gom.em.Rd
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.
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,...)
dat | Data frame with dichotomous responses |
---|---|
K | Number of classes (only applies for |
problevels | Vector containing probability levels for membership functions
(only applies for |
weights | Optional vector of sampling weights |
model | The type of grade of membership model. The default |
theta0.k | Vector of \(\tilde{\theta}_k\) grid (applies only for |
xsi0.k | Vector of \(\xi_p\) grid (applies only for |
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 |
progress | Display iteration progress? Default is |
object | Object of class |
file | Optional file name for summary output |
... | Further arguments to be passed |
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).
A list with following entries:
Deviance
Information criteria
Data frame with item parameters
Data frame with person parameters
EAP reliability (only applies for model="GOMRasch"
)
Maximum aposteriori estimate of the membership function
EAP estimate for individual membership scores
Descriptives for class membership
Estimated response probabilities \(\lambda_{ik}\)
Standard error for estimated response probabilities \(\lambda_{ik}\)
Mean of the distribution of \((\theta_p, \xi_p)\)
(only applies for model="GOMRasch"
)
Covariance matrix of \((\theta_p, \xi_p)\)
(only applies for model="GOMRasch"
)
Estimated item difficulties (only applies for model="GOMRasch"
)
Standard error of estimated difficulties
(only applies for model="GOMRasch"
)
Individual likelihood
Individual posterior
Array with response probabilities
Expected counts
Number of iterations
Number of items
Number of classes
Number of discrete integration points for \((g_{p1},...,g_{pK})\)
Used grid of membership functions
Further values
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.
For joint maximum likelihood estimation of the grade of membership model
see gom.jml
.
See also the mixedMem package for estimating mixed membership models by a variational EM algorithm.
The C code of Erosheva et al. (2007) can be downloaded from http://projecteuclid.org/euclid.aoas/1196438029#supplemental.
Code from Manrique-Vallier can be downloaded from http://pages.iu.edu/~dmanriqu/software.html.
See http://users.ics.aalto.fi/ella/publications/aspect_bernoulli.m for a Matlab implementation of the algorithm in Bingham, Kaban and Fortelius (2009).