Goodness of fit of trend curves and significance of trend differences

Experimental studies of the successive changes (frequently represented by curves describing laws of learning and other similar functional relationships) in a criterion variable accompanying experimental variations in a given treatment, and experimental comparisons of such changes for different populations or for different treatments, constitute a large and important class of psychological experiments. In most such experiments, no attempt has been made to analyze or to make allowance for errors of sampling or of observation. In many others, the techniques of error analysis that have been employed have been inefficient, inexact, or inappropriate. This paper suggests tests, using the methods of analysis of variance, of certain hypotheses concerning trends and trend differences in sample means obtained in experiments of this general type. For means of successive independent samples, tests are provided of the hypotheses: (H ₁) that there is no trend, or that the trend is a horizontal straight line, (H ₃) that there is a linear trend, (H ₅) that the trend is as described by a line not derived from the observed means, and (H ₇) that the trend is as described by a line fitted to the observed means. Tests are also provided of similar hypotheses (H ₂,H ₄,H ₆, andH ₈, respectively) for means of successive measurements of the same sample. Finally, tests are provided of the null hypotheses that there is no difference in trend in two series of means: (H ₉) when each mean in each series is based on an independent sample, (H ₁₀) when each pair of corresponding means is based on an independent sample, (H ₁₁) when each series of means is based on an independent sample, and (H ₁₂) when both series are based on a single sample.