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I'm using a poisson-binominal logsitic regression model to analyze a list experiment (item count technique) where the outcome variable is a binary response of the respondent to a sensitive item (e.g., whether or not a black family moving next door to you will make you angry). I'm using a poisson-binomial distribution rather than the binominal distribution because the probability of giving an affirmative answer to control items differs across items, hence I'm looking at the sum of independent but heterogenous bernoulli random variables, each with different success probabilities.

I'm interested in explanatory variable A (also dichotomous). From this model, the estimated coefficient of explanatory variable A is -2.275 and the standard error is 0.972 (i.e., the coefficient is significant at the 95% level). However, when I calculate the mean difference in predicted values across two datasets where explanatory variable A = 1 and A= 0 (all other covariates are set to their observed values), the mean difference is -0.093 and the standard error is 0.287 (i.e., the 95% confidence intervals cross 0).

What would explain this difference between a significant coefficient estimate and a not significant mean difference in predicted values?

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After some searching and consulting with colleagues, I realize that the discrepancy between a significant coefficient estimate and a not significant mean difference in predicted values can happen anytime there are non-linearities in the link function or via the distribution. In this case there are non-linearities in the link function. A simpler case would be a standard logit where a coefficient statistically different from zero may not have a statistically observable shift in the predicted values in the tails.

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    $\begingroup$ This could actually be due to the fact that the two estimates generalized to different populations. The coefficient in the model is a conditional odds ratio (presumably; in standard logistic regression it is), whereas the difference in probability with all observations set at their observed values is a marginal difference in means. In nonlinear models, marginal and conditional effects tend to differ. I would expect a marginal odds ratio to lead to the same conclusions as would the marginal difference in probability. $\endgroup$
    – Noah
    Commented Feb 14, 2020 at 22:04

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