I would like to conduct a power analysis for a linear mixed model with fixed effects for Treatment(two levels) and Time (four time points: pre, mid, post treatment, 3 months post treatment; each three months apart) and correlated random effects for children (intercepts and slopes). I am using the following code in R using the lmer function:

expdat <- expand.grid(kid = factor(1:500), Time = factor(1:4), Treat = c("XTx", "BAU"))
expdat$obs <- factor(seq(nrow(expdat)))
set.seed(101)
nsim <- 20
beta <- c(100, -7, 8, 15, 20, 0, 0, 0)
theta <- c(15.000000, 7.500000, 7.500000, 7.500000, 12.990381, 4.330127, 4.330127, 
           12.247449, 3.061862, 11.858541)
ss <- simulate(~Treat*Time + (1+Time | kid), nsim = nsim, family = gaussian, 
      weights = rep(25, nrow(expdat)), newdata = expdat, newparams = 
      list(theta = theta, beta = beta, sigma = 1))
expdat$Outcome <- ss[, 1]
fit1 <- lmer(Outcome ~ Treat*Time + (1+Time | kid), data = expdat)
I have the following questions:

In the output I see the variance associated with the random intercepts at each time point and their correlation. However, I cannot find in the output the variance associated with the slopes, and I cannot find information about the correction between the intercepts and the slopes. Where is that information?
How does the lmer know that Time has levels and that it should estimate a unique effect for each level of the Time factor instead of a general slope parameter?
I am attaching the summary statement of the model:
Linear mixed model fit by REML ['lmerMod']
Formula: Outcome ~ Treat * Time + (1 + Time | kid)
Data: expdat

REML criterion at convergence: 22951.8

Scaled residuals: 
Min       1Q   Median       3Q      Max 
-2.55473 -0.47186 -0.00007  0.47268  2.57786 

Random effects:
Groups   Name        Variance Std.Dev. Corr          
 kid      (Intercept) 235.4531 15.3445                
          Time2       230.9249 15.1962  0.43          
          Time3       220.6076 14.8529  0.52 0.53     
          Time4       213.2725 14.6039  0.48 0.51 0.52
 Residual               0.9749  0.9874                
Number of obs: 4000, groups: kid, 500

Fixed effects:
                Estimate Std. Error t value
(Intercept)    100.92006    0.68765  146.76
TreatBAU        -6.94796    0.06245 -111.26
Time2            7.39371    0.68246   10.83
Time3           15.17849    0.66717   22.75
Time4           19.27623    0.65608   29.38
TreatBAU:Time2  -0.16052    0.08831   -1.82
TreatBAU:Time3  -0.05291    0.08831   -0.60
TreatBAU:Time4  -0.14215    0.08831   -1.61

Correlation of Fixed Effects:
            (Intr) TrtBAU Time2  Time3  Time4  TBAU:T2 TBAU:T3
TreatBAU    -0.045                                            
Time2        0.422  0.046                                     
Time3        0.510  0.047  0.528                              
Time4        0.468  0.048  0.506  0.520                       
TretBAU:Tm2  0.032 -0.707 -0.065 -0.033 -0.034                
TretBAU:Tm3  0.032 -0.707 -0.032 -0.066 -0.034  0.500         
TretBAU:Tm4  0.032 -0.707 -0.032 -0.033 -0.067  0.500   0.500

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Gerasimos Fergadiotis, PhD., CCC-SLP
Assistant Professor
Speech & Hearing Sciences
724 SW Harrison St., RM 84B
Portland State University
Portland, Oregon 97201
Phone: 503.725.2217
Fax: 503.725.9171
Web page: http://aaldresearch.wix.com/aald
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


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