A Quick Distribution-Free Test for Trend That Contributes Evidence of Construct Validity
Sawilowsky, Shlomo S., Measurement and Evaluation in Counseling and Development
A quick distribution-free test that contributes evidence a/construct validity/mm D. T Campbell and D. W. Fisk's (19S9) Multitrait--Multimethod Matrix is presented. The procedure is carried by reducing the heterotrait--heteromethod and heterotrait--monomethod triangles and the validity and reliability diagonals into a matrix of 4 levels containing the minimum, median, and maximum values. The null hypothesis states that the values are unordered, which is tested against the alternative hypothesis of an increasing trend. The test statistic, I, is the number of inversions. The easily remembered critical values/or [alpha] = .05 and .01 are 10 and 14, respectively.
Since 1959, the Multitrait--Multimethod Matrix (MTMM Matrix) by Campbell and Fiske (1959) has become one of the most frequently used approaches in demonstrating evidence of construct validity. In an American Psychological Association Centennial feature, Sternberg (1992) publicized Psychological Bulletin's top ten list of the most cited articles from the early 1940s to the early 1990s (p. 387). The most frequently cited article of that half century was by Campbell and Fiske (1959), with over 2,000 citations. (The second most cited article, with over 900 citations, was also on construct validity.)
An excerpt of a sample matrix provided by Campbell and Fiske (1959) is depicted in Table I. Construct validity evidence within the matrix is obtained on the basis of the desiderata that (a) values in the reliability diagonal are as high as possible, (b) the validity coefficients are sufficiently high to warrant further examination, (c) the validity coefficients are in turn larger than heterotrait--monomethod values, for which in turn the latter are (d) larger than values in the heterotrait--heteromethod triangles. (For a critique of these criteria, see Jackson, 1969, pp. 31-32, and Widaman, 1985, pp. 1-2.)
Campbell and Fiske (1959, p. 82) applied these rules to the synthetic matrix. Their analysis indicated construct validity considerations were met, save "perhaps" values in one of the diagonals, for which they concluded that it "probably" was not met (p. 93). It is instructive that the quoted terms indicate that the interpretation of the matrix is subjective. Campbell and Fiske (1959) considered a variety of real studies, applying heuristic arguments based on the criteria mentioned in the initial paragraph, to form conclusions on the presence or absence of construct validation evidence. By their choice of wording, it is obvious they believed their arguments supporting or failing to support construct validity for those data sets met with varying degrees of success.
Indeed, the interpretation of the matrix is not amenable to straightforward interpretation. The original authors, Fiske and Campbell (1992), noted it is "not surprising that the question of the appropriate statistical analysis of these matrices has no consensual answer" (p. 393) and "the problem is with us 33 years and more than 2,000 citations later" (p. 394). The data contained within the diagonals and triangles often conflict, requiring considerable skill in making the case for or against evidence of construct validity. A perusal of the literature indicates that researchers continue to struggle in making sense of the data contained within the matrix.
Initial attempts at resolving the data analysis problem were based on analysis of variance (ANOVA) models (e.g., Boruch, Larkin, Wolins, & MacKinney, 1970; Boruch & Wolins, 1970; Stanley, 1961). Schmitt and Stults (1986) discussed many criticisms of the ANOVA approach, including the "number of restrictive assumptions" (p. 3) that must be made. Also, Stanley pointed out that important aspects of the matrix cannot be evaluated without repeated data.
A nonparametric analog to the ANOVA approach was presented by Huber and Baker (1978). …