| Title: | Polytomous Variable Latent Class Analysis |
|---|---|
| Description: | Latent class analysis and latent class regression models for polytomous outcome variables. Also known as latent structure analysis. |
| Authors: | Drew Linzer <[email protected]>, Jeffrey Lewis <[email protected]>. |
| Maintainer: | Drew Linzer <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.6.0.1 |
| Built: | 2024-10-26 06:27:48 UTC |
| Source: | CRAN |
Dichotomous ratings by seven pathologists of 118 slides for the presence or absence of carcinoma in the uterine cervix. Pathologists are labeled A through G. There were 20 different observed response patterns. This data set appears in Agresti (2002, p. 542) as Table 13.1.
data(carcinoma)data(carcinoma)
A data frame with 118 observations on 7 variables representing pathologist ratings with 1 denoting "no" and 2 denoting "yes".
Agresti, Alan. 2002. Categorical Data Analysis, second edition. Hoboken: John Wiley \& Sons.
## ## Replication of latent class models in Agresti (2002, p. 543), ## Table 13.2 and Table 13.3. ## data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca2 <- poLCA(f,carcinoma,nclass=2) # log-likelihood: -317.2568 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 lca4 <- poLCA(f,carcinoma,nclass=4,nrep=10,maxiter=5000) # log-likelihood: -289.2858## ## Replication of latent class models in Agresti (2002, p. 543), ## Table 13.2 and Table 13.3. ## data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca2 <- poLCA(f,carcinoma,nclass=2) # log-likelihood: -317.2568 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 lca4 <- poLCA(f,carcinoma,nclass=4,nrep=10,maxiter=5000) # log-likelihood: -289.2858
Dichotomous responses by 319 undergraduates to four questions about cheating behavior, and each student's academic GPA.
Students responded either (1) no or (2) yes as to whether they had ever lied to avoid taking an exam (LIEEXAM), lied to avoid handing a term paper in on time (LIEPAPER), purchased a term paper to hand in as their own or had obtained a copy of an exam prior to taking the exam (FRAUD), or copied answers during an exam from someone sitting near to them (COPYEXAM).
The GPA variable is partitioned into five groups: (1) 2.99 or less; (2) 3.00-3.25; (3) 3.26-3.50; (4) 3.51-3.75; (5) 3.76-4.00.
This data set appears in Dayton (1998, pp. 33 and 85) as Tables 3.4 and 7.1.
data(cheating)data(cheating)
A data frame with 319 observations on 5 variables. Note: GPA data were not available for four students who reported never cheating.
Dayton, C. Mitchell. 1998. Latent Class Scaling Analysis. Thousand Oaks, CA: SAGE Publications.
## ## Replication of latent class models in Dayton (1998) ## ## Example 1. Two-class LCA. (Table 3.3, p. 32) ## data(cheating) f <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~1 ch2 <- poLCA(f,cheating,nclass=2) # log-likelihood: -440.0271 ## ## Example 2. Two-class latent class regression using ## GPA as a covariate to predict class membership as ## "cheaters" vs. "non-cheaters". ## (Table 7.1, p. 85, and Figure 7.1, p. 86) ## f2 <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~GPA ch2c <- poLCA(f2,cheating,nclass=2) # log-likelihood: -429.6384 GPAmat <- cbind(1,c(1:5)) exb <- exp(GPAmat %*% ch2c$coeff) matplot(c(1:5),cbind(1/(1+exb),exb/(1+exb)),type="l",lwd=2, main="GPA as a predictor of persistent cheating", xlab="GPA category, low to high", ylab="Probability of latent class membership") text(1.7,0.3,"Cheaters") text(1.7,0.7,"Non-cheaters") ## ## Compare results from Example 1 to Example 2. ## Non-simultaneous estimation of effect of GPA on latent class ## membership biases the estimated effect in Example 1. ## cheatcl <- which.min(ch2$P) predcc <- sapply(c(1:5),function(v) mean(ch2$posterior[cheating$GPA==v,cheatcl],na.rm=TRUE)) ## Having run Ex.2, add to plot: matplot(c(1:5),cbind(1-predcc,predcc),type="l",lwd=2,add=TRUE) text(4,0.14,"Cheaters\n (non-simul. estimate)") text(4,0.87,"Non-cheaters\n (non-simul. estimate)")## ## Replication of latent class models in Dayton (1998) ## ## Example 1. Two-class LCA. (Table 3.3, p. 32) ## data(cheating) f <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~1 ch2 <- poLCA(f,cheating,nclass=2) # log-likelihood: -440.0271 ## ## Example 2. Two-class latent class regression using ## GPA as a covariate to predict class membership as ## "cheaters" vs. "non-cheaters". ## (Table 7.1, p. 85, and Figure 7.1, p. 86) ## f2 <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~GPA ch2c <- poLCA(f2,cheating,nclass=2) # log-likelihood: -429.6384 GPAmat <- cbind(1,c(1:5)) exb <- exp(GPAmat %*% ch2c$coeff) matplot(c(1:5),cbind(1/(1+exb),exb/(1+exb)),type="l",lwd=2, main="GPA as a predictor of persistent cheating", xlab="GPA category, low to high", ylab="Probability of latent class membership") text(1.7,0.3,"Cheaters") text(1.7,0.7,"Non-cheaters") ## ## Compare results from Example 1 to Example 2. ## Non-simultaneous estimation of effect of GPA on latent class ## membership biases the estimated effect in Example 1. ## cheatcl <- which.min(ch2$P) predcc <- sapply(c(1:5),function(v) mean(ch2$posterior[cheating$GPA==v,cheatcl],na.rm=TRUE)) ## Having run Ex.2, add to plot: matplot(c(1:5),cbind(1-predcc,predcc),type="l",lwd=2,add=TRUE) text(4,0.14,"Cheaters\n (non-simul. estimate)") text(4,0.87,"Non-cheaters\n (non-simul. estimate)")
Survey data from the 2000 American National Election Study. Two sets of six questions with four responses each, asking respondents' opinions of how well various traits (moral, caring, knowledgable, good leader, dishonest, intelligent) describe presidential candidates Al Gore and George W. Bush. The responses are (1) Extremely well; (2) Quite well; (3) Not too well; (4) Not well at all. Many respondents have varying numbers of missing values on these variables.
The data set also includes potential covariates VOTE3, the respondent's 2000 vote choice (when asked); AGE, the respondent's age; EDUC, the respondent's level of education; GENDER, the respondent's gender; and PARTY, the respondent's Democratic-Republican partisan identification.
VOTE3 is coded as (1) Gore; (2) Bush; (3) Other.
EDUC is coded as (1) 8 grades or less; (2) 9-11 grades, no further schooling; (3) High school diploma or equivalency; (4) More than 12 years of schooling, no higher degree; (5) Junior or community college level degree; (6) BA level degrees, no advanced degree; (7) Advanced degree.
GENDER is coded as (1) Male; (2) Female.
PARTY is coded as (1) Strong Democrat; (2) Weak Democrat; (3) Independent-Democrat; (4) Independent-Independent; (5) Independent-Republican; (6) Weak Republican; (7) Strong Republican.
data(election)data(election)
A data frame with 1785 observations on 17 survey variables. Of these, 1311 individuals provided responses on all twelve candidate evaluations.
The National Election Studies (https://electionstudies.org/). THE 2000 NATIONAL ELECTION STUDY [dataset]. Ann Arbor, MI: University of Michigan, Center for Political Studies [producer and distributor].
# Latent class models with one (loglinear independence) to three classes data(election) f <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~1 nes1 <- poLCA(f,election,nclass=1) # log-likelihood: -18647.31 nes2 <- poLCA(f,election,nclass=2) # log-likelihood: -17344.92 nes3 <- poLCA(f,election,nclass=3) # log-likelihood: -16714.66 # Three-class model with a single covariate (party) f2a <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~PARTY nes2a <- poLCA(f2a,election,nclass=3,nrep=5) # log-likelihood: -16222.32 pidmat <- cbind(1,c(1:7)) exb <- exp(pidmat %*% nes2a$coeff) matplot(c(1:7),(cbind(1,exb)/(1+rowSums(exb))),ylim=c(0,1),type="l", main="Party ID as a predictor of candidate affinity class", xlab="Party ID: strong Democratic (1) to strong Republican (7)", ylab="Probability of latent class membership",lwd=2,col=1) text(5.9,0.35,"Other") text(5.4,0.7,"Bush affinity") text(1.8,0.6,"Gore affinity")# Latent class models with one (loglinear independence) to three classes data(election) f <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~1 nes1 <- poLCA(f,election,nclass=1) # log-likelihood: -18647.31 nes2 <- poLCA(f,election,nclass=2) # log-likelihood: -17344.92 nes3 <- poLCA(f,election,nclass=3) # log-likelihood: -16714.66 # Three-class model with a single covariate (party) f2a <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~PARTY nes2a <- poLCA(f2a,election,nclass=3,nrep=5) # log-likelihood: -16222.32 pidmat <- cbind(1,c(1:7)) exb <- exp(pidmat %*% nes2a$coeff) matplot(c(1:7),(cbind(1,exb)/(1+rowSums(exb))),ylim=c(0,1),type="l", main="Party ID as a predictor of candidate affinity class", xlab="Party ID: strong Democratic (1) to strong Republican (7)", ylab="Probability of latent class membership",lwd=2,col=1) text(5.9,0.35,"Other") text(5.4,0.7,"Bush affinity") text(1.8,0.6,"Gore affinity")
Attitudes towards survey taking across two dichotomous and two trichotomous items among 1202 white respondents to the 1982 General Social Survey. Respondents give their opinion of the purpose of surveys (PURPOSE; good/depends/waste of time and money), the accuracy of surveys (ACCURACY; mostly true/not true), their understanding of survey questions (UNDERSTA; good/fair, poor), and how well they cooperated with the interviewer (COOPERAT; interested/cooperative/impatient, hostile). This data set appears in McCutcheon (1987, p. 30) as Table 3.1.
data(gss82)data(gss82)
A data frame with 1202 observations on 4 survey variables.
McCutcheon, A.L. 1987. Latent class analysis. Newbury Park: SAGE Publications.
data(gss82) f <- cbind(PURPOSE,ACCURACY,UNDERSTA,COOPERAT)~1 gss.lc2 <- poLCA(f,gss82,nclass=2) # log-likelihood = -2783.268 # Could also try: # gss.lc3 <- poLCA(f,gss82,nclass=3,maxiter=3000,nrep=10) # log-likelihood = -2754.545 # gss.lc4 <- poLCA(f,gss82,nclass=4,maxiter=15000,nrep=10,tol=1e-7) # log-likelihood = -2746.621data(gss82) f <- cbind(PURPOSE,ACCURACY,UNDERSTA,COOPERAT)~1 gss.lc2 <- poLCA(f,gss82,nclass=2) # log-likelihood = -2783.268 # Could also try: # gss.lc3 <- poLCA(f,gss82,nclass=3,maxiter=3000,nrep=10) # log-likelihood = -2754.545 # gss.lc4 <- poLCA(f,gss82,nclass=4,maxiter=15000,nrep=10,tol=1e-7) # log-likelihood = -2746.621
Estimates latent class and latent class regression models for polytomous outcome variables.
poLCA(formula, data, nclass = 2, maxiter = 1000, graphs = FALSE, tol = 1e-10, na.rm = TRUE, probs.start = NULL, nrep = 1, verbose = TRUE, calc.se = TRUE)poLCA(formula, data, nclass = 2, maxiter = 1000, graphs = FALSE, tol = 1e-10, na.rm = TRUE, probs.start = NULL, nrep = 1, verbose = TRUE, calc.se = TRUE)
formula |
A formula expression of the form |
data |
A data frame containing variables in |
nclass |
The number of latent classes to assume in the model. Setting |
maxiter |
The maximum number of iterations through which the estimation algorithm will cycle. |
graphs |
Logical, for whether |
tol |
A tolerance value for judging when convergence has been reached. When the one-iteration change in the estimated log-likelihood is less than |
na.rm |
Logical, for how |
probs.start |
A list of matrices of class-conditional response probabilities 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 |
nrep |
Number of times to estimate the model, using different values of |
verbose |
Logical, indicating whether |
calc.se |
Logical, indicating whether |
Latent class analysis, also known as latent structure analysis, is a technique for the analysis of clustering among observations in multi-way tables of qualitative/categorical variables. The central idea is to fit a model in which any confounding between the manifest variables can be explained by a single unobserved "latent" categorical variable. poLCA uses the assumption of local independence to estimate a mixture model of latent multi-way tables, the number of which (nclass) is specified by the user. Estimated parameters include the class-conditional response probabilities for each manifest variable, the "mixing" proportions denoting population share of observations corresponding to each latent multi-way table, and coefficients on any class-predictor covariates, if specified in the model.
Model specification: Latent class models have more than one manifest variable, so the response variables are cbind(dv1,dv2,dv3...) where dv# refer to variable names in the data frame. For models with no covariates, the formula is cbind(dv1,dv2,dv3)~1. For models with covariates, replace the ~1 with the desired function of predictors iv1,iv2,iv3... as, for example, cbind(dv1,dv2,dv3)~iv1+iv2*iv3.
poLCA treats all manifest variables as qualitative/categorical/nominal – NOT as ordinal.
poLCA returns an object of class poLCA; a list containing the following elements:
y |
data frame of manifest variables. |
x |
data frame of covariates, if specified. |
N |
number of cases used in model. |
Nobs |
number of fully observed cases (less than or equal to |
probs |
estimated class-conditional response probabilities. |
probs.se |
standard errors of estimated class-conditional response probabilities, in the same format as |
P |
sizes of each latent class; equal to the mixing proportions in the basic latent class model, or the mean of the priors in the latent class regression model. |
P.se |
the standard errors of the estimated |
posterior |
matrix of posterior class membership probabilities; also see function |
predclass |
vector of predicted class memberships, by modal assignment. |
predcell |
table of observed versus predicted cell counts for cases with no missing values; also see functions |
llik |
maximum value of the log-likelihood. |
numiter |
number of iterations until reaching convergence. |
maxiter |
maximum number of iterations through which the estimation algorithm was set to run. |
coeff |
multinomial logit coefficient estimates on covariates (when estimated). |
coeff.se |
standard errors of coefficient estimates on covariates (when estimated), in the same format as |
coeff.V |
covariance matrix of coefficient estimates on covariates (when estimated). |
aic |
Akaike Information Criterion. |
bic |
Bayesian Information Criterion. |
Gsq |
Likelihood ratio/deviance statistic. |
Chisq |
Pearson Chi-square goodness of fit statistic for fitted vs. observed multiway tables. |
time |
length of time it took to run the model. |
npar |
number of degrees of freedom used by the model (estimated parameters). |
resid.df |
number of residual degrees of freedom. |
attempts |
a vector containing the maximum log-likelihood values found in each of the |
eflag |
Logical, error flag. |
probs.start |
A list of matrices containing the class-conditional response probabilities used as starting values in the estimation algorithm. If the algorithm needed to restart (see |
probs.start.ok |
Logical. |
call |
function call to |
poLCA uses EM and Newton-Raphson algorithms to maximize the latent class model log-likelihood function. Depending on the starting parameters, this algorithm may only locate a local, rather than global, maximum. This becomes more and more of a problem as nclass increases. It is therefore highly advisable to run poLCA multiple times until you are relatively certain that you have located the global maximum log-likelihood. As long as probs.start=NULL, each function call will use different (random) initial starting parameters. Alternatively, setting nrep to a value greater than one enables the user to estimate the latent class model multiple times with a single call to poLCA, thus conducting the search for the global maximizer automatically.
The term "Latent class regression" (LCR) can have two meanings. In this package, LCR models refer to latent class models in which the probability of class membership is predicted by one or more covariates. However, in other contexts, LCR is also used to refer to regression models in which the manifest variable is partitioned into some specified number of latent classes as part of estimating the regression model. It is a way to simultaneously fit more than one regression to the data when the latent data partition is unknown. The flexmix function in package flexmix will estimate this other type of LCR model. Because of these terminology issues, the LCR models this package estimates are sometimes termed "latent class models with covariates" or "concomitant-variable latent class analysis," both of which are accurate descriptions of this model.
A more detailed presentation is available online at https://www.jstatsoft.org/article/view/v042i10.
Agresti, Alan. 2002. Categorical Data Analysis, second edition. Hoboken: John Wiley \& Sons.
Bandeen-Roche, Karen, Diana L. Miglioretti, Scott L. Zeger, and Paul J. Rathouz. 1997. "Latent Variable Regression for Multiple Discrete Outcomes." Journal of the American Statistical Association. 92(440): 1375-1386.
Hagenaars, Jacques A. and Allan L. McCutcheon, eds. 2002. Applied Latent Class Analysis. Cambridge: Cambridge University Press.
McLachlan, Geoffrey J. and Thriyambakam Krishnan. 1997. The EM Algorithm and Extensions. New York: John Wiley \& Sons.
## ## Three models without covariates: ## M0: Loglinear independence model. ## M1: Two-class latent class model. ## M2: Three-class latent class model. ## data(values) f <- cbind(A,B,C,D)~1 M0 <- poLCA(f,values,nclass=1) # log-likelihood: -543.6498 M1 <- poLCA(f,values,nclass=2) # log-likelihood: -504.4677 M2 <- poLCA(f,values,nclass=3,maxiter=8000) # log-likelihood: -503.3011 ## ## Three-class model with a single covariate. ## data(election) f2a <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~PARTY nes2a <- poLCA(f2a,election,nclass=3,nrep=5) # log-likelihood: -16222.32 pidmat <- cbind(1,c(1:7)) exb <- exp(pidmat %*% nes2a$coeff) matplot(c(1:7),(cbind(1,exb)/(1+rowSums(exb))),ylim=c(0,1),type="l", main="Party ID as a predictor of candidate affinity class", xlab="Party ID: strong Democratic (1) to strong Republican (7)", ylab="Probability of latent class membership",lwd=2,col=1) text(5.9,0.35,"Other") text(5.4,0.7,"Bush affinity") text(1.8,0.6,"Gore affinity")## ## Three models without covariates: ## M0: Loglinear independence model. ## M1: Two-class latent class model. ## M2: Three-class latent class model. ## data(values) f <- cbind(A,B,C,D)~1 M0 <- poLCA(f,values,nclass=1) # log-likelihood: -543.6498 M1 <- poLCA(f,values,nclass=2) # log-likelihood: -504.4677 M2 <- poLCA(f,values,nclass=3,maxiter=8000) # log-likelihood: -503.3011 ## ## Three-class model with a single covariate. ## data(election) f2a <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~PARTY nes2a <- poLCA(f2a,election,nclass=3,nrep=5) # log-likelihood: -16222.32 pidmat <- cbind(1,c(1:7)) exb <- exp(pidmat %*% nes2a$coeff) matplot(c(1:7),(cbind(1,exb)/(1+rowSums(exb))),ylim=c(0,1),type="l", main="Party ID as a predictor of candidate affinity class", xlab="Party ID: strong Democratic (1) to strong Republican (7)", ylab="Probability of latent class membership",lwd=2,col=1) text(5.9,0.35,"Other") text(5.4,0.7,"Bush affinity") text(1.8,0.6,"Gore affinity")
Calculates the entropy of a cross-classification table produced as a density estimate using a latent class model.
poLCA.entropy(lc)poLCA.entropy(lc)
lc |
A model object estimated using the |
Entropy is a measure of dispersion (or concentration) in a probability mass function. For multivariate categorical data it is calculated
where is the share of the probability in the cth cell of the cross-classification table. A fitted latent class model produces a smoothed density estimate of the underlying distribution of cell percentages in the multi-way table of the manifest variables. This function calculates the entropy of that estimated probability mass function.
A number taking a minumum value of 0 (representing complete concentration of probability on one cell) and a maximum value equal to the logarithm of the total number of cells in the fitted cross-classfication table (representing complete dispersion, or equal probability for outcomes across every cell).
data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca2 <- poLCA(f,carcinoma,nclass=2) # log-likelihood: -317.2568 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 lca4 <- poLCA(f,carcinoma,nclass=4,nrep=10,maxiter=5000) # log-likelihood: -289.2858 # Maximum entropy (if all cases equally dispersed) log(prod(sapply(lca2$probs,ncol))) # Sample entropy ("plug-in" estimator, or MLE) p.hat <- lca2$predcell$observed/lca2$N H.hat <- -sum(p.hat * log(p.hat)) H.hat # 2.42 # Entropy of fitted latent class models poLCA.entropy(lca2) poLCA.entropy(lca3) poLCA.entropy(lca4)data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca2 <- poLCA(f,carcinoma,nclass=2) # log-likelihood: -317.2568 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 lca4 <- poLCA(f,carcinoma,nclass=4,nrep=10,maxiter=5000) # log-likelihood: -289.2858 # Maximum entropy (if all cases equally dispersed) log(prod(sapply(lca2$probs,ncol))) # Sample entropy ("plug-in" estimator, or MLE) p.hat <- lca2$predcell$observed/lca2$N H.hat <- -sum(p.hat * log(p.hat)) H.hat # 2.42 # Entropy of fitted latent class models poLCA.entropy(lca2) poLCA.entropy(lca3) poLCA.entropy(lca4)
Calculates the posterior probability that cases belong to each latent class.
poLCA.posterior(lc,y,x=NULL)poLCA.posterior(lc,y,x=NULL)
lc |
A model object estimated using the |
y |
A vector or matrix containing series of responses on the manifest variables in |
x |
An optional vector or matrix of covariate values, if |
From the parameters estimated by the latent class model, this function calculates the "posterior" probability that a specified case – characterized by values of the manifest variables y, and, if a latent class regression model, concomitant variables x – "belongs to" each latent class in lc. For observed cases, this information is also contained in the lc model object as lc$posterior. The added benefit of this function is that it can calculate posterior class membership probabilities for arbitrary values of x and y, whether or observed or not.
A matrix containing posterior probabilities corresponding to the specified sets of responses y, based on the estimated latent class model lc. For each row (one case), the first column gives the posterior probability of being in class 1, the second column gives the posterior probability of being in class 2, and so forth. Across rows, these probabilities sum to one.
data(election) ## Basic latent class model with three classes f1 <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~1 lc1 <- poLCA(f1,election,nclass=3) # log-likelihood: -16714.66 # The first observed case lc1$y[1,] lc1$posterior[1,] poLCA.posterior(lc=lc1,y=as.numeric(lc1$y[1,])) # A hypothetical case poLCA.posterior(lc=lc1,y=rep(2,12)) # Entering y as a matrix lc1$posterior[1:10,] poLCA.posterior(lc=lc1,y=mapply(as.numeric,lc1$y[1:10,])) ## Latent class regression model with three classes f2 <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~AGE+EDUC+GENDER lc2 <- poLCA(f2,election,nclass=3) # log-likelihood: -16598.38 # Posteriors for case number 97 (poorly classified) lc2$y[97,] lc2$x[97,] lc2$posterior[97,] poLCA.posterior(lc=lc2,y=as.numeric(lc2$y[97,]),x=c(41,6,1)) # If x is not specified, the posterior is calculated using the population average poLCA.posterior(lc=lc2,y=as.numeric(lc2$y[97,])) # Entering y and x as matrices round(lc2$posterior[95:100,],2) round(poLCA.posterior(lc=lc2,y=mapply(as.numeric,lc2$y[95:100,]), x=as.matrix(lc2$x[95:100,-1])),2)data(election) ## Basic latent class model with three classes f1 <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~1 lc1 <- poLCA(f1,election,nclass=3) # log-likelihood: -16714.66 # The first observed case lc1$y[1,] lc1$posterior[1,] poLCA.posterior(lc=lc1,y=as.numeric(lc1$y[1,])) # A hypothetical case poLCA.posterior(lc=lc1,y=rep(2,12)) # Entering y as a matrix lc1$posterior[1:10,] poLCA.posterior(lc=lc1,y=mapply(as.numeric,lc1$y[1:10,])) ## Latent class regression model with three classes f2 <- cbind(MORALG,CARESG,KNOWG,LEADG,DISHONG,INTELG, MORALB,CARESB,KNOWB,LEADB,DISHONB,INTELB)~AGE+EDUC+GENDER lc2 <- poLCA(f2,election,nclass=3) # log-likelihood: -16598.38 # Posteriors for case number 97 (poorly classified) lc2$y[97,] lc2$x[97,] lc2$posterior[97,] poLCA.posterior(lc=lc2,y=as.numeric(lc2$y[97,]),x=c(41,6,1)) # If x is not specified, the posterior is calculated using the population average poLCA.posterior(lc=lc2,y=as.numeric(lc2$y[97,])) # Entering y and x as matrices round(lc2$posterior[95:100,],2) round(poLCA.posterior(lc=lc2,y=mapply(as.numeric,lc2$y[95:100,]), x=as.matrix(lc2$x[95:100,-1])),2)
Calculates the predicted cell percentages from a latent class model, for specified values of the manifest variables.
poLCA.predcell(lc,y)poLCA.predcell(lc,y)
lc |
A model object estimated using the |
y |
A vector or matrix containing series of responses on the manifest variables in |
The parameters estimated by a latent class model can be used to produce a density estimate of the underlying probability mass function across the cells in the multi-way table of manifest variables. This function calculates cell percentages for that density estimate, corresponding to selected sets of responses on the manifest variables, y.
A vector containing cell percentages corresponding to the specified sets of responses y, based on the estimated latent class model lc.
data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 # Only 20 out of 32 possible response patterns are observed lca3$predcell # Produce cell probabilities for one sequence of responses poLCA.predcell(lc=lca3,y=c(1,1,1,1,1,1,1)) # Estimated probabilities for a cell with zero observations poLCA.predcell(lc=lca3,y=c(1,1,1,1,1,1,2)) # Cell probabilities for both cells at once; y entered as a matrix poLCA.predcell(lc=lca3,y=rbind(c(1,1,1,1,1,1,1),c(1,1,1,1,1,1,2)))data(carcinoma) f <- cbind(A,B,C,D,E,F,G)~1 lca3 <- poLCA(f,carcinoma,nclass=3) # log-likelihood: -293.705 # Only 20 out of 32 possible response patterns are observed lca3$predcell # Produce cell probabilities for one sequence of responses poLCA.predcell(lc=lca3,y=c(1,1,1,1,1,1,1)) # Estimated probabilities for a cell with zero observations poLCA.predcell(lc=lca3,y=c(1,1,1,1,1,1,2)) # Cell probabilities for both cells at once; y entered as a matrix poLCA.predcell(lc=lca3,y=rbind(c(1,1,1,1,1,1,1),c(1,1,1,1,1,1,2)))
A helper function to simplify the reordering of latent classes returned by poLCA.
poLCA.reorder(probs, o.new)poLCA.reorder(probs, o.new)
probs |
a list of class-conditional response probabilities previously used as start values to estimate a particular latent class model using |
o.new |
a vector of length equal to the number of latent classes in |
Because the latent classes outputted by poLCA are unordered categories, the numerical order of the classes is arbitrary, and is determined solely by the initial values of the EM algorithm. If probs.start is set to NULL (the default) when calling poLCA, then the function generates the starting values randomly in each run, typically rearranging the latent class labels. The poLCA.reorder function is a convenient way to manually adjust the order of the latent classes, by changing the order of the probs.start. Refitting the latent class model using these reordered start values will produce a model having the desired category labels.
The function returns a list of matrices containing the rearranged (by row) class-conditional response probabilities.
## ## Using the "cheating" sample data set, make the larger ## non-cheater class the first ("reference") class in a ## latent class regression model. The coefficient on GPA ## now maintains a consistent interpretation. ## data(cheating) f2 <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~GPA lc.ch <- poLCA(f2,cheating,nclass=2,verbose=FALSE) probs.start.new <- poLCA.reorder(lc.ch$probs.start,order(lc.ch$P,decreasing=TRUE)) lc.ch <- poLCA(f2,cheating,nclass=2,probs.start=probs.start.new)## ## Using the "cheating" sample data set, make the larger ## non-cheater class the first ("reference") class in a ## latent class regression model. The coefficient on GPA ## now maintains a consistent interpretation. ## data(cheating) f2 <- cbind(LIEEXAM,LIEPAPER,FRAUD,COPYEXAM)~GPA lc.ch <- poLCA(f2,cheating,nclass=2,verbose=FALSE) probs.start.new <- poLCA.reorder(lc.ch$probs.start,order(lc.ch$P,decreasing=TRUE)) lc.ch <- poLCA(f2,cheating,nclass=2,probs.start=probs.start.new)
Uses the latent class model's assumed data-generating process to create a simulated dataset that can be used to test the properties of the poLCA latent class and latent class regression estimator.
poLCA.simdata(N = 5000, probs = NULL, nclass = 2, ndv = 4, nresp = NULL, x = NULL, niv = 0, b = NULL, P = NULL, missval = FALSE, pctmiss = NULL)poLCA.simdata(N = 5000, probs = NULL, nclass = 2, ndv = 4, nresp = NULL, x = NULL, niv = 0, b = NULL, P = NULL, missval = FALSE, pctmiss = NULL)
N |
number of observations. |
probs |
a list of matrices of dimension |
nclass |
number of latent classes. If |
ndv |
number of manifest variables. If |
nresp |
number of possible outcomes for each manifest variable. If |
x |
a matrix of concomicant variables with |
niv |
number of concomitant variables (covariates). Setting |
b |
when using covariates, an |
P |
a vector of mixing proportions (class population shares) of length |
missval |
logical. If |
pctmiss |
percentage of values to be dropped as missing, if |
Note that entering probs overrides nclass, ndv, and nresp. It also overrides P if the length of the P vector is not equal to the length of the probs list. Likewise, if probs=NULL, then length(nresp) overrides ndv and length(P) overrides nclass. Setting niv>1 causes any user-entered value of P to be disregarded.
dat |
a data frame containing the simulated variables. Variable names for manifest variables are Y1, Y2, etc. Variable names for concomitant variables are X1, X2, etc. |
probs |
a list of matrices of dimension |
nresp |
a vector containing the number of possible outcomes for each manifest variable. |
b |
coefficients on covariates, if used. |
P |
mixing proportions corresponding to each latent class. |
pctmiss |
percent of observations missing. |
trueclass |
|
# Create a sample data set with 3 classes and no covariates # and run poLCA to recover the specified parameters. probs <- list(matrix(c(0.6, 0.1, 0.3, 0.6, 0.3, 0.1, 0.3, 0.1, 0.6), ncol = 3,byrow = TRUE), # conditional resp prob to Y1 matrix(c(0.2, 0.8, 0.7, 0.3, 0.3, 0.7), ncol = 2, byrow = TRUE), # conditional resp prob to Y2 matrix(c(0.3, 0.6, 0.1, 0.1, 0.3, 0.6, 0.3, 0.6, 0.1), ncol = 3,byrow = TRUE), # conditional resp prob to Y3 matrix(c(0.1, 0.1, 0.5, 0.3, 0.5, 0.3, 0.1, 0.1, 0.3, 0.1, 0.1, 0.5), ncol = 4,byrow = TRUE), # conditional resp prob to Y4 matrix(c(0.1, 0.1, 0.8, 0.1, 0.8, 0.1, 0.8, 0.1, 0.1), ncol = 3, byrow = TRUE)) # conditional resp prob to Y5 simdat <- poLCA.simdata(N=1000,probs,P=c(0.2,0.3,0.5)) f1 <- cbind(Y1,Y2,Y3,Y4,Y5)~1 lc1 <- poLCA(f1,simdat$dat,nclass=3) table(lc1$predclass,simdat$trueclass) # Create a sample dataset with 2 classes and three covariates. # Then compare predicted class memberships when the model is # estimated "correctly" with covariates to when it is estimated # "incorrectly" without covariates. simdat2 <- poLCA.simdata(N=1000,ndv=7,niv=3,nclass=2,b=matrix(c(1,-2,1,-1))) f2a <- cbind(Y1,Y2,Y3,Y4,Y5,Y6,Y7)~X1+X2+X3 lc2a <- poLCA(f2a,simdat2$dat,nclass=2) f2b <- cbind(Y1,Y2,Y3,Y4,Y5,Y6,Y7)~1 lc2b <- poLCA(f2b,simdat2$dat,nclass=2) table(lc2a$predclass,lc2b$predclass)# Create a sample data set with 3 classes and no covariates # and run poLCA to recover the specified parameters. probs <- list(matrix(c(0.6, 0.1, 0.3, 0.6, 0.3, 0.1, 0.3, 0.1, 0.6), ncol = 3,byrow = TRUE), # conditional resp prob to Y1 matrix(c(0.2, 0.8, 0.7, 0.3, 0.3, 0.7), ncol = 2, byrow = TRUE), # conditional resp prob to Y2 matrix(c(0.3, 0.6, 0.1, 0.1, 0.3, 0.6, 0.3, 0.6, 0.1), ncol = 3,byrow = TRUE), # conditional resp prob to Y3 matrix(c(0.1, 0.1, 0.5, 0.3, 0.5, 0.3, 0.1, 0.1, 0.3, 0.1, 0.1, 0.5), ncol = 4,byrow = TRUE), # conditional resp prob to Y4 matrix(c(0.1, 0.1, 0.8, 0.1, 0.8, 0.1, 0.8, 0.1, 0.1), ncol = 3, byrow = TRUE)) # conditional resp prob to Y5 simdat <- poLCA.simdata(N=1000,probs,P=c(0.2,0.3,0.5)) f1 <- cbind(Y1,Y2,Y3,Y4,Y5)~1 lc1 <- poLCA(f1,simdat$dat,nclass=3) table(lc1$predclass,simdat$trueclass) # Create a sample dataset with 2 classes and three covariates. # Then compare predicted class memberships when the model is # estimated "correctly" with covariates to when it is estimated # "incorrectly" without covariates. simdat2 <- poLCA.simdata(N=1000,ndv=7,niv=3,nclass=2,b=matrix(c(1,-2,1,-1))) f2a <- cbind(Y1,Y2,Y3,Y4,Y5,Y6,Y7)~X1+X2+X3 lc2a <- poLCA(f2a,simdat2$dat,nclass=2) f2b <- cbind(Y1,Y2,Y3,Y4,Y5,Y6,Y7)~1 lc2b <- poLCA(f2b,simdat2$dat,nclass=2) table(lc2a$predclass,lc2b$predclass)
Calculates predicted cell frequencies based on an estimated latent class model.
poLCA.table(formula, condition, lc)poLCA.table(formula, condition, lc)
formula |
A formula expression of the form |
condition |
A list containing the values of the manifest variables to hold fixed when creating the table specified by the |
lc |
A model object previously estimated using the |
This function outputs predicted cell counts for user-specified combinations of the manifest variables, based on a latent class model estimated by the poLCA function. The predcell table outputted automatically by poLCA also contains predicted cell frequencies, but only for cells containing at least one observation. In contrast, poLCA.table will calculate predicted cell counts for all cells, including those with zero observations.
A vector or table containing the specified frequency distribution.
data(gss82) f <- cbind(PURPOSE,ACCURACY,UNDERSTA,COOPERAT)~1 gss.lc2 <- poLCA(f,gss82,nclass=2) gss.lc2$predcell poLCA.table(formula=COOPERAT~1,condition=list(PURPOSE=3,ACCURACY=1,UNDERSTA=2),lc=gss.lc2) poLCA.table(formula=COOPERAT~UNDERSTA,condition=list(PURPOSE=3,ACCURACY=1),lc=gss.lc2) poLCA.table(formula=COOPERAT~UNDERSTA,condition=list(),lc=gss.lc2)data(gss82) f <- cbind(PURPOSE,ACCURACY,UNDERSTA,COOPERAT)~1 gss.lc2 <- poLCA(f,gss82,nclass=2) gss.lc2$predcell poLCA.table(formula=COOPERAT~1,condition=list(PURPOSE=3,ACCURACY=1,UNDERSTA=2),lc=gss.lc2) poLCA.table(formula=COOPERAT~UNDERSTA,condition=list(PURPOSE=3,ACCURACY=1),lc=gss.lc2) poLCA.table(formula=COOPERAT~UNDERSTA,condition=list(),lc=gss.lc2)
One random draw from a multinomial distribution or list of multinomial distributions.
rmulti(p)rmulti(p)
p |
matrix of dimension |
Returns a vector of length n. Each item represents one draw from the multinomial distribution parameterized by the outcome probabilities in each row of p.
Each row of matrix p must sum to 1 or rmulti will not work properly.
## ## One draw from a three-category multinomial distribution. ## p1 <- c(0.7,0.2,0.1) rmulti(p1) ## ## 10,000 draws from a three-category multinomial distribution. ## n <- 10000 p2 <- matrix(p1,nrow=n,ncol=length(p1),byrow=TRUE) rmdraws <- rmulti(p2) table(rmdraws)/n # should be approximately 0.7, 0.2, 0.1 ## ## 10,000 draws from a mixture of three groups of a ## four-category multinomial distribution. ## group.p <- matrix(c(0.5,0.3,0.2),nrow=n,ncol=3,byrow=TRUE) group <- rmulti(group.p) p3 <- t(matrix(NA,nrow=n,ncol=4)) p3[,group==1] <- c(0.7,0.1,0.1,0.1) p3[,group==2] <- c(0.1,0.7,0.1,0.1) p3[,group==3] <- c(0.1,0.1,0.1,0.7) p3 <- t(p3) rmdraws3 <- rmulti(p3) table(group,rmdraws3) table(group,rmdraws3)/rowSums(table(group,rmdraws3))## ## One draw from a three-category multinomial distribution. ## p1 <- c(0.7,0.2,0.1) rmulti(p1) ## ## 10,000 draws from a three-category multinomial distribution. ## n <- 10000 p2 <- matrix(p1,nrow=n,ncol=length(p1),byrow=TRUE) rmdraws <- rmulti(p2) table(rmdraws)/n # should be approximately 0.7, 0.2, 0.1 ## ## 10,000 draws from a mixture of three groups of a ## four-category multinomial distribution. ## group.p <- matrix(c(0.5,0.3,0.2),nrow=n,ncol=3,byrow=TRUE) group <- rmulti(group.p) p3 <- t(matrix(NA,nrow=n,ncol=4)) p3[,group==1] <- c(0.7,0.1,0.1,0.1) p3[,group==2] <- c(0.1,0.7,0.1,0.1) p3[,group==3] <- c(0.1,0.1,0.1,0.7) p3 <- t(p3) rmdraws3 <- rmulti(p3) table(group,rmdraws3) table(group,rmdraws3)/rowSums(table(group,rmdraws3))
Dichotomous survey responses from 216 respondents to four questions (A, B, C, D) measuring tendencies towards "universalistic" or "particularistic" values. This data set appears in Goodman (2002, p. 14) as Table 4, and previously appeared in Goodman (1974) and Stouffer and Toby (1951).
data(values)data(values)
A data frame with 216 observations on 4 variables representing survey responses to dichotomous questions, with 1 denoting the "particularistic" values response and 2 denoting the "universalistic" values response.
Stouffer, S.A. and J. Toby. 1951. "Role conflict and personality." American Journal of Sociology. 56: 395:406.
Goodman, Leo A. 1974. "Exploratory Latent-Structure Analysis Using Both Identifiable and Unidentifiable Models." Biometrika. 61(2): 215-231.
Goodman, Leo A. 2002. "Latent Class Analysis; The Empirical Study of Latent Types, Latent Variables, and Latent Structures." in Jacques A. Hagenaars and Allan L. McCutcheon, eds. Applied Latent Class Analysis. Cambridge: Cambridge University Press.
## ## Replication of latent class models in Goodman (2002), ## Tables 5b, 5c, and 6. ## data(values) f <- cbind(A,B,C,D)~1 M0 <- poLCA(f,values,nclass=1) # log-likelihood: -543.6498 M1 <- poLCA(f,values,nclass=2) # log-likelihood: -504.4677 M2 <- poLCA(f,values,nclass=3,maxiter=8000) # log-likelihood: -503.3011## ## Replication of latent class models in Goodman (2002), ## Tables 5b, 5c, and 6. ## data(values) f <- cbind(A,B,C,D)~1 M0 <- poLCA(f,values,nclass=1) # log-likelihood: -543.6498 M1 <- poLCA(f,values,nclass=2) # log-likelihood: -504.4677 M2 <- poLCA(f,values,nclass=3,maxiter=8000) # log-likelihood: -503.3011