README.md
In zrmacc/BinReg: BinReg
title: "README"
author: "Zachary McCaw"
date: "2018-09-16"
output:
html_document:
keep_md: TRUE
Package Vignette
Contents
Introduction
This package provides implementations of regression models for a binary outcome. Throughout denote the outcome $y_{i}\in{0,1}$, the covariates $x_{i}\in\mathbb{R}^{p}$, and the linear predictor $\eta_{i} = x_{i}'\beta$.
Logistic
Model
In logistic regression, the latent variable $z_{i}$ follows a logistic distribution with center $\eta_{i}$ and unit scale:
$$
z_{i}|x_{i} \sim \text{Logistic}\big(\eta_{i},1) \
y_{i} = I[z_{i}>0]
$$
Estimation
Below, data are simulated for $n=10^{3}$ subjects. The model matrix X includes an intercept and four standard normal covariates. The function rBinReg is provided for simulating from binary regression models. The function Fit.BinReg with model="logistic" specifies a logistic model.
set.seed(101);
library(BinReg);
# Subjects
n = 1e3;
# Design matrix
X = cbind(1,matrix(rnorm(n=4*n),nrow=n));
# Regression coefficient
b = c(1,-1,2,-1,0);
# Logistic outcome
y = rBinReg(X,b,model="logistic");
# Estimate regression parameters
M = Fit.BinReg(y=y,X=X,model="logistic");
show(M);
## Objective increment: 163
## Objective increment: 28.3
## Objective increment: 3.23
## Objective increment: 0.0653
## Objective increment: 3.39e-05
## Objective increment: 9.55e-12
## 5 update(s) performed before tolerance limit.
##
## Fitted logistic Model
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.0200 0.1020 0.825 1.220 7.65e-24
## 2 x2 -1.0000 0.1060 -1.210 -0.794 3.28e-21
## 3 x3 2.0400 0.1390 1.760 2.310 2.26e-48
## 4 x4 -1.0800 0.1080 -1.300 -0.874 6.49e-24
## 5 x5 0.0755 0.0931 -0.107 0.258 4.17e-01
Inference
Wald tests for individual coefficients are stored in M@Coefficients and displayed using the show method. The function Score.BinReg performs score tests on subsets of the regression coefficients. The test is specified using a logical vector L with as many elements as columns in the model matrix X. An element of L is set to TRUE if the regression coefficient for that column of X is fixed under $H_{0}$. An element of L is set to FALSE if the regression coefficient for that column requires estimation under $H_{0}$. At least one element of L must be TRUE (i.e. a test must be specified) and at least one element of L must be FALSE (i.e. a null model must be estimable).
Below, various hypothses are tested on the example data. The first is an overall test that the coefficients for all covariates, other than the intercept, are zero. The null hypothesis is $H_{0}:\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0$, which is false. The second is an individual test that the coefficient for the fourth covariate is zero. The null hypothesis is $H_{0}:\beta_{4}=0$, which is true. The third is a joint test that the coefficient for the second covariate is zero, and the coefficient for the fourth covariate is two. The null hypothesis is $H_{0}:\beta_{2}=0,\beta_{4}=2$, which if false. The fourth is a joint test that the coefficients for the first and third covariates are equal to negative one. The null hypothesis is $H_{0}:\beta_{1}=\beta_{3}=-1$, which is true. All models include an intercept $\beta_{0}$, since in each case L[1]=FALSE.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="logistic",report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="logistic",report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="logistic",report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="logistic",report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 4.271668e+02 4.000000e+00 3.745524e-91
##
## Test of b4=0:
## Score df p
## 0.6583181 1.0000000 0.4171544
##
## Test of b2=0, b4=2:
## Score df p
## 1665.101 2.000 0.000
##
## Test of b1=b3=-1:
## Score df p
## 0.6456440 2.0000000 0.7241027
Probit
Model
In probit regression, the latent variable $z_{i}$ follows a normal distribution with center $\eta_{i}$ and unit scale:
$$
z_{i}|x_{i} \sim N\big(\eta_{i},1) \
y_{i} = I[z_{i}>0]
$$
Estimation
The function Fit.BinReg with model="probit" specifies a probit model.
# Probit outcome
y = rBinReg(X,b,model="probit");
# Estimate regression parameters
M = Fit.BinReg(y=y,X=X,model="probit");
show(M);
## Objective increment: 196
## Objective increment: 50
## Objective increment: 17.5
## Objective increment: 3.38
## Objective increment: 0.185
## Objective increment: 0.00134
## Objective increment: 3.19e-06
## Objective increment: 8.95e-09
## 7 update(s) performed before tolerance limit.
##
## Fitted probit Model
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.1000 0.0888 0.929 1.2800 2.06e-35
## 2 x2 -1.0800 0.0913 -1.250 -0.8960 5.16e-32
## 3 x3 2.1800 0.1450 1.900 2.4700 1.58e-51
## 4 x4 -1.0500 0.0892 -1.220 -0.8720 9.40e-32
## 5 x5 -0.0456 0.0684 -0.180 0.0884 5.05e-01
Inference
The same hypotheses considered previously are tested for the probit model.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="probit",report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="probit",report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="probit",report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="probit",report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 5.399379e+02 4.000000e+00 1.537766e-115
##
## Test of b4=0:
## Score df p
## 0.4245797 1.0000000 0.5146607
##
## Test of b2=0, b4=2:
## Score df p
## 14293.39 2.00 0.00
##
## Test of b1=b3=-1:
## Score df p
## 0.6870218 2.0000000 0.7092758
Robit
Model
In robit regression, the latent variable $z_{i}$ follows a $t_{\nu}$ distribution with center $\eta_{i}$, unit scale, and specified degrees of freedom $\nu$:
$$
z_{i}|x_{i} \sim t_{\nu}(\eta_{i}) \
y_{i} = I[z_{i}>0]
$$
Estimation
The function Fit.BinReg with model="robit" specifies a robit model. Note that the degrees of freedom requires specification. The robit model with df=7 is similar to the logistic model.
# Degrees of freedom
df = 7;
# Robit outcome
y = rBinReg(X,b,model="robit",df=df);
# Recover regression parameters
M = Fit.BinReg(y=y,X=X,model="robit",df=df);
show(M);
## Objective increment: 199
## Objective increment: 52.1
## Objective increment: 14.8
## Objective increment: 1.8
## Objective increment: 0.0339
## Objective increment: 6.67e-06
## Objective increment: 5.46e-10
## 6 update(s) performed before tolerance limit.
##
## Fitted robit Model with 7 degrees of freedom
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.1900 0.1050 0.9860 1.400 6.32e-30
## 2 x2 -1.2400 0.1100 -1.4500 -1.020 3.62e-29
## 3 x3 2.3500 0.1720 2.0100 2.690 1.40e-42
## 4 x4 -1.1900 0.1060 -1.4000 -0.986 2.72e-29
## 5 x5 0.0816 0.0776 -0.0705 0.234 2.93e-01
Inference
The same hypotheses considered previously are tested for the robit model. Note that the degrees of freedom requires specification.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="robit",df=df,report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="robit",df=df,report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="robit",df=df,report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="robit",df=df,report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 5.390587e+02 4.000000e+00 2.382912e-115
##
## Test of b4=0:
## Score df p
## 1.1064617 1.0000000 0.2928524
##
## Test of b2=0, b4=2:
## Score df p
## 3598.797 2.000 0.000
##
## Test of b1=b3=-1:
## Score df p
## 5.52201878 2.00000000 0.06322791
zrmacc/BinReg documentation built on May 9, 2019, 8:08 a.m.
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title: "README" author: "Zachary McCaw" date: "2018-09-16" output: html_document: keep_md: TRUE
Package Vignette
Contents
Introduction
This package provides implementations of regression models for a binary outcome. Throughout denote the outcome $y_{i}\in{0,1}$, the covariates $x_{i}\in\mathbb{R}^{p}$, and the linear predictor $\eta_{i} = x_{i}'\beta$.
Logistic
Model
In logistic regression, the latent variable $z_{i}$ follows a logistic distribution with center $\eta_{i}$ and unit scale:
$$ z_{i}|x_{i} \sim \text{Logistic}\big(\eta_{i},1) \ y_{i} = I[z_{i}>0] $$
Estimation
Below, data are simulated for $n=10^{3}$ subjects. The model matrix X includes an intercept and four standard normal covariates. The function rBinReg is provided for simulating from binary regression models. The function Fit.BinReg with model="logistic" specifies a logistic model.
set.seed(101);
library(BinReg);
# Subjects
n = 1e3;
# Design matrix
X = cbind(1,matrix(rnorm(n=4*n),nrow=n));
# Regression coefficient
b = c(1,-1,2,-1,0);
# Logistic outcome
y = rBinReg(X,b,model="logistic");
# Estimate regression parameters
M = Fit.BinReg(y=y,X=X,model="logistic");
show(M);
## Objective increment: 163
## Objective increment: 28.3
## Objective increment: 3.23
## Objective increment: 0.0653
## Objective increment: 3.39e-05
## Objective increment: 9.55e-12
## 5 update(s) performed before tolerance limit.
##
## Fitted logistic Model
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.0200 0.1020 0.825 1.220 7.65e-24
## 2 x2 -1.0000 0.1060 -1.210 -0.794 3.28e-21
## 3 x3 2.0400 0.1390 1.760 2.310 2.26e-48
## 4 x4 -1.0800 0.1080 -1.300 -0.874 6.49e-24
## 5 x5 0.0755 0.0931 -0.107 0.258 4.17e-01
Inference
Wald tests for individual coefficients are stored in M@Coefficients and displayed using the show method. The function Score.BinReg performs score tests on subsets of the regression coefficients. The test is specified using a logical vector L with as many elements as columns in the model matrix X. An element of L is set to TRUE if the regression coefficient for that column of X is fixed under $H_{0}$. An element of L is set to FALSE if the regression coefficient for that column requires estimation under $H_{0}$. At least one element of L must be TRUE (i.e. a test must be specified) and at least one element of L must be FALSE (i.e. a null model must be estimable).
Below, various hypothses are tested on the example data. The first is an overall test that the coefficients for all covariates, other than the intercept, are zero. The null hypothesis is $H_{0}:\beta_{1}=\beta_{2}=\beta_{3}=\beta_{4}=0$, which is false. The second is an individual test that the coefficient for the fourth covariate is zero. The null hypothesis is $H_{0}:\beta_{4}=0$, which is true. The third is a joint test that the coefficient for the second covariate is zero, and the coefficient for the fourth covariate is two. The null hypothesis is $H_{0}:\beta_{2}=0,\beta_{4}=2$, which if false. The fourth is a joint test that the coefficients for the first and third covariates are equal to negative one. The null hypothesis is $H_{0}:\beta_{1}=\beta_{3}=-1$, which is true. All models include an intercept $\beta_{0}$, since in each case L[1]=FALSE.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="logistic",report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="logistic",report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="logistic",report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="logistic",report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 4.271668e+02 4.000000e+00 3.745524e-91
##
## Test of b4=0:
## Score df p
## 0.6583181 1.0000000 0.4171544
##
## Test of b2=0, b4=2:
## Score df p
## 1665.101 2.000 0.000
##
## Test of b1=b3=-1:
## Score df p
## 0.6456440 2.0000000 0.7241027
Probit
Model
In probit regression, the latent variable $z_{i}$ follows a normal distribution with center $\eta_{i}$ and unit scale:
$$ z_{i}|x_{i} \sim N\big(\eta_{i},1) \ y_{i} = I[z_{i}>0] $$
Estimation
The function Fit.BinReg with model="probit" specifies a probit model.
# Probit outcome
y = rBinReg(X,b,model="probit");
# Estimate regression parameters
M = Fit.BinReg(y=y,X=X,model="probit");
show(M);
## Objective increment: 196
## Objective increment: 50
## Objective increment: 17.5
## Objective increment: 3.38
## Objective increment: 0.185
## Objective increment: 0.00134
## Objective increment: 3.19e-06
## Objective increment: 8.95e-09
## 7 update(s) performed before tolerance limit.
##
## Fitted probit Model
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.1000 0.0888 0.929 1.2800 2.06e-35
## 2 x2 -1.0800 0.0913 -1.250 -0.8960 5.16e-32
## 3 x3 2.1800 0.1450 1.900 2.4700 1.58e-51
## 4 x4 -1.0500 0.0892 -1.220 -0.8720 9.40e-32
## 5 x5 -0.0456 0.0684 -0.180 0.0884 5.05e-01
Inference
The same hypotheses considered previously are tested for the probit model.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="probit",report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="probit",report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="probit",report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="probit",report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 5.399379e+02 4.000000e+00 1.537766e-115
##
## Test of b4=0:
## Score df p
## 0.4245797 1.0000000 0.5146607
##
## Test of b2=0, b4=2:
## Score df p
## 14293.39 2.00 0.00
##
## Test of b1=b3=-1:
## Score df p
## 0.6870218 2.0000000 0.7092758
Robit
Model
In robit regression, the latent variable $z_{i}$ follows a $t_{\nu}$ distribution with center $\eta_{i}$, unit scale, and specified degrees of freedom $\nu$:
$$ z_{i}|x_{i} \sim t_{\nu}(\eta_{i}) \ y_{i} = I[z_{i}>0] $$
Estimation
The function Fit.BinReg with model="robit" specifies a robit model. Note that the degrees of freedom requires specification. The robit model with df=7 is similar to the logistic model.
# Degrees of freedom
df = 7;
# Robit outcome
y = rBinReg(X,b,model="robit",df=df);
# Recover regression parameters
M = Fit.BinReg(y=y,X=X,model="robit",df=df);
show(M);
## Objective increment: 199
## Objective increment: 52.1
## Objective increment: 14.8
## Objective increment: 1.8
## Objective increment: 0.0339
## Objective increment: 6.67e-06
## Objective increment: 5.46e-10
## 6 update(s) performed before tolerance limit.
##
## Fitted robit Model with 7 degrees of freedom
## Estimated Coefficients:
## Coeff Point SE L U p
## 1 x1 1.1900 0.1050 0.9860 1.400 6.32e-30
## 2 x2 -1.2400 0.1100 -1.4500 -1.020 3.62e-29
## 3 x3 2.3500 0.1720 2.0100 2.690 1.40e-42
## 4 x4 -1.1900 0.1060 -1.4000 -0.986 2.72e-29
## 5 x5 0.0816 0.0776 -0.0705 0.234 2.93e-01
Inference
The same hypotheses considered previously are tested for the robit model. Note that the degrees of freedom requires specification.
cat("Test of b1=b2=b3=b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,T,T,T,T),model="robit",df=df,report=F);
cat("\n");
cat("Test of b4=0:\n");
Score.BinReg(y=y,X=X,L=c(F,F,F,F,T),model="robit",df=df,report=F);
cat("\n");
cat("Test of b2=0, b4=2:\n");
Score.BinReg(y=y,X=X,L=c(F,F,T,F,T),b10=c(0,2),model="robit",df=df,report=F);
cat("\n");
cat("Test of b1=b3=-1:\n");
Score.BinReg(y=y,X=X,L=c(F,T,F,T,F),b10=c(-1,-1),model="robit",df=df,report=F);
cat("\n");
## Test of b1=b2=b3=b4=0:
## Score df p
## 5.390587e+02 4.000000e+00 2.382912e-115
##
## Test of b4=0:
## Score df p
## 1.1064617 1.0000000 0.2928524
##
## Test of b2=0, b4=2:
## Score df p
## 3598.797 2.000 0.000
##
## Test of b1=b3=-1:
## Score df p
## 5.52201878 2.00000000 0.06322791
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