Package 'squeezy'

Group-Adaptive Elastic Net Penalised Generalised Linear Models

Description

Fit linear and logistic regression models penalised with group-adaptive elastic net penalties. The group penalties correspond to groups of covariates defined by a co-data group set. The method accommodates inclusion of unpenalised covariates and overlapping groups. See Van Nee et al. (2021) <arXiv:2101.03875>.

Details

See squeezy for example code.

Author(s)

Mirrelijn M. van Nee [aut, cre], Tim van de Brug [aut], Mark A. van de Wiel [aut]

Maintainer: Mirrelijn M. van Nee <[email protected]>

References

Mirrelijn M. van Nee, Tim van de Brug, Mark A. van de Wiel, "Fast marginal likelihood estimation of penalties for group-adaptive elastic net.", 2021

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Partial derivatives of -log(ML) of ridge penalised GLMs

Description

Returns the partial derivatives (w.r.t. 'loglambdas') of the minus log Laplace approximation (LA) of the marginal likelihood of ridge penalised generalised linear models. Note: currently only implemented for linear and logistic regression.

Usage

dminML.LA.ridgeGLM(loglambdas, XXblocks, Y, sigmasq = 1, 
                  Xunpen = NULL, intrcpt = TRUE, model, minlam = 0, 
                  opt.sigma = FALSE)

Arguments

loglambdas	Logarithm of the ridge penalties as returned by ecpc or squeezy; Gx1 vector.
XXblocks	List of sample covariance matrices X_g %*% t(X_g) for groups g = 1,..., G.
Y	Response data; n-dimensional vector (n: number of samples) for linear and logistic outcomes.
sigmasq	(linear model only) Noise level (Y~N(X*beta,sd=sqrt(sigmasq))).
Xunpen	Unpenalised variables; nxp_1-dimensional matrix for p_1 unpenalised variables.
intrcpt	Should an intercept be included? Set to TRUE by default.
model	Type of model for the response; linear or logistic.
minlam	Minimum value of lambda that is added to exp(loglambdas); set to 0 as default.
opt.sigma	(linear model only) TRUE/FALSE if log(sigmasq) is given as first argument of loglambdas for optimisation purposes

Value

Partial derivatives of the Laplace approximation of the minus log marginal likelihood to the model parameters 'loglambdas';

For opt.sigma=FALSE: Gx1-dimensional vector for the G log(group ridge penalties).

For opt.sigma=TRUE (linear model only): (G+1)x1-dimensional vector for the partial derivative to log(sigmasq) (first element) and for the G log(group ridge penalties).

Examples

#Simulate toy data
n<-100
p<-300
X <- matrix(rnorm(n*p),n,p)
Y <- rnorm(n)
groupset <- list(1:(p/2),(p/2+1):p)
sigmahat <- 2
alpha <- 0.5
tauMR <- c(0.01,0.005)

XXblocks <- lapply(groupset, function(x)X[,x]%*%t(X[,x]))

#compute partial derivatives of the minus log marginal likelihood to the penalties only
dminML.LA.ridgeGLM(loglambdas = log(sigmahat/tauMR), 
                  XXblocks, Y, sigmasq = sigmahat, 
                  model="linear",opt.sigma=FALSE)

#additionally, compute the partial derivative to the linear regression noise parameter sigma^2
dminML.LA.ridgeGLM(loglambdas = log(c(sigmahat,sigmahat/tauMR)), 
                   XXblocks, Y, sigmasq = sigmahat, 
                   model="linear",opt.sigma=TRUE)

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Marginal AIC of a multi-group, ridge penalised GLM

Description

Compute the marginal AIC for the marginal likelihood (ML) of multi-group, ridge penalised generalised linear models. Note: currently only implemented for linear and logistic regression.

Usage

mAIC.LA.ridgeGLM(loglambdas, XXblocks, Y, sigmasq = 1, 
                Xunpen = NULL, intrcpt = TRUE, model, minlam = 0)

Arguments

loglambdas	Logarithm of the ridge penalties as returned by ecpc or squeezy; Gx1 vector.
XXblocks	List of sample covariance matrices X_g %*% t(X_g) for groups g = 1,..., G.
Y	Response data; n-dimensional vector (n: number of samples) for linear and logistic outcomes.
sigmasq	(linear model only) Noise level (Y~N(X*beta,sd=sqrt(sigmasq))).
Xunpen	Unpenalised variables; nxp_1-dimensional matrix for p_1 unpenalised variables.
intrcpt	Should an intercept be included? Set to TRUE by default.
model	Type of model for the response; linear or logistic.
minlam	Minimum value of lambda that is added to exp(loglambdas); set to 0 as default.

Value

mAIC	mAIC of the model

Examples

#Simulate toy data
n<-100
p<-300
X <- matrix(rnorm(n*p),n,p)
Y <- rnorm(n)
groupset <- list(1:(p/2),(p/2+1):p)
sigmahat <- 2
alpha <- 0.5
tauMR <- c(0.01,0.005)

XXblocks <- lapply(groupset, function(x)X[,x]%*%t(X[,x]))

#compute the mAIC of a co-data model with multiple groups
mAIC.LA.ridgeGLM(loglambdas=log(sigmahat/tauMR), XXblocks=XXblocks, 
                 Y = Y, sigmasq = sigmahat, model="linear")

#compute the mAIC of a co-data agnostic model, i.e. only one group of covariates
mAIC.LA.ridgeGLM(loglambdas=log(sigmahat/median(tauMR)), 
                 XXblocks=list(X%*%t(X)), 
                 Y = Y, sigmasq = sigmahat, model="linear")

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-log(ML) of ridge penalised GLMs

Description

Returns the Laplace approximation (LA) of the minus log marginal likelihood of ridge penalised generalised linear models. Note: currently only implemented for linear and logistic regression.

Usage

minML.LA.ridgeGLM(loglambdas, XXblocks, Y, sigmasq = 1, 
                  Xunpen = NULL, intrcpt = TRUE, model, minlam = 0, 
                  opt.sigma = FALSE)

Arguments

loglambdas	Logarithm of the ridge penalties as returned by ecpc or squeezy; Gx1 vector.
XXblocks	List of sample covariance matrices X_g %*% t(X_g) for groups g = 1,...,G.
Y	Response data; n-dimensional vector (n: number of samples) for linear and logistic outcomes.
sigmasq	(linear model only) Noise level (Y~N(X*beta,sd=sqrt(sigmasq))).
Xunpen	Unpenalised variables; nxp_1-dimensional matrix for p_1 unpenalised variables.
intrcpt	Should an intercept be included? Set to TRUE by default.
model	Type of model for the response; linear or logistic.
minlam	Minimum value of lambda that is added to exp(loglambdas); set to 0 as default.
opt.sigma	(linear model only) TRUE/FALSE if log(sigmasq) is given as first argument of loglambdas for optimisation purposes

Value

Laplace approximation of the minus log marginal likelihood for the ridge penalised GLM with model parameters 'loglambdas' and 'sigmasq' (for linear regression).

Examples

#Simulate toy data
n<-100
p<-300
X <- matrix(rnorm(n*p),n,p)
Y <- rnorm(n)
groupset <- list(1:(p/2),(p/2+1):p)
sigmahat <- 2
alpha <- 0.5
tauMR <- c(0.01,0.005)

XXblocks <- lapply(groupset, function(x)X[,x]%*%t(X[,x]))

#compute minus log marginal likelihood
minML.LA.ridgeGLM(loglambdas = log(sigmahat/tauMR), 
                  XXblocks, Y, sigmasq = sigmahat, 
                  model="linear")

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Visual posterior check of multivariate normality of the linear predictors

Description

Produce a qq-plot to visually check whether the assumption of multivariate normality of the linear predictors is valid for the data and model fit with 'squeezy'.

Usage

normalityCheckQQ(X,groupset,fit.squeezy,nSim=500)

Arguments

X	Observed data; (nxp)-dimensional matrix (p: number of covariates) with each row the observed high-dimensional feature vector of a sample.
groupset	Co-data group set; list with G groups. Each group is a vector containing the indices of the covariates in that group.
fit.squeezy	Model fit obtained by the function squeezy.
nSim	Number of simulated vectors of linear predictors.

Value

The qqplot of the empirical versus theoretical quantiles is plotted. If ‘ggplot2’ is installed, the plot is returned as ‘ggplot’ object.

Examples

#Simulate toy data
n<-100
p<-300
X <- matrix(rnorm(n*p),n,p)
Y <- rnorm(n)
groupset <- list(1:(p/2),(p/2+1):p)
sigmahat <- 2
alpha <- 0.5
tauMR <- c(0.01,0.005)

#Fit group-regularised elastic net model with squeezy
fit.squeezy <- squeezy(Y,X,groupset,alpha=alpha,
                       lambdas=sigmahat/tauMR,sigmasq=sigmahat,
                       lambdaglobal=mean(sigmahat/tauMR))
                       
#Check qq-plot
normalityCheckQQ(X,groupset,fit.squeezy)

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Fit a group-adaptive elastic net penalised linear or logistic model

Description

Estimate group-specific elastic net penalties and fit a linear or logistic regression model.

Usage

squeezy(Y, X, groupset, alpha = 1, model = NULL, X2 = NULL, 
        Y2 = NULL, unpen = NULL, intrcpt = TRUE, 
        method = c("ecpcEN", "MML", "MML.noDeriv", "CV"), 
        fold = 10, compareMR = TRUE, selectAIC = FALSE, fit.ecpc = NULL, 
        lambdas = NULL, lambdaglobal = NULL, lambdasinit = NULL, 
        sigmasq = NULL, ecpcinit = TRUE, SANN = FALSE, minlam = 10^-3, 
        standardise_Y = NULL, reCV = NULL, opt.sigma = NULL, 
        resultsAICboth = FALSE, silent=FALSE)

Arguments

Y	Response data; n-dimensional vector (n: number of samples) for linear and logistic outcomes.
X	Observed data; (nxp)-dimensional matrix (p: number of covariates) with each row the observed high-dimensional feature vector of a sample.
groupset	Co-data group set; list with G groups. Each group is a vector containing the indices of the covariates in that group.
alpha	Elastic net penalty mixing parameter.
model	Type of model for the response; linear or logistic.
X2	(optional) Independent observed data for which response is predicted.
Y2	(optional) Independent response data to compare with predicted response.
unpen	Unpenalised covariates; vector with indices of covariates that should not be penalised.
intrcpt	Should an intercept be included? Included by default for linear and logistic, excluded for Cox for which the baseline hazard is estimated.
method	Which method should be used to estimate the group-specific penalties? Default MML.
fold	Number of folds used in inner cross-validation to estimate (initial) global ridge penalty lambda (if not given).
compareMR	TRUE/FALSE to fit the multi-ridge model and return results for comparison.
selectAIC	TRUE/FALSE to select the single-group model or multi-group model.
fit.ecpc	(optional) Model fit obtained by the function ecpc (from the ecpc R-package)
lambdas	(optional) Group-specific ridge penalty parameters. If given, these are transformed to elastic net penalties.
lambdaglobal	(optional) Global ridge penalty parameter used for initialising the optimisation.
lambdasinit	(optional) Group-specific ridge penalty parameters used for initialising the optimisation.
sigmasq	(linear model only) If given, noise level is fixed (Y~N(X*beta,sd=sqrt(sigmasq))).
ecpcinit	TRUE/FALSE for using group-specific ridge penalties as given in ‘fit.ecpc’ for initialising the optimisation.
SANN	('method'=MML.noDeriv only) TRUE/FALSE to use simulated annealing in optimisation of the ridge penalties.
minlam	Minimal value of group-specific ridge penalty used in the optimisation.
standardise_Y	TRUE/FALSE should Y be standardised?
reCV	TRUE/FALSE should the elastic net penalties be recalibrated by cross-validation of a global rescaling penalty?
opt.sigma	(linear model only) TRUE/FALSE to optimise sigmasq jointly with the ridge penalties.
resultsAICboth	(selectAIC=TRUE only) TRUE/FALSE to return results of both the single-group and multi-group model.
silent	Should output messages be suppressed (default FALSE)?

Value

betaApprox	Estimated regression coefficients of the group-adaptive elastic net model; p-dimensional vector.
a0Approx	Estimated intercept of the group-adaptive elastic net model; scalar.
lambdaApprox	Estimated group penalty parameters of the group-adaptive elastic net model; G-dimensional vector.
lambdapApprox	Estimated elastic net penalty parameter of the group-adaptive elastic net model for all covariates; p-dimensional vector.
tauMR	Estimated group variances of the multi-ridge model; G-dimensional vector.
lambdaMR	Estimated group penalties of the multi-ridge model; G-dimensional vector.
lambdaglobal	Estimated global ridge penalty; scalar. Note: only optimised if selectAIC=TRUE or compareMR=TRUE, else the returned crude estimate is sufficient for initialisation of squeezy.
sigmahat	(linear model) Estimated sigma^2; scalar.
MLinit	Min log marginal likelihood value at initial group penalties; scalar.
MLfinal	Min log marginal likelihood value at estimated group penalties; scalar.
alpha	Value used for the elastic net mixing parameter alpha; scalar.
glmnet.fit	Fit of the ‘glmnet’ function to obtain the regression coefficients.

If ‘compareMR’=TRUE, multi-ridge model is returned as well:

betaMR	Estimated regression coefficients of the multi-ridge model; p-dimensional vector.
a0MR	Estimated intercept of the multi-ridge model; scalar.

If independent test set ‘X2’ is given, predictions and MSE are returned:

YpredApprox	Predictions for the test set of the estimated group-adaptive elastic net model.
MSEApprox	Mean squared error on the test set of the estimated group-adaptive elastic net model.
YpredMR	Predictions for the test set of the estimated group-adaptive multi-ridge model.
MSEMR	Mean squared error on the test set of the estimated group-adaptive multi-ridge model.

If ‘selectAIC’=TRUE, the multi-group or single-group model with best AIC is selected. Results in ‘betaApprox’, ‘a0Approx’, ‘lambdaApprox’ contain those results of the best model. Summary results of both models are included as well:

AICmodels	List with elements “multigroup" and “onegroup".- Each element is a list with results of the multi-group or single-group model, containing the group penalties (‘lambdas’), sigma^2 (‘sigmahat’, linear model only), and AIC (‘AIC’). If besides ‘selectAIC’=TRUE, also ‘resultsAICboth’=TRUE, the fit of both the single-group model and multi-group model as obtained with squeezy are returned (‘fit’).
modelbestAIC	Either “onegroup" or “multigroup" for the selected model.

Author(s)

Mirrelijn M. van Nee, Tim van de Brug, Mark A. van de Wiel

References

Mirrelijn M. van Nee, Tim van de Brug, Mark A. van de Wiel, "Fast marginal likelihood estimation of penalties for group-adaptive elastic net", arXiv preprint, arXiv:2101.03875 (2021).

Examples

#####################
# Simulate toy data #
#####################
p<-100 #number of covariates
n<-50 #sample size training data set
n2<-100 #sample size test data set
G<- 5 #number of groups

taugrp <- rep(c(0.05,0.1,0.2,0.5,1),each=p/G) #ridge prior variance
groupIndex <- rep(1:G,each=p/G) #groups for co-data
groupset <- lapply(1:G,function(x){which(groupIndex==x)}) #group set with each element one group
sigmasq <- 2 #linear regression noise
lambda1 <- sqrt(taugrp/2) #corresponding lasso penalty
#A Laplace(0,b) variate can also be generated as the difference of two i.i.d.
#Exponential(1/b) random variables
betas <-   rexp(p, 1/lambda1) -  rexp(p, 1/lambda1) #regression coefficients
X <- matrix(rnorm(n*p),n,p) #simulate training data
Y <- rnorm(n,X%*%betas,sd=sqrt(sigmasq))
X2 <- matrix(rnorm(n*p),n,p) #simulate test data
Y2 <- rnorm(n,X2%*%betas,sd=sqrt(sigmasq))

###############
# Fit squeezy #
###############
#may be fit directly..
res.squeezy <- squeezy(Y,X,groupset=groupset,Y2=Y2,X2=X2,
                       model="linear",alpha=0.5)


  #..or with ecpc-fit as initialisation
  if(requireNamespace("ecpc")){
    res.ecpc <- ecpc::ecpc(Y,X, #observed data and response to train model
                     groupsets=list(groupset), #informative co-data group set
                     Y2=Y2,X2=X2, #test data
                     model="linear",
                     hypershrinkage="none",postselection = FALSE)
    res.squeezy <- squeezy(Y,X, #observed data and response to train model
                           groupset=groupset, #informative co-data group set
                           Y2=Y2,X2=X2, #test data
                           fit.ecpc = res.ecpc, #ecpc-fit for initial values
                           model="linear", #type of model for the response
                           alpha=0.5) #elastic net mixing parameter
  }



summary(res.squeezy$betaApprox) #estimated elastic net regression coefficients
summary(res.squeezy$betaMR) #estimated multi-ridge regression coefficients
res.squeezy$lambdaApprox #estimated group elastic net penalties
res.squeezy$tauMR #multi-ridge group variances
res.squeezy$MSEApprox #MSE group-elastic net model
res.squeezy$MSEMR #MSE group-ridge model

#once fit, quickly find model fit for different values of alpha:
res.squeezy2 <- squeezy(Y,X, #observed data and response to train model
                        groupset=groupset, #informative co-data groupset
                        Y2=Y2,X2=X2, #test data
                        lambdas = res.squeezy$lambdaMR, #fix lambdas at multi-ridge estimate
                        model="linear", #type of model for the response
                        alpha=0.9) #elastic net mixing parameter


  #Select single-group model or multi-group model based on best mAIC
  res.squeezy <- squeezy(Y,X, #observed data and response to train model
                         groupset=groupset, #informative co-data group set
                         Y2=Y2,X2=X2, #test data
                         fit.ecpc = res.ecpc, #ecpc-fit for initial values
                         model="linear", #type of model for the response
                         alpha=0.5, #elastic net mixing parameter
                         selectAIC = TRUE,resultsAICboth = TRUE)
  
  res.squeezy$modelbestAIC #selected model
  res.squeezy$AICmodels$multigroup$fit$MSEApprox #MSE on test set of multi-group model
  res.squeezy$AICmodels$onegroup$fit$MSEApprox #MSE on test set of single-group model

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