lm.interaction: Interaction linear adaptive index model
In AIM: AIM: adaptive index model

Description Usage Arguments Details Value Author(s) References Examples

Estimate adpative index model for continuous outcomes in the context of linear regression. The resulting index characterizes the interactions between the covariates and treatment.

1	lm.interaction(x, trt, y, nsteps=8, backfit=F, mincut=.1, maxnumcut=1, dirp=0)

`x`	n by p matrix. The covariate matrix
`trt`	n vector. The treatment indicator
`y`	n vector. The continuous response variable
`nsteps`	the maximum number of binary rules to be included in the index
`backfit`	T/F. Whether the existing split points are adjusted after including a new binary rule
`mincut`	The minimum cutting proportion for the binary rule at either end. It typically is between 0 and 0.2.
`maxnumcut`	The maximum number of binary splits per predictor
`dirp`	p vector. The given direction of the binary split for each of the p predictors. 0 represents "no pre-given direction"; 1 represents "(x>cut)"; -1 represents "(x<cut)". Alternatively, "dirp=0" represents that there is no pre-given direction for any of the predictor.

lm.interaction sequentially estimates a sequence of adaptive index models with up to "nsteps" terms for continuous outcomes. The aglorithm seeks the index having the strong interaction with the treatment in the linear model. The appropriate number of binary rules can be selected via K-fold cross-validation (cv.lm.interaction).

lm.interaction returns maxsc, which is the observed score test statistics for the index*treatment interaction in the fitted model and res, which is a list with components

`jmaa`	number of predictors
`cutp`	split points for the binary rules
`maxdir`	direction of split: 1 represents "(x>cut)" and -1 represents "(x<cut)"
`maxsc`	observed score test statistics for the interaction

Lu Tian and Robert Tibshirani

Lu Tian and Robert Tibshirani (2010) "Adaptive index models for marker-based risk stratification", Tech Report, available at http://www-stat.stanford.edu/~tibs/AIM.

## generate data
set.seed(1)

n=400
p=10
x=matrix(rnorm(n*p), n, p)
z=(x[,1]<0.2)+(x[,5]>0.2)
trt=rbinom(n, 1, 0.5)
beta=1
y=trt+beta*trt*z+rnorm(n)


## fit the interaction linear AIM 
a=lm.interaction(x, trt, y, nsteps=10)
 
## examine the model sequence 
print(a)

## compute the index based on the 2nd model of the sequence, using data x 
z.prd=index.prediction(a$res[[2]],x)

## compute the index based on the 2nd model of the sequence using new data xx, and compare the result with the true index
nn=10
xx=matrix(rnorm(nn*p), nn, p)
zz=(xx[,1]<0.2)+(xx[,5]>0.2)
zz.prd=index.prediction(a$res[[2]],xx) 
cbind(zz, zz.prd)