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Goodness of Fit of Trend Curves and Significance of Trend Differences

Published online by Cambridge University Press:  01 January 2025

E. F. Lindquist
E. F. Lindquist*
Affiliation:
State University of Iowa
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Abstract

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: (H1) that there is no trend, or that the trend is a horizontal straight line, (H3) that there is a linear trend, (H5) that the trend is as described by a line not derived from the observed means, and (H7) that the trend is as described by a line fitted to the observed means. Tests are also provided of similar hypotheses (H2, H4, H6, and H8, 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: (H9) when each mean in each series is based on an independent sample, (H10) when each pair of corresponding means is based on an independent sample, (H11) when each series of means is based on an independent sample, and (H12) when both series are based on a single sample.

Type
Original Paper
Information
Psychometrika , Volume 12 , Issue 2 , June 1947 , pp. 65 - 78
Copyright
Copyright © 1947 The Psychometric Society

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References

* Lindquist, E. F. Statistical analysis in educatinal research, Boston: Houghton-Mifflin, 1940, pp. 180-196.

† Ibid., pp. 235-238.

‡ Ibid., pp. 180-186, 196-203, 235-238.

§ This article (somewhat differently organized), was first used in lithoprinted form in the author’s classes in March, 1945, and is now being published in response te numerous requests that it be made more generally available in permanent form. H. W. Alexander has since published an article on “A General Test for Trend,” Psychological Bulletin, 1946, 43, 533-557, which contains tests for the hypotheses here designated as H2 and H1. Alexander’s tests, while considerably more complex, are superior in that they provide for the possibility of individual differences in regression. Where there is no good reason to suspect heterogeneous regression, the simpler tests here suggested may perhaps be safely employed.

* For methods of fitting curved regression lines, see Fisher, R. A. Statistical methods for research workers, Edinburgh and London: Oliver and Boyd, 1938, Secs. 27-29, pp. 148-177 and Snedecor, George W. Statistical methods, Ames, Iowa: Iowa State College Press, 1940, Chapter 14 (Curvilinear regression), pp. 308-335. The methods there described are for fitting curved regression lines to individual observations, but of course are applicable to means as well. Being concerned with individual observations, however, these discussions do not suggest any tests based upon within-groups variance, such as are necessary if the tests are to be regarded as tests of adequacy of fit.