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.

Dynamics of the Arctic Fox Population on St. Lawrence Island, Bering Sea

English abstract: We hypothesized that the arctic fox, Alopex lagopus (Linnaeus), population on St. Lawrence Island was cyclic and that its fluctuations in size. structure, and productivity were correlated with the relative size of the population of northern voles, Microtus oeconomus Pallas, the primary prey. Based on a nine-year study, we determined that the variations in size of the fox and vole populations were similar, but they both were of low amplitude and not closely correlated. The high pregnancy rate (mean, 86%/yr) and numbers of young conceived (mean, 11.5/pregnancy) did not vary significantly among years, probably because of the consistently abundant and diverse food supply available to the foxes. The age composition of the trappers' catch of foxes each winter also was comparatively stable, but it was closely correlated with the size of the vole population in the previous summer. The survival of the young foxes during the summer probably was dependent on the availability of the voles, The composition of the catch also appeared to be influenced by immigration of faxes from the adjacent continents via the pack ice. French abstract: Nous avons émis I'hypothèse que la population du renard arctique, Alopex lagopus (Linnaeus), sur I'île Saint Lawrence était cyclique el que les fluctuations concernant sa tailIe, sa structure et sa productivité étaient corrélées à la taille relative de la population du campagnol nordique. Microtus oeconomus Pallas, sa principale proie. En nous appuyant sur une étude menée sur neuf ans, nous avons déterminé que les variations dans la taille des populations du renard et du campagnol étaient semblables. mais que toutes deux avaient une faible amplitude et n'étaient pas corrélées de façon étroite. Le taux de grossesse élevé (moyenne 86 p. cent/an) et Ie nombre dc petits conçus (moyenne 11,5/grossesse) ne variaient pas de façon significative au cours des ans, probablement à cause de I'abondance et de la variété de sources de nourriture pour les renards. La composition d'âge des prises des trappeurs était également stable d'un hiver à I'autre, mais elle était corrélée de façon étroite avec la taille de la population dc campagnols au cours de I'été précédent. La survie des renardeaux au cours de I'été dépendait probablement de la disponibilité des campagnols. La composition des prises semblait aussi être influencée par I'immigration des renards venant des terres continentales adjacentes par la voie de la banquise.