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Say, I am trying to predict likelihood to buy ice cream (event =1 , non-event = 0) using logistic regression.

I have only one variable "Gender" (2 value= Male/Female), so, the formula would be like this.

$logit(p)=β_0+β_1∗female$

Odd ratio for female with reference to $male (A) = \frac{odd(female)}{odd(male)}$

Next, I added in "TodayWeather" (2 $value = \frac{Sun}{Rain}$).

$logit(p)=β_0+β_1∗female+β_2*SUN$

Q1) Why does the Odd ratio for A with reference to male change? Shouldn't my Odd ratio for female still be the same as (A) above? Since it is still comparing female and male (i.e. the odd of female to buy ice cream if increase by one unit)?

Q2) Say (A) above is < 1, is it possible that once i added "TodayWeather", the odd ratio become > 1? Why?

Ferdi
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Mei
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  • Your first $\beta_1$ may or may not the same as the second $\beta_1$ since your first $\beta_1$ might be confounded by SUN (the second variable), such as males are more exposed to SUN and expose to SUN may relate to your outcome variable, buying ice cream. You may read something on cofounding, I think. – Deep North Nov 28 '17 at 04:58

1 Answers1

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I suspect that what puzzles you is that you make the fair assumption that someone's sex and the weather are uncorrelated; someone does not become more female when the sun [shines][1]. In linear regression adding a uncorrelated explanatory variable does not change the effects of other explanatory variables. In logistic (or probit, or other non-linear models) it does change these effects. Many consider that a problem, but I don't.

A brief hint of why that is the case is that with logistic regression you model how some numerical representation of how likely we think it is that a certain event happens (the odds). Such an assessment of how likely something is naturally depends on the available information. The available information in logistic regression is represented by the variables included in your model. More on that here.

[1]: It may be possible that when the sun shines women are more likely to go out than men, and you collected data outside, so that my end up causing correlation in your data, but for argument sake, lets assume that is not the case.

Maarten Buis
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