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?

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    $\begingroup$ CAn you show us the output of the selected model? $\endgroup$ – user2974951 Jan 9 '19 at 10:04

A couple of points:

  • When you fit polynomials, you typically look at the highest order term. If this seems to be important, you also keep all lower order terms, even if they may seem not to contribute statistically. Moreover, there is no requirement that when the quadratic term of Time is significant that the linear term will also be.
  • 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().
  • You wrote that the questionnaire was administered at 5 points in time. If these time points were not the same for all participants from the start of the study, it would be better to use the actual time points at which it was administered.
  • If your fit is nonlinear and you either use polynomials or splines, the coefficients you obtain do not have a straightforward interpretation. In such cases, it is better to visualize the model using an effects plot, for example, using the effects package in R.
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It is not entirely clear what is being asked. However, a few points are worth mentioning. The first point is although such things can occur, that it is rare that a polynomial model or even a linear one, is a physical process in time. What that often means is that although one can use polynomials to fit any data acquired over a limited length of time, that model only really can be trusted to agree with the data during the time period over which the data was collected, it will not generally extrapolate to predict any withheld past or future data. If you think about what that means, it implies that the such models (when nonphysical) are not reliable as a method of explaining the time based occurrences.

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.

The first step in modeling is to ask what answers do I wish to impose on the data? That is, what do I want to know? Next, one examines the data, (presuming that the questions one asks can be posited with that data), and a model that can represent the presumed physical properties of that data can then be chosen. Finally, interpretation of the results is only a theory, not something that one can draw firm conclusions from, just establish evidence to support or refute some tentative conclusions. None of this is particularly clear in the question. Other models to consider include exponential decay models, logarithms of time, power functions of time and so forth.

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