Convert Standardized Beta Coefficient Estimates to Raw Data Scale to Interpret Odds Ratios--Logistic Regression

Question

I'm currently working on an analysis where I standardized all data values by subtracting the mean and dividing by the standard deviation (Z-score). However, when I interpret the odds ratios then the scale is unit less (or lacks a tie to my original data values). Therefore, how would you calculate the odds ratios based on standardized coefficient estimates but on the raw variable scale? In other words, my raw variables are distance to features (e.g., 100-m to an agricultural field) but the z-scores are now unit less so I can't make statements currently like: for every 100-meter increase, turkeys are 1.38 times more likely to occur. I would greatly appreciate your assistance!

results<-glmer(ROA1~MP_Scaled_Z*Week1 + MPHW_Scaled_Z*Week1 + HW_Scaled_Z*Week1 + YP_Scaled_Z*Week1 + AG_Scaled_Z*Week1 + SB_Scaled_Z*Week1 + WET_Scaled_Z*Week1 + UB_Scaled_Z*Week1 + (1|Collar_ID),data=turkey.3rdorder,family=binomial)

summary(results)

Answer 1

To convert a standardized beta coefficient estimate to raw data scale in a logistic regression you need the standard deviation of the original raw variable. The the conversion is $OR_{raw} = exp( log(OR_{std})/SD_{raw})$, where $SD_{raw}$ is the standard deviation of the raw variable, $OR_{std}$ is the odds-ratio calculated from the standardized variable (i.e. z-score). You don't need to worry about the subtraction of the original variable's mean because it only affects the interpretation of the intercept.

Answer 2

If you have only standardised the predictor variables (and it is hard to see how you standardise the outcome for logistic regression) then assuming you have the original standard deviations you can back calculate the OR on the raw scale. Suppose for concreteness that the original standard deviation was 10. Your current OR is therefor for a unit increase on the standardised scale which corresponds to 10 units on the original scale. So if you divide by 10 you are back where you started on the raw scale.

Answer 3

It seems that it is not possible to convert back to get Odds for the raw data. The standardized odds ratio should be interpreted as the odds ratio for an increase of one standard deviation in the predictor.