(Simulating Wald and Likelihood Ratio Tests)

In this section we will investigate the distributions of hypothesis tests for logistic regression. For this exercise, we will use the following predictors.

sample_size = 150 set.seed(420) x1 = rnorm(n = sample_size) x2 = rnorm(n = sample_size) x3 = rnorm(n = sample_size) 

Recall that

p(x)=P[Y=1X=x]

p(x)=P[Y=1X=x]

Consider the true model

log(p(x)

1p(x)

)=

0

+

1

x

1

log(p(x)1p(x))=0+1x1

where

• 0
• =0.4
• 0=0.4
• 1
• =0.35
• 1=0.35

(a)To investigate the distributions, simulate from this model 2500 times. To do so, calculate

P[Y=1X=x]

P[Y=1X=x]

for an observation, and then make a random draw from a Bernoulli distribution with that success probability. (Note that a Bernoulli distribution is a Binomial distribution with parametern=1

n=1. There is no direction function inRfor a Bernoulli distribution.)

Each time, fit the model:

log(p(x)

1p(x)

)=

0

+

1

x

1

+

2

x

2

+

3

x

3

log(p(x)1p(x))=0+1x1+2x2+3x3

Store the test statistics for two tests:

• The Wald test forH
• 0
• :
• 2
• =0
• H0:2=0, which we say follows a standard normal distribution for "large" samples
• The likelihood ratio test forH
• 0
• :
• 2
• =
• 3
• =0
• H0:2=3=0, which we say follows a
• 2
• 2distribution (with some degrees of freedom) for "large" samples

(b)Plot a histogram of the empirical values for the Wald test statistic. Overlay the density of the true distribution assuming a large sample.

(c)Use the empirical results for the Wald test statistic to estimate the probability of observing a test statistic larger than 1. Also report this probability using the true distribution of the test statistic assuming a large sample.

(d)Plot a histogram of the empirical values for the likelihood ratio test statistic. Overlay the density of the true distribution assuming a large sample.

(e)Use the empirical results for the likelihood ratio test statistic to estimate the probability of observing a test statistic larger than 5. Also report this probability using the true distribution of the test statistic assuming a large sample.

(f)Repeat(a)-(e)but with simulation using a smaller sample size of 10. Based on these results, is this sample size large enough to use the standard normal and

2

2distributions in this situation? Explain.

sample_size = 10 set.seed(420) x1 = rnorm(n = sample_size) x2 = rnorm(n = sample_size) x3 = rnorm(n = sample_size) 

looking for R coding help.