lcda {lcda}

Latent Class Discriminant Analysis (LCDA)

Description

Local Discrimination via Latent Class Models

Usage

lcda(x, ...)


## Default S3 method:
lcda(x, grouping=NULL, prior=NULL,
                       probs.start=NULL, nrep=1, m=3, 
                       maxiter = 1000, tol = 1e-10,
                       subset, na.rm = FALSE, ...)

## S3 method for class 'formula'
lcda(formula, data, ...)

Arguments

x

Matrix or data frame containing the explanatory variables. Manifest variables must contain only integer values, and must be coded with consecutive values from 1 to the maximum number of outcomes for each variable. All missing values should be entered as NA.

grouping

A factor specifying the class for each observation; if not specified, the first column of data is taken. The class must be coded by integer values with consecutive values from 1 to the maximum number of classes.

formula

Formula of the form 'groups ~ x1 + x2 + ...'.

data

Data frame from which variables specified in formula are to be taken.

prior

The prior probabilities of class membership. If unspecified, the class proportions for the training set are used. If present, the probabilities should be specified in the order of the factor levels.

probs.start

A list (per class) of lists of matrices (per variable) of response probabilities \theta_{mkdr} to be used as the starting values for the estimation algorithm. Each matrix in the list corresponds to one manifest variable, with one row for each latent class, and one column for each outcome. The default is NULL, producing random starting values. Note that if nrep>1, then any user-specified probs.start values are only used in the first of the nrep attempts.

nrep

Number of times to estimate the model, using different random values of probs.start. The default is one. Setting nrep>1 automates the search for the global – rather than just a local – maximum of the log-likelihood function. lcda uses the parameter estimates corresponding to the model with the greatest log-likelihood.

m

The number of subclasses per class. Can be either a vector containing the number of subclasses per class or a number of subclasses for all classes. Default is m=3.

maxiter

The maximum number of iterations through which the estimation algorithm will cycle.

tol

A tolerance value for judging when convergence has been reached. When the one-iteration change in the estimated log-likelihood is less than tol, the estimation algorithm stops updating and considers the maximum log-likelihood to have been found.

subset

An index vector specifying the cases to be used in the training sample.

na.rm

Logical, for how lcda handles cases with missing values on the manifest variables. If TRUE, those cases are removed (listwise deleted) before estimating the model. If FALSE, cases with missing values are retained. Cases with missing covariates are always removed. The default is TRUE.

...

Further arguments to be passed to lcda.

Details

The lcda-function performs a Latent Class Discriminant Analysis (LCDA). A Latent Class Modell will be estimated for each class by the poLCA-function (see poLCA). The class conditional model is given by

f_k(x) = \sum_{m=1}^{M_k} w_{mk} \prod_{d=1}^D\prod_{r=1}^{R_d} \theta_{mkdr}^{x_{kdr}},

where k is the class index, m is the latent subclass index, d is the variable index and r is the observation index. The variable x_{kdr} is 1 if the variable d of this observation is r and in class k. The parameter w_{mk} is the class conditional mixture weight and \theta_{mkdr} is the probability for outcome r of variable d in subclass m of class k.

These Latent Class Models use the assumption of local independence to estimate a mixture model of latent multi-way tables. The mixture models are estimated by the EM-algorithm. The number of mixture components (m) is specified by the user. Estimated parameters include the latent-class conditional response probabilities for each manifest variable \theta_{mkdr} and the class conditional mixing proportions w_{mk} denoting the population share of observations corresponding to each latent multi-way table.

Posterior class probabilities and class memberships can be estimated with the predict method.

Value

A list of class lcda containing the following components:

call

The (matched) function call.

lca.theta

The estimated class conditional response probabilities of the LCA given as a list of matrices like probs.start.

lca.w

The estimated mixing proportions of the LCA.

prior

Prior probabilites.

m

Number of latent subclasses per class.

r

Number of possible responses per variable.

k

Number of classes.

d

Number of variables.

aic

Value of the AIC for each class conditional Latent Class Model.

bic

Value of the BIC for each class conditional Latent Class Model.

Gsq

The likelihood ratio/deviance statistic for each class conditional model.

Chisq

The Pearson Chi-square goodness of fit statistic for fitted vs. observed multiway tables for each class conditional model.

Note

If the number of latent classes per class is unknown a model selection must be accomplished to determine the value of m. For this goal there are some model selection criteria implemented. The AIC, BIC, likelihood ratio statistic and the Chi-square goodness of fit statistic are taken from the poLCA-function (see poLCA). For each class these criteria can be regarded separately and for each class the number of latent classes can be determined.

Author(s)

Michael B\"ucker

See Also

predict.lcda, cclcda, predict.cclcda, cclcda2, predict.cclcda2, poLCA

Examples

# response probabilites for class 1
probs1 <- list()
probs1[[1]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1), 
                      nrow=2, byrow=TRUE)
probs1[[2]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=2, byrow=TRUE)
probs1[[3]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7),
                      nrow=2, byrow=TRUE)
probs1[[4]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=2, byrow=TRUE)

# response probabilites for class 2
probs2 <- list()
probs2[[1]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7),
                      nrow=2, byrow=TRUE)
probs2[[2]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=2, byrow=TRUE)
probs2[[3]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1),
                      nrow=2, byrow=TRUE)
probs2[[4]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=2, byrow=TRUE)

# generation of data
simdata1 <- poLCA.simdata(N = 500, probs = probs1, nclass = 2,
              ndv = 4, nresp = 4, missval = FALSE)

simdata2 <- poLCA.simdata(N = 500, probs = probs2, nclass = 2,
              ndv = 4, nresp = 4, missval = FALSE)

data1 <- simdata1$dat
data2 <- simdata2$dat

data <- cbind(rbind(data1, data2), rep(c(1,2), each=500))
names(data)[5] <- "grouping"
data <- data[sample(1:1000),]
grouping <- data[[5]]
data <- data[,1:4]

# lcda-procedure
object <- lcda(data, grouping=grouping, m=2)
object