First there is the question of whether it is OK to use percent change as the outcome. In a regression model with baseline as a regressor this is a very bad idea because the outcome is mathematically coupled to the regressor which will induce correlation (ie statistically significant associations) where none is actually present (or mask actual change). This is easy to show with a simulation:
We simulate 2 goups of 100 each where in the first instance there is no change from baseline in either group:
set.seed(15)
N <- 200
x1 <- rnorm(N, 50, 10)
trt <- c(rep(0, N/2), rep(1, N/2)) # allocate to 2 groups
x2 <- rnorm(N, 50, 10) # no change from baseline
So we expect to find nothing of any interest:
summary(lm(x2 ~ x1 * trt))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 45.75024 5.37505 8.512 4.43e-15 ***
x1 0.06776 0.10342 0.655 0.513
trt 3.25135 7.12887 0.456 0.649
x1:trt -0.01689 0.13942 -0.121 0.904
as expected. But now we create a percent change variable and use it as the outcome:
pct.change <- 100*(x2 - x1)/x1
summary(lm(pct.change ~ x1 * trt))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 97.5339 12.7814 7.631 9.93e-13 ***
x1 -1.9096 0.2459 -7.765 4.44e-13 ***
trt 45.1394 16.9519 2.663 0.00839 **
x1:trt -0.7662 0.3315 -2.311 0.02188 *
Everything is significant ! So we would interpet this as: the expected percent change in weight for a subject in the control group with zero baseline weight is 97; the expected change in the percent change in weight for a subject in the control group for each additional unit of baseline weight is -1.91; the expected difference in the percent change in weight between the control and treatment group for a subject with zero baseline weight is 45; and the expected difference in the percent change in weight between the treatment and control groups for each additional unit of baseline weight is -0.77.... All completely suprious !!!! Note also that with a "percent change" variable, then we have to use language like "expected change in the percent change" which does not help with understanding.
Now let's introduce an actual treatment effect of 10,
x3 <- x1 + rnorm(N, 0, 1) + trt*10
summary(lm(x3 ~ x1 * trt))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.95933 0.54404 -1.763 0.0794 .
x1 1.01921 0.01047 97.365 <2e-16 ***
trt 10.78643 0.72156 14.949 <2e-16 ***
x1:trt -0.01126 0.01411 -0.798 0.4260
...all good.
Now again, we create a percent change variable and use it as the outcome:
pct.change.trt <- 100*(x3 - x1)/x1
summary(lm(pct.change.trt ~ x1 * trt))
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.77928 1.23337 -1.443 0.151
x1 0.03439 0.02373 1.449 0.149
trt 49.11734 1.63580 30.027 <2e-16 ***
x1:trt -0.54947 0.03199 -17.175 <2e-16 ***
..more spurious results.
As to the specific models :
Repeated measures ANOVA (with "Weight" as outcome, ["Group", "Time"] as within-factors and adjusting for "subject").
This is one option that could work.
ANCOVA (with "Percent reduction in weight" as outcome, "Group" as between-factor and "baseline weight" as a covariate)
Besides the mathematical coupling problem, this would not control for repeated measures
Linear mixed effects method with "Weight" as outcome, [group, time, group*time] as fixed effects and [subject] as random effect. Again, can we use "Percent reduction in weight" here?
This would be my preferred option, but again, not with percent reduction. This should be equivalent to repeated measures ANOVA. For example with your data:
lmer(wt ~ group*time + age + gender + (1 |Subject, data=mydata)
lme(wt ~ group*time + age + gender, random= ~ 1 | Subject, data=mydata)
You may want to add random slopes by placing one or more of the fixed effects that vary within subjects (only time in this case) to the left of the |, if justified by the theory, study design, and supported by the data. Personally I always start from a model with only random intercepts.
Linear model with interaction: "Percent reduction in weight" ~ "Group" * "Baseline weight"
This should be avoided due to the mathematical coupling problem. Even if baseline was removed as a regressor, this would then just an ANOVA model, and while repeated measures are handled by the percent variable, the residuals may not be close to normal, so inference may be affected.
tl;dr your analysis removes missing data. You lose a bit of power, but the analysis remains unbiased when model assumptions are met. If an assumption is violated, there's not much a missing data method can do for you.
LMER in R will remove missing observations, i.e. a complete case analysis is performed. As with complete case analysis in a likelihood based procedure, depending on the nature of missingness, the resulting analysis is not biased, but is slightly inefficient.
I'm defining a cluster to be repeated measures within subject. Imbalance in cluster size would matter if you used methods for independent data. The resulting analysis would be anti-conservative meaning your 95% CIs would be too narrow to accurately describe the uncertainty. This is because those later observations have a larger residual correlation due to the subject-level uncertainty that is only correctly measured with a mixed model. The mixed model uses a random intercept; this imposes an exchangeable correlation structure meaning repeated measures within subject are positively correlated. The impact of this positive correlation is that repeated measures within subjects are relatively downweighted. This will adjust your power downwards for detection trends in time. Consequently the time by group interaction will be appropriately conservative if model assumptions are not met.
If the power is too low to perform a final analysis, some power may be recovered with integrated likelihood methods, i.e. expectation maximization (EM) or multiple imputation (MI) methods. Accounting for missingness with EM is more tractable than MI, but both are theoretically possible only on the basis that it would improve the efficiency of the analysis. They are complicated to do, require many assumptions, and would raise the eyebrows of a few reviewers.
To underscore all of this, the matter of bias creeps in only when we consider a rather grave type of violation. A model assumption of acute interest is informative loss-to-follow-up. Subjects may drop out early when Group doesn't have the desired effect. This is an issue that may cause subjects contributing more measurements per time to "exaggerate" their specific effects as one which is population-level. The problem with this is that there really is no way to handle this.
One may argue for pessimistic types of imputation such as last observation carried forward (LOCF) or worst observation carried forward (WOCF), each presuming there was at least one negative observation prior to a subject's withdrawal from study.
Best Answer
You have participants nested within doctors, so the random structure of the model should reflect this by estimating random intercepts for doctor and participants within doctors, provided that you have sufficient numbers of both (10 is a typical rule of thumb). I would start with the model:
glmer(outcome ~ intervention*time + (1|Doctor/ID), data=mydata, family=binomial(link=logit)where
intervention*timeis shorthand forintervention + time + intervention:time. I don't know what you mean by "I want time to be within ID". From your description,timeis simply a pre/post indicator variable. You could allow for the effect oftimeto differ among participants (and/or doctors) by adding a random coefficient for time:glmer(outcome ~ intervention*time + (time|Doctor/ID), data=mydata, family=binomial(link=logit)In this formulation, the model will estimate
timerandom slopes for both doctors and participants. If you wantedtimerandom slopes for only participants you would use:glmer(outcome ~ intervention + time + intervention:time + (1|Doctor) + (time|Doctor:ID), data=mydata, family=binomial(link=logit)where this uses the fact that
(1|Doctor/ID)is just shorthand notation for(1|Doctor) + (1|Doctor:ID