Allow crossed designs

[rpsychologist]

Cluster can be crossed with treatments, e.g. therapists that deliver two interventions. These models are currently not supported by powerlmm.

[rpsychologist]

I'm a bit unsure how to best specify standardized parameters, in a way that's not completely unintuitive.

The lmer model would be
lmer(y ~ time*treatment + (1 + time | subject) + (1 + time + treatment + time:treatment | cluster))

For the raw parameters we have

crossed <- study_design(nested = FALSE)
p <- study_parameters(design = crossed,
                      n1 = 3,
                      n2 = 500,
                      n3 = 100,
                      T_end = 2,
                      fixed_intercept = 4,
                      fixed_tx = 0,
                      fixed_slope = -1,
                      sigma_subject_intercept = 0,
                      sigma_subject_slope = 0,
                      sigma_cluster_intercept = 10, # cc intercept
                      sigma_cluster_slope = 2, # time
                      sigma_cluster_intercept_x = 20, # treatment
                      sigma_cluster_slope_x = 1, # time:treatment
                      cor_cluster_intercept_slope = 0, # cor among 3-lvl RE effects
                      cor_cluster_intercept_intercept_x = 0,
                      cor_cluster_intercept_slope_x = 0,
                      cor_cluster_slope_intercept_x = 0.5,
                      cor_cluster_slope_slope_x = 0.5,
                      cor_cluster_intercept_x_slope_x = 0.5,
                      sigma_error = 10,
                      cor_subject = 0,
                      effect_size = -1
)

However, most of the variance components have no impact on the variance of the fixed time:treatment interaction. Extending the syntax I used for nested 3-lvl models would give something like

p <- study_parameters(design = crossed,
    n1 = 11,
    n2 = 10,
    n3 = 4,
    var_ratio = 0.03,
    icc_pre_subject = 0.5,
    icc_pre_cluster = 0.1,
    icc_pre_cluster_x = 0.25,
    icc_slope = 0.05,
    icc_slope_x = 0.25,
    sigma_error = 10,
    cor_cluster_* = ..., # as above
    cor_subject = 0,
    effect_size = 0.5
)

Here icc_slope is the proportion of slope variance at the cluster-level, whereas icc_slope_x is the proportion of the cluster-level random slopes

 u1 <- object$sigma_subject_slope
 v2 <- object$sigma_cluster_slope
 v3 <- object$sigma_cluster_slope_x

# ICC_slope_x
 v3^2/(v2^2 + v3^2)

# ICC_slope
(v2^2 + v3^2)/(u1^2 + v2^2 + v3^2)

The problem with this approach is that it's not an ICC and it's not the actual amount of slope variance - as that depends on the covariance between some of the RE effects.

For a balanced design, the variance of the fixed time:treatment interaction depend on sigma_cluster_slope_x and not on sigma_cluster_slope

Any input on this issue is welcome!