An R Introduction to Statistics

Significance Test for Logistic Regression

We can decide whether there is any significant relationship between the dependent variable y and the independent variables xk (k = 1, 2, ..., p) in the logistic regression equation. In particular, if any of the null hypothesis that βk = 0 (k = 1, 2, ..., p) is valid, then xk is statistically insignificant in the logistic regression model.

Problem

At .05 significance level, decide if any of the independent variables in the logistic regression model of vehicle transmission in data set mtcars is statistically insignificant.

Solution

We apply the function glm to a formula that describes the transmission type (am) by the horsepower (hp) and weight (wt). This creates a generalized linear model (GLM) in the binomial family.

> am.glm = glm(formula=am ~ hp + wt, 
+              data=mtcars, 
+              family=binomial)

We then print out the summary of the generalized linear model and check for the p-values of the hp and wt variables.

> summary(am.glm) 
 
Call: 
glm(formula = am ~ hp + wt, family = binomial, data = mtcars) 
 
Deviance Residuals: 
    Min       1Q   Median       3Q      Max 
-2.2537  -0.1568  -0.0168   0.1543   1.3449 
 
Coefficients: 
            Estimate Std. Error z value Pr(>|z|) 
(Intercept)  18.8663     7.4436    2.53   0.0113 * 
hp            0.0363     0.0177    2.04   0.0409 * 
wt           -8.0835     3.0687   -2.63   0.0084 ** 
--- 
Signif. codes:  0 *** 0.001 ** 0.01 * 0.05 . 0.1   1 
 
(Dispersion parameter for binomial family taken to be 1) 
 
    Null deviance: 43.230  on 31  degrees of freedom 
Residual deviance: 10.059  on 29  degrees of freedom 
AIC: 16.06 
 
Number of Fisher Scoring iterations: 8

Answer

As the p-values of the hp and wt variables are both less than 0.05, neither hp or wt is insignificant in the logistic regression model.

Note

Further detail of the function summary for the generalized linear model can be found in the R documentation.

> help(summary.glm)