Topics

• Introduce multinomial logistic regression

• Interpret model coefficients

• Inference for a coefficient βjk

Generalized Linear Models (GLM)

• In practice, there are many different types of response variables including:

  • Binary: Win or Lose
  • Nominal: Democrat, Republican or Third Party candidate
  • Ordered: Movie rating (1 - 5 stars)
  • and others...

• These are all examples of generalized linear models, a broader class of models that generalize the multiple linear regression model

• See Generalized Linear Models: A Unifying Theory for more details about GLMs

Binary Response (Logistic)

• Given P(yi=1|xi)=ˆπi and P(yi=0|xi)=1−ˆπi

log(ˆπi1−ˆπi)=ˆβ0+ˆβ1xi

• We can calculate ˆπi by solving the logit equation:

ˆπi=exp{ˆβ0+ˆβ1xi}1+exp{ˆβ0+ˆβ1xi}

Binary Response (Logistic)

Suppose we consider y=0 the baseline category such that

P(yi=1|xi)=ˆπi1 and P(yi=0|xi)=ˆπi0

Then the logistic regression model is

log(ˆπi11−ˆπi1)=log(ˆπi1ˆπi0)=ˆβ0+ˆβ1xi

Slope, ˆβ1: When x increases by one unit, the odds of y=1 versus the baseline y=0 are expected to multiply by a factor of exp{ˆβ1}

Intercept, ˆβ0: When x=0, the predicted odds of y=1 versus the baseline y=0 are exp{ˆβ0}

Multinomial response variable

• Suppose the response variable y is categorical and can take values 1,2,…,K such that (K>2)

• Multinomial Distribution:

P(y=1)=π1,P(y=2)=π2,…,P(y=K)=πK

such that K∑k=1πk=1

Multinomial Logistic Regression

• If we have an explanatory variable x, then we want to fit a model such that P(y=k)=πk is a function of x

• Choose a baseline category. Let's choose y=1. Then,

• In the multinomial logistic model, we have a separate equation for each category of the response relative to the baseline category
  • If the response has K possible categories, there will be K−1 equations as part of the multinomial logistic model

Multinomial Logistic Regression

• Suppose we have a response variable y that can take three possible outcomes that are coded as "A", "B", "C"

• Let "A" be the baseline category. Then

log(πiBπiA)=β0B+β1Bxilog(πiCπiA)=β0C+β1Cxi

NHANES Data

• National Health and Nutrition Examination Survey is conducted by the National Center for Health Statistics (NCHS)

• The goal is to "assess the health and nutritional status of adults and children in the United States"

• This survey includes an interview and a physical examination

NHANES Data

• We will use the data from the NHANES R package

• Contains 75 variables for the 2009 - 2010 and 2011 - 2012 sample years

• The data in this package is modified for educational purposes and should not be used for research

• Original data can be obtained from the NCHS website for research purposes

• Type ?NHANES in console to see list of variables and definitions

Health Rating vs. Age & Physical Activity

• Question: Can we use a person's age and whether they do regular physical activity to predict their self-reported health rating?

• We will analyze the following variables:

  • HealthGen: Self-reported rating of participant's health in general. Excellent, Vgood, Good, Fair, or Poor.

  • Age: Age at time of screening (in years). Participants 80 or older were recorded as 80.

  • PhysActive: Participant does moderate to vigorous-intensity sports, fitness or recreational activities

The data

## Rows: 6,710
## Columns: 4
## $ HealthGen  <fct> Good, Good, Good, Good, Vgood, Vgood, Vgood, Vgood, Vgood, …
## $ Age        <int> 34, 34, 34, 49, 45, 45, 45, 66, 58, 54, 50, 33, 60, 56, 56,…
## $ PhysActive <fct> No, No, No, No, Yes, Yes, Yes, Yes, Yes, Yes, Yes, No, No, …
## $ obs_num    <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, …

Exploratory data analysis

Exploratory data analysis

Model in R

• Use the multinom() function in the nnet package

library(nnet)
health_m <- multinom(HealthGen ~ Age + PhysActive, 
                     data = nhanes_adult)

• Put results = "hide" in the code chunk header to suppress convergence output

Output results

tidy(health_m, conf.int = TRUE, exponentiate = FALSE) %>%
  kable(digits = 3, format = "markdown")

Model output

Fair vs. Excellent Health

The baseline category for the model is Excellent.

The model equation for the log-odds a person rates themselves as having "Fair" health vs. "Excellent" is

log(ˆπFairˆπExcellent)=0.915+0.003 age−1.645 PhysActive

Interpretations

For each additional year in age, the odds a person rates themselves as having fair health versus excellent health are expected to multiply by 1.003 (exp(0.003)), holding physical activity constant.

The odds a person who does physical activity will rate themselves as having fair health versus excellent health are expected to be 0.193 (exp(-1.645 )) times the odds for a person who doesn't do physical activity, holding age constant.

Interpretations

The odds a 0 year old person who doesn't do physical activity rates themselves as having fair health vs. excellent health are 2.497 (exp(0.915)).

⚠️ Need to mean-center age for the intercept to have a meaningful interpretation!

Hypothesis test for βjk

The test of significance for the coefficient βjk is

Confidence interval for βjk

• We can calculate the C% confidence interval for βjk using the following:

ˆβjk±z∗SE(ˆβjk)

Interpreting confidence intervals for βjk

We are 95% confident, that for each additional year in age, the odds a person rates themselves as having fair health versus excellent health will multiply by 0.997 (exp(-0.003)) to 1.009 (exp(0.009)) , holding physical activity constant.

Interpreting confidence intervals for βjk

We are 95% confident that the odds a person who does physical activity will rate themselves as having fair health versus excellent health are 0.156 (exp(-1.856 )) to 0.238 (exp(-1.435)) times the odds for a person who doesn't do physical activity, holding age constant.

Recap

• Introduce multinomial logistic regression

• Interpret model coefficients

• Inference for a coefficient βjk