Thank you for checking this post out!
I'll use the following example to discuss the challenge I'm facing:
My logistic-exposure model asks whether a study species' nest success (1/0) can be explained by the density of the overstory (a proportion) around each nest, and the distance (meters, continuous) from the nest to the edge of the habitat patch, i.e.:
NestSuccess ~ OverstoryDen + DisToEdge
The independent variables are scaled - mean subtracted, divided by standard deviation, using R function scale().
The model output gives me:
OverstoryDen estimate = 2.91
DisToEdge estimate = 0.87
I am interested in interpreting the output in real, useful terms, but I'm not sure I'm getting this right. This is what I've done:
Let's look at OverstoryDen first: Odds Ratio = e^2.91 = 18.36
So, this implies that the probability of a nest succeeding is 18.36 greater in areas where overstory density if one standard deviation greater, correct?
Now, here's the part that stumps me: The standard deviation of OverstoryDen is 0.146, and, recall, this parameter is a proportion, i.e. 0 - 1. So, can I say anything more general/relevant about the relationship between nest success and OverstoryDen? i.e. would it be prudent to divide 18.36 by 0.146 and say that for every 1% increase in density the probability of success increases by a factor of 2.26? Or is there a linearity issue here?
Similarly, for DisToEdge, the odds ratio = 2.39, sd of the variable = 14 meters, so would it be prudent to divide per meter and say that success increases by 17.1% with each added meter of distance?
Thanks so much for any help you can offer!
First, you should not use the term "probability" as a synonym for "odds." Their numerical values will be similar only when very low (and identical only when both are zero). So an odds ratio of 18 does not say the PROBABILITY of nest success is 18 times greater for each one s.d. increase in overstory density. Rather, it says the ODDS are 18 times greater. Only the latter interpretation is always accurate, because the coefficients from your logit model are linear for the logit function (the log of the odds), not for the probabilities.
Second, if your objective is to interpret the effects in relation to the original units of the variables rather than in relation to their standard deviations, a much simpler approach would be to use the original, unscaled variables in your logit model. With DistToEdge scored in meters, its odds ratio would be directly interpretable as the multiplicative effect on the odds of nest success for each additional meter of distance. You could check to see if that gives you the same effect as the more round-about approach you suggest. If I'm understanding your approach correctly, I think it might, but your approach is not entirely clear to me.
For OverstoryDen, I think it would be handy to first convert it from a proportion to a percentage, by multiplying by 100. Then the odds ratio would be directly interpretable as the multiplicative effect on the odds of nest succes for each additional one percentage point in density. (That's another nuance of terminology worth paying attention to: a one percentage point change is not the same as a 1% change. For example, a one PERCENTAGE POINT change from 19 to 20 is roughly a 5 PER CENT increase.)
Basically agree with Burke D. Grandjean
Add: Are you sure your data matrix needs a logit model?
Burke D. Grandjean Thank you for the kind and thoughtful answer!
(All responses are written with courteousness and a kind tone, even if phrased with brevity)
I agree that it would be convenient for interpretation to use unscaled variables, but I am scaling them because some of the candidate models include variables with different scales and magnitudes. I am scaling to help make all variables similar in range and variance. Are you suggesting this might not be necessary?
To clarify my statistical approach - I'm performing a multi-scale spatial analysis to understand the spatio-structural drivers of nest success/failure. The simplified model I refer to in the original post has independent variables of 2 different spatial scales.
Gioacchino de Candia I am not sure about this, but I'm not a statistician by training and this sounds a little complex to me. Generally, with binary response, and specifically with nest success and similar enquiries, investigators in my field use logistic regression models, or logistic-exposure models (modified link function) when exposure data is available.
Thanks again
You are up against an inherent trade-off between using scaled (aka "standardized") variables vs. the original, raw metrics. Standardizing is convenient for comparing the relative effects of predictors in the model, but inconvenient for interpreting those effects in real-world terms. I generally prefer the ease of interpretation of metric variables, but that's merely my personal taste.
If you generally prefer standardized variables, I suggest you run a (perhaps simplified) model once using standardized variables, and again using the same variables in their metric forms. Use the results to check whether your proposed hand calculations on the standardized results do indeed generate the same conclusions as direct interpretation of the metric results. If so, then you can run all of your candidate models with the standardized variables, and then only do the hand calculations at the end, on those effects that you wish to interpret in their real-world metrics.
Logarithm is used for large and very small numbers with far-flung values
Most folks do not understand odds, and think of odds ratios as being probability ratios. I try to be very careful when describing odds ratios. Regarding whether or not to standardize, consider whether or not your audience will understand how large a one point change in X is, and keep in mind that this may well differ across X variables in your model.