

Arguments

Value

A list with three elements, est, se, vcm for the parameter estimates, their standard errors, and the variance-covariance matrix per item. The first two are matrices with one row per item, and three columns: [,1] item discrimination a, [,2] item difficulty b, and [,3] asymptote c. For the 1PL and 2PL models, all asymptotes are equal to 0; for the 1PL, the discriminations are all equal but not necessarily equal to 1.

Author(s)

Ivailo Partchev

Read in parameter estimates

Description

From ICL output, read in estimates of item parameters. Invoked automatically when the model is estimated via irtoys. If that is not the case, file must be a file produced with the write_item_param file command in ICL.

Usage

read.ip.icl(file)

Arguments

Value

A list with three elements, est, se, vcm for the parameter estimates, their standard errors, and the variance-covariance matrix per item. Because ICL does not compute standard errors, se and vcm will be NULL. est is a matrix with one row per item, and three columns: [,1] item discrimination a, [,2] item difficulty b, and [,3] asymptote c. For the 1PL and 2PL models, all asymptotes are equal to 0; for the 1PL, the discriminations are all equal but not necessarily equal to 1.

Author(s)

Ivailo Partchev

Read in quadrature

Description

From ICL output, read in estimates of item parameters. file must be a file produced with the write_latent_dist file command in ICL. Quadrature objects are used when estimating abilities with eap and for some of the scaling methods in sca.

Usage

read.qu.icl(file)

Arguments

Value

A list of:

Author(s)

Ivailo Partchev

See Also

normal.qu, eap, sca

Read responses from a file

Description

Reads responses to a questionnaire from a text file

Usage

read.resp(file, na = ".")

Arguments

Details

Included for those who are too faint-hearted to write as.matrix(read.table(file, head=F)). Of course, data can be entered into R in many other ways.

The data values in the file must be separated with blanks.

Responses are the empirical data used in IRT. Note that irtoys deals with models for dichotomous data, and typically expects data consisting of zeroes and ones, without any missing values (non-responses are considered as wrong responses). In fact, there are only two commands in irtoys that accept other kinds of data: sco and tgf.

read.resp does accept missing data and values other than 0 and 1. Use sco and a key to score multiple choice responses to 0/1. If you have dichotomous data that contains NAs, you can use sco without a key to change all NA to 0.

Value

A matrix, typically of zeroes and ones, representing the correct or wrong responses given by persons (rows) to items (columns).

Author(s)

Ivailo Partchev

See Also

sco, tgf,

Examples

## Not run: 
  r <- read.resp("c:/myfiles/irt.dat")

## End(Not run)

Item-total regressions for the Rasch vs. the interaction model

Description

Compare the item-total regressions fit by the Rasch model and the interaction model, for one, several, or all items in the test.

Usage

rim(x, items = NULL, showData = FALSE, shade = 10, ncol = 3, ...)

Arguments

Author(s)

Ivailo Partchev, using theory and C code by Gunter Maris

See Also

interactionModel

Examples

rim(Scored, items=2)

Linear transformation of the IRT scale

Description

Linearly transform a set of IRT parameters to bring them to the scale of another set of parameters. Four methods are implemented: Mean/Mean, Mean/Sigma, Lord-Stocking, and Haebara.

Usage

sca(
  old.ip,
  new.ip,
  old.items,
  new.items,
  old.qu = NULL,
  new.qu = NULL,
  method = "MS",
  bec = FALSE
)

Arguments

Value

A list of:

Author(s)

Ivailo Partchev and Tamaki Hattori

References

Kolen, M.J. & R.L. Brennan (1995) Test Equating: Methods and Practices. Springer.

Haebara, T. (1980) Equating logistic ability scales by a weighted lest squares method. Japanese Psychological Research, 22, p.144–149

Examples

## Not run: 
# a small simulation to demonstrate transformation to a common scale
# fake 50 2PL items
pa <- cbind(runif(50,.8,2), runif(50,-2.4,2.4), rep(0,50))
# simulate responses with  two samples of different ability levels
r.1 <- sim(ip=pa[1:30,],  x=rnorm(1000,-.5))
r.2 <- sim(ip=pa[21:50,], x=rnorm(1000,.5))
# estimate item parameters
p.1 <- est(r.1, engine="ltm")
p.2 <- est(r.2, engine="ltm")
# plot difficulties to show difference in scale
plot(c(-3,3), c(-3,3), ty="n", xlab="True",ylab="Estimated",
   main="Achieving common scale")
text(pa[1:30,2],  p.1$est[,2], 1:30)
text(pa[21:50,2], p.2$est[,2], 21:50, co=2)
# scale with the default Mean/Sigma method
pa.sc = sca(old.ip=p.1$est, new.ip=p.2$est, old.items=21:30, new.items=1:10)
# add difficulties of scaled items to plot
text(pa[21:50,2], pa.sc$scaled.ip[,2], 21:50, co=3)

## End(Not run)

Score a multiple choice test

Description

Given a key, score a multiple choice test, i.e. recode the original choices to right (1) or wrong (0). Missing responses are treated as wrong.

Usage

sco(choices, key, na.false = FALSE)

Arguments

Value

A matrix of responses scored 0=wrong 1=correct, and possibly NA

Author(s)

Ivailo Partchev

Examples

res <- sco(Unscored, key=c(2,3,1,1,4,1,2,1,2,3,3,4,3,4,2,2,4,3))

Sum score metric

Description

From a set of estimated item parameters, compute an item response function in sum score metric, i.e., the probabilities of a correct response predicted by the model at abilities corresponding to the possible (raw) sum scores.

Usage

scoreMetric(ip)

Arguments

Value

An object with the same structure as the output of irf

Author(s)

Ivailo Partchev

Plot observed and predicted scores against ability

Description

Produces a plot of IRT true scores (test response function at the estimated ability) with a confidence band (plus/minus standard error). The observed sum scores are shown in red.

Usage

scp(resp, ip, theta = NULL)

Arguments

Value

None

Author(s)

Ivailo Partchev

See Also

mlebme, eap, tsc, trf

Examples

scp(Scored, Scored2pl)

Simulate response data

Description

Simulate responses from the 1PL, 2PL, or 3PL model

Usage

sim(ip, x = NULL)

Arguments

Value

A matrix of responses: persons as rows, items as columns, entries are either 0 or 1, no missing data

Author(s)

Ivailo Partchev

Examples

pa <- cbind(runif(20,.8,2), runif(20,-2.4,2.4), rep(0,50))
rs <- sim(ip=pa, x=rnorm(1000))

Approximate tetchoric correlation matrix

Description

Matrix of tetchoric correlations using the approximation by Bonett and Price (2005).

Usage

tet(d)

Arguments

Value

A matrix of approximate tetchoric correlations.

Author(s)

Ivailo Partchev

References

Douglas G. Bonett and Robert M. Price (2005). Inferential Methods for the Tetrachoric Correlation Coefficient. Journal of Educational and Behavioral Statistics, Vol. 30, No. 2, pp. 213–225

Examples

tets <- tet(Scored)

Non-parametric option curves

Description

A plotting function producing non-parametric analogues of the IRF for each option in a multiple choice item not unlike those in Jim Ramsay's TestGraf program.

Usage

tgf(
  choices,
  key,
  item,
  main = "Non-parametric response function",
  co = 1,
  label = FALSE
)

Arguments

Author(s)

Ivailo Partchev

References

James O. Ramsay (2000). TestGraf: A program for the graphical analysis of multiple choice test and questionnaire data. McGill University, Montreal, Canada

See Also

qrs, irf, plot.irf

Examples

key=c(2,3,1,1,4,1,2,1,2,3,3,4,3,4,2,2,4,3)
tgf(choices=Unscored, key=key, item=4, co=NA, label=TRUE)

Elementary test-item analysis

Description

Elementary analysis of the items in a test and the test sumscores based on Classical Test Theory.

Usage

tia(choices, key, ...)

Arguments

Value

A list with three elements:

testlevel

A list of statistics at test level (currently, only Cronbach's alpha, may be extended in future)

itemlevel

A matrix showing, for each item, the proportion of correct responses, the correlation with the sum score, and the alpha that the test would have if the item were dropped.

optionlevel

A matrix showing, for each possible choice in the multiple-choice item, the proportion of responses given, and the correlation with the sum score for the test (including the item). The correct response is highlighted with asterisks.

Author(s)

Ivailo Partchev

Examples

itemsum <- tia(Unscored, key=c(2,3,1,1,4,1,2,1,2,3,3,4,3,4,2,2,4,3))

Test information function

Description

Returns the test information function (TIF) of the 3PL (1PL, 2PL) model. The TIF is the sum of the item information functions (IIF) in a test, and indicates the precision of measurement that can be achieved with the test at any value of the latent variable, bein inversely related to measurement variance.

Usage

tif(ip, x = NULL)

Arguments

Details

A common use of this function would be to obtain a plot of the TIF.

Value

A list of:

Author(s)

Ivailo Partchev

See Also

plot.tif, iif

Examples

plot(trf(Scored2pl))

Test response function

Description

Returns the test response function (TRF) of the 3PL (1PL, 2PL) model. The TRF is the sum of the item response functions (IRF) in a test, and represents the expected score as a function of the latent variable \theta.

Usage

trf(ip, x = NULL)

Arguments

Details

A common use of this function would be to obtain a plot of the TRF.

Value

A list of:

Author(s)

Ivailo Partchev

See Also

plot.trf, irf

Examples

plot(trf(Scored2pl))

True scores with standard errors

Description

Computes the IRT true scores (test response function at the estimated ability) and an estimate of their standard error via the delta theorem, treating item parameters as known).

Usage

tsc(ip, theta)

Arguments

Value

A matrix with the true scores in column 1, and their standard errors of measurement (SEM) in column 2

Author(s)

Ivailo Partchev

See Also

mlebme, eap, trf

Examples

th <- mlebme(resp=Scored, ip=Scored2pl)
tsc(Scored2pl, th)

Bias-corrected (Warm's) estimates of ability

Description

Weighted likelihood estimates (WLE) of ability, designed to remove the first order bias term from the ML estimates. WLE are finite for response patterns consisting of either uniformly wrong or uniformly correct responses.

Usage

wle(resp, ip)

Arguments

Value

A matrix with the ability estimates in column 1, and their standard errors of measurement (SEM) in column 2, and the number of non-missing reponses in column 3

Author(s)

Ivailo Partchev

References

Warm T.A. (1989) Weighted Likelihood Estimation of Ability in Item Response Theory. Psychometrika, 54, 427-450.

See Also

mlebme, eap

Examples

th.bce <- wle(resp=Scored, ip=Scored2pl)