Author(s): Ricker WE
A number of regression situations in fish and fishery biology are examined, in which both of the variates are subject to error of measurement, or inherent variability, or both. For most of these situations a functional regression line is more suitable than the ordinary predictive regressions that have usually been employed, so that many estimates now in use are in some degree biased. Examples are (1) estimation of the exponent in the weight:length relationship, where almost all published values are somewhat too small; and (2) estimating the regression of logarithm of metabolic rate on log body weight of fish, where the best average figure proves to be 0.85 rather than 0.80. In the very common situation where the distribution of the variates is non-normal and open-ended, a functional regression is the most appropriate one even for purposes of prediction. Two ways of estimating the functional regression are (1) from arithmetic means of segments of the distribution, when computed symmetrically; and (2) from the geometric mean of one predictive regression and the reciprocal of the other. The GM regression gives a more accurate estimate when it is applicable; it is appropriate in all situations where the variability is mainly inherent in the material (little of it due to errors of measurement), or where the measurement variances are approximately proportional to the total variance of each variate; and it is the best estimate available for short series with moderate or large variability even when neither of these conditions applies. When error in X results solely from the measuring process the predictive regression of Y on X is also the functional regression if observations of X are not taken at random but rather have pre-established values, as is usual in experimental work. The uses of the various regressions are summarized in Table 8.
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