Modelling longitudinal pre-/post-intervention data

Question

I have longitudinal data of a number of patients for which some biomarker was measured at irregular time intervals over the course of several years before treatment and over a shorter period of time (a couple of months) after treatment. I am interested in finding out if there is a significant difference in the biomarker progression rates (slopes) between the pre- and post-treatment phases

My current approach includes one single linear mixed effects model that accounts for the data’s underlying hierarchical structure (estimates per patient, per eye), for both time periods (pre-/post-treatment) as indicated by the binary variable “pre_Treatment”. By doing so, my interpretation of the result is that the estimate for the variable “Time” represents the rate of change (or slope) in biomarker pre-treatment (as this is the reference), and the interaction of Time and pre_Treamtent gives an estimate of the change in slope from pre- to post-treatment, respectively:

lmer(Biomarker ~  Time:(pre_Treatment) + Time:(post_Treatment)  + 
    (Time*pre_Treatment| PatientID:Laterality) + 
    (Time*post_Treatment| PatientID:Laterality)

This framework seems quite suitable for modeling the situation, but I noticed that commonly people are also using two separate models, one for estimating the slope pre-, and one for modeling the slope post-treatment.

• Are there any downsides to this modeling approach I should be aware of / is there anything else I should incorporate?
• Are there any good justifications for why two models are fine/good enough, or is the right thing to do to bring it all into one model?
• Is there some “gold standard” method to handle this type of situation that is commonly accepted?

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One major concern is the imbalance my dataset since although I have up to 5 years of follow up pre-treatment, post-treatment it is much less. In some cases only two data points. This was originally the reason why I opted for mixed models, since it allows the slopes to be drawn from the same distribution and thereby information from all patients/eyes can be used to improve parameter estimation for each individual patient/eye.

However, I noticed that the two distributions from which the slopes per treatment phase (pre- and post-treatment) are drawn from presumably influence each other as well. Since I have much more pre-treatment follow-up data, the slope estimate for the post-treatment phase is being influenced. In particular, the post-treatment slope is more similar to the pre-treatment slope as compared to when I estimate the two slope individually in two separate mixed models (one for pre-, one for post-treatment phase).

Answer 1

From what you wrote, it sounds like you are describing regression discontinuity designs (RDD) and/or regression kink designs (RKD). The plot of a RDD appears as if there is a vertical jump/drop in the y values after treatment/intervention. E.g. measuring testosterone levels before and after hormone therapy would present that way. RKD's plot does not have a jump, but instead presents as an inflection point where the slopes change drastically before and after treatment. E.g, the impact of opioid drug policies on prescription rates and opioid overdose deaths (overall, illicit, and Rx) over time does not have any vertical jumps at the critical points, but they do show drastic changes in slopes that are centered around various factors (most notably the start of the war on pills in 2011).

I have linked a few papers that cover each of the approaches. Are these the gold standard? Not necessarily. It depends largely on the kind of analysis and what data is available. While interrupted time series (ITS) analysis is often viewed as superior, it requires that measurements be obtained at equal time intervals. RDD and RKD, however, do not require this, which is why I opted for these options over ITS.

Feel free to ask any follow up questions if it is not clear. Method Paper: Identification and Estimation of Treatment Effects with a Regression -Discontinuity Design By Hahn et al. DOI: 10.1111/1468-0262.00183

Real World Example: A Regression Discontinuity Test of Strategic Voting and Duverger’s Law∗ https://www.princeton.edu/~fujiwara/papers/duverger_site.pdf

Method Paper: REGRESSION KINK DESIGN: THEORY AND PRACTICE https://www.nber.org/system/files/working_papers/w22781/w22781.pdf

Method Paper 2: Inference on Causal Effects in a Generalized Regression Kink Inference on Causal Effects in a Generalized Regression Kink Design https://research.upjohn.org/cgi/viewcontent.cgi?article=1235&context=up_workingpapers

Real World Example: Assessing the Welfare Effects of Unemployment
Benefits Using the Regression Kink Design DOI: 10.1257/pol.20130248

Answer 2

You seem to be trying to answer a causal question (i.e. did the treatment have an effect). What you describe seems like an unsuitable analysis to do that since you do not have randomized decisions to initialize treatment vs. not. There would a serious concern that treatment e.g. gets started due to particularly concerning high values or a few in a row, which would potentially result in regression to the mean thereafter. All the other issues of non-randomized observational studies are present here, too, and simple pre-/post-comparisons are just not the way to address these. There are many different potential causal inference techniques one could use here to address this.

Besides that, you are assuming that the variability is the same at all measurement timepoints (incl. pre- and post-treatment). The way you set things up, it's hard to say what the random effects you have do in terms of the correlation structure you assume between observations. Presumably, one would thing that there's higher correlation between observations that are closer in time, but with non-standardized observation times this is hard to reflect (but there relatively complex techniques to do so such a Gaussian-Process methods). Why is there no time main effect in the model, just in case that's all that's needed? Some biomarkers may need log-transformation, but without knowing more, I don't know whether that's the case here.

Regarding a single model vs. two models (pre- and post-): two models allow extreme flexibility (no assumption of same variability, factors you adjust for etc.) and in a sense makes fewer assumptions (unless you make the single model sufficiently flexible, at which point the approaches become very similar).