# How should I interpret a factor that is significant both (1) in linear form in interaction with another factor and (2) in its quadratic form?

Our study aims to identify and understand how several ecological factors relate to parasite abundance in a colonial animal. However, we are uncertain on how to interpret a factor (density) that is significant both as an interaction (density*spatial location) and as a quadratic term (density2). Please note that the interaction with density in its quadratic form (density2*spatial location) is not significant.

As we understand, when a factor is significant in both its linear form (e.g., synchrony) and in interaction (synchrony*temperature), the interpretation of its relationship with the response variable should always take into account the other interacting variable, in this example, temperature. In our case, however, the factor in issue is significant in its quadratic form and in its linear form when in interaction. Our main interest is to correctly interpret the relationship between density and parasite abundance. Can we ignore density´s quadratic form and focus only on density's interaction with spatial location? Or do we need to consider both of these relationships for interpretation?

EDIT I analyzed data with generalized linear mixed models (GLMM).

• What sort of analytic framework are you using? ANOVA? Regression? Sep 16, 2014 at 17:15
• We used generalized linear mixed models
– argo
Sep 16, 2014 at 20:57

A good place to start is visualizing your data, e.g. plotting each cell of your design. If both $density$ and $spatialLocation$ are continuous, plot one as continuous along the X axis and the other as separate lines $\bar{x}-s, \bar{x}$, and $\bar{x}+s$. In doing so, keep your axes the same between plots and only draw the line within the range of values observed for the variable you've plotted on the x axis AND for which there are x values in that range for individuals with scores in some reasonable distance relative to the line estimation points (this bit AFAIK doesn't have as much developed in terms of accepted practice but is quite important as the correlation between the first continuous variable and second continuous variable increase). Do this both ways, i.e. having $density$ on the X axis and $spatialLocation$ on the X axis. This exercise alone will make it more clear to you what the appropriate interpretation of your data is. In the context of the GLMM, make sure you've centered your variables and just plot your fixed effects.
Ignoring the data visualization advice I just gave: in regression generally, each slope refers to the case where the variability of all other sources has been controlled for. That is, your quadratic term represents the magnitude of the effect after controlling for the linear effect. Naively, the pattern of results you are describing sounds as as though parasite abundance is a function of density ($density$) but that the magnitude of that function of density varies depending on spatial location ($density * spatialLocation$). There is a non-linear effect of density ($density^2$) but the magnitude of that non-linear effect of density does not vary to a statistically significant degree between spatial locations ($density^2*spatial location$).