逻辑回归的 glm() 结果

Question

This might be a trivial question but I don't know where to find answers. I'm wondering when using glm() for logistic regression in R, if the response variable Y has factor values 1 or 2, does the result of glm() correspond to logit(P(Y=1)) or logit(P(Y=2)) ? What if Y has logical values TRUE or FALSE ?

Answer 1

Why not just test it yourself?

output_bool <- c(rep(c(TRUE, FALSE), c(25, 75)), rep(c(TRUE, FALSE), c(75, 25)))
output_num <- c(rep(c(2, 1), c(25, 75)), rep(c(2, 1), c(75, 25)))
output_fact <- factor(output_num)
var <- rep(c("unlikely", "likely"), each = 100)

glm(output_bool ~ var, binomial)
#> 
#> Call:  glm(formula = output_bool ~ var, family = binomial)
#> 
#> Coefficients:
#> (Intercept)  varunlikely  
#>       1.099       -2.197  
#> 
#> Degrees of Freedom: 199 Total (i.e. Null);  198 Residual
#> Null Deviance:       277.3 
#> Residual Deviance: 224.9     AIC: 228.9
glm(output_num ~ var, binomial)
#> Error in eval(family$initialize): y values must be 0 <= y <= 1
glm(output_fact ~ var, binomial)
#> 
#> Call:  glm(formula = output_fact ~ var, family = binomial)
#> 
#> Coefficients:
#> (Intercept)  varunlikely  
#>       1.099       -2.197  
#> 
#> Degrees of Freedom: 199 Total (i.e. Null);  198 Residual
#> Null Deviance:       277.3 
#> Residual Deviance: 224.9     AIC: 228.9

So, we get the correct answer if we use TRUE and FALSE, an error if we use 1 and 2 as numbers, and the correct result if we use 1 and 2 as a factor with two levels provided the TRUE value has a higher factor level than the FALSE. However, we have to be careful in how our factors are ordered or we will get the wrong result:

output_fact <- factor(output_fact, levels = c("2", "1"))
glm(output_fact ~ var, binomial)
#> 
#> Call:  glm(formula = output_fact ~ var, family = binomial)
#> 
#> Coefficients:
#> (Intercept)  varunlikely  
#>      -1.099        2.197  
#> 
#> Degrees of Freedom: 199 Total (i.e. Null);  198 Residual
#> Null Deviance:       277.3 
#> Residual Deviance: 224.9     AIC: 228.9

(Notice the intercept and coefficient have flipped signs)

^{Created on 2020-06-21 by the reprex package (v0.3.0)}

Answer 2

Testing is good. If you want the documentation, it's in ?binomial (which is the same as ?family ):

For the 'binomial' and 'quasibinomial' families the response can be specified in one of three ways:

1. As a factor: 'success' is interpreted as the factor not having the first level (and hence usually of having the second level).

2. As a numerical vector with values between '0' and '1', interpreted as the proportion of successful cases (with the total number of cases given by the 'weights').

3. As a two-column integer matrix: the first column gives the number of successes and the second the number of failures.

It doesn't explicitly say what happens in the logical ( TRUE / FALSE ) case; for that you have to know that, when coercing logical to numeric values, FALSE → 0 and TRUE → 1.