3PL Structured Item Response Model in TAM

This estimates a 3PL model with design matrices for item slopes and item intercepts. Discrete distributions of the latent variables are allowed which can be log-linearly smoothed.

tam.mml.3pl(resp, Y=NULL, group=NULL, formulaY=NULL, dataY=NULL, ndim=1,
  pid=NULL, xsi.fixed=NULL, xsi.inits=NULL, xsi.prior=NULL,
  beta.fixed=NULL, beta.inits=NULL, variance.fixed=NULL, variance.inits=NULL,
  est.variance=TRUE, A=NULL, notA=FALSE, Q=NULL, Q.fixed=NULL, E=NULL,
  gammaslope.des="2PL", gammaslope=NULL, gammaslope.fixed=NULL,
  est.some.slopes=TRUE, gammaslope.max=9.99, gammaslope.constr.V=NULL,
  gammaslope.constr.c=NULL, gammaslope.center.index=NULL,  gammaslope.center.value=NULL,
  gammaslope.prior=NULL, userfct.gammaslope=NULL, gammaslope.constr.Npars=0,
  est.guess=NULL, guess=rep(0, ncol(resp)),
  guess.prior=NULL, max.guess=0.5, skillspace="normal", theta.k=NULL,
  delta.designmatrix=NULL, delta.fixed=NULL, delta.inits=NULL, pweights=NULL,
  item.elim=TRUE, verbose=TRUE, control=list(), Edes=NULL )

# S3 method for tam.mml.3pl
summary(object,file=NULL,...)

# S3 method for tam.mml.3pl
print(x,...)

Arguments

resp: Data frame with polytomous item responses $k=0,...,K$. Missing responses must be declared as NA.
Y: A matrix of covariates in latent regression. Note that the intercept is automatically included as the first predictor.
group: An optional vector of group identifiers
formulaY: An R formula for latent regression. Transformations of predictors in $Y$ (included in dataY) can be easily specified, e. g. female*race or I(age^2).
dataY: An optional data frame with possible covariates $Y$ in latent regression. This data frame will be used if an R formula in formulaY is specified.
ndim: Number of dimensions (is not needed to determined by the user)
pid: An optional vector of person identifiers
xsi.fixed: A matrix with two columns for fixing $\xi$ parameters. 1st column: index of $\xi$ parameter, 2nd column: fixed value
xsi.inits: A matrix with two columns (in the same way defined as in xsi.fixed with initial value for $\xi$.
xsi.prior: Item-specific prior distribution for $\xi$ parameters. It is assumed that $\xi_k \sim N( \mu_k, \sigma_k^2 )$. The first column in xsi.prior is $\mu_k$, the second is $\sigma_k$.
beta.fixed: A matrix with three columns for fixing regression coefficients. 1st column: Index of $Y$ value, 2nd column: dimension, 3rd column: fixed $\beta$ value.
If no constraints should be imposed on $\beta$, then set beta.fixed=FALSE (see Example 2, Model 2_4).
beta.inits: A matrix (same format as in beta.fixed) with initial $\beta$ values
variance.fixed: An optional matrix. In case of a single group it is a matrix with three columns for fixing entries in covariance matrix: 1st column: dimension 1, 2nd column: dimension 2, 3rd column: fixed value of covariance/variance. In case of multiple groups, it is a matrix with four columns 1st column: group index (from $1,\ldots,G$, 2nd column: dimension 1, 3rd column: dimension 2, 4th column: fixed value of covariance
variance.inits: Initial covariance matrix in estimation. All matrix entries have to be specified and this matrix is NOT in the same format like variance.fixed.
est.variance: Should the covariance matrix be estimated? This argument applies to estimated item slopes in tam.mml.2pl. The default is FALSE which means that latent variables (in the first group) are standardized in 2PL estimation.
A: An optional array of dimension $ I \times (K+1) \times N_\xi$. Only $\xi$ parameters are estimated, entries in $A$ only correspond to the design.
notA: An optional logical indicating whether all entries in the $A$ matrix are set to zero and no item intercept $\xi$ should be estimated.
Q: An optional $I \times D$ matrix (the Q-matrix) which specifies the loading structure of items on dimensions.
Q.fixed: Optional $I \times D$ matrix of the same dimensions like Q. Non NA entries contain values at which item loadings should be fixed to.
E: Optional design array for item slopes $\gamma$. It is a four dimensional array of size $I \times (K+1) \times D \times N_\gamma$ containing items, categories, dimensions, $\gamma$ parameter.
gammaslope.des: Optional string indicating type of item slope parameter to be estimated. gammaslope.des="2PL" estimates a slope parameter for an item, gammaslope.des="2PLcat" for an item and a
gammaslope: Initial or fixed vector of $\gamma$ parameters
gammaslope.fixed: An optional matrix containing fixed values in the $\gamma$ vector. First column: parameter index; second colunmn: fixed value.
est.some.slopes: An optional logical indicating whether some item slopes should be estimated.
gammaslope.max: Value for absolute entries in $\gamma$ vector
gammaslope.constr.V: An optional constraint matrix $V$ for item slope parameters $\gamma$
gammaslope.constr.c: An optional constraint vector $c$ for item slope parameters $\gamma$
gammaslope.center.index: Indices of gammaslope parameters which should be fixed to sum specified in gammaslope.center.value (see Example 7).
gammaslope.center.value: Specified values of sum of subset of gammaslope parameters.
gammaslope.prior: Item-specific prior distribution for $\gamma$ parameters. It is assumed that $\gamma_k \sim N( \mu_k, \sigma_k^2 )$. The first column in gammaslope.prior is $\mu_k$, the second is $\sigma_k$.
userfct.gammaslope: A user specified function for constraints or transformations of the $\gamma$ parameters within the algorithm. See Example 17 in tam.mml.
gammaslope.constr.Npars: Number of constrained $\gamma$ parameters in userfct.gammaslope
est.guess: An optional vector of integers indicating for which items a guessing parameter should be estimated. Same integers correspond to same estimated guessing parameters. A value of 0 denotes an item for which no guessing parameter is estimated.
guess: Fixed or initial guessing parameters
guess.prior: Item-specific prior distribution for guessing parameters $c_i$. It is assumed that $c_i \sim Beta(a_i, b_i)$. The first column in gammaslope.prior is $a_i$, the second is $b_i$.
max.guess: Upper bound for guessing parameters
skillspace: Skill space: normal distribution ("normal") or discrete distribution ("discrete").
theta.k: A matrix of the $\bold{\theta}$ skill space in case of a discrete distribution (skillspace="discrete").
delta.designmatrix: A design matrix of the log-linear model for the skill space in case of a discrete distribution (skillspace="discrete").
delta.fixed: Fixed $\delta$ values of the log-linear skill space. delta.fixed must be a matrix with three columns. First column: $\delta$ parameter index, Second column: Group index, Third column: Fixed $\delta$ parameter value.
delta.inits: Optional initial matrix of $\delta$ parameters.
pweights: Optional vector of person weights.
item.elim: Optional logical indicating whether an item with has only zero entries should be removed from the analysis. The default is TRUE.
verbose: Logical indicating whether output should be printed during iterations. This argument replaces control$progress.
control: See tam.mml for more details.
Edes: Compact form of design matrix. This argument is only defined for convenience for models with random starting values to avoid recalculations.
object: Object of class tam.mml.3pl
file: A file name in which the summary output will be written
x: Object of class tam.mml.3pl
...: Further arguments to be passed

Details

The item response model for item $i$ and category $h$ and no guessing parameters can be written as $$ P( X_{i}=h | \bold{\theta} ) \propto \exp( \sum_d b_{ihd} \theta_d + \sum_k a_{ih} \xi_k ) $$ For item slopes, a linear decomposition is allowed $$ b_{ihd}=\sum_k e_{ihdk} \gamma_k $$ In case of a guessing parameter, the item response function is $$ P( X_{i}=h | \bold{\theta} )=c_i + ( 1 - c_i ) \cdot ( 1 + \exp( - \sum_d b_{ihd} \theta_d - \sum_k a_{ih} \xi_k ) )^{-1} $$ For possibilities of definitions of the design matrix $E=(e_{ihdk})$ see Formann (2007), for the strongly related linear logistic latent class model see Formann (1992). Linear constraints on $\gamma$ can be specified by equations $V \gamma=c$ and using the arguments gammaslope.constr.V and gammaslope.constr.c.

Like in tam.mml, a multivariate linear regression $$ \bold{\theta}=Y \beta + \bold{\epsilon}$$ assuming a multivariate normal distribution on the residuals $\bold{\epsilon}$ can be specified (skillspace="normal").

Alternatively, a log-linear distribution of the skill classes $P(\theta)$ (skillspace="discrete") is performed $$\log P(\theta )=D_{ \delta } \delta $$ See Xu and von Davier (2008). The design matrix $D_{\delta}$ can be specified in delta.designmatrix. The theta grid $\bold{\theta}$ of the skill space can be defined in theta.k.

Value

The same output as in tam.mml and additional entries

delta: Parameter vector $\delta$
gammaslope: Estimated $\gamma$ item slope parameters
se.gammaslope: Standard errors $\gamma$ item slope parameters
E: Used design matrix $E$
Edes: Used design matrix $E$ in compact form
guess: Estimated $c$ guessing parameters
se.guess: Standard errors $c$ guessing parameters

References

Formann, A. K. (1992). Linear logistic latent class analysis for polytomous data. Journal of the American Statistical Association, 87, 476-486. doi:10.2307/2290280

Formann, A. K. (2007). (Almost) Equivalence between conditional and mixture maximum likelihood estimates for some models of the Rasch type. In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models (pp. 177-189). New York: Springer. doi:10.1007/978-0-387-49839-3_11

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS. doi:10.1002/j.2333-8504.2008.tb02113.x

Note

The implementation of the model builds on pieces work of Anton Formann. See http://www.antonformann.at/ for more information.