The Goal

We have been working with linear mixed effects (LME) models. Today, we are going to put together everything we have learned to perform an analysis on a data set.

The Data

Today we will be working with data from a study on teen drinking behavior. The data may be found on Canvas.

Our data contain information on 82 teenagers. Each teenager was given the same survey 3 times: once at age 14, once at age 15, and once at age 16. The goal was to monitor \(Y=\) alcohol use in these teenagers and to explore what factors might be related to higher alcohol use in teens.

The data contain information on the following variables:

• id: numerical identifier for the teenager taking the survey.
• age: the age of the teenager when they took the survey: 14, 15, or 16
• coa: 1 if the teen is a child of an alcoholic parent; 0 otherwise.
• male: 1 if the teen identifies as male; 0 otherwise.
• peer: a measure of peer alcohol use, taken when each teen was 14. This is the square root of the sum of two 6-point items about the proportion of their friends who drink occasionally or regularly.

We also have information on the response variable, \(Y\) which is recorded in the alcuse column in the data set. This variable is a numeric measure of alcohol use. Four items were used to obtain this measure:

• 1. how often did the teen drink beer or wine ?
• 2. how often did the teen drink hard liquor?
• 3. how often did the teen drink 5 or more drinks in a row?
• 4. how often did the teen get drunk?

Responses to each of the 4 questions above were scored on an 8-point scale, where 0=“not at all” and 8=“every day”. Our response variable \(Y\) is the square root of the sum of these four items.

Researcher 1

Researcher 1 comes to you and asks you to use linear regression to answer the following research question:

• What is the relationship between the alcohol use of their peers (X = peer) and the alcohol use of the teen (Y = alcuse)?

They ask you to build a linear regression model using only these two variables, \(X\) and \(Y\).

Researcher 2

Researcher 2 is interested in the same research question as Researcher 1, but they believe that it is important to control for (1) whether the teen identifies as male, (2) whether or not a parent of teenager is an alcoholic, and (3) the age of the teenager when looking at the relationship between peer drinking and \(Y\) = the drinking habits of a teen (alcuse).

Researcher 2 knows that male and coa are binary variables, but they are uncertain about how to treat age in the model. Age is a discrete numeric variable, but should they include it in the model as a numeric variable or a categorical variable?

Now that Researcher 2 has a model and Researcher 1 has a model, they ask you which of the two models you would recommend that they use to answer their research question.

Researcher 3

A third researcher enters this discussion, and they believe both Researcher 1 and Researcher 2 have failed to notice something important - this is a repeated measures study.

We now have three Researchers and three different models!

Inference with the model of Researcher 3

Researcher 3 has further questions about their model. They know that they will need to report a confidence interval rather than just a sample statistic as a result of their study. Their data is only a sample, and they want to be able to draw conclusions about teenage drinking as a whole. The model we used for Researcher 3 is an LME model. To build confidence intervals with an LME model, we often need to use a technique called bootstrapping to construct confidence intervals. Let’s explore this technique to wrap up our lab.

dataSmall <- alcohol[1:5,]

bootSample <- dataSmall[ c(2, 2, 4, 1, 4), ]
bootSample

# Makes sure everyone gets the same bootstrap sample
set.seed(271)

# Choose which rows are in the bootstrap sample
rowsChosen <-sample(1:246, 246, replace = TRUE)

# Select those rows to create the bootstrap sample
bootSample1 <- alcohol[ rowsChosen, ]

# Compute n, the number of rows 
n <- nrow(alcohol)

# Set the Seed
set.seed(214) 

B = 100

betas <- data.frame("beta1" = rep(0,B))

for(i in 1:B){

  # Tell R to sample those rows with replacement 
  bootSample <- alcohol[ sample(1:n, n, replace = TRUE), ] 

  # Build Researcher 3's model
  model_boot <- lmer( alcuse ~ peer + coa + male + age + (1|id), data = bootSample)
                   
  # Store the output 
  betas$beta1[i] <- summary(model_boot)$coefficients[2,1]
}

summary(betas)

# Find the lower bound
quantile(betas$beta1, .025)

# Find the upper bound
quantile(betas$beta1, .975 )