lca {lcca}R Documentation

Latent-class analysis

Description

Fits a latent-class model to a set of categorical items using an EM algorithm.

Usage

lca( formula, data, freq, groups, nclass = 2,
   constrain.rhos = F, constrain.gammas = F, iseeds = NULL,
   iter.max = 5000, tol = 1e-06, starting.values = NULL,
   flatten.rhos = 0, flatten.gammas = 0, se.method = "STANDARD",
   weights, clusters, strata, subpop)

Arguments

formula

an object of class "formula" that determines the model to be fit; see DETAILS.

data

an optional data frame, list or environment containing the variables in the model. If data is not given, or if variables are not found in data, the variables are taken from environment(formula), typically the environment from which lca is called.

freq

optional frequency or count variable, to be used when data are aggregated.

groups

optional categorical grouping variable for multi-group analysis; see DETAILS.

nclass

number of latent classes to be fit. Response items are assumed to be conditionally independent within classes.

constrain.rhos

if T, then rho-parameters (item-response probabilities) are assumed to be equal across groups. If no groups variable is given, this argument is ignored.

constrain.gammas

if T, then gamma-parameters (class prevalences) are assumed to be equal across groups. If no groups variable is given, this argument is ignored.

iseeds

two integers to initialize the random number generator; see DETAILS.

iter.max

maximum number of iterations to be performed. Each iteration consists of an Expectation or E-step followed by a Maximization or M-step. The procedure halts if it has not converged by this many iterations.

tol

convergence criterion. The procedure halts if the maximum absolute change in all parameters (gammas and rhos) from one iteration to the next falls below this value.

starting.values

optional starting values for the model parameters. This must be a list with two named components, gamma and rho, which are numeric arrays with correct dimensions; see DETAILS.

flatten.rhos

optional flattening constant for estimation of rho-paramaters (item-response probabilities); see DETAILS.

flatten.gammas

optional flattening constant for estimation of gamma-parameters (class prevalences); see DETAILS.

se.method

method to use for computing standard errors; see DETAILS.

weights

optional numeric variable containing sampling weights for survey data; see DETAILS.

clusters

optional integer or factor variable containing sampling cluster identifiers; see DETAILS.

strata

optional integer or factor variable containing sampling stratum identifiers; see DETAILS.

subpop

optional logical variable indicating a subpopulation to which the model is to be fit; see DETAILS.

Details

This function fits a standard latent-class model without covariates. Under this model, polytomous response variables are assumed to be conditionally independent given a latent classification variable. For models with covariates, use lcacov.

The formula argument should have the form cbind(Y1,Y2,...)~1. The term on the left-hand side of ~ is a matrix (not a data frame) of variables to be used as responses. Each response variable should consist of integer codes 1,2,.... Response variables may also be factors, in which case they will be automatically converted to integer codes (as in the function unclass), and the levels of the factors will be ignored. The right-hand side of formula should be 1, indicating that the only predictor is a constant; any other predictors in the model formula will be ignored.

Missing values in response variables are allowed and should be conveyed by the R missing value code NA. Cases with missing responses are retained in the fitting procedure, and the missing values are assumed to be ignorably missing or missing at random.

The number of latent classes to be fit is determined by nclass, with nclass=1 indicating that the response variables are jointly independent.

By default, each case (row) of data or the model environment is assumed to represent one observational unit or individual. Data may also be aggegated, with individuals bearing identical responses to all variables (including NA's, if present) collapsed into a single case, with frequencies conveyed through the numeric variable freq.

The unknown parameters to be estimated are the class prevalences, which we call gamma's and item-response probabilities, which we call rho's. Estimation proceeds by an EM algorithm which, by default, computes maximum-likelihood estimates of these parameters. EM may converge to a global or local maximum, possibly on the boundary of the parameter space where one or more gamma- or rho-parameters are zero.

This function supports multi-group analyses in which the gamma- or rho-parameters are allowed to vary across levels of a categorical variable groups. This feature is useful for testing invariance of measurement (i.e., equality of rho-paraneters) across groups. The groups variable, if present, should be integers coded as 1,2,...,ngroups, where ngroups is the number of distinct groups. The groups variable may also be a factor, and the levels of this factor will appear in the printed output summaries. If groups is not present, then ngroups is taken to be 1. Constraints on the group-specific parameters are conveyed by constrain.rhos and constrain.gammas.

Optional starting values for parameters are provided through starting.values. This argument should be a list with two named components, rho and gamma. The component rho should be an array of dimension c(nitems,maxlevs,nclass,ngroups), where nitems is the number of response variables on the left-hand side of formula, maxlevs is the maximum number of levels (distinct response categories) among the response variables, nclass is the number of latent classes, and ngroups is the number of groups (equal to 1 if groups is not supplied). The element starting.values$rho[j,k,c,g] is the probability that an individual in group g and class c supplies a response of k to item j. The component gamma should be a matrix with nclass rows and ngroups columns, with starting.values$gamma[c,g] containing the prevalence of class c within group g.

If starting.values is not supplied, or if starting.values$rho=NULL, then starting values will be randomly generated. This function uses its own internal random number generator which is seeded by two integers, for example, seeds=c(123,456), which allows results to be reproduced in the future. If seeds=NULL then the function will seed itself with two random integers from R. Therefore, results can also be made reproducible by calling set.seed beforehand and taking seeds=NULL. Different starting values for the rhos's may lead to solutions in which the classes have different orderings. To reorder the classes in the printed output summaries, use the function permute.class. If starting.values is not supplied, or if starting.values$gamma=NULL, then starting values for the gamma's will be uniform across the classes within each group.

Rho-parameters at or near a boundary, which are commonplace in latent-class analysis, create difficulty when computing derivative-based standard errors, because the likelihood function at the solution may not be log-concave. The argument flatten.rhos allows the user to supply a positive flattening constant to smooth the estimated rho's toward the interior of the parameter space. A value of flatten.rhos=1, which should be adequate in most cases, supplies information equivalent to one prior observation for each response item in each class and each group, distributed fractionally across the response categories in equal amounts. Estimated gamma-parameters at or near zero are also problematic, because rho-parameters for an empty group are not identified. A value of flatten.gammas=1, which should be adequate for most purposes, adds information equivalent to one prior observation in each group, distributed fractionally across the classes in equal amounts. Starting values that are poor in the sense that they are far away from the global maximum (and random starting values are often poor) may lead to a local maximum or a sub-optimal solution on the boundary, and supplying small flattening constants for the rho's and the gamma's often helps the EM algorithm to find a better solution.

Acceptable values for the argument se.method are "STANDARD","FAST", "SANDWICH" or "NONE". If "STANDARD", then standard errors are obtained by inverting the matrix of (minus one times) the second derivatives of the loglikelihood function (plus penalty terms for flattening constants, if present) at the solution. If "FAST", then the matrix of second derivatives is approximated by the sum of the outer products of individuals' contributions to the score functions (first derivatives). If "SANDWICH", then the sum of the score outer-products is pre- and post-multiplied by the inverse of the second derivatives. If "NONE", then computation of first- and second-derivatives and standard errors is suppressed.

Survey weights may be supplied via the argument weights. Weights should not be confused with frequencies as supplied by freq. A frequency of 10 indicates that ten individuals in the sample exhibited the given pattern of responses, but a survey weight of 10 indicates that one sampled individual is representing ten individuals in the population. The same variable supplied as freq or weights will lead to identical estimates, but the standard errors may be drastically different. You cannot supply both freq and weights; data from a survey with unequal probabilities of selection must be supplied in disaggregated form. If a weights variable is present, then se.method is automatically set to "SANDWICH", and the inner matrix of the sandich formula (i.e., the meat of the sandwich) is an estimated covariance matrix for the total quasi-score.

Weights provided with large, nationally-representative survey datasets are often very large, because their sum estimates the size of the population. Many data analysts are accustomed to rescaling survey weights to have a mean of one, so that they sum to the sample size rather than the population size. Rescaling a weights variable will have no effect on estimates or standard errors, because the scale factor cancels out when se.method="SANDWICH".

Additional information about the sample design may be supplied via clusters and strata. If weights is supplied, then the sampling plan is assumed to fall within the general class of with-replacement (WR) designs. At the first stage, clusters are drawn with replacement. Then individuals are selected within clusters, possibly with unequal probabilities, possibly in multiple stages. The clusters and strata variables should be integers or factors. The integers serve merely as identifiers; the actual values are unimportant. Cluster identifiers are assumed to be unique within strata, so that cluster 1 in stratum 1 and cluster 1 in stratum 2 are assumed to be different. If clusters is not supplied, then each sampled individual is assumed to be a cluster. If strata is not supplied, then one stratum is assumed for the whole population. Note a weight variable is required for complex survey data; if weights=NULL, then clusters and strata will be ignored.

It is often useful to fit a model that describes only a part of the full population (e.g., females). With a simple random sample, it is acceptable to remove the sampled individuals who are not in this subpopulation (e.g, males) from the data frame or model environent, because those who remain will be then a simple random sample of the subpopulation of interest. With a complex survey design, however, discarding individuals who do not belong to the subpopulation may lead to incorrect standard errors, because the overall design may not scale down to the subpopulation. To fit a model to a subpopulation when weights is present, supply a logical variable subpop whose elements are TRUE for members of the subpopulation. If subpop is supplied and weights=NULL, individuals outside of the subpopulation will be ignored when computing estimates and standard errors.

Value

A list whose class attribute has been set to "lca". A nicely formatted summary of this object may be seen by applying the summary method, but its components may also be accessed directly. Components which may be of interest include:

ncases

number of ncases (rows) from the data frame or model environment used in the procedure.

nitems

number of response variables appearing in the model.

nlevs

integer vector of length nitems indicating the number of levels or response categories for each item.

ngroups

number of groups present in a multi-group analysis, or 1 if groups=NULL.

iter

number of EM iterations performed.

converged

logical value indicating whether the algorithm converged by iter iterations.

loglik

vector of length iter reporting the value of the loglikelihood function at the beginning of each EM iteration. If weights is present, then this will be a weighted pseudo-loglikelihood.

loglik.final

value of the loglikelihood or pseudo-loglikelihood function after the final iteration.

logpost

vector of length iter reporting the value of the log-posterior density (loglikelihood or pseudo-loglikelihood plus penalty terms, if flattening constants are used) at the beginning of each EM iteration.

logpost.final

value of the log-posterior density after the final iteration.

AIC

Akaike's information criterion (smaller is better). Will be NA if weights is present, because AIC for pseudo-likelihood is not defined.

BIC

Bayesian information criterion (smaller is better). Will be NA if weights is present, because BIC for pseudo-likelihood is not defined.

param

estimated parameters after the final iteration. This is a list with two named components, rho and gamma, with same format as starting.values; see DETAILS.

post.probs

matrix of estimated posterior probabilities of class membership given the observed items. This is a matrix with rows corresponding to cases or rows of the dataset and columns corresponding to classes.

se.fail

logical value equal to TRUE if the computation of standard errors failed for any reason. Failure usually indicates that the model is under-identified or that one or more estimated parameters lie on or near a boundary, and the problem can often be resolved by increasing the value of flatten.rhos or flatten.gammas.

se.rho

array with same dimensions as param$rho containing the standard errors for param$rho.

se.gamma

array with same dimensions as param$gamma containing the standard errors for param$gamma.

dim.theta

number of free parameters estimated in the model. The free parameters correspond to the elements of param$gamma and param$rho, eliminating those that are redundant due to the usual sum-to-one probability constraints.

theta

vector of length dim.theta containing the final estimates of the free parameters. Examine names(theta) to see the correspondence between the elements of theta and the contents of param.

cov.theta

estimated covariance matrix for the parameters in theta. How this is computed depends on se.method.

score

vector of first derivatives of the loglikelihood of pseudo-loglikelihood (plus penalty terms for flattening constants, if present) with respect to the free parameters in theta.

hessian

matrix of (minus one times) the second derivatives of the loglikelihood or pseudo-loglikehood (plus penalty terms for flattening constants, if present) with respect to the free parameters in theta.

sandwich.meat

inner matrix of the sandwich variance formula. This is an empirical estimated covariance matrix for score.

deff.trace

the design-effect trace, if weights is present. This is the sum of the diagonal elements of solve(hessian) multiplied by sandwich.meat. This quantity is used by compare.fit when comparing the pseudo-loglikelihood values for two models.

Author(s)

Joe Schafer

Send questions to mchelpdesk@psu.edu

References

For more information on using this function and other functions in the LCCA package, see the manual LCCA Package for R, Version 1 in the subdirectory doc.

See Also

compare.fit, lcacov, lca.datasim, permute.class, summary.lca

Examples

## fit a two-class model to HIV test data
data(hivtest)
set.seed(123)
fit <- lca( cbind(A,B,C,D)~1, data=hivtest, freq=COUNT )
summary(fit, show.all=TRUE)

[Package lcca version 2.0.0 Index]