Improved Regression Estimation of a Multivariate Relationship with Population Data on the Bivariate Relationship

Regression coefficients specify the partial effect of a regressor on the dependent variable. Sometimes the bivariate or limited multivariate relationship of that regressor variable with the dependent variable is known from population-level data. We show here that such population- level data can be used to reduce variance and bias about estimates of those regression coefficients from sample survey data. The method of constrained MLE is used to achieve these improvements. Its statistical properties are first described. The method constrains the weighted sum of all the covariate-specific associations (partial effects) of the regressors on the dependent variable to equal the overall association of one or more regressors, where the latter is known exactly from the population data. We refer to those regressors whose bivariate or limited multivariate relationships with the dependent variable are constrained by population data as being ‘‘directly constrained.’’ Our study investigates the improvements in the estimation of directly constrained variables as well as the improvements in the estimation of other regressor variables that may be correlated with the directly constrained variables, and thus ‘‘indirectly constrained’’ by the population data. The example application is to the marital fertility of black versus white women. The difference between white and black women’s rates of marital fertility, available from population-level data, gives the overall association of race with fertility. We show that the constrained MLE technique both provides a far more powerful statistical test of the partial effect of being black and purges the test of a bias that would otherwise distort the estimated magnitude of this effect. We find only trivial reductions, however, in the standard errors of the parameters for indirectly constrained regressors.

Wild Boar and Red Deer Display High Prevalences of Tuberculosis-Like Lesions in Spain

We describe the distribution of tuberculosis-like lesions (TBL) in wild boar (Sus scrofa) and red deer (Cervus elaphus) in Spain. Animals with TBL were confirmed in 84.21% of mixed populations (n = 57) of red deer and wild boar and in 75% of populations of wild boar alone (n = 8) in central and southern Spain (core area). The prevalence of TBL declined towards the periphery of this region. In the core area, the prevalence ranged up to 100% in local populations of wild boar (mean estate prevalence 42.51%) and up to 50% in red deer (mean estate prevalence 13.70%). We carried out exploratory statistical analyses to describe the epidemiology of TBL in both species throughout the core area. Prevalence of TBL increased with age in both species. Wild boar and red deer mean TBL prevalence at the estate level were positively associated, and lesion scores were consistently higher in wild boars than in red deer. The wild boar prevalence of TBL in wild boar did not differ between populations that were or were not cohabiting with red deer. Amongst the wild boars with TBL, 61.19% presented generalized lesions, and the proportion of generalized cases was similar between sex and age classes. In red deer, 57.14% of TBL-positive individuals presented generalized lesions, and the percentage of generalized cases increased with age class, but did not differ between the sexes. These results highlight the potential importance of wild boar and red deer in the maintenance of tuberculosis in south central Spain.