The applicability of restricted maximum likelihood (REML) in genetic analyses of categorical data was evaluated using simulation and field data. Four genetic models were used to simulate underlying phenotypic variates, which were derived as the sum of additive genetic and environmental effects (Model 1A and 1B) or additive genetic and permanent and temporary environmental effects (Model 2A and 2B). Fifty-eight replicates were simulated, each of which contained 5000 ewes by 500 sires and 5000 dams and with up to five records per ewe. The usual transformation of heritability estimated on the categorical scale to the normal scale for fertility and litter size performed better for a simple animal model than for a repeatability model. Genetic correlation estimates between the two categorical traits for Model 1B and 2B were .49 ± .01 and .48 ± .04, respectively, and were close to the expected value of .50. However, permanent and temporary environmental correlations whose input values were each .50 were underestimated with estimates of .41 ± .05 and .26 ± .03, respectively for Model 2B, and .33 ± .02 for the temporary environmental correlation for Model 1B. Bivariate genetic analyses of litter size with growth and fleece traits were carried out by REML for the data of Suffolk, Targhee and Polypay. Direct heritabilities for most growth traits in all the breeds were low (<.20). Maternal genetic and maternal permanent environmental effects were important for all body weights except for the weaning weight at 120 d for Polypay sheep. Estimates of heritability and permanent environmental effects for litter size for these breeds ranged from .09 to .12 and .00 to .05, respectively. Heritabilities for grease fleece weight and fiber diameter were high for Targhee and Polypay sheep. Direct genetic correlations between growth and litter size were favorable for Suffolk and Targhee but weak for Polypay sheep. Genetic correlations between maternal effects for growth and direct effects for litter size for the breeds were generally small. Within-trait maternal-direct genetic correlations for growth in the breeds were variable and generally negative. Direct genetic correlations of litter size with grease fleece weight and fiber diameter were variable across the breeds. / Ph. D.