cv.pgee {pgee.mixed}

Cross validation for Penalized Generalized Estimating Equations

Description

Performs k-fold cross-validation for Penalized Generalized Estimating Equations (PGEEs) over grid(s) of tuning parameters lambda. Linear and binary logistic models are supported. In particular, can handle the case of bivariate correlated mixed outcomes, in which each cluster consists of one continuous outcome and one binary outcome.

Usage

cv.pgee(N, m, X, Z = NULL, y = NULL, yc = NULL, yb = NULL, K = 5,
  grid1, grid2 = NULL, wctype = "Ind", family = "Gaussian", eps = 1e-06,
  maxiter = 1000, tol.coef = 0.001, tol.score = 0.001, init = NULL,
  standardize = TRUE, penalty = "SCAD", warm = TRUE, weights = rep(1,
  N), type_c = "square", type_b = "deviance", marginal = 0, FDR = FALSE,
  fdr.corr = NULL, fdr.type = "all")

Arguments

N

Number of clusters.

m

Cluster size. Assumed equal across all clusters. Should be set to 2 for family=="Mixed".

X

Design matrix. If family=="Mixed", then design matrix for continuous responses. For family!="Mixed", should have N*m rows. For family=="Mixed", should have N rows.

Z

Design matrix for binary responses for family=="Mixed". Should not be provided for other family types. If not provided for family=="Mixed", is set equal to X. For family!="Mixed", should have N*m rows. For family=="Mixed", should have N rows.

y

Response vector. Don't use this argument for family == "Mixed". Instead, use arguments yc and yb. Since the cluster size is assumed equal across clusters, the vector is assumed to have the form c(y_1, y_2,...,y_N), with y_i = c(y_i1,...,y_im).

yc

Continuous response vector. Use only for family=="Mixed".

yb

Binary (0/1) response vector. Use only for family=="Mixed".

K

Number of folds.

grid1

For family!="Mixed", the grid of tuning parameters. For family=="Mixed", the grid of tuning parameters for coefficients corresponding to the continuous outcomes.

grid2

For family=="Mixed", the grid of tuning parameters for coefficients corresponding to the binary outcomes. Not used for family!="Mixed".

wctype

Working correlation type; one of "Ind", "CS", or "AR1". For family=="Mixed", "CS" and "AR1" are equivalent.

family

"Gaussian", "Binomial", or "Mixed" (use the last for bivariate mixed outcomes). Note that for "Binomial", currently only binary outcomes are supported.

eps

Disturbance in the Linear Quadratic Approximation algorithm.

maxiter

The maximum number of iterations the Newton algorithm tries before declaring failure to converge.

tol.coef

Converge of the Newton algorithm is declared if two conditions are met: The L1-norm of the difference of successive iterates should be less than tol.coef AND the L1-norm of the penalized score should be less than tol.score.

tol.score

See tol.coef.

init

Vector of initial values for regression coefficients. For family=="Mixed", should be c(init_c, init_b). Defaults to glm values.

standardize

Standardize the design matrices prior to estimation?

penalty

"SCAD", "MCP", or "LASSO".

warm

Use warm starts?

weights

Vector of cluster weights. All observations in a cluster are assumed to have the same weight.

type_c

Loss function for continuous outcomes. "square" (square error loss, the default) or "absolute" (absolute error loss).

type_b

Loss function for binary outcomes. "deviance" (binomial deviance, the default) or "classification" (prediction error).

marginal

For the mixed outcomes case, set to 0 (the default) to account for both the continuous loss and the binary loss, set to 1 to only account for the continuous loss, and set to 2 to only account for the binary loss.

FDR

Should the false discovery rate be estimated for family=="Mixed"? Currently, FDR cannot be estimated for other family types.

fdr.corr

Association parameter to use in FDR estimation. The default is to use the association parameter estimated from the PGEEs.

fdr.type

Estimate the FDR for only the coefficients corresponding to the continuous outcomes ("continuous"), for only the coefficients corresponding to the binary outcomes ("binary"), or for all coefficients ("all", the default).

Details

The function calls pgee.fit K times, each time leaving out 1/K of the data. The cross-validation error is determined by the arguments type_c and type_b. For family=="Mixed", the cross-validation error is (by default) the sum of the continuous error and the binary error.

Value

A list

coefficients

Vector of estimated regression coefficients. For family=="Mixed", this takes the form c(coef_c, coef_b).

vcov

Sandwich formula based covariance matrix of estimated regression coefficients (other than the intercept(s)). The row/column names correspond to elements of coefficients.

phi

Estimated dispersion parameter.

alpha

Estimated association parameter.

num_iterations

Number of iterations the Newton algorithm ran.

converge

0=converged, 1=did not converge.

PenScore

Vector of penalized score functions at the estimated regression coefficients. If the algorithm converges, then these should be close to zero.

FDR

Estimated FDR for family=="Mixed", if requested.

lambda.loss

Cross validation loss (error) for the optimal tuning parameter(s) lambda, averaged across folds.

LossMat

Matrix of cross validation losses. Rows denote tuning parameter values, columns denote folds.

Examples

## Not run: 
# Gaussian
N <- 100
m <- 10
p <- 50
y <- rnorm(N * m)
# If you want standardize = TRUE, you must provide an intercept.
X <- cbind(1, matrix(rnorm(N * m * (p - 1)), N * m, p - 1))
gr1 <- seq(0.001, 0.1, length.out = 100)
fit <- cv.pgee(X = X, y = y, N = N, m = m, grid1 = gr1, wctype = "CS",
            family = "Gaussian")

# Binary
y <- sample(0:1, N*m, replace = TRUE)
fit <- cv.pgee(X = X, y = y, N = N, m = m, grid1 = gr1, wctype = "CS",
            family = "Binomial")

# Bivariate mixed outcomes
# Generate some data
Bc <- c(2.0, 3.0, 1.5, 2.0, rep(0,times=p-4))
Bb <- c(0.7, -0.7, -0.4, rep(0,times=p-3))
dat <- gen_mixed_data(Bc, Bc, N, 0.5)
# We require two grids of tuning parameters
gr2 <- seq(0.0001, 0.01, length.out = 100)
# Estimate regression coefficients and false discovery rate
fit <- cv.pgee(X = dat$X, Z = dat$Z, yc = dat$yc, yb = dat$yb, N = N, m = 2,
               wctype = "CS", family = "Mixed", grid1 = gr1, grid2 = gr2,
               FDR = TRUE)

## End(Not run)