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Are random intercepts a theoretical/practical prerequisite to random slopes? Why?

I have a three level (rep measures) mixed model where I wouldn't expect lvl 3 variation in initial status of outcome variable. That is, patients were randomly assigned to doctors, so their shouldn't be variation in mean of DV across doctors (all have equal chance of getting patients with high and low initial DV).

Variation exists in intercepts of patients, this is ok.

Question is, is a random slopes only model at lvl 3 allowed. Every example I see always has random intercepts first. Is this because it must or because I just got unlucky with examples?

Also, does anyone know how different numerical structures of the subject/ID variables affect mixed model? I'd. Having a unique identifier for both levels vs having a unique identifier within each group at lvl 2 (individuals).

Also, can someone please explain the benefits, reasons and differences behind mean centered and group centered DVs?

Hope this is clear enough for discussion. Thanks

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  • $\begingroup$ What do you measure at level 1 for each patient? Repeated values of the DV at multiple time points? Also, does each patient see one and only one doctor? $\endgroup$
    – Isabella Ghement
    Commented Apr 4, 2019 at 0:51
  • $\begingroup$ @IsabellaGhement thanks for commenting. In this case the DV is a scale score made from the average of responses on 8 likert scale items. Yes, it's a rep measures design, 3 time points. Yes, one and only one doctor for all three time points. $\endgroup$
    – Forevertrip
    Commented Apr 4, 2019 at 9:27

3 Answers 3

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This all depends on the nature of your study.

When you fit random intercepts, without random slopes, this assumes that each subject has the same response to the treatment, but each subject has a different baseline value:

enter image description here

When you add random slopes, then you allow each subject to have a different response to the treatment, and each subject still has a different baseline value:

enter image description here

It you don't fit random intercepts, but retain the slopes then you assume that each subject has a different response to the treatment, and each subject has the SAME baseline value:

enter image description here

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    $\begingroup$ Excellent response, Wayne! $\endgroup$
    – Isabella Ghement
    Commented Apr 4, 2019 at 14:53
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A classic must-read on some aspects is Bill Venables (the Jekyll subset of Ripley/Venables) https://www.stats.ox.ac.uk/pub/MASS3/Exegeses.pdf.

And as a rare example where fixed zero offsets might be useful: Assume you measure "new-bone growth" after some surgery. By definition newbone is exactly zero at time of surgery, later it is measured and varies.

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Question is, is a random slopes only model at lvl 3 allowed. Every example I see always has random intercepts first. Is this because it must or because I just got unlucky with examples?

Yes, it is certainly allowed, but as illustrated in the plots in the answer by @Wayne, it is making an assumption about the baseline observations being equal at that level.

does anyone know how different numerical structures of the subject/ID variables affect mixed model? I'd. Having a unique identifier for both levels vs having a unique identifier within each group at lvl 2 (individuals).

This will depend on the software that you are using to fit the model and how you specify the model. In lme4 for R, for example, it should not make any difference provided that you specify the random intercepts correctly. See this answer and this answer for more information.

can someone please explain the benefits, reasons and differences behind mean centered and group centered DVs?

Grand-mean centering is often desirable for interpretation purposes, especially when interactions are involved, where the main effects show the effect of a covariate when the other one (in the case of a two-way interaction) is held at zero (which is often implausible in real world cases). In the case of group-mean centering, the random intercepts will be the unadjusted group means. The group means can then be used as a group-level predictor, and the coefficient for this predictor will be a between-group predictor. Personally I have never found the need to do group-mean centering - though I have seen it done, particularly in the multilevel modelling literature.

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  • $\begingroup$ Thanks. How would the first response see change,if any, with the predictor as a factor rather than a continuous. Time is a factor since they are rep measures $\endgroup$
    – Forevertrip
    Commented Apr 5, 2019 at 19:53
  • $\begingroup$ If I understand your point correctly it would make no difference. If time is a fixed effect then whether you model it as continuous or discrete is a modelling decision, and if you have a question about that, you should ask a new question here, or refer to other answers on that topic. $\endgroup$
    – Robert Long
    Commented Apr 5, 2019 at 20:11
  • $\begingroup$ Ok, so the random intercepts specification only ever applies to the initial status. $\endgroup$
    – Forevertrip
    Commented Apr 7, 2019 at 21:15
  • $\begingroup$ We seem to be veering away from the substance of your question, so it would be better if you asked a new question about that, especially since I am not totally clear what you are driving at now, and I wouldn't want to give you incorrect advice. In the meantime please consider marking one of these answers as accepted. Comments are not meant for discussion on substantive issues. $\endgroup$
    – Robert Long
    Commented Apr 7, 2019 at 21:52
  • $\begingroup$ "Yes, it is certainly allowed, but as illustrated in the plots in the answer by @Wayne, it is making an assumption about the baseline observations being equal at that level." "Ok, so the random intercepts specification only ever applies to the initial status." this is all I meant $\endgroup$
    – Forevertrip
    Commented Apr 7, 2019 at 21:58

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