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I conducted an experiment where people either endorsed or did not endorse a statement at the commencement of a one-week training session, at the end of the one-week training session, and at a follow-up interview one week later. I hypothesised that the odds of endorsing the statement would increase between pre-training, and post-training measurement, but would decrease between post-training and follow-up. Here is the graph of the proportions endorsing the statement at each time point (pre-training = pre, post-training = post, follow-up = fu).

enter image description here

I turned the three-level time predictor into an ordered factor (pre < post < fu) in order to estimate linear and quadratic trends across the three time points and ran a repeated-measures logistic regression (i.e. a mixed effects model).

Here is the output of that analysis

            Estimate Std.Err z-value  p-value
(Intercept)   2.9262  0.5042  5.8036  < 1e-04
timeOrd.L     0.8594  0.4039  2.1278 0.033354
timeOrd.Q    -1.6452  0.7628 -2.1566 0.031034

After exponentiating the log-odds coefficients and confidence intervals these are the results.

                        or lowCI   hiCI     p
linear trend          2.36  1.07   5.21 0.033
quadratic trend       0.19  0.04   0.86 0.031

So for the linear trend I gather the interpretation would be "the odds of endorsing the statement at each time point increase an estimated 2.36 times, compared to the previous time point."

However, I am struggling with how to interpret the quadratic trend. This post has some clues but that is for a model where the time predictor is continuous rather than a factor with three levels.

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The coefficients for the linear and quadratics terms cannot be interpreted in isolation. The main point of using ordered factors is to see if you in you can advantage of the natural ordering of the levels and use less coefficients to describe the relationship with your outcome.

For example, say that you have a factor with a natural ordering of its levels, e.g., none, low, medium and high. If you treat it as an ordinary factor, then you need three coefficients to describe the relationship with the outcome. You could also treat as an ordered factor, in which case you get the coefficients for the linear, quadratic and cubic terms. This fit will be exactly equivalent with the fit when you treat as an ordinary factor. But perhaps you can see from the output that the quadratic and cubic terms are very small and statistically significant. Then you see that you described the relationship of this factor with the outcome with just one coefficient and not spend three.

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    $\begingroup$ Thanks you for your answer @Dimitris Rizopoulos. So you are saying that you cannot interpret the coefficients (either as log odds or odds ratios) in any direct way, the way you could if you were comparing odds of endorsement between individual levels of the time predictor? So what could you say with my results. Could you say that the positive linear log-odds coefficient suggests odds of endorsement increase over time, but that the negative quadratic log odds coefficient suggests that the odds peak at post and then decrease again? Something like that? $\endgroup$
    – llewmills
    Mar 21 '20 at 11:43

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