README.md

In zrmacc/LiST: Linear Score Tests

title: "README" author: "Zachary McCaw" date: "2018-09-21" output: html_document: keep_md: TRUE

Package Vignette

Contents

• Setting
• Example Data
• Score Tests

Setting

Consider the standard linear model:

$$ y = X\beta + \epsilon,\ \epsilon\sim(0,\sigma^2) $$

Partition the regression coefficient $\beta=(\beta_{1},\beta_{2})$. Suppose interest lies in testing whether $\beta_{1}$ is fixed at a reference value. This package implements score tests of $H_{0}:\beta_{1}=\beta_{1}^{\dagger}$.

Example Data

Below, data are simulated for $n=10^{3}$ subjects. The outcome $y_{i}\in\mathbb{R}$ is normally distributed with unit variance. The subject-specific means depend on an intercept and four covariates. The function Fit.LinReg provides an implementation of linear regression.

library(LiST);
# Generate data
set.seed(102);
# Observations
n = 1e3;
# Design
X = cbind(1,matrix(rnorm(4*n),nrow=n));
colnames(X) = c("int",paste0("x",seq(1:4)));
# Regression coefficient
b = c(1,-0.1,0.2,-0.1,0);
# Linear predictor
h = as.numeric(X%*%b);
# Outcome
y = h+rnorm(n);
# Fitted linear model
M = Fit.LinReg(y=y,X=X);
show(M);

## Fitted Linear Model
## Estimated Coefficients:
##   Coeff   Point     SE       L       U         p
## 1   int  1.0200 0.0315  0.9600  1.0800 2.51e-231
## 2    x1 -0.1430 0.0328 -0.2070 -0.0786  1.34e-05
## 3    x2  0.1780 0.0310  0.1170  0.2380  1.01e-08
## 4    x3 -0.0839 0.0317 -0.1460 -0.0218  8.09e-03
## 5    x4 -0.0130 0.0322 -0.0761  0.0500  6.85e-01

Score Tests

Standard Test

Score conducts a standard score test, using either the asymptotic variance or perturbation to estimate a $p$-value. The hypothesis test is specified using a logical vector L. Elements of L that are constrained under the null are set to TRUE, those which are estimated under the null are set to FALSE. At least one element of L must be TRUE (a test must be specified), and at least one element of L must be FALSE (the null model must be estimable). Below, hypotheses are tested on the example data.

• The first test assesses $H_{0}:\beta_{4}=0$, which is true.
• The second test assesses $H_{0}:\beta_{2}=0$, which is false.
• The third test assesses $H_{0}:\beta_{1}=\beta_{3}=-0.1$, which is true.

cat("Score test of b4=0\n");
Score(y=y,X=X,L=c(F,F,F,F,T),method="asymptotic");
Score(y=y,X=X,L=c(F,F,F,F,T),method="perturbation");
cat("\n");
cat("Score test of b2=0\n");
Score(y=y,X=X,L=c(F,F,T,F,F),method="asymptotic");
Score(y=y,X=X,L=c(F,F,T,F,F),method="perturbation");
cat("\n");
cat("Score test of b1=b3=-0.1\n");
Score(y=y,X=X,L=c(F,T,F,T,F),b10=c(-0.1,-0.1),method="asymptotic");
Score(y=y,X=X,L=c(F,T,F,T,F),b10=c(-0.1,-0.1),method="perturbation");
cat("\n");

## Score test of b4=0
##     Score        df         p 
## 0.1647543 1.0000000 0.6848166 
##     Score        df         p 
## 0.1647543 1.0000000 0.6543457 
## 
## Score test of b2=0
##        Score           df            p 
## 3.180192e+01 1.000000e+00 1.707236e-08 
##        Score           df            p 
## 31.801923115  1.000000000  0.000999001 
## 
## Score test of b1=b3=-0.1
##     Score        df         p 
## 2.0582910 2.0000000 0.3573121 
##     Score        df         p 
## 2.0582910 2.0000000 0.3396603

Weighted Score Test

wScore conducts a score test that affords different components of the hypothesis different weights. The following assesses $H_{0}:\beta_{3}=\beta_{4}=0$, equivalently $H_{0}:(\beta_{3}=0)\land(\beta_{4}=0)$, giving different weights to the different components. The first component $(\beta_{3}=0)$ is false, while the second component $(\beta_{4}=0)$ is true. The overall hypothesis is false. Whether or not the weighted test rejects depends on which component of the hypothesis is afforded more weight.

cat("Weighted score test of b3=b4=0\n");
wScore(y=y,X=X,L=c(F,F,F,T,T),w=c(10,1),method="asymptotic");
wScore(y=y,X=X,L=c(F,F,F,T,T),w=c(10,1),method="perturbation");
cat("\n");
cat("Weighted score test of b3=b4=0\n");
wScore(y=y,X=X,L=c(F,F,T,F,T),w=c(1,10),method="asymptotic");
wScore(y=y,X=X,L=c(F,F,T,F,T),w=c(1,10),method="perturbation");

## Weighted score test of b3=b4=0
##       Score          df           p 
## 1.000000000 2.000000000 0.008175889 
##       Score          df           p 
## 1.000000000 2.000000000 0.006993007 
## 
## Weighted score test of b3=b4=0
##     Score        df         p 
## 1.0000000 2.0000000 0.4700908 
##     Score        df         p 
## 1.0000000 2.0000000 0.4445554

zrmacc/LiST documentation built on May 9, 2019, 8:20 a.m.