Structured Latent Class Analysis (SLCA)

This function implements a structured latent class model for polytomous item responses (Formann, 1985, 1992). Lasso estimation for the item parameters is included (Chen, Liu, Xu & Ying, 2015; Chen, Li, Liu & Ying, 2017; Sun, Chen, Liu, Ying & Xin, 2016).

slca(data, group=NULL, weights=rep(1, nrow(data)), Xdes,
  Xlambda.init=NULL, Xlambda.fixed=NULL, Xlambda.constr.V=NULL,
  Xlambda.constr.c=NULL,  delta.designmatrix=NULL,
  delta.init=NULL, delta.fixed=NULL, delta.linkfct="log",
  Xlambda_positive=NULL, regular_type="lasso", regular_lam=0, regular_w=NULL,
  regular_n=nrow(data), maxiter=1000, conv=1e-5, globconv=1e-5, msteps=10,
  convM=5e-04, decrease.increments=FALSE, oldfac=0, dampening_factor=1.01,
  seed=NULL, progress=TRUE, PEM=TRUE, PEM_itermax=maxiter, ...)

# S3 method for slca
summary(object, file=NULL, ...)

# S3 method for slca
print(x, ...)

# S3 method for slca
plot(x, group=1, ... )

Arguments

data	Matrix of polytomous item responses
group	Optional vector of group identifiers. For `plot.slca` it is a single integer group identified.
weights	Optional vector of sample weights
Xdes	Design matrix for $x_{ijh}$ with $ q_{ihjv}$ entries. Therefore, it must be an array with four dimensions referring to items ($i$), categories ($h$), latent classes ($j$) and $\lambda$ parameters ($v$).
Xlambda.init	Initial $\lambda_x$ parameters
Xlambda.fixed	Fixed $\lambda_x$ parameters. These must be provided by a matrix with two columns: 1st column -- Parameter index, 2nd column: Fixed value.
Xlambda.constr.V	A design matrix for linear restrictions of the form $V_x \lambda_x=c_x$ for the $\lambda_x$ parameter.
Xlambda.constr.c	A vector for the linear restriction $V_x \lambda_x=c_x$ of the $\lambda_x$ parameter.
delta.designmatrix	Design matrix for delta parameters $\delta$ parameterizing the latent class distribution by log-linear smoothing (Xu & von Davier, 2008)
delta.init	Initial $\delta$ parameters
delta.fixed	Fixed $\delta$ parameters. This must be a matrix with three columns: 1st column: Parameter index, 2nd column: Group index, 3rd column: Fixed value
delta.linkfct	Link function for skill space reduction. This can be the log-linear link (`log`) or the logistic link function (`logit`).
Xlambda_positive	Optional vector of logical indicating which elements of $\bold{\lambda}_x$ should be constrained to be positive.
regular_type	Regularization method which can be `lasso`, `scad` or `mcp`. See `gdina` for more information and references.
regular_lam	Numeric. Regularization parameter
regular_w	Vector for weighting the regularization penalty
regular_n	Vector of regularization factor. This will be typically the sample size.
maxiter	Maximum number of iterations
conv	Convergence criterion for item parameters and distribution parameters
globconv	Global deviance convergence criterion
msteps	Maximum number of M steps in estimating $b$ and $a$ item parameters. The default is to use 4 M steps.
convM	Convergence criterion in M step
decrease.increments	Should in the M step the increments of $a$ and $b$ parameters decrease during iterations? The default is `FALSE`. If there is an increase in deviance during estimation, setting `decrease.increments` to `TRUE` is recommended.
oldfac	Factor $f$ between 0 and 1 to control convergence behavior. If $x_t$ denotes the estimated parameter in iteration $t$, then the regularized estimate $x_t^{\ast}$ is obtained by $x_t^{\ast}=f x_{t-1} + (1-f) x_t$. Therefore, values of `oldfac` near to one only allow for small changes in estimated parameters from in succeeding iterations.
dampening_factor	Factor larger than one defining the specified decrease in decrements in iterations.
seed	Simulation seed for initial parameters. The default of `NULL` corresponds to a random seed.
progress	An optional logical indicating whether the function should print the progress of iteration in the estimation process.
PEM	Logical indicating whether the P-EM acceleration should be applied (Berlinet & Roland, 2012).
PEM_itermax	Number of iterations in which the P-EM method should be applied.
object	A required object of class `slca`
file	Optional file name for a file in which `summary` should be sinked.
x	A required object of class `slca`
...	Optional parameters to be passed to or from other methods will be ignored.

Details

The structured latent class model allows for general constraints of items $i$ in categories $h$ and classes $j$. The item response model is $$P( X_{i}=h | j )=\frac{ \exp( x_{ihj} ) }{ \sum_l \exp( x_{ilj} ) }$$ with linear constraints on the class specific probabilities $$ x_{ihj}=\sum_v q_{ihjv} \lambda_{xv} $$

Linear restrictions on the $\lambda_x$ parameter can be specified by a matrix equation $V_x \lambda_x=c_x$ (see Xlambda.constr.V and Xlambda.constr.c; Neuhaus, 1996).

The latent class distribution can be smoothed by a log-linear link function (Xu & von Davier, 2008) or a logistic link function (Formann, 1992). For class $j$ in group $g$ employing a link function $h$, it holds that $$ h [ P( j| g) ] \propto \sum_w r_{jw} \delta_{gw} $$ where group-specific distributions are allowed. The values $r_{jw}$ are specified in the design matrix delta.designmatrix.

This model contains classical uni- and multidimensional latent trait models, latent class analysis, located latent class analysis, cognitive diagnostic models, the general diagnostic model and mixture item response models as special cases (see Formann & Kohlmann, 1998; Formann, 2007).

The function also allows for regularization of $\lambda_{xv}$ parameters using the lasso approach (Sun et al., 2016). More formally, the penalty function can be written as $$pen( \bold{\lambda}_x )=p_\lambda \sum_v n_v w_v | \lambda_{xv} | $$ where $p_\lambda$ can be specified with regular_lam, $w_v$ can be specified with regular_w, and $n_v$ can be specified with regular_n.

Value

An object of class slca. The list contains the following entries:

item

Data frame with conditional item probabilities

deviance

Deviance

Information criteria, number of estimated parameters

Xlambda

Estimated $\lambda_x$ parameters

se.Xlambda

Standard error of $\lambda_x$ parameters

pi.k

Trait distribution

pjk

Item response probabilities evaluated for all classes

n.ik

An array of expected counts $n_{cikg}$ of ability class $c$ at item $i$ at category $k$ in group $g$

Number of groups

Number of items

Number of persons

delta

Parameter estimates for skillspace representation

covdelta

Covariance matrix of parameter estimates for skillspace representation

MLE.class

Classified skills for each student (MLE)

MAP.class

Classified skills for each student (MAP)

data

Original data frame

group.stat

Group statistics (sample sizes, group labels)

p.xi.aj

Individual likelihood

posterior

Individual posterior distribution

K.item

Maximal category per item

time

Info about computation time

skillspace

Used skillspace parametrization

iter

Number of iterations

seed.used

Used simulation seed

Xlambda.init

Used initial lambda parameters

delta.init

Used initial delta parameters

converged

Logical indicating whether convergence was achieved.

References

Berlinet, A. F., & Roland, C. (2012). Acceleration of the EM algorithm: P-EM versus epsilon algorithm. Computational Statistics & Data Analysis, 56(12), 4122-4137.

Chen, Y., Liu, J., Xu, G., & Ying, Z. (2015). Statistical analysis of Q-matrix based diagnostic classification models. Journal of the American Statistical Association, 110, 850-866.

Chen, Y., Li, X., Liu, J., & Ying, Z. (2017). Regularized latent class analysis with application in cognitive diagnosis. Psychometrika, 82, 660-692.

Formann, A. K. (1985). Constrained latent class models: Theory and applications. British Journal of Mathematical and Statistical Psychology, 38, 87-111.

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

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.

Formann, A. K., & Kohlmann, T. (1998). Structural latent class models. Sociological Methods & Research, 26, 530-565.

Neuhaus, W. (1996). Optimal estimation under linear constraints. Astin Bulletin, 26, 233-245.

Sun, J., Chen, Y., Liu, J., Ying, Z., & Xin, T. (2016). Latent variable selection for multidimensional item response theory models via $L_1$ regularization. Psychometrika, 81(4), 921-939.

Xu, X., & von Davier, M. (2008). Fitting the structured general diagnostic model to NAEP data. ETS Research Report ETS RR-08-27. Princeton, ETS.

Note

If some items have differing number of categories, appropriate class probabilities in non-existing categories per items can be practically set to zero by loading an item for all skill classes on a fixed $\lambda_x$ parameter of a small number, e.g. -999.

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