STA 210 - Fall 2023 – Multinomial Logistic Regression

Model in R

Use the multinom_reg() function with the "nnet" engine:

health_fit <- multinom_reg() |>
  set_engine("nnet") |>
  fit(HealthGen ~ Age + PhysActive, data = nhanes_adult)

Model result

health_fit

parsnip model object

Call:
nnet::multinom(formula = HealthGen ~ Age + PhysActive, data = data, 
    trace = FALSE)

Coefficients:
      (Intercept)           Age PhysActiveYes
Vgood   1.2053460  0.0009101848    -0.3209047
Good    1.9476261 -0.0023686122    -1.0014925
Fair    0.9145492  0.0030462534    -1.6454297
Poor   -1.5211414  0.0221905681    -2.6556343

Residual Deviance: 17588.88 
AIC: 17612.88

Model output

What function do we use to get the model summary, i.e., coefficient estimates.

tidy(health_fit)

# A tibble: 12 × 6
   y.level term           estimate std.error statistic  p.value
   <chr>   <chr>             <dbl>     <dbl>     <dbl>    <dbl>
 1 Vgood   (Intercept)    1.21       0.145       8.33  8.42e-17
 2 Vgood   Age            0.000910   0.00246     0.369 7.12e- 1
 3 Vgood   PhysActiveYes -0.321      0.0929     -3.45  5.51e- 4
 4 Good    (Intercept)    1.95       0.141      13.8   1.39e-43
 5 Good    Age           -0.00237    0.00242    -0.977 3.29e- 1
 6 Good    PhysActiveYes -1.00       0.0901    -11.1   1.00e-28
 7 Fair    (Intercept)    0.915      0.164       5.57  2.61e- 8
 8 Fair    Age            0.00305    0.00288     1.06  2.90e- 1
 9 Fair    PhysActiveYes -1.65       0.107     -15.3   5.69e-53
10 Poor    (Intercept)   -1.52       0.290      -5.24  1.62e- 7
11 Poor    Age            0.0222     0.00491     4.52  6.11e- 6
12 Poor    PhysActiveYes -2.66       0.236     -11.3   1.75e-29

Model output, with CI

tidy(health_fit, conf.int = TRUE)

# A tibble: 12 × 8
   y.level term         estimate std.error statistic  p.value conf.low conf.high
   <chr>   <chr>           <dbl>     <dbl>     <dbl>    <dbl>    <dbl>     <dbl>
 1 Vgood   (Intercept)   1.21e+0   0.145       8.33  8.42e-17  0.922     1.49   
 2 Vgood   Age           9.10e-4   0.00246     0.369 7.12e- 1 -0.00392   0.00574
 3 Vgood   PhysActiveY… -3.21e-1   0.0929     -3.45  5.51e- 4 -0.503    -0.139  
 4 Good    (Intercept)   1.95e+0   0.141      13.8   1.39e-43  1.67      2.22   
 5 Good    Age          -2.37e-3   0.00242    -0.977 3.29e- 1 -0.00712   0.00238
 6 Good    PhysActiveY… -1.00e+0   0.0901    -11.1   1.00e-28 -1.18     -0.825  
 7 Fair    (Intercept)   9.15e-1   0.164       5.57  2.61e- 8  0.592     1.24   
 8 Fair    Age           3.05e-3   0.00288     1.06  2.90e- 1 -0.00260   0.00869
 9 Fair    PhysActiveY… -1.65e+0   0.107     -15.3   5.69e-53 -1.86     -1.43   
10 Poor    (Intercept)  -1.52e+0   0.290      -5.24  1.62e- 7 -2.09     -0.952  
11 Poor    Age           2.22e-2   0.00491     4.52  6.11e- 6  0.0126    0.0318 
12 Poor    PhysActiveY… -2.66e+0   0.236     -11.3   1.75e-29 -3.12     -2.19

Model output, with CI

y.level	term	estimate	std.error	statistic	p.value	conf.low	conf.high
Vgood	(Intercept)	1.205	0.145	8.325	0.000	0.922	1.489
Vgood	Age	0.001	0.002	0.369	0.712	-0.004	0.006
Vgood	PhysActiveYes	-0.321	0.093	-3.454	0.001	-0.503	-0.139
Good	(Intercept)	1.948	0.141	13.844	0.000	1.672	2.223
Good	Age	-0.002	0.002	-0.977	0.329	-0.007	0.002
Good	PhysActiveYes	-1.001	0.090	-11.120	0.000	-1.178	-0.825
Fair	(Intercept)	0.915	0.164	5.566	0.000	0.592	1.237
Fair	Age	0.003	0.003	1.058	0.290	-0.003	0.009
Fair	PhysActiveYes	-1.645	0.107	-15.319	0.000	-1.856	-1.435
Poor	(Intercept)	-1.521	0.290	-5.238	0.000	-2.090	-0.952
Poor	Age	0.022	0.005	4.522	0.000	0.013	0.032
Poor	PhysActiveYes	-2.656	0.236	-11.275	0.000	-3.117	-2.194

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

$\log (\frac{{\hat{π}}_{F a i r}}{{\hat{π}}_{E x c e l l e n t}}) = 0.915 + 0.003 age - 1.645 PhysActive$

Interpretations

$\log (\frac{{\hat{π}}_{F a i r}}{{\hat{π}}_{E x c e l l e n t}}) = 0.915 + 0.003 age - 1.645 PhysActive$

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

$\log (\frac{{\hat{π}}_{F a i r}}{{\hat{π}}_{E x c e l l e n t}}) = 0.915 + 0.003 age - 1.645 PhysActive$

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)).

Warning

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

Hypothesis test for $β_{j k}$

The test of significance for the coefficient $β_{j k}$ is

Hypotheses: $H_{0} : β_{j k} = 0 vs H_{a} : β_{j k} \neq 0$

Test Statistic: $z = \frac{{\hat{β}}_{j k} - 0}{S E (\hat{β_{j k}})}$

P-value: $P (| Z | > | z |)$ ,

where $Z \sim N (0, 1)$ , the Standard Normal distribution

Confidence interval for $β_{j k}$

We can calculate the C% confidence interval for $β_{j k}$ using ${\hat{β}}_{j k} \pm z^{*} S E ({\hat{β}}_{j k})$ , where $z^{*}$ is calculated from the $N (0, 1)$ distribution.
Interpretation: We are $C %$ confident that for every one unit change in $x_{j}$ , the odds of $y = k$ versus the baseline will multiply by a factor of $\exp {{\hat{β}}_{j k} - z^{*} S E ({\hat{β}}_{j k})}$ to $\exp {{\hat{β}}_{j k} + z^{*} S E ({\hat{β}}_{j k})}$ , holding all else constant.

Interpreting CIs for $β_{j k}$

tidy(health_fit, conf.int = TRUE) |>
  filter(y.level == "Fair") |>
  kable(digits = 3)

y.level	term	estimate	std.error	statistic	p.value	conf.low	conf.high
Fair	(Intercept)	0.915	0.164	5.566	0.00	0.592	1.237
Fair	Age	0.003	0.003	1.058	0.29	-0.003	0.009
Fair	PhysActiveYes	-1.645	0.107	-15.319	0.00	-1.856	-1.435

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 CIs for $β_{j k}$

tidy(health_fit, conf.int = TRUE) |>
  filter(y.level == "Fair") |>
  kable(digits = 3)

y.level	term	estimate	std.error	statistic	p.value	conf.low	conf.high
Fair	(Intercept)	0.915	0.164	5.566	0.00	0.592	1.237
Fair	Age	0.003	0.003	1.058	0.29	-0.003	0.009
Fair	PhysActiveYes	-1.645	0.107	-15.319	0.00	-1.856	-1.435

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 $β_{j k}$

Multinomial Logistic Regression

Announcements

Topics

Application exercise

Computational setup

Generalized Linear Models

Generalized Linear Models (GLMs)

Binary outcome (Logistic)

Binary outcome (Logistic)

Multinomial outcome variable

Multinomial Logistic Regression

Multinomial Logistic Regression

Data

NHANES Data

NHANES Data

Variables

The data

Exploratory data analysis

Exploratory data analysis

Fitting a multinomial logistic regression model

Model in R

Model result

Model output

Model output, with CI

Model output, with CI

Fair vs. Excellent Health

Interpretations

Interpretations

Hypothesis test for $β_{j k}$

Confidence interval for $β_{j k}$

Interpreting CIs for $β_{j k}$

Interpreting CIs for $β_{j k}$

Recap