[R-sig-ME] Reformat: Assistance with specification of crossover design model

[Ben Bolker]

Adam Smith <raptorbio at ...> writes:

> 
> Apologies for the formatting in the initial post.
> 
> Hello list,
 
> 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).

  One way to see whether you might have missed something would
be to draw (e.g.) boxplots of residuals by whatever groupings
you might be concerned about.

 
> 2 - Is an interaction between treatment and period prescribed?
> Because the design is uniform within periods, I believe that period
> and treatment effects are likewise not confounded.

  I think so.
 
> 3 - Does the random effect for each individual captures variation
> due to individual, sequence, and baseline measurement?

  I think so.
 
> Thanks very much for your assistance,
> 
> Adam Smith
> Department of Natural Resources Science
> University of Rhode Island
> 
> # Download data
> datURL <- "https://dl.dropboxusercontent.com/u/23278690/xover_test.csv"
> dat <- repmis::source_data(datURL, sep = ",", header = TRUE)
> dat <- within(dat, {
>      id = factor(id)
>      trt = factor(trt)
>      period = factor(period)
>      cov1 = factor(cov1)
>      cov2 = factor(cov2)
> })
> 
> require(lme4)
> # Proposed linear mixed model analysis
> mod1 <- lmer(y ~ trt + period + cov1 + cov2 + (1|id), data = dat)

  I think you do want trt*period (possibly as a random effect?)
  In a setting where I had access to some regularization (e.g.
blme) I would be tempted to treat period as random -- but I wouldn't
do it in this vanilla model, because there aren't enough periods to
estimate it reliably.

  While you don't _need_ to incorporate pre-treatment covariate,
carryover effects, etc., it might be interesting -- I think these
would be partitioned out of the among-individual variation ...

> # Test global treatment effect, for example...
> mod2 <- update(mod1, . ~ . - trt)
> anova(mod1, mod2)
> 
> sessionInfo()
> 
> R version 3.0.3 (2014-03-06)
> 
> Platform: x86_64-w64-mingw32/x64 (64-bit)
 
> attached base packages:[1] stats graphics grDevices utils datasets
  methods base
 
> other attached packages:[1] car_2.0-19 AICcmodavg_1.35 lme4_1.1-5
> Rcpp_0.11.1 Matrix_1.1-3 plyr_1.8.1