alr: Alternating Logistic Regression for multivariate binary data...

In alr: alternating logistic regression

Usage

alr(x, y, id, z=0, zmast=0, zid=0, zlocs=0, binit, ainit, tol=0.001,
       bweight="full", depmodel="general", subclust=0)

Arguments

x	matrix of predictors of logit probability of marginal outcome. should include a column of 1 for estimation of "baseline" probability if desired: Nxp, p includes intercept
y	vector of binary outcomes corresponding to rows of x: Nx1
id	cluster discriminator: N-vector
z	matrix of predictors for pairwise log odds ratio regression. this may be omitted for certain choices of "depmodel", see below. if used, this matrix may take one of two forms. it may specify directly, for each cluster, the form of the dependency design. in this case, it has dimension (sum_i(n_i*(n_i - 1))) x q. on the other hand, if the data to be analyzed are replicated (possibly subject to missingness), then if n is the size of a complete cluster, z has dimension (n*(n-1)) x q. see zmast and zlocs below
zmast	if 1, then z is a "master" design for pairwise log oddsratio regression. each cluster is a replicate of this master design, but clusters may have missing elements, thus missing pairs. the missing information is extracted from zlocs, below. if 0, then z has dimension (sum_i(n_i*(n_i-1))) x q
zid	cluster discriminators for the z matrix, used only if zmast == 0. if used, has dimension (sum_i(n_i *(n_i-1))) x 1.
zlocs	used only if zmast == 1. this is an N-vector of "locations" in the dependency design. if replication is perfect and complete, and n is the cluster size, and there are C clusters, then this vector should have the value rep(1:n,C).
binit	p-vector of initial values for beta – required; should be obtained from a logistic regression fit assuming independence
ainit	q-vector of initial values for alpha – difficult to recommend starting values here, but rep(.01,q) often works
tol	maximum relative change in parameter tolerated before asserting convergence
bweight	currently takes one of two values: "full" or "independence". if "independence", then the estimating equation for estimating regression parameters beta uses a diagonal weighting matrix, and the estimated beta should be identical to the GLIM estimate. if "full", then the estimating equation for beta uses the estimated covariance (nxn) matrix of the outcomes as weight. the "independence" setting may be useful for very large clusters (n>50?) with sparse outcomes, in which the weighting matrix can tend to singularity.
depmodel	currently takes one of three values: "general", "exchangeable", "1-nested". The "general" setting uses a Z matrix (see above) to specify the dependency model. The "exchangeable" setting uses no Z matrix, but assumes a common pairwise log-odds ratio for all cluster elements. The "1-nested" setting estimates a log-odds ratio regression with first parameter the log pairwise odds ratio for any two elements in a cluster, the second parameter is the incremental dependency between members of the same subcluster. subclusters are identified through the "subclust" vector; see below.
subclust	N-vector discriminating subclusters within clusters. Thus two 8-clusters might have id=c(1,1,1,1,1,1,1,2,2,2,2,2,2,2,2), subclust=c(10,10,11,11,11,11,11,11,20,20,20,20,21,21,21,21). the first cluster falls into 2 subclusters, the first is size 2, the second size 6; the second has 2 subclusters each of size 4. the 1-nested model says that the log-pairwise odds ratios between elements of the same cluster is a0+a1 if the two elements are ALSO of the same subcluster, and a0 if they are from different subclusters.

Value

a structure including estimates of beta, alpha, and the parameter covariance matrix

Note

The most complicated idea in using this program concerns the structure and modelling of the pairwise odds-ratio data. We deal with the "general" case first. If n (clustersize) = 4, then there are 6 = (n choose 2) possible pairwise odds ratios: O12, O13, O14, O23, O24, O34. However, to estimate these, we use the 12 (n*n-1) redundant pairings (Y1,Y2), (Y1,Y3), (Y1,Y4), (Y2,Y1), (Y2,Y3), ..., (Y4,Y3). It is in this order that we index the covariate vectors for regression analysis : Zij, ordered (i = 1:n, j = 1:n, j!=i ). The zloc argument tells uses the indices 1:n to tell which pairing element each outcome corresponds to. Thus suppose the design is a replication with n = 4. Complete data has zloc = c(1,2,3,4); if a cluster is missing, say, the outcome for the "second" subject, zloc = c(1,3,4), indicating that the cluster contributes to the estimation only of O13, O14, O34. Of course, the odds-ratio parameterization may be more parsimonious; if the model O12=O13=...=O34 is adopted, the specific values of zloc are not relevant to estimation of the common pairwise log odds ratio. But in the event that q>1, zloc tells which subjects are to be used for estimation of which combination of odds ratio parameters.

The simplest model to fit with alr is the "exchangeable" model, in that no information is needed other than x,y, and id.

The "1-nested" model requires a subclust vector in addition to x, y, and id.

Examples

data(alrset)
a1 <- alr(alr.y ~ alr.x - 1, id=alr.id, depm="exchangeable", ainit=0.01)
summary(a1)

#using a master z matrix for a balanced design
ZMAST <- rep(1,12)
ZMAST <- cbind(ZMAST,c(0,0,0,0,1,1,0,1,1,0,1,1))
ZIND <- rep(1:4,125)
Y <- as.vector(alr.y)
X <- as.vector(alr.x[,2])
NY <- split(alr.y,alr.id)
NY <- unlist(lapply(NY,function(x)rev(x)))
NX <- split(X,alr.id)
NX <- unlist(lapply(NX,function(x)rev(x)))
Y <- as.vector(NY)
X <- as.vector(NX)
mast.out <- alr(Y ~ X, id=alr.id, depmod="general", z=ZMAST,
zmast=1, zloc=ZIND,
               ainit = c(0.01,0.01) )
summary(mast.out)

alr documentation built on May 2, 2019, 5:21 p.m.