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I am looking for suggestions on how to deal with uneven time points and missing data in a single-group repeated-measures design.

BACKGROUND: A group of 220 subjects went through a 4-month weight loss treatment, and were given wireless scales to weigh themselves daily at home. The primary outcome is change in weight from pre- to post-treatment (for which I would normally use a paired samples t-test on).

HOW TO DETERMINE POST-TREATMENT WEIGHT?
We have pre-treatment weights for everyone, but post-treatment weight is hard to determine since not all the subjects weighed-in at the same time after treatment (and some dropped out altogether).

APPROACH A: Creating "Time Windows" to Determine Post-Treatment Weight.
I've tried creating arbitrary time windows (1, 2, 4, or 6 weeks around the treatment end date), and then taking the median weight to determine "post-treatment weight." I found (all p's < 0.01):
1 week window: $n = 124$, $-9.55$% weight loss
2-week window: $n = 143$, $-7.52$% weight loss
4-week window: $n = 161$, $-6.39$% weight loss
6-week window: $n = 168$, $-6.30$% weight loss

However, this is problematic because it excludes dropouts. Furthermore, the bigger the window, the bigger the "n", but smaller the effect size (because subjects who did not weigh in regularly also lost the least weight) and less valid the weight in terms of being measured at the same time between subjects.

APPROACH B: Intention-to-Treat with Last Observation Carried Forward. Just using the last observation of all the subjects, up to 2 weeks post-treatment:
$n = 220$, $-4.22$% weight loss

However, this is problematic because the dropouts are depressing the effect size (though retains the largest "n").

APPROACH C: MIXED MODELS?
I've read that mixed models can handle uneven time points and missing data, though I'm not super familiar with this procedure, and am not sure what are the fixed and random are effects here. My attempts to run it in SPSS 21 caused an "insufficient memory" error.

APPROACH D: SOMETHING ELSE? (Multiple Imputation/Regression/ANCOVA, etc.)

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Industry standard for weight loss trials is often last-observation-carried-forward, though (as you've found), there are limitations to that type of data analysis. Mixed models are likely your best bet; I have run them in R with similar data though, and have not tried them with SPSS.

Random effects and fixed effects (though there is some debate about whether they should be called as such) address whether the particular parameter (e.g., intercept, slope) can vary by case (random) or not (fixed). Gelman & Hill's Data Analysis Using Regression and Multilevel/Hierarchical Models provides a fantastic overview and introduction to these types of models, which do handle missing data and uneven time points, as well as R code.

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