# Why is there a difference between chi-square and logistic regression?

I have two categorical variables: gender (male & female) and eye color (blue, brown, & other). And I also have age.

I used the chi-square test and the multinomial logistic regression.

When doing the chi-squared test, I set gender vs eye color. The Pearson Chi-Square and Likelihood Ratio p-values were not significant, meaning there is no association between the two. The Linear-by-Linear Association, was significant though, meaning there is an association between the two. Could this be explained to me, I'm not sure why these are different.

In addition, I also ran the multinomial logistic regression. Eye color was my dependent variable, while gender and age were my independent variables.

Eye color = Gender + Age

When looking through the Parameter Estimates table (other and male are the reference categories), I see that female is significant in relation to blue, but it's not significant in relation to brown. Is this normal to have the chi-square say there is no association between the categorical variables, but the logistic regression say that there is a significant association? If not, what is happening?

{ I am using SPSS }

• Is the difference large? If it's a marginal difference it's probably just the different way the tests are being computed, which is normal. – Paze Jan 29 at 10:13
• @Paze The Pearson Chi-Square p-value is 0.112, the Linear-by-Linear Association p-value is 0.037, and the significance value for the multinomial logistic regression for blue eyes in comparison to gender is 0.013 – Cynical F Jan 29 at 18:25
• Consider uploading your data in CSV/Excel so we can better interpret what is going on. These sound like more than marginal differences. – Paze Jan 29 at 20:42

High $$p$$-values are no guarantees that there is no association between two variables. The high $$p$$-value just means that the evidence is not strong enough to indicate an association. In other words, the lack of evidence for a claim is not the same as evidence for the opposite of the claim. Also, it is not unusual for two tests to say differing things about a statistic; after all, statistics are probabilistic, and it's perfectly possible that unprobable events occur, especially if you are conducting multiple tests.