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Factorial Design and Covariance in the Study of Individual Educational Development

Published online by Cambridge University Press:  01 January 2025

Palmer O. Johnson  and
Fei Tsao
Palmer O. Johnson
Affiliation:
University of Minnesota
Fei Tsao
Affiliation:
University of Minnesota
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Abstract

This is the report of the application of the principles of factorial design to an investigation of individual educational development. The specific type of factorial design formulated was a 2 Χ 3 Χ 3 Χ 3 arrangement, that is, the effect of sex, grade location, scholastic standing, and individual order, singly and in all possible combinations was studied in relation to educational development as measured by the Iowa Tests of Educational Development. An application of the covariance method was introduced which resulted in increased precision of this type of experimental design by significantly reducing experimental error. The two concomitant measures used to increase the sensitiveness of the experiment were initial status of individual development and mental age. Without these statistical controls all main effects and two first-order interactions would have been accepted as significant. With their use only sex (doubtful), scholastic standing, and individual order demonstrated significant effects. The chief beauty of the analysis of variance and covariance as an integral part of a self-contained experiment is demonstrated in the complete single analysis of the data. The statistical utilization of the experimental results has also been developed for purposes of estimation and prediction. The mathematical statistician is being continuously required to develop and analyze experimental designs of increasing complexity since the introduction of the analysis of variance and covariance. The mathematical formulation and solution of the problem of this investigation is carried out. The methods illustrated and explained in this study, and modifications and extensions of them are capable of very wide application. The general principles can be used to various degrees and in a number of ways.

Type
Original Paper
Information
Psychometrika , Volume 10 , Issue 2 , June 1945 , pp. 133 - 162
Copyright
Copyright © 1945 The Psychometric Society

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Footnotes

*

For the research grant to finance this study, grateful acknowledgment is given to the Graduate School, the University of Minnesota.

†

Johnson, Palmer O. and Tsao, Fei. Factorial design in the determination of differential limen values. Psychometrika, 1944, 9, 107-145.

References

* Ibid., p. 108.

† Fisher, R. A. The design of experiments. London: Oliver and Boyd, 1944, pp. 111-113.

* College of Education, State University of Iowa, Iowa City, Iowa, 1942.

† We designed the investigation so as to secure equal representation in the sub-classes. While this equality is not an essential condition, it is highly desirable in order to avoid rather complicated mathematical formulation thus leading to a more laborious statistical analysis

* The chronological age o f every individual in grade 10 s 15 years: grade 11, 16 years: grade 12, 17 years. A hundred has been subtracted from Mental has been subtracted from Mental Age (in terms of months). Moreover it has been assumed the mental test was admimstered on the 21st of May in 1939.

* Fisher, R. A., and Yates, F. Statistical tables. London: Oliver and Boyd, 1938, pp. 82-87.

† It is customary to call the interaction involving two factors, an interaction of first order; to call the interaction involving three factors, an interaction of second order; etc. Sometimes for the sake of brevity, the terms single, double, and triple, interactions are used to denote the interactions of first order, second order, and third order, respectively. For instance, we may call the interaction sex Х grade Х scholastic Х individual, an interaction of third order. Or we may call it a triple interaction. Particularly in this study, this interaction may be regarded as of highest order.

* The hyvothesis tested is a null hypothesis regarding the variation in the same row. For instance, the hypothesis regrading is that there is no significant difference between grade-means.

* Where b = .64

† Near the borderline of 5% significance and insignificance.

‡ The hypothesis tested is a null hypothcsis concerning the variation in the same row. For instance, the hypothesis regarding sex is that there is no significant difference between sex-means when the effect of initial sectors has been partialled out.

* See pp. 15960, Part II, for explanation.

† See pp. 160-162, Part II, for the mathematical formulation of the procedures employed.

‡ By ordinary methods, we obtain the following three zero-order correlations from the scores of all the individuals: r1y = .96, r2y = .82, r12 = .81, where y denotes the final score; 1 denotes he initial score; and 2 denotes the mental age score. The first-order correlations with the third variable partialled.

* In the equations that follow we should like to state here the distinction in notation used: ∼ is used as the symbol of prediction and ^ of estimation.

† Errors of estimation can be calculated in the usual manner.

‡ See page 160, Part II for the development of this equation. The values of b1 and b2 were calculated from the residual component (see Table 8).

† The details of the method of calculating the regression equations are not described here. As we wish to use the analysis of variance for testing the goodness of fit, it is convenient to use orthogonal polynomials. The interested reader is recommended to refer to the following sources:

Goulden, C. H. Methods of statistical analysis. New York: John Wiley, 1939, pp. 219-246.

Anderson, R. L., and Houseman, E. E. Tables of orthogonal polvnomial values extended to N = 104. Agricultural Experiment Station. Iowa State College of Agricutture and Mechanics Arts, Research Bulletin 297, 1942, pp. 595-606.

* Fisher, R. A., and Yates, F. Statistical tables. London: Oliver and Boyd, 1938, pp. 54-59.

* Johnson and Tsao , op. cit., pp. 128-143.

* Fisher, R. A. Statistical methods for research workers, (7th edition). London: Oliver and Boyd, 1938, pp. 281-294.