Reformat: Assistance with specification of crossover design model

Fri, Jun 13, 2014 10:33 AM

Thanks very much for your thoughts, Ben. I've added some additional thoughts below
that I hope you'll correct me on. ?

Best,
Adam

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I'm analyzing data (variable names in brackets) generated by a
4-period [period], 3-treatment [trt] crossover design. The design
was strongly balanced (all treatments preceded all others, including
itself) and uniform within period (all treatments occurred the same
# of times in each of the 4 periods).

This produced 18 distinct sequences of treatments (e.g., ABCC, CAAB,
etc.), to which a single individual [id] was randomly assigned. Two
covariates [cov1, cov2] were also measured on each individual.

The response [y] was continuous, and each individual was associated

with a pre-study baseline measure [y0].

Because a single individual was assigned to each sequence, I believe
that a random effect for each individual will capture individual,
baseline, and sequence effects (they're perfectly confounded).

The question is simple: does the response differ among treatments?
I'm hoping the correct specification is as simple as I think it is,
illustrated below with mock data. My specific questions:


Thanks for a very clear question. I'm going to take a stab at
these, but haven't thought *very* deeply about them; hopefully
this will inspire corrections/alternative answers.

1 - Does this model correctly capture treatment, period, and
potential carryover effects? As I understand it, in a strongly
balanced design such as this, carryover and treatment effects are
not confounded, so my assumption is that I don't have to specify any
addition variable to capture this effect (e.g., a variable
indicating the treatment in the preceding period). I'm happy to be
corrected.

I think you're right that the individual-level random effect
controls for pre-treatment and sequence/carryover effects. However,
I'm not sure it will capture period or treatment-by-period effects
(i.e. residuals within the same period might be correlated, and
residuals from individuals who received the treatment in the same
period might be correlated).

On further thought, I think the carryover effects should be modeled
explicitly. In fact, we can distinguish between two types of carryover
effects - so-called 'self' (treatment follows itself) and 'mixed'?
(treatment follows a different treatment) carryover effects. ?A common?
model to analyze in this?case is:

Ydks = mu + Ak + Td(k,s) + I*Sd(k-1, s) + (1-I)*Md(k-1, s) + Us + eks

where:

Ydks = response to treatment d in period k for subject s
Ak = effect of period
Td = effect of treatment
I = indicator if treatment in period k-1 was the same as in period k
Sd = Self carryover effect of treatment d
Md = Mixed carryover effect of treatment d
? ?(NOTE: no carryover effect in period 1; i.e., Sd and Md = 0 in period 1)
? ?(I'm not sure how to distinguish this from treatment "0"; currently in?
? ? period 1, I've set Sd = Md = NA, but this is incorrect b/c it drops from
? ? consideration the observations from period 1)
Us = subject effect (random)
eks = error term

The corresponding LMM is specified, I think, as follows:

datURL <- "https://www.dropbox.com/s/qyer7qrl1ay2h22/xover_test.csv"

# Download data
dat <- repmis::source_data(datURL, sep = ",", header = TRUE)
dat <- within(dat, {
? ? ? ? ? ? ? id = factor(id)
? ? ? ? ? ? ? period = factor(period)
? ? ? ? ? ? ? cov1 = factor(cov1)
? ? ? ? ? ? ? cov2 = factor(cov2)
? ? ? ? ? ? ? selfco = factor(self * prevtrt)
? ? ? ? ? ? ? mixedco = factor((1-self) * prevtrt)
? ? ? ? ? ? ? trt = factor(trt)
? ? ? ? ? ? ? })
str(dat)


# Proposed linear mixed model analysis
require(lme4)
mod1 <- lmer(y ~ trt + period + selfco + mixedco + cov1 + cov2 +?
? ? ? ? ? ? ? ? (1|id), data = dat)
summary(mod1)

I tried including the baseline measurement (y0) as you suggested, but it?
reduced the variance estimate of (1|id) to zero. ?I'm inclined, then, to?
drop the baseline measurement and leave the random intercept for individuals. ?
In fact, the variance is very near zero even when excluding y0.?

I'm still unsure about the trt*period interaction. ?It will eat several degrees of
freedom as a fixed effect. ?Including it as a random effect (1|trt:period) produces
an estimate of zero variance. ?What is my interpretation if treating it as a random
effect?