linear models for longitudinal observational study before - after surgery I'd need an help from you for an analysis.
I'm a physician with an interest in statistics.
A colleague of mine asked me to analyse a dataset.
He had the impression that symptoms get worse after surgery in patients with cancer.
So, he measured patients twice: before (t0) and after (t1) surgery.
He collected a number of measures such as gender, age, cancer type, questionnaires assessing symptoms severity and several instrumental measures.
Some of these measures were collected at t0 only (e.g. age, gender) and others are collected at t0 and t1 (e.g. symptoms, instrumental measures).
He asked me: which are the determinants of symptoms after surgery?
In a sense, he wants to know why patients get worse after surgery and he thinks that at least part of this worsening can be explained by the worsening of some instrumental measures.
Now, consider a symptom S measured at t0 and t1, some baseline characteristics (e.g. gender) and several instrumental measures M measured at t0 and t1.
I see three analyses options and three different models, which all seem reasonable to me (analysis with R).
Analysis 1
I could calculate the deltas DS and DM:
DS = St1 - St0
DM = Mt1 - Mt0
and then run this model:
model_delta <- lm(DS ~ DM)
Analysis 2
model_lm <- lm(St1 ~ Mt1 + Mt0 + St0)
This model evaluates if symptoms at t1 can be explained by the value of an instrumental measure M at t1, after taking into account its baseline value and the baseline symptom severity.
St1 could depend on some baseline value of M, rather than on the new value of M after surgery.
In other words, M could act like a prognostic factor (a patient intrinsic charcteristic), rather than as the "cause" of symptoms after surgery.
For this reason, I think is important to include both Mt1 and Mt0 as regressors (or covariates, in the ANCOVA parlance).
Analysis 3
model_lmer <- lmer(S ~ M + time + (1|ID))
with time, a categorical variable with two levels (t0 and t1) and ID, a categorical variable identifying the different participants.
In analysis 1 and 2, data are in wide format, while they are arranged in long format for analysis 3.
After running these models for each I variable, I would build a multivariate model so that to evaluate the relationship between different instrumental and clinical measures with symptoms. I would choose the variables to include in the multivariate model according to the p < 0.1 rule of thumb.
I would prefer to use model_lm, but I'm not completely sure that this is correct. Does adding Mt1 and Mt0 as covariates in the same model causes a repeated measurement problem?
Also model_lmer looks ok to me, but I think there would be problems with the next step of the analysis. I think it would not possible to run a multivariate model with instrumental measures (which are measured twice) and clinical characteristics such as gender (which are measured only once).
model_delta is the analysis I like the least because I expect that the relationship between delta simptoms (DS) and the baseline simptoms (St0) is non linear and I've read that in this case Mt0 should not be added as regressor to the model.
Feedback from this community about this would be great. In particular I'd ask your help to choose the analysis that is "valid" from a statistical point of view.
2021 06 26
I add an example of the data.frame in long format, after Robert's comment.




ID
time
M
gender




1
t0
3
M


2
t0
4
F


3
t0
5
F


...
...
...
...


1
t1
2
NA


2
t1
2
NA


3
t1
3
NA




My perplexity with the mixed model is that I don't know how to deal with baseline variables such as gender (which have been collected at t0 only) in a model like this:
model_lmer_multivariate <- lmer(S ~ M + gender + time + (1|ID))
Does gender NA at t1 are ok in the model?
Or should the t0 values be inputted also at t1?
Is the 0 variance between t0 and t1 for the gender variable a problem in the model estimation?

The ancova model will be biased if the study was not randomized, and the change score approach does not properly measure causal effects in observational studies.
So the mixed model might be the best approach. The question boils down to:

Does gender NA at t1 are ok in the model? Or should the t0 values be inputted also at t1? Is the 0 variance between t0 and t1 for the gender variable a problem in the model estimation?

gender should not be NA (missing) at t1. A subject's gender at t1 should be the same as their gender at t0. If you have missing values then you need to reshape your data, so that their gender status (and any other variable that does not vary within subjects) is consistent accross both time periods:




ID
time
M
gender




1
t0
3
M


2
t0
4
F


3
t0
5
F


...
...
...
...


1
t1
2
M


2
t1
2
F


3
t1
3
F



