# 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 '19 at 10:04

• A better way to model nonlinear relationships is via splines instead of polynomials because they have a local nature. If you happen to work in R, you could, for example, load the splines package and use either function ns() or bs().
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.