Discrete (Rasch) Grade of Membership Model

This function estimates the grade of membership model (Erosheva, Fienberg & Joutard, 2007; also called mixed membership model) by the EM algorithm assuming a discrete membership score distribution. The function is restricted to dichotomous item responses.

Usage

gom.em(dat, K=NULL, problevels=NULL, weights=NULL, model="GOM", theta0.k=seq(-5,5,len=15),
    xsi0.k=exp(seq(-6, 3, len=15)), max.increment=0.3, numdiff.parm=1e-4,
    maxdevchange=1e-6, globconv=1e-4, maxiter=1000, msteps=4, mstepconv=0.001,
    theta_adjust=FALSE, lambda.inits=NULL, lambda.index=NULL, pi.k.inits=NULL,
    newton_raphson=TRUE, optimizer="nlminb", progress=TRUE)

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

# S3 method for gom
anova(object,...)

# S3 method for gom
logLik(object,...)

# S3 method for gom
IRT.irfprob(object,...)

# S3 method for gom
IRT.likelihood(object,...)

# S3 method for gom
IRT.posterior(object,...)

# S3 method for gom
IRT.modelfit(object,...)

# S3 method for IRT.modelfit.gom
summary(object,...)

Arguments

dat: Data frame with dichotomous responses
K: Number of classes (only applies for model="GOM")
problevels: Vector containing probability levels for membership functions (only applies for model="GOM"). If a specific space of probability levels should be estimated, then a matrix can be supplied (see Example 1, Model 2a).
weights: Optional vector of sampling weights
model: The type of grade of membership model. The default "GOM" is the nonparametric grade of membership model. A parametric multivariate normal representation can be requested by "GOMnormal". The probabilities and membership functions specifications described in Details are called via "GOMRasch".
theta0.k: Vector of $\tilde{\theta}_k$ grid (applies only for model="GOMRasch")
xsi0.k: Vector of $\xi_p$ grid (applies only for model="GOMRasch")
max.increment: Maximum increment
numdiff.parm: Numerical differentiation parameter
maxdevchange: Convergence criterion for change in relative deviance
globconv: Global convergence criterion for parameter change
maxiter: Maximum number of iterations
msteps: Number of iterations within a M step
mstepconv: Convergence criterion within a M step
theta_adjust: Logical indicating whether multivariate normal distribution should be adaptively chosen during the EM algorithm.
lambda.inits: Initial values for item parameters
lambda.index: Optional integer matrix with integers indicating equality constraints among $\lambda$ item parameters
pi.k.inits: Initial values for distribution parameters
newton_raphson: Logical indicating whether Newton-Raphson should be used for final iterations
optimizer: Type of optimizer. Can be "optim" or "nlminb".
progress: Display iteration progress? Default is TRUE.
object: Object of class gom
file: Optional file name for summary output
...: Further arguments to be passed

Details

The item response model of the grade of membership model (Erosheva, Fienberg & Junker, 2002; Erosheva, Fienberg & Joutard, 2007) with $K$ classes for dichotomous correct responses $X_{pi}$ of person $p$ on item $i$ is as follows (model="GOM") $$ P(X_{pi}=1 | g_{p1}, \ldots, g_{pK} )=\sum_k \lambda_{ik} g_{pk} \quad, \quad \sum_{k=1}^K g_{pk}=1 \quad, \quad 0 \leq g_{pk} \leq 1 $$

In most applications (e.g. Erosheva et al., 2007), the grade of membership function $\{g_{pk}\}$ is assumed to follow a Dirichlet distribution. In our gom.em implementation the membership function is assumed to be discretely represented by a grid $u=(u_1, \ldots, u_L)$ with entries between 0 and 1 (e.g. seq(0,1,length=5) with $L=5$). The values $g_{pk}$ of the membership function can then only take values in $\{ u_1, \ldots, u_L \}$ with the restriction $\sum_k g_{pk} \sum_l \bold{1}(g_{pk}=u_l )=1$. The grid $u$ is specified by using the argument problevels.

The Rasch grade of membership model (model="GOMRasch") poses constraints on probabilities $\lambda_{ik}$ and membership functions $g_{pk}$. The membership function of person $p$ is parameterized by a location parameter $\theta_p$ and a variability parameter $\xi_p$. Each class $k$ is represented by a location parameter $\tilde{\theta}_k$. The membership function is defined as $$ g_{pk} \propto \exp \left[ - \frac{ (\theta_p - \tilde{\theta}_k)^2 }{2 \xi_p^2 } \right] $$

The person parameter $\theta_p$ indicates the usual 'ability', while $\xi_p$ describes the individual tendency to change between classes $1,\ldots,K$ and their corresponding locations $\tilde{\theta}_1, \ldots,\tilde{\theta}_K$. The extremal class probabilities $\lambda_{ik}$ follow the Rasch model $$ \lambda_{ik}=invlogit( \tilde{\theta}_k - b_i )= \frac{ \exp( \tilde{\theta}_k - b_i ) }{ 1 + \exp( \tilde{\theta}_k - b_i ) }$$

Putting these assumptions together leads to the model equation $$ P(X_{pi}=1 | g_{p1}, \ldots, g_{pK} )= P(X_{pi}=1 | \theta_p, \xi_p )= \sum_k \frac{ \exp( \tilde{\theta}_k - b_i ) }{ 1 + \exp(\tilde{\theta}_k - b_i ) } \cdot \exp \left[ - \frac{ (\theta_p - \tilde{\theta}_k)^2 }{2 \xi_p^2 } \right] $$

In the extreme case of a very small $\xi_p=\varepsilon > 0$ and $\theta_p=\theta_0$, the Rasch model is obtained

$$ P(X_{pi}=1 | \theta_p, \xi_p )= P(X_{pi}=1 | \theta_0, \varepsilon )= \frac{ \exp( \theta_0 - b_i ) }{ 1 + \exp( \theta_0 - b_i ) } $$

See Erosheva et al. (2002), Erosheva (2005, 2006) or Galyart (2015) for a comparison of grade of membership models with latent trait models and latent class models.

The grade of membership model is also published under the name Bernoulli aspect model, see Bingham, Kaban and Fortelius (2009).

Value

A list with following entries:

deviance: Deviance
ic: Information criteria
item: Data frame with item parameters
person: Data frame with person parameters
EAP.rel: EAP reliability (only applies for model="GOMRasch")
MAP: Maximum aposteriori estimate of the membership function
EAP: EAP estimate for individual membership scores
classdesc: Descriptives for class membership
lambda: Estimated response probabilities $\lambda_{ik}$
se.lambda: Standard error for estimated response probabilities $\lambda_{ik}$
mu: Mean of the distribution of $(\theta_p, \xi_p)$ (only applies for model="GOMRasch")
Sigma: Covariance matrix of $(\theta_p, \xi_p)$ (only applies for model="GOMRasch")
b: Estimated item difficulties (only applies for model="GOMRasch")
se.b: Standard error of estimated difficulties (only applies for model="GOMRasch")
f.yi.qk: Individual likelihood
f.qk.yi: Individual posterior
probs: Array with response probabilities
n.ik: Expected counts
iter: Number of iterations
I: Number of items
K: Number of classes
TP: Number of discrete integration points for $(g_{p1},...,g_{pK})$
theta.k: Used grid of membership functions
...: Further values

References

Bingham, E., Kaban, A., & Fortelius, M. (2009). The aspect Bernoulli model: multiple causes of presences and absences. Pattern Analysis and Applications, 12(1), 55-78.

Erosheva, E. A. (2005). Comparing latent structures of the grade of membership, Rasch, and latent class models. Psychometrika, 70, 619-628.

Erosheva, E. A. (2006). Latent class representation of the grade of membership model. Seattle: University of Washington.

Erosheva, E. A., Fienberg, S. E., & Junker, B. W. (2002). Alternative statistical models and representations for large sparse multi-dimensional contingency tables. Annales-Faculte Des Sciences Toulouse Mathematiques, 11, 485-505.

Erosheva, E. A., Fienberg, S. E., & Joutard, C. (2007). Describing disability through individual-level mixture models for multivariate binary data. Annals of Applied Statistics, 1, 502-537.

Galyardt, A. (2015). Interpreting mixed membership models: Implications of Erosheva's representation theorem. In E. M. Airoldi, D. Blei, E. A. Erosheva, & S. E. Fienberg (Eds.). Handbook of Mixed Membership Models (pp. 39-65). Chapman & Hall.