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Let's say I run the following regression:

fit <- glm(outcome ~ satisfaction, data = df, family = binomial)

------------------------------------------------------------------
                        exp(Est.)    2.5%   97.5%   z val.       p
--------------------- ----------- ------- ------- -------- -------
(Intercept)                 0.413   0.279   0.611   -4.429   0.000
satisfaction                0.744   0.644   0.860   -4.007   0.000
------------------------------------------------------------------

I am interpreting this to mean that a 1-unit change in satisfaction is associated with a decrease in the odds of the outcome by (1-.744)*100 = 25.6%. My question is: is this effect constant at every value of X? That is, can you say "for every 1-unit change in satisfaction the odds decrease by...?" Or is this only true at the mean of satisfaction?

For comparison:

If we were to compute an average marginal effect (e.g. via the margins command), we get the following:

factor          AME 
satisfaction    -0.03943611

Which I'm interpreting as a 1 unit change in satisfaction (when satisfaction is at its mean) results in a 3.9 percentage point decrease in the probability of the outcome. Notice that I now say "when satisfaction is at its mean", because the probability will be different depending where on the logistic curve you are.

Why do we not say "when satisfaction is at its mean" when interpreting the odds ratio. I understand that the log-odds are linear in X in the logit model, but the odds shouldn't be linear... right?

Parseltongue
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    Your initial interpretation is not linear in the odds: it's exactly the same thing to say a one-unit change in $x$ causes a the log odds to change by $-0.256$ as to say the odds are multiplied by $\exp(-0.256).$ Does the explanation at https://stats.stackexchange.com/questions/133623 perhaps answer your questions? – whuber Feb 25 '20 at 20:43
  • @whuber, but is it necessary to say "a one-unit change in x AT THE MEAN OF X" causes a exp(−0.256) decrease... or is this effect true regardless of where on X the one-unit change happens? – Parseltongue Feb 25 '20 at 20:46
  • In other words, why is that, when talking about margins, the interpretation is about a 1 unit change in satisfaction *when satisfaction is at its mean*, but you don't seem to need to make the same proviso with odds. – Parseltongue Feb 25 '20 at 20:47
  • Because the interpretation in terms of odds is *multiplicative,* not additive. E.g., 25.6% of 1 is not 25.6% of 10. – whuber Feb 25 '20 at 20:48
  • Hmm, I'm not following sorry. What do you mean that an interpretation in terms of odds is multiplicative? – Parseltongue Feb 25 '20 at 20:50
  • Please read my answer in the question I referenced: it explains that, with detailed examples and calculations. – whuber Feb 25 '20 at 20:56
  • Thanks so much! – Parseltongue Feb 25 '20 at 21:05

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