# class2z: Pairwise log OR regression design generator In alr: alternating logistic regression

## Description

Regression estimation for clustered binary outcomes can be facilitated by regression analysis of pairwise intracluster log odds ratios; such regressions form the “extension” to extended GEE methods GEE2 (Liang, Zeger, Qaqish, 1992) and ALR (Carey, Zeger, Diggle, 1993). Specification of pairwise log odds ratio regression designs can be cumbersome; this program assists in this task. There are assumed to be N observations in all, falling into M = length(unique(id)) clusters; each observation is a member of one of k classes.

## Usage

 ```1 2``` ```class2z(cvec, id, k, dmat) ~move the above line to just above the first optional argument ```

## Arguments

 `cvec` class vector: an N vector with values in 1:k; cvec[i] is the class in which the ith outcome lies `id` cluster vector: an N vector with M distinct values `k` the number of classes `dmat` design generator. Let q denote the number of terms on the right hand side of the pairwise log OR regression equation, inclusive of intercept. dim(dmat) = c(n*(n+1)/2,q). The rows of dmat are in lexicographic order corresponding to non-redundant enumeration of pairs: (1,1), (1,2), ... , (1,k), (2,2), ..., (2,k), ... ,(k,k), and the rows of this matrix describe how the q regression parameters are used to describe association for pairs of elements. Specifically, let t denote the row of dmat corresponding to the class-pair (r,s), and let Y1 be the outcome from a member of class r and Y2 be the outcome from a member of class s. Then log odds ratio(Y1,Y2)=a1*dmat[t,1]+...+ aq*dmat[t,q]. See example below.

## Details

A simple example: a family of size 4 with 2 parents and 2 children; cvec = c(1,1,2,2), k=2, dmat = rbind(c(1,0),c(1,1),c(1,1)). The dmat expresses the idea that first-order relations (parent-child, sibling-sibling) have equivalent pairwise associations in the outcome, distinct from the spouse-spouse association. The spouse-spouse log OR is then alpha[1], whilst the first-order relative pairwise log OR is alpha[1]+alpha[2].

## Value

a self-describing list with components z, zid, dmat, and flag.

## References

Heagerty, P, Ph.D. diss, Dept Biostat, Johns Hopkins University School of Public Health.

Heagerty P, Zeger S, "Marginal regression models for clustered ordinal measurements", JASA Sept 1996, v91, n435, 1024–1036.

Carey V, Zeger S, Diggle P, "Modelling multivariate binary data with alternating logistic regressions", Biometrika 1993 p517.

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37``` ```# 12 observations in clusters of size 3, 5, 4 id <- c(rep(2,3),rep(4,5),rep(6,4)) # each observation is in one of two classes k <- 2 cvec <- c(1,1,2,1,1,2,2,2,1,1,2,2) # alpha1 is LPWOR for outcomes in the same cluster # when both are in class 1 # alpha1+alpha2 = LPWOR for pairs with # one element from class 1 and one from class 2 # alpha1+alpha3 = LPWOR for pairs with both elements # from class 2 dmat <- rbind(c(1,0,0),c(1,1,0),c(1,0,1)) out <- class2z(cvec,id,k,dmat) # Data: # id 2 2 2 4 4 4 4 4 6 6 6 6 #class 1 1 2 1 1 2 2 2 1 1 2 2 out\$z[1:6,] # [,1] [,2] [,3] #[1,] 1 0 0 #[2,] 1 1 0 #[3,] 1 0 0 #[4,] 1 1 0 #[5,] 1 1 0 #[6,] 1 1 0 out\$zid[1:6] # 21 22 23 24 25 26 # 2 2 2 2 2 2 out\$dmat # a1 a2 a3 #(1,1) 1 0 0 #(1,2) 1 1 0 #(2,2) 1 0 1 # ```