# Interpreting linear and polynomial predictors in LMM

I am using linear mixed models to investigate change over time on the score on a questionnaire, which was administered at 5 points in time. While I hypothesized a decline, I had no a priori assumptions as to how this decline would look (whether it's linear or not). After visually checking the trajectory, I did see the slope become steeper after the second time point, and thus I thought I'd need to add polynomials and see what model fits the data better—but correct me if I'm wrong.

I created an initial model (Model 1) with just the variable Time as a predictor of Score. Here, Time was a significant predictor (β = -3.8, p < .001).

Next, I wanted to know whether a quadratic or cubic slope model fitted my data better, so I added a second-order polynomial (Time * Time), along with Time, in Model 2. Model fit significantly increased (comparing -2LL between Models 1 and 2 with a chi square test). Here, the polynomial was significant (β = -4.6, p = .007), which I believe suggests a non-linear slope for decline in Score over time. However, Time was no longer significant (β = 0.2, p = .905), yet in all examples I could find, both the linear and quadratic predictors were significant.

My question is, how should I interpret the fact that Time is not a significant predictor in this model? Can I still conclude that a quadratic slope best represents my data, even if the linear slope is no longer significant?

• CAn you show us the output of the selected model? – user2974951 Jan 9 at 10:04

The second point is that time itself need not be an unmodified explanatory variable for a time based process. For example, for radioactive decay, one tends to use exponential models ($$M_0 e^{-\lambda t}$$). When you say, time is no longer a variable, presumably the time squared term is still being used, and a time based process does not have to have time itself as a bare naked variable. However, this is not entirely clear in what you related and I agree with user2974951's statement in so far as showing us the model may help. More important is to state what the properties of the data are; for example, if any of the data can have negative values. If the data cannot have negative values, then a model with a negative linear slope cannot properly model it, in a physical sense, because there will come a time when the linear model will predict negative amplitudes.