fusionmixed {FusionLearn}

Fusion learning algorithm for mixed data

Description

fusionmixed conducts the group penalization with a specified penalty value learning from multiple generalized linear models with mixed continuous and binary responses. fusionmixed.fit can be used to search the best candidate model based on the pseudo Bayesian information criterion with a sequence of penalty values.

Usage

fusionmixed(x, y, lambda, N, p, m1, m2, beta=0.1, thresh=0.1, 
            maxiter=100, methods="scad", link="logit", Complete=TRUE)

fusionmixed.fit(x, y, lambda, N, p, m1, m2, beta=0.1, thresh=0.1, 
                maxiter=100, methods="scad",link="logit", Complete=TRUE, 
                depen ="IND", a=1)

Arguments

x

List. Listing matrices of the predictors from different platforms. The first m1 data sets in the list are the ones of continuous responses, and the following m2 data sets are the ones of binary responses.

y

List. A list of the responses vectors from different platforms following the same order as in x. The values m1 and m2 must be specified.

lambda

Numeric or vector. For fusionmixed, lambda is a numeric value for the penalty; for fusionmixed.fit, lambda is a vector with a list of penalty values.

N

Numeric or vector. If only one numeric value is provided, equal sample size will be assumed for each data set. If a vector is provided, then the elements are the sample sizes for all the platforms.

p

Numeric. The number of predictors.

m1

Numeric. Number of platforms whose response variables are continuous.

m2

Numeric. Number of platforms whose response variables are binary.

beta

Numeric. An initial value for the estimated parameters with dimensions nvars x nplatforms. The default value is 0.1.

thresh

Numeric. The stopping criteria. The default value is 0.1.

maxiter

Numeric. Maximum number of iterations. The default value is 100.

methods

Character ("lass" or "scad"). lass: LASSO; scad: SCAD.

link

Character ("logit" or "probit"). Link functions: logistic or probit.

Complete

Logic input. If Complete == TRUE, the predictors M_1,...,M_p are measured in all platforms. If Compelte == FALSE, in some platforms, not all of the predictors \{M_1,M_2,...,M_p\} are measured. The values of the corresponding estimated coefficients for the missing predictors will be NA.

depen

Character. Input only for function fusionmixed.fit. "IND" means the observations across different platforms are independent; "CORR" means the observations are correlated, and the sample sizes should be equal for different platforms.

a

Numeric. Input only for function fusionmixed.fit. The free multiplicative constant used in \gamma_n. The default value is 1.

Details

fusionmixed is designed for a more complex data structure by aggregating information from continuous and binary responses. More details regarding the algorithm can be found in FusionLearn.

Value

fusionmixed returns a list that has components:

beta

A matrix (nvars x nplatforms) containing estimated coefficients of each linear model. If some data sets do not have the complete set of predictors, the corresponding coefficients are output as NA.

method

Penalty function LASSO or SCAD.

link

The link function used in the estimation.

threshold

The numeric value shows the difference in the estimates between the successive updates upon convergence.

iteration

The numeric value shows the number of iterations upon convergence.

fusionmixed.fit provides the results in a table:

lambda

The sequence of penalty values.

BIC

The pseudolikelihood Bayesian information criterion evaluated at the sequence of the penalty values.

-2Loglkh

Minus twice the pseudo loglikelihood of the chosen model.

Est_df

The estimated degrees of freedom quantifying the model complexity.

fusionmixed.fit also returns a model selection plot showing the results above.

Note

The range of the penalty values should be carefully chosen. For some penalty values, the resulting models may have singular information matrix or the fitting of the glm cannot converge.

Author(s)

Xin Gao, Yuan Zhong, and Raymond J. Carroll

References

Gao, X and Carroll, R. J. (2017) Data integration with high dimensionality. Biometrika, 104, 2, pp. 251-272

See Also

fusionbase,fusionbinary,

Examples

##Analysis of the index data

#Responses contain indices "VIX","GSPC", and "DJI",  
#"DJI" is dichotomized into "increasing" or "decreasing"
y <- list(stockindexVIX[,1],stockindexGSPC[,1],stockindexDJI[,1]>0)

#Predictors include 46 stocks
x <- list(stockindexVIX[,2:47],stockindexGSPC[,2:47],stockindexDJI[,2:47])  
##Implementing the model selection based on psuedolikelihood 
##information criteria
model <- fusionmixed.fit(x,y,seq(0.03,5,length.out = 10),232,46,2,1,depen="CORR")
lambda <- model[which.min(model[,2]),1]  
result <- fusionmixed(x,y,lambda,232,46,2,1)

##Identify the significant predictors for three indices
id <- which(result$beta[,1]!=0)+1
colnames(stockindexVIX)[id]