HLM FAQ #5: Confidence Intervals for Random Effects

Question:

How do I obtain a 95% confidence interval for the random effects in my hierarchical linear model?

Answer:

A confidence interval can be obtained through using the standard deviations of the errors of the parameters in which you are interested. For example, you could obtain a 95% confidence interval of a random effect by multiplying the standard deviation of its error by -1.96 and 1.96, as 95% of the values comprising a random effect should be 1.96 standard deviations above or below the value of the random effect. The example below is the unrestricted model, which contains only a random intercept, using the hsb.ssm dataset in the HLM Examples directory:

The outcome variable is MATHACH

Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
For       INTRCPT1, B0
    INTRCPT2, G00          12.636972   0.243628    51.870       159    0.000
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square P-value
                         Deviation     Component
-----------------------------------------------------------------------------
INTRCPT1,       U0        2.93501       8.61431   159    1660.23264    0.000
level-1,       R         6.25686      39.14831
-----------------------------------------------------------------------------

Using the notation from the above table, the level-1 intercept is being estimated from the level-2 equation as follows:

B0 = G00 + U0

The output here shows a value of 12.64 for G00, which is the intercept for the equation used to predict the level-1 intercept, in the Final estimation of fixed effects table. The error term for this parameter is U0 and it has a standard deviation of 2.94. Multiplying this value by -1.96 and 1.96 provides us with values of -5.76 and 5.76. Next, add these values to 12.64 to obtain the confidence range: 6.88 to 18.40.

If you have further questions, send E-mail to stats@ssc.utexas.edu.