A robust approach for estimating standard errors of variance components by using quantitative phenotypes from families ascertained through a proband with an extreme phenotypic value is presented. Estimators that use the multivariate normal distribution as a "working likelihood" are obtained by computing conditional ln-likelihoods, conditional first and second derivatives in a Newton-Raphson approach. Robust estimates of standard errors about the estimators are also provided. Tests of hypotheses are based on a modification of the score test, which allows the assumption of multivariate normality to be relaxed. Conditional goodness-of-fit statistics are proposed that can be used to examine the fit of separate pedigrees to the overall model. This robust approach for estimating the standard errors for variance components by conditioning on the proband's phenotype will allow general inferences to be made from the analysis of families ascertained through probands with extreme or unusual phenotypes and should be most appropriate for studying many physiological traits that may be intrinsically nonnormal.