Browsing by Author "Gross, Alan L."

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    The correction for restriction of range and nonlinear regressions: An analytic study
    (1987) Gross, Alan L.; Fleischman, Lynn E.
    The effect of a nonlinear regression function on the accuracy of the restriction of range correction formula was investigated using analytic methods. Expressions were derived for the expected mean square error (EMSE) of both the correction formula and the squared correlation computed in the selected group, with respect to their use as estimators of the population relationship. The relative accuracy of these two estimators was then studied as a function of the form of the regression, the form of the marginal distribution of x scores, the strength of the relationship, sample size, and the degree of selection. Although the relative accuracy of the correction formula was comparable for both linear and concave regression forms, the correction formula performed poorly when the regression form was convex. Further, even when the regression is linear or concave, it may not be advantageous to employ the correction formula unless the xy relationship is strong and sample size is large.
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    Restriction of range corrections when both distribution and selection assumptions are violated
    (1983) Gross, Alan L.; Fleischman, Lynn E.
    In validating a selection test (x) as a predictor of y, an incomplete xy data set must often be dealt with. A well-known correction formula is available for estimating the xy correlation in some total group using the xy data of the selected cases and x data of the unselected cases. The formula yields the r[subscript yх] correlation (1) when the regression of y on x is linear and homoscedastic and (2) when selection can be assumed to be based on x alone. Although previous research has considered the accuracy of the correction formula when either Condition 1 or 2 is violated, no studies have considered the most realistic case where both Conditions 1 and 2 are simultaneously violated. In the present study six real data sets and five simulated selection models were used to investigate the accuracy of the correction formula when neither assumption is satisfied. Each of the data sets violated the linearity and/or homogeneity assumptions. Further, the selection models represent cases where selection is not a function of x alone. The results support two basic conclusions. First, the correction formula is not robust to violations in Conditions 1 and 2. Reasonably small errors occur only for very modest degrees of selection. Secondly, although biased, the correction formula can be less biased than the uncorrected correlation for certain distribution forms. However, for other distribution forms, the corrected correlation can be less accurate than the uncorrected correlation. A description of this latter type of distribution form is given.