# 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