[R-sig-ME] SE and CI for ICC

[Martin Maechler]

>>>>> Mohd Masood <drmasoodmohd at gmail.com>
>>>>>     on Sat, 13 Sep 2014 00:20:34 +0800 writes:

    > I am using random intercept logistic model (in lme4) to calculated
    > Intraclass correlation coefficient (ICC). lme4 only provides point
    > estimates and standard deviation (not standard errors) of variance
    > estimates.

Hmm, this is not true:

  > require(lme4)
  > summary(fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy))
  Linear mixed model fit by REML ['lmerMod']
  Formula: Reaction ~ Days + (Days | Subject)
     Data: sleepstudy

  REML criterion at convergence: 1743.6

  Scaled residuals: 
      Min      1Q  Median      3Q     Max 
  -3.9536 -0.4634  0.0231  0.4634  5.1793 

  Random effects:
   Groups   Name        Variance Std.Dev. Corr
   Subject  (Intercept) 612.09   24.740       
	    Days         35.07    5.922   0.07
   Residual             654.94   25.592       
  Number of obs: 180, groups:  Subject, 18

  Fixed effects:
	      Estimate Std. Error t value
  (Intercept)  251.405      6.825   36.84
  Days          10.467      1.546    6.77

  Correlation of Fixed Effects:
       (Intr)
  Days -0.138
  > pfm1 <- profile(fm1) # the main computation for the confint() below:
  > confint(pfm1)
		    2.5 %     97.5 %
  .sig01       14.3814761  37.715996
  .sig02       -0.4815007   0.684986
  .sig03        3.8011641   8.753383
  .sigma       22.8982669  28.857997
  (Intercept) 237.6806955 265.129515
  Days          7.3586533  13.575919
  > confint(pfm1, level = 0.99)
		   0.5 %      99.5 %
  .sig01       11.697963  43.9249121
  .sig02       -0.611183   0.8620229
  .sig03        3.313161  10.1306220
  .sigma       22.149064  30.0282535
  (Intercept) 232.619539 270.1906680
  Days          6.212281  14.7222899
  > 

does provide confidence intervals (CI) also for all variance
parameters, not just the fixed effects 

and you should really also look at

  require(lattice)
  xyplot(pfm1)  

which shows you confidence intervals to a couple of levels simultaneously,
and notably also visualizes how (un)reasonable a Gaussian
approximation to the sigma's would be.

Martin Maechler, ETH Zurich

    > These point estimates can be used to calculated point estimates for ICC. The
    > problem is how can I calculate standard error and confidence interval for
    > ICC. I couldn't find any literature showing formula to calculate confidence
    > interval around ICC.  Or is it not possible to calculate SE and CI for ICC
    > due to skewed sampling distribution (Please see PMCID: PMC3426610).

    > Thanks
    > Masood

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