interaction between continuous variables

[heather]

I would like to determine if two continuous variables interact to influence bird occupancy. Is it possible to do this or should interactions in PRESENCE only involve one categorical and one continuous variable?

My variables of interest are temperature (TEMP, range 15.2-21.2) and exurban development (DEV, range 0-4.2). I know that to model and an interaction in PRESENCE I need to include in the design matrix TEMP, DEV, and TEMP*DEV (each standardized by mean and SE), but I am not sure if I have calculated the TEMP*DEV variable correctly. Because the ranges of the two were so different I used the following formula to calculate TEMP*DEV: (TEMP-min(TEMP))*DEV. Then I standardized DEV, TEMP, and TEMP*DEV. Is this a valid way to model this interaction?

Thank you for your help,

Heather

[darryl]

It's absolutely fine to have interactions between 2 (or more) continuous variables (at least theoretically), just like any other regression-type problem. What I suggest you do though is to standarise your TEMP and DEV variables first, that take the product of the standarised variables for your interaction. You don't want to standardise the interaction values separately.
Darryl

[heather]

Hi Darryl,

Thank you for your suggestion. My concern with standardizing TEMP and DEV first is that after doing so each will have both positive and negative values. The product of these will result in a very different interaction term from what I am ecologically interested in. With this method high interaction values will be obtained in sites where TEMP and DEV are very low (negative when standardized) as well as in sites where TEMP and DEV are high (positive when standardized). With the previous method that I mentioned high interaction values are only obtained when TEMP and DEV are high and it seems that this is a more ecologically relevant interaction to examine. Which argument seems stronger to you?

Thanks!
Heather

[darryl]

Don't forget that when you're including interaction terms in a model, you also need to include the main effect terms (unless you have a really good reason for not doing so), so you can't consider just the value of the interaction product in isolation. If you fit 2 regressions to the data, in one case using the raw variables values, and a second one using the standardization I suggested, theoretically you should be able to take 1 set of estimated regression coefficients and get the second set with a bit of algebra and knowing the standardization used. In practice, this doesn't always work because things like round-off error cause problems with the optimisation algorithms used in the software, particularly when you have large raw values.

Recall that the purpose of including the interaction term in a model is allow the effect of X1 to be different for different values of X2. My argument is stronger to me.

[heather]

I have been careful to make sure that I include the main effect terms in each model where I include an interaction term. The terms which I included in the model are:
TEMP = (rawTEMP-mean(rawTEMP))/SD(rawTEMP)
DEV = (rawDEV-mean(rawDEV))/SD(rawDEV)
DEV*TEMP = (rawDEV*(rawTEMP-min(rawTEMP))) - this variable was then standardized by its own mean and SD as done above with TEMP and DEV

Below you said that I don't want to standardize the interaction values separately. Is this what I am doing here when I standardize the DEV*TEMP variable by its own mean and SD? I assumed it would be necessary to do this in order to make it similar to all of the other variables (all standardized) that went into the model and I'm not sure that I understand why it is not a good idea.

[darryl]

It probably doesn't HAVE to be done separately, but the estimated regression coefficients would certainly be more interpretable if you did (unless you want manually rescale the interaction coefficient after the analysis). I'm not claiming to know everything, but my suggestion is that your interaction variable should be the product of the 2 standardised variables.
Darryl

[heather]

Hi Darryl,

I tried your suggestion to use the product of two standardized variables and it looks like it is going to work great. Thank you so much for your help!

Heather