Commentary on McAllister and Gillis (1996) concerning "Estimation of selection differentials from fish scales: a step towards evaluating genetic alteration of fish size in exploited populations

Abstract

We wish to discuss the differences between the approaches of Sheridan (1996) and McAllister and Gillis (1996) in deriving formulae for predicting the response to selection using correlated selection differential. McAllister and Gillis replace Sx of their eq. 2b, namely Rx = hx Sx, with the expected value of Sx given the observed value for S ′y, which they obtain from the regression of Sx on Sy. Sheridan (1996), on the other hand, uses the regression of Sy on Sx, but inverts this relationship when making the substitution. The resolution of the contradiction between the two approaches thus depends on which is the appropriate regression to use. The advice given in the statistical literature is that the appropriate regression to use depends on which quantities are assumed known and which are in error. Thus, for instance, Williams (1959, p. 90) states, “One point on which there is sometimes uncertainty is the regression equation to be employed, that is, which variable is to be treated as the dependent variable. It must first be emphasized that the dependent variable has to be subject to random error in order that the theory on which the method of estimation of the regression equation rests may be applicable; a variable which is errorless . . . cannot be chosen as dependent variable.” Which variable then should be assumed to be in error? The direction of the regression used by McAllister and Gillis (1996) implicitly assumes that the appropriate error is the error made when inferring the selected trait from the measured correlated trait, i.e., that the error term relates to a measurement or prediction process. The method of Sheridan (1996), on the other hand, considers the underlying biological process as the one for which the regression and the associated error term is relevant; i.e., the independent variable is the selection differential of the selected trait, while the dependent variable with the associated error is the apparent selection differential of the correlated trait. As the selection pressure is applied on the actual (though unknown) selection trait, it seems more appropriate to regard it as the independent variable, taken to be without error in the regression. (The very term “correlated trait” in fact also argues for the correlated trait as being considered to be the dependent term, with error). Suppose our model is