@pink_poodle: You're welcome
@pink_poodle wrote:
Yes, in both models (with and without intercept) this number estimates the difference between those logits, i.e., the log-odds ratio of level 1 vs. level 2,everything else(i.e. the value of the second predictor)being the same.
Do you mean everything else being zero, like you answered for Scenario 2 ("the first predictor being at its reference level and the second predictor being equal to zero")?
No, I meant that the logit difference is calculated at the same level of the second predictor, regardless of what level this may be. When you look at your equation (1), let D_level1 be 1 (let's call this equation "1a"), then change it to 0 (equation "1b") and finally subtract equation 1b from equation 1a, then you see that the right-hand side equals coefficient_1 while the left-hand side is the difference of two logits: The logit pertaining to level 1 minus the logit pertaining to level 2 of the first predictor. You also see that this doesn't change if you include the intercept into the model: the intercept cancels out when subtracting 1b from 1a. Now do the same with equation (2): To obtain the analogous result the continuous predictor must not change, it has to be "the same" in equations "2a" and "2b". Otherwise the difference "2a − 2b" would yield "coefficient_1 plus coefficient_2 times the difference between the two values of continuous_predictor" on the RHS and not just "coefficient_1" as desired. But it doesn't matter what the common value of the continuous predictor in 2a and 2b is, because it will cancel out in all cases: x − x = 0.
But, in fact, coefficient_1 changes with the addition of the continuous predictor. Also with the intercept, coefficient_1 changes with the addition of the continuous predictor. How would you explain that?
Adding a new predictor changes the model. Hence, all coefficients are newly estimated considering the additional information introduced by the values of the new predictor (and their relationship to the other predictor values and the values of the dependent variable). So, it's quite common that coefficient_1 changes in this situation, especially if the new predictor is correlated with the existing predictor.
Is NOINT only appropriate for a model that contains only the continuous predictors?
Even if all predictors were continuous you would need a good reason for using the NOINT option. I think I've very rarely used it in practice with PROC LOGISTIC (although I did a lot of logistic regressions over the years), rather with linear models, e.g. in PROC REG. Not surprisingly: There are examples of linear regression models where the conclusion "all [predictors] xi are zero ==> [dependent variable] y must be zero" is plausible. It's more difficult to contrive a real-world example of a logistic regression model where the conclusion "all xi are zero ==> the probability of the event must be 1/2" makes sense.
How would you tell if a true intercept is not zero? When would you say it is inappropriate to use the NOINT option?
As mentioned, in most cases it's quite natural to expect that the (true) intercept is not necessarily zero. So, you would omit the NOINT option and let the maximum-likelihood method do it's work without that restriction. Suppose, the estimate of the intercept turns out to be strikingly close to zero (unlike the other estimated coefficients) and, thinking about the model, you realize that the above conclusion ("... probability ... must be 1/2") is sensible based on subject-matter considerations, then you may want to consider a no-intercept model. As a rule, if you're analyzing data from a planned experiment, you will use the type of model that has been defined in advance.
