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I'm conducting some multivariate analyses and I'm wondering why MANOVAs have two test statistics. For example, pasted below is an image of output usually produced by (I think) SPSS software. From what I understand, the various values in the second column ("Pillai's Trace", "Wilks' Lambda", etc.) are test statistics for the multivariate effect. However, what I'm unsure about is why there is a value of "F" (AKA another test statistic) in the third column. That is, I'm wondering why we need two test statistics here.

Any answers or especially good references would be helpful.

enter image description here

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  • $\begingroup$ This question is completely confusing, especially its title. I think what you want to ask is: "What does the column named "F" mean in the MANOVA output in SPSS?" Did I understand your question correctly? $\endgroup$
    – amoeba
    Dec 10, 2016 at 17:20
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    $\begingroup$ If i understand the question correctly the reason each statistic (ie Pillai, Wilk's Hotelling, Roy) has an associated F is that their distribution is complicated and transforming to F makes life simpler. $\endgroup$
    – mdewey
    Dec 10, 2016 at 17:23
  • $\begingroup$ When there is 1 dependent variable and I want to know if an independent variable (e.g., in a one-way ANOVA), I examine 1 test statistic (a value of F). In a MANOVA case, there are values like Wilks, Pillai etc, which are in fact test statistics. There are also values of F, which is also a test statistic. Thus, why do we need 2? $\endgroup$ Dec 10, 2016 at 17:27
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    $\begingroup$ It is the same as for the Mann-Whitney $U$ (for example, there are other examples) where it is transformed into a $z$ to enable use of the readily accessible normal tables. Here each of the four tests is converted to an $F$ to enable use of the $F$ tables. $\endgroup$
    – mdewey
    Dec 11, 2016 at 15:51
  • $\begingroup$ @mdewey (+1) Consider writing this up as an answer. $\endgroup$
    – amoeba
    Dec 12, 2016 at 12:48

1 Answer 1

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The primary test statistics are the four values given: Pillai's trace, Wilks' lambda, Hotelling's trace and Roy's largest root. The distribution of each of these is, in general, complicated and so workers derived from each of them an approximation using the $F$ statistic. under some circumstances the approximation is exact and you will note these are flagged in the output. The $F$ is not an extra statistic but only a transformation for convenience. Compare this with other situations like, for example, Mann-Whitney $U$ where it is transformed to a $z$ to enable use of the normal. That $z$ is not a second statistic but the first transformed. In the good old days before electronic computers all we had were tables of the normal, $t$, $\chi^2$ and $F$ so this sort of transformation was essential.

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