selects a latent class model for pedigree data

Description

Performs selection of a latent class model for phenotypic measurements in pedigrees based on one of two possible methods: likelihood-based cross-validation or Bayesian Information Criterion (BIC) selection. This is the top-level function to perform a Latent Class Analysis (LCA), which calls the model fitting function lca.model. Model selection is performed among models within one of two types: with and without familial dependence. Two families of distributions are currently implemented: product multinomial for discrete (or ordinal) data and mutivariate normal for continuous data.

Usage

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model.select(ped, distribution, trans.const = TRUE, optim.param,
             optim.probs.indic = c(TRUE, TRUE, TRUE, TRUE), 
             famdep = TRUE, selec = "bic", H = 5, K.vec = 1:7, 
             tol = 0.001, x = NULL, var.list = NULL)

Arguments

ped

a matrix containing variables coding the pedigree structure and the phenotype measurements: ped[,1] family ID, ped[,2] subjects ID, ped[,3] dad ID, ped[,4] mom ID, ped[,5] sex, ped[,6] symptom status (2: symptomatic, 1: without symptoms, 0: missing), ped[,7:ncol(ped)] measurements, each column corresponds to a phenotypic measurement,

distribution

a character variable taking the value "normal" for multivariate normal measurements and "multinomial" for ordinal or discrete multinomial measurements,

trans.const

a logical variable indicating if the parental constraint is used. Parental constraint means that the class of a subject must be one of his parents classes. Default is TRUE,

optim.param

a variable indicating how the measurement distribution parameter optimization is performed (see below for more details),

optim.probs.indic

a vector of logical values indicating which probability parameters to estimate (see below for more details),

famdep

a logical variable indicating if the familial dependence model is used or not. Default is TRUE. In models without familial dependence, individuals are treated as independent and pedigree structure is meaningless. In models with familial dependence, a child class depends in his parents classes via a triplet transition probability,

selec

a character variables taking the value bic if BIC selection is used and the value cross if cross-validation is used,

H

an integer giving the number of equal parts into which data will be splitted for the likelihood-based cross-validation model selection (see below for more details),

K.vec

a vector of integers, the number of latent classes of candidate models, if K.vec has one value, only models with that number of classes will be fitted,

tol

a small number governing the stopping rule of the EM algorithm. Default is 0.001,

x

a matrix of covariates (optional), default is NULL,

var.list

a list of integers indicating the columns of x containing the covariates to use for a given phenotypic measurement, default is NULL.

Details

In the case of cross-validation based-likelihood method, data is splitted into H parts: H-1 parts as a training set and one part as a test set. For each model, a validation log-likelihood is obtained by evaluating the log-likelihood of the test set data using the parameter values estimated in the training set. This is repeated H times using a different part as training set each time, and a total validation log-likelihood is obtained by summation over the H test sets. The best model is the one having the largest validation log-likelihood. In the case of BIC selection method, the BIC is computed for each candidate model. The model with the smallest BIC is selected.

The symptom status vector (column 6 of ped) takes value 1 for subjects that have been examined and show no symptoms (i.e. completely unaffected subjects). When applying the LCA to measurements available on all subjects, the status vector must take the value of 2 for every individual with measurements. If covariates are used, covariate values must be provided for subjects with symptom status 0 (missing) but not for subjects with symptom status 1 (if covariate values are provided, they will be ignored).

optim.param is a variable indicating how the measurement distribution parameter optimization of the M step is performed. Two possibilities, optim.noconst.ordi and optim.const.ordi, are now available in the case of discrete or ordinal measurements, and four possibilities, optim.indep.norm (measurements are independent, diagonal variance-covariance matrix), optim.diff.norm (general variance-covariance matrix but equal for all classes), optim.equal.norm (variance-covariance matrices are different for each class but equal variance and equal covariance for a class) and optim.gene.norm (general variance-covariance matrices for all classes), in the case of continuous measurements. One of the allowed values of optim.param must be entered without quotes.

optim.probs.indic is a vector of logical values of length 4 for models with familial dependence and 2 for models without familial dependence indicating which probability parameters to estimate. See the help page for lca.model for a definition of the parameters.

For models with familial dependence:

optim.probs.indic[1]

indicates whether p0 will be estimated or not,

optim.probs.indic[2]

indicates whether p0connect will be estimated or not,

optim.probs.indic[3]

indicates whether p.found will be estimated or not,

optim.probs.indic[4]

indicates whether p.connect will be estimated or not.

For models without familial dependence:

optim.probs.indic[1]

indicates whether p0 will be estimated or not,

optim.probs.indic[2]

indicates whether p.aff will be estimated or not.

All defaults are TRUE.

Value

The function returns a list of 5 elements, the first 3 elements are common for BIC and cross-validation model selection methods and are:

param

the Maximum Likelihood Estimator (MLE) of the measurement distribution parameters of the selected model,

probs

the Maximum Likelihood Estimator (MLE) of the probability parameters of the selected model,

weight

an array of dimension n (the number of individuals) times 2 times K+1 (K being the number of latent classes in the selected model and the K+1th class being the unaffected class) giving the individual posterior probabilities. weight[i,s,c] is the posterior probability that individual i belongs to class c when his affection status is s, where s takes two values: 1 for symptomatic and 2 for without symptom. In particular, all weight[,2,] are 0 for symptomatic individuals and all weight[,1,] are 0 for individuals without symptoms. For missing individuals (unkown symptom status), both weight[,1,] and weight[,2,] may be greater than 0.

If the cross-validation selection method is used, the function returns also

ll

the value of the maximum log-likelihood (log-ML) of the selected model,

ll.valid

the total cross-validation log-likelihood of all candidate models,

and if the Bayesian Information Criterion selection method is used, the function returns also

ll

the value of maximum log-likelihood (log-ML) of all candidate models,

bic

the Bayesian Information Criterion BIC=-2*log(ll)+m*log(n) of all candidate models, where m is the number of free parameters of the model and n the total number of individuals.

References

TAYEB, A. LABBE, A., BUREAU, A. and MERETTE, C. (2011) Solving Genetic Heterogeneity in Extended Families by Identifying Sub-types of Complex Diseases. Computational Statistics, 26(3): 539-560. DOI: 10.1007/s00180-010-0224-2,

LABBE, A., BUREAU, A. et MERETTE, C. (2009) Integration of Genetic Familial Dependence Structure in Latent Class Models. The International Journal of Biostatistics, 5(1): Article 6.

See Also

See also lca.model.

Examples

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#data
data(ped.cont)
fam <- ped.cont[,1]
#the function applied for the two first families of ped.cont
model.select(ped.cont[fam%in%1:2,],distribution="normal",trans.const=TRUE,
             optim.indep.norm,optim.probs.indic=c(TRUE,TRUE,TRUE,TRUE),
             famdep=TRUE,selec="bic",K.vec=1:3,tol=0.001,x=NULL,var.list=NULL)

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