Common Components Latent Class Discriminant Analysis (CCLCDA)

Description

Local Discrimination via Latent Class Models with common components.

Usage

cclcda(x, ...)


## Default S3 method:
cclcda(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'
cclcda(formula, data, ...)

Arguments

Details

The cclcda-function performs a Common Components Latent Class Discriminant Analysis (CCLCDA). The model to estimate is

f(x) = \sum_{m=1}^{M} w_{m} \prod_{d=1}^D\prod_{r=1}^{R_d} \theta_{mdr}^{x_{dr}},

where m is the latent subclass index, d is the variable index and r is the observation index. The variable x_{dr} is 1 if the variable d of this observation is r. This common Latent Class Modell will be estimated for all classes by the poLCA-function (see poLCA) and class conditional mixing proportions w_{mk} are computed afterwards. These weights are computed by

\frac{1}{N_k} \sum_{n=1}^{N_k}\hat{P}(m, k|X=x_n),

where k is the class index and N_k the number of observations in class k.

The LCA uses the assumption of local independence to estimate a mixture model of latent multi-way tables, the number of which (m) is specified by the user. Estimated parameters include the latent-class-conditional response probabilities for each manifest variable \theta_{mdr} and the class conditional mixing proportions w_{mk} denoting population share of observations corresponding to each latent multi-way table per class.

Posterior class probabilities can be estimated with the predict method.

Value

A list of class cclcda containing the following components:

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).

Additionally cclcda provides quality criteria which should give insight into the model's classification potential. These criteria are similar to the splitting criteria of classification trees. The impurity measures are

– Weighted entropy: The weighted entropy is given by

H := - \sum_{m=1}^M P(m) \sum_{k=1}^K \left(P(k|m) \cdot \log_{K}{P(k|m)}\right).

– Weighted Gini coefficient: The weighted Gini coefficient is given by

G := \sum_{m=1}^M P(m) \left[ 1- \sum_{k=1}^{K} \left( P(k|m) \right)^2 \right].

– Pearson's Chi-square test: A Pearson's Chi-square test is performed to test the independence of latent class membership and class membership.

Author(s)

Michael B\"ucker

See Also

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

Examples

# response probabilites
probs1 <- list()

probs1[[1]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1,
                        0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7), 
                      nrow=4, byrow=TRUE)
probs1[[2]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1,
                        0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[3]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7,
                        0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[4]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1,
                        0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=4, byrow=TRUE)

prior <- c(0.5,0.5)
wmk <- matrix(c(0.45,0.45,0.05,0.05,0.05,0.05,0.45,0.45),
              ncol=4, nrow=2, byrow=TRUE)
wkm <- apply(wmk*prior, 2, function(x) x/sum(x))

# generation of training data
data_temp <- poLCA.simdata(N = 1000, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data <- data_temp$dat
lclass <- data_temp$trueclass
grouping <- numeric()
for (i in 1:length(lclass))
{
grouping[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# generation of test data
data_temp <- poLCA.simdata(N = 500, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data.test <- data_temp$dat
lclass <- data_temp$trueclass
grouping.test <- numeric()
for (i in 1:length(lclass))
{
grouping.test[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# cclcda-procedure
object <- cclcda(data, grouping, m=4)
object

Common Components Latent Class Discriminant Analysis 2 (CCLCDA2)

Description

Local Discrimination via Latent Class Models with common components

Usage

cclcda2(x, ...)

## Default S3 method:
cclcda2(x, grouping=NULL, prior=NULL,
                          probs.start=NULL, wmk.start=NULL, 
                          nrep=1, m=3, maxiter = 1000, 
                          tol = 1e-10, subset, na.rm = FALSE, ...)

## S3 method for class 'formula'
cclcda2(formula, data, ...)

Arguments

Details

The cclcda2-function performs a Common Components Latent Class Discriminant Analysis 2 (CCLCDA2). The class conditional model to estimate is

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

where m is the latent subclass index, d is the variable index and r is the observation index. The variable x_{dr} is 1 if the variable d of this observation is r. This Latent Class Modell will be estimated. The class conditional mixing proportions w_{mk} and the parameters \theta_{mdr} are computed in every step of the EM-Algorithm.

The LCA uses the assumption of local independence to estimate a mixture model of latent multi-way tables, the number of which (m) is specified by the user. Estimated parameters include the latent-class-conditional response probabilities for each manifest variable \theta_{mdr} and the class conditional mixing proportions w_{mk} denoting population share of observations corresponding to each latent multi-way table per class.

Posterior class probabilities can be estimated with the predict method.

Value

A list of class cclcda2 containing the following components:

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).

Additionally cclcda2 provides quality criteria which should give insight into the model's classification potential. These criteria are similar to the splitting criteria of classification trees. The impurity measures are

– Weighted entropy: The weighted entropy is given by

H := - \sum_{m=1}^M P(m) \sum_{k=1}^K \left(P(k|m) \cdot \log_{K}{P(k|m)}\right).

– Weighted Gini coefficient: The weighted Gini coefficient is given by

G := \sum_{m=1}^M P(m) \left[ 1- \sum_{k=1}^{K} \left( P(k|m) \right)^2 \right].

– Pearson's Chi-square test: A Pearson's Chi-square test is performed to test the independence of latent class membership and class membership.

Author(s)

Michael B\"ucker

See Also

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

Examples

# response probabilites
probs1 <- list()

probs1[[1]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1,
                        0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7), 
                      nrow=4, byrow=TRUE)
probs1[[2]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1,
                        0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[3]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7,
                        0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[4]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1,
                        0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=4, byrow=TRUE)

prior <- c(0.5,0.5)
wmk <- matrix(c(0.45,0.45,0.05,0.05,0.05,0.05,0.45,0.45),
              ncol=4, nrow=2, byrow=TRUE)
wkm <- apply(wmk*prior, 2, function(x) x/sum(x))

# generation of training data
data_temp <- poLCA.simdata(N = 1000, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data <- data_temp$dat
lclass <- data_temp$trueclass
grouping <- numeric()
for (i in 1:length(lclass))
{
grouping[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# generation of test data
data_temp <- poLCA.simdata(N = 500, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data.test <- data_temp$dat
lclass <- data_temp$trueclass
grouping.test <- numeric()
for (i in 1:length(lclass))
{
grouping.test[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# cclcda-procedure
object <- cclcda2(data, grouping, m=4)
object

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

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:

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

Predict method for Common Components Latent Class Discriminant Analysis (CCLCDA)

Description

Classifies new observations using parameters determined by the cclcda-function.

Usage

## S3 method for class 'cclcda'
predict(object, newdata, ...)

Arguments

Details

Posterior probabilities for new observations using parameters determined by the cclcda-function are computed. The classification of the new data is done by the Bayes decision function.

Value

A list with components:

Author(s)

Michael B\"ucker

See Also

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

Examples

# response probabilites
probs1 <- list()

probs1[[1]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1,
                        0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7), 
                      nrow=4, byrow=TRUE)
probs1[[2]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1,
                        0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[3]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7,
                        0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[4]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1,
                        0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=4, byrow=TRUE)

prior <- c(0.5,0.5)
wmk <- matrix(c(0.45,0.45,0.05,0.05,0.05,0.05,0.45,0.45),
              ncol=4, nrow=2, byrow=TRUE)
wkm <- apply(wmk*prior, 2, function(x) x/sum(x))

# generation of training data
data_temp <- poLCA.simdata(N = 1000, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data <- data_temp$dat
lclass <- data_temp$trueclass
grouping <- numeric()
for (i in 1:length(lclass))
{
grouping[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# generation of test data
data_temp <- poLCA.simdata(N = 500, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data.test <- data_temp$dat
lclass <- data_temp$trueclass
grouping.test <- numeric()
for (i in 1:length(lclass))
{
grouping.test[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# cclcda-procedure
object <- cclcda(data, grouping, m=4)
pred <- predict(object, data.test)$class
1-(sum(pred==grouping.test)/500)

Predict method for Common Components Latent Class Discriminant Analysis 2 (CCLCDA2)

Description

Classifies new observations using parameters determined by the cclcda2-function.

Usage

## S3 method for class 'cclcda2'
predict(object, newdata, ...)

Arguments

Details

Posterior probabilities for new observations using parameters determined by the cclcda2-function are computed. The classification of the new data is done by the Bayes decision function.

Value

A list with components:

Author(s)

Michael B\"ucker

See Also

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

Examples

# response probabilites
probs1 <- list()

probs1[[1]] <- matrix(c(0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1,
                        0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7), 
                      nrow=4, byrow=TRUE)
probs1[[2]] <- matrix(c(0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1,
                        0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[3]] <- matrix(c(0.1,0.1,0.7,0.1,0.1,0.1,0.1,0.7,
                        0.7,0.1,0.1,0.1,0.1,0.7,0.1,0.1),
                      nrow=4, byrow=TRUE)
probs1[[4]] <- matrix(c(0.1,0.1,0.1,0.7,0.7,0.1,0.1,0.1,
                        0.1,0.7,0.1,0.1,0.1,0.1,0.7,0.1),
                      nrow=4, byrow=TRUE)

prior <- c(0.5,0.5)
wmk <- matrix(c(0.45,0.45,0.05,0.05,0.05,0.05,0.45,0.45),
              ncol=4, nrow=2, byrow=TRUE)
wkm <- apply(wmk*prior, 2, function(x) x/sum(x))

# generation of training data
data_temp <- poLCA.simdata(N = 1000, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data <- data_temp$dat
lclass <- data_temp$trueclass
grouping <- numeric()
for (i in 1:length(lclass))
{
grouping[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# generation of test data
data_temp <- poLCA.simdata(N = 500, probs = probs1,
                           nclass = 2, ndv = 4, nresp = 4,
                           P=rep(0.25,4))
data.test <- data_temp$dat
lclass <- data_temp$trueclass
grouping.test <- numeric()
for (i in 1:length(lclass))
{
grouping.test[i] <- sample(c(1,2),1, prob=wkm[,lclass[i]])
}

# cclcda2-procedure
object <- cclcda2(data, grouping, m=4)
pred <- predict(object, data.test)$class
1-(sum(pred==grouping.test)/500)

Predict method for Latent Class Discriminant Analysis (LCDA)

Description

Classifies new observations using the parameters determined by the lcda-function.

Usage

## S3 method for class 'lcda'
predict(object, newdata, ...)

Arguments

Details

Posterior probabilities for new observations using parameters determined by the lcda-function are computed. The classification of the new data is done by the Bayes decision function.

Value

A list with components:

Author(s)

Michael B\"ucker

See Also

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)
pred.class <- predict(object, newdata=data)$class
sum(pred.class==grouping)/length(pred.class)