categorical random effects correlation in lme4

Hi,

I'm currently investigating a question of relative effectiveness of
therapists, and the particular question is whether some therapists are
differentially effective with white versus racial/ethnic minority clients
(this is coded as a binary variable called "white" in this data). We have
conceptualized this as a cross-level random effect, so the model has one
random effect for therapist intercept and one effect for the difference in
effectiveness between their white and nonwhite clients.

I am relatively new to lme4, but I think I have specified the model
correctly (the fixed effects represent client pretreatment severity and the
nonsignificant fixed effect of binary race; they don't seem to impact the
estimation problem). Here's the model of interest:

print(fm1_ml <- lmer(DI ~ first_di + white + (0 +

factor(white)|primary_ther), rem3post, REML=F), corr=F)

The problem is that the two random effects are appearing to correlate at r
= 1.000. I think this is an estimation problem, and probably indicates that
the random variables aren't accounting for all that much variance. I'm
dubious of interpreting this model, therefore. However, when comparing it
to the random intercepts only model using the LRT, there is a significant
difference, suggesting that even though the explained variance is (very)
small, it may be worth including:

anova(fm1_a_ml, fm1_ml)

Data: rem3post
Models:
fm1_a_ml: DI ~ first_di + factor(white) + (1 | primary_ther)
fm1_ml: DI ~ first_di + factor(white) + (0 + factor(white) | primary_ther)
                 Df    AIC      BIC      logLik     Chisq    Chi Df
Pr(>Chisq)
fm1_a_ml   5    4982.7  5011.3  -2486.3
fm1_ml      7    4979.9   5019.9  -2482.9  6.7871      2        0.03359 *

My question is basically this: How should I interpret these results? There
are significant differences between therapists in terms of their relative
effectiveness with white vs. nonwhite clients, but they're just small? Or
is even this not justified? Would it be safer to say that there are likely
no estimable differences? Am I missing something else?

Thanks a lot,
Andrew McAleavey

Here's the model of interest output:

print(fm1_ml <- lmer(DI ~ first_di + factor(white) + (0 +

factor(white)|primary_ther), rem3post, REML=F), corr=F)
Linear mixed model fit by maximum likelihood
Formula: DI ~ first_di + factor(white) + (0 + factor(white) | primary_ther)
    Data: rem3post
   AIC  BIC logLik deviance REMLdev
  4980 5020  -2483     4966    4983
Random effects:
  Groups       Name              Variance        Std.Dev.   Corr
  primary_ther factor(white)    0 0.0417050  0.204218
                  factor(white)1     0.0044086     0.066397   1.000
  Residual                            0.5099596     0.714115
Number of obs: 2263, groups: primary_ther, 192

Fixed effects:
                      Estimate Std. Error t value
(Intercept)        0.45138    0.06007   7.514
first_di             0.48009    0.02360   20.338
factor(white)1   0.01140    0.03298   0.346