ASSORTATIVE MATING IN THE JEWEL WASP : 2 . SEQUENTIAL CANONICAL ANALYSIS AS AN EXPLORATORY FORM OF PATH ANALYSIS

This paper discusses the conceptual and mathematical relationships between three statistical models: 1) traditional, or “Simultaneous”, Canonical Analysis, 2) hierarchical, or “Sequential”, Canonical Analysis, and 3) Structural Equations Modeling, or confirmatory path analysis. The advantages of Sequential over Simultaneous Canonical Analysis are reviewed. The relationships between Sequential Canonical Analysis and Path Analysis are explored. The need for a legitimate exploratory form of path analysis, analogous to existing exploratory forms of both multiple regression and factor analysis, is discussed. A logical extension of Sequential Canonical Analysis is proposed as adequately serving the function of an exploratory path analysis. Empirical data from psychological research is used to illustrate and qualitatively compare and contrast the results of these three approaches. FIGUEREDO, A. J., AND R. GORSUCH. 2007. ASSORTATIVE MATING IN THE JEWEL WASP: 2. SEQUENTIAL CANONICAL ANALYSIS AS AN EXPLORATORY FORM OF PATH ANALYSIS. JOURNAL OF THE ARIZONA-NEVADA ACADEMY OF SCIENCE 39(2):59-64. INTRODUCTION It is often the case that the multiple outcomes of an experimental treatment or manipulation need to be assessed. Moreover, these dependent variables are often selected to measure the impact of the treatment on several conceptually distinct outcomes, rather than converge upon a single construct. Thus, multivariate data reduction methods, such as common factor modeling, may not be appropriate. On the other hand, because the intervention may exert a common causal influence on these outcomes, the dependent variables are likely to be at least spuriously correlated with each other. In addition, the multiple outcomes may also subsequently exert various causal influences on each other. Thus, separate causal analyses for each of these dependent variables may also not be appropriate. Fortunately, there exist several statistical procedures for the analysis of multiple correlated dependent variables. One of these is structural equations modeling (SEM), or confirmatory path analysis, in which the hypothesized causal network between outcomes can be fully specified, estimated, and tested (Bentler 1989). This method, however, requires the guidance of a strong causal theory which is often not available. Another multivariate method is simultaneous canonical analysis (SIMCA), which requires little theory. This method, however, produces empirically-derived linear composites of dependent variables, or “canonical variates”, which are often difficult to interpret pragmatically in terms of the concrete outcomes that we are interested in. The linear composites are statistically defined, but are dependent upon the exact data set and so may not be replicated if the next study uses a slightly different set of variables. A third alternative is sequential canonical analysis (SEQCA), which combines some of the advantages of both (Gorsuch 1991, Gorsuch and Figueredo 1991). Rather than combining all dependent variables into uninterpretable linear composites, as a simultaneous regression does for independent variables, it partitions their covariance sequentially, as a hierarchical regression does, while maintaining their separate identity. This method isolates the direct effects of the independent variables, or interventions, sequentially on each of the dependent variables, or outcomes, controlling for all indirect effects through the prior dependent variables, or outcomes. Because the only theoretical guidance required is a tentative specification of a meaningful order among the dependent variables, we propose that this sequential method can be developed into an exploratory form of path analysis. The purpose of this paper is to explore the properties of sequential canonical analysis both in relation to those of confirmatory path analysis and simultaneous canonical analysis, and as a potential model for exploratory path analysis. To do this, we will use an example from insect psychobiology which has the advantage of clearly illustrating these principles with very concrete, “brass tacks”, numbers. ASSORTATIVE MATING IN THE JEWEL WASP The natural history of the Jewel wasp and the psychobiological rationale for this study is described in Figueredo and Sage (2007). Basically, this experiment was designed to test the effects of juvenile feeding experience on adult mate choice in the Jewel wasp. To ASSORTATIVE MATING IN THE JEWEL WASP 2 FIGUEREDO AND GORSUCH 60 accurately determine the female choice of male wasps as sires for their offspring, genetically marked wasps were used. Two recessive mutant genotypes were used, one for white eyes and one for red eyes, which, when crossed, produce hybrids with brown eyes. Each female was given a choice of two males, one of each eye-color genotype (which were counterbalanced for the particular prey species, either housefly or blowfly, that they had been fed on as larvae). Females of independently crossed eye-color genotype and juvenile feeding experience were also used. Because these different genotypes might themselves influence mate choice, the experiment was designed to test the relative influence of two factors, 1) inherited eye-color genotype, and 2) conditioned foraging phenotype, on assortative mating in the Jewel wasp. We wound up with a slight, though nonsignificant, negative correlation between the independent variables, female genotype (FG) and female juvenile host (FJ), due to postoperative differential subject mortality. Such correlated independent variables are easily modeled by hierarchical partitioning of variance. To be conservative in our testing of the effects of female host, we assigned causal priority to female genotype. Moreover, we did not hypothesize an interaction between these two factors. The dependent variables, however, presented problems that could not be handled by unaided multiple regression. First, female Jewel wasps that feed on housefly hosts are significantly less fertile than those that feed on blowfly hosts (Figueredo 1987, 1989). This means that we could expect significantly different total numbers of offspring (TOT), regardless of paternity, from subjects reared on different juvenile hosts. Second, a variable number of Jewel wasp offspring remain immature in a dormant condition, called diapause, for up to six months past the normal fourteen days (and, thus, the end of our experiment). The total number of such dormant immature offspring (TIO) is known to vary with certain environmental factors, such as ambient temperatures and photoperiods, but might also be affected in some unknown way by the local mating conditions experienced by the mother. Because these dormant larvae had not yet developed eyes of either color, it was not possible to determine their paternity, perhaps producing nonrandom missing data. Third, male Jewel wasps are always produced asexually by the mother and, therefore, have no biological father at all. Thus, a female Jewel wasp that does not mate with either of the males provided might produce a clutch of all male offspring. The total number of male offspring (TMO) could not be simply ignored in this study because it might have indicated female rejection of both males, and perhaps constituted more nonrandom missing data. Finally, the critical information in this study was provided by the paternity of the female offspring, but had to be controlled for all of the above causally prior offspring outcomes. Thus, the two dependent variables of principal interest were the total number of female offspring sired by red-eyed, rather than white-eyed, males (TRFO), and the total number of female offspring sired by blowfly-reared, rather than housefly-reared, males (TBFO). In the interests of brevity, these female offspring were, respectively, referred to as “red-sired female offspring”, “white-sired female offspring”, “blowflysired female offspring”, and “housefly-sired female offspring”. Of course, the specified eye colors and juvenile hosts referred to are those of the fathers, rather than those of the daughters, because all genetic hybrids had brown eyes, and all posttest offspring were reared on the larger blowfly hosts for convenience. Because paternal genotypes and juvenile hosts were also slightly correlated, by the postoperative mortality noted above, eye color was modeled as causally prior to juvenile host in these two dependent variables, for the same reasons that they were in the two corresponding independent variables. ALTERNATIVE METHODS OF ANALYSIS The question then becomes, precisely how do we accomplish the requisite feats of multivariate statistical control? To expedite our discussion, we report the results of the most uninformative statistical model first, namely, simultaneous canonical analysis (SIMCA). This analysis was performed using PROC CANCORR in SAS (SAS Institute 1989). SIMCA constructed two pairs of canonical variates, V1 and W1, and V2 and W2, to represent our manifest variables. Pillai’s Trace for the whole model was . 21, F(10,158)= 1.856, p= . 055. Furthermore, V1 was correlated to W1 by a canonical correlation of . 42, which was statistically significant (F(10,156)= 1.89, p= . 05); V2, however, was correlated to W2 by a canonical correlation of . 17, which was statistically nonsignificant (F(4,79)= 0.61, p= . 66). Presumably, we could therefore proceed to interpret V1 and W1, and forget about V2 and W2. Fortunately, V1 and W1 were defined by the following linear equations, using the standardized canonical coefficients, as seen in Table 1 above. Using SIMCA, we were also helpfully provided with the resulting indirect correlations between W1 and the independent manifest variables (FG and FJ) and between V1 and the dependent manifest variables (TOT through TBFO), as well as all the corresponding estimates for the nonsignificant pair of canonical variates, V2 and W2, in both their raw and standardized forms. Unfortunately, in spite of this apparent