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I am trying to fit a multilevel model for a longitudinal repeated measure design with two levels: I have 50 participants with a continuous dependent variable that was measured at three time points. When I add "time" as a predictor in the model, its fixed effect is not significant (which means there is no change over time) but when I add both time and time squared (quadratic), both are significant. By plotting the data it is quite clear that the growth curve is u-shaped and I can see reduction at the second timepoint and then increase in the dependent variable which returns to the baseline values. I wanted to know whether it is correct to add "time squared" to the model even when "time" is not significant, and then to report that there is no change over time but there is a significant reduction at the second time point?

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First: "its fixed effect is not significant (which means there is no change over time)" - this is not correct. It is a common misinterpretation of non-significance.

Second, there's nothing special about multilevel models when it comes to non-linear effects. The way that you should consider them is the same way as you should in least squares regression. So yes, it's correct to add time squared, even though time is not statistically significant.

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  • $\begingroup$ Thanks for your reply. One more question. If I add time squared to the model, is it a necessity to add it to all interaction terms, too? I need to add a cross-level interaction to my model (between time and group (coded 0 and 1)) but I am not sure if I must also add interaction between group and time squared term. $\endgroup$
    – Sanam
    Commented Oct 23, 2021 at 5:28
  • $\begingroup$ You don't have to add the non-linear interaction. $\endgroup$ Commented Oct 25, 2021 at 21:36
  • $\begingroup$ The linear interaction says "Does the slope change", the non-linear says "Does the rate of change of the slope change." $\endgroup$ Commented Oct 25, 2021 at 21:36

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