Logistic regression: anova chi-square test vs. significance of coefficients (anova() vs summary() in R)

I have a logistic GLM model with 8 variables. I ran a chi-square test in R anova(glm.model,test='Chisq') and 2 of the variables turn out to be predictive when ordered at the top of the test and not so much when ordered at the bottom. The summary(glm.model) suggests that their coefficients are insignificant (high p-value). In this case it seems that the variables are not significant.

I wanted to ask which is a better test of variables significance - the coefficient significance in the model summary or the chi-square test from anova(). Also - when is either one better over the other?

I guess it's a broad question but any pointers on what to consider will be appreciate.

In addition to @gung's answer, I'll try to provide an example of what the anova function actually tests. I hope this enables you to decide what tests are appropriate for the hypotheses you are interested in testing.

Let's assume that you have an outcome $y$ and 3 predictor variables: $x_{1}$, $x_{2}$, and $x_{3}$. Now, if your logistic regression model would be my.mod <- glm(y~x1+x2+x3, family="binomial"). When you run anova(my.mod, test="Chisq"), the function compares the following models in sequential order:

1. glm(y~1, family="binomial") vs. glm(y~x1, family="binomial")
2. glm(y~x1, family="binomial") vs. glm(y~x1+x2, family="binomial")
3. glm(y~x1+x2, family="binomial") vs. glm(y~x1+x2+x3, family="binomial")

So it sequentially compares the smaller model with the next more complex model by adding one variable in each step. Each of those comparisons is done via a likelihood ratio test (LR test; see example below). To my knowledge, these hypotheses are rarely of interest, but this has to be decided by you.

Here is an example in R:

mydata$rank <- factor(mydata$rank)

my.mod <- glm(admit ~ gre + gpa + rank, data = mydata, family = "binomial")
summary(my.mod)

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.989979   1.139951  -3.500 0.000465 ***
gre          0.002264   0.001094   2.070 0.038465 *
gpa          0.804038   0.331819   2.423 0.015388 *
rank2       -0.675443   0.316490  -2.134 0.032829 *
rank3       -1.340204   0.345306  -3.881 0.000104 ***
rank4       -1.551464   0.417832  -3.713 0.000205 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# The sequential analysis
anova(my.mod, test="Chisq")

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev  Pr(>Chi)
NULL                   399     499.98
gre   1  13.9204       398     486.06 0.0001907 ***
gpa   1   5.7122       397     480.34 0.0168478 *
rank  3  21.8265       394     458.52 7.088e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

# We can make the comparisons by hand (adding a variable in each step)

# model only the intercept
mod1 <- glm(admit ~ 1,                data = mydata, family = "binomial")
# model with intercept + gre
mod2 <- glm(admit ~ gre,              data = mydata, family = "binomial")
# model with intercept + gre + gpa
mod3 <- glm(admit ~ gre + gpa,        data = mydata, family = "binomial")
# model containing all variables (full model)
mod4 <- glm(admit ~ gre + gpa + rank, data = mydata, family = "binomial")

anova(mod1, mod2, test="LRT")

Model 1: admit ~ 1
Model 2: admit ~ gre
Resid. Df Resid. Dev Df Deviance  Pr(>Chi)
1       399     499.98
2       398     486.06  1    13.92 0.0001907 ***

anova(mod2, mod3, test="LRT")

Model 1: admit ~ gre
Model 2: admit ~ gre + gpa
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       398     486.06
2       397     480.34  1   5.7122  0.01685 *

anova(mod3, mod4, test="LRT")

Model 1: admit ~ gre + gpa
Model 2: admit ~ gre + gpa + rank
Resid. Df Resid. Dev Df Deviance  Pr(>Chi)
1       397     480.34
2       394     458.52  3   21.826 7.088e-05 ***

The $p$-values in the output of summary(my.mod) are Wald tests which test the following hypotheses (note that they're interchangeable and the order of the tests does not matter):

• For coefficient of x1: glm(y~x2+x3, family="binomial") vs. glm(y~x1+x2+x3, family="binomial")
• For coefficient of x2: glm(y~x1+x3, family="binomial") vs. glm(y~x1+x2+x3, family="binomial")
• For coefficient of x3: glm(y~x1+x2, family="binomial") vs. glm(y~x1+x2+x3, family="binomial")

So each coefficient against the full model containing all coefficients. Wald tests are an approximation of the likelihood ratio test. We could also do the likelihood ratio tests (LR test). Here is how:

mod1.2 <- glm(admit ~ gre + gpa,  data = mydata, family = "binomial")
mod2.2 <- glm(admit ~ gre + rank, data = mydata, family = "binomial")
mod3.2 <- glm(admit ~ gpa + rank, data = mydata, family = "binomial")

anova(mod1.2, my.mod, test="LRT") # joint LR test for rank

Model 1: admit ~ gre + gpa
Model 2: admit ~ gre + gpa + rank
Resid. Df Resid. Dev Df Deviance  Pr(>Chi)
1       397     480.34
2       394     458.52  3   21.826 7.088e-05 ***

anova(mod2.2, my.mod, test="LRT") # LR test for gpa

Model 1: admit ~ gre + rank
Model 2: admit ~ gre + gpa + rank
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       395     464.53
2       394     458.52  1   6.0143  0.01419 *

anova(mod3.2, my.mod, test="LRT") # LR test for gre

Model 1: admit ~ gpa + rank
Model 2: admit ~ gre + gpa + rank
Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       395     462.88
2       394     458.52  1   4.3578  0.03684 *

The $p$-values from the likelihood ratio tests are very similar to those obtained by the Wald tests by summary(my.mod) above.

Note: The third model comparison for rank of anova(my.mod, test="Chisq") is the same as the comparison for rank in the example below (anova(mod1.2, my.mod, test="Chisq")). Each time, the $p$-value is the same, $7.088\cdot 10^{-5}$. It is each time the comparison between the model without rank vs. the model containing it.

• +1, this is a good, comprehensive explanation. 1 small point: I believe that when test="Chisq", you are not running a likelihood ratio test, you need to set test="LRT" for that, see ?anova.glm. – gung May 23 '13 at 21:49
• @gung Thanks for the compliment. test="LRT" and test="Chisq" are synonymous (it says it on the page you linked). – COOLSerdash May 23 '13 at 21:52
• No problem, but I think it's actually a good point. test="LRT" is better as it is immediately clear that it is a likelihood ratio test. I changed it. Thanks. – COOLSerdash May 23 '13 at 22:03
• +1 I'm impressed with your rapid progress here in just one month and your ability to provide a well-worked, clear explanation. Thanks for your efforts! – whuber May 23 '13 at 22:27
• Great answer. May I ask how the p-values (7.088e-05, 0.01419, 00.03684) should be interpreted? – TheSimpliFire Jan 24 '18 at 20:19