I have an independent variable with three levels. And I have two covariates - age and gender. I want to compare cognitive performance of my study participants (total sample size = 58) across the three levels while controlling the effect of the two covariates. I checked the Normal distribution of the cognitive performance measures under each level and found they are not normally distributed. So, I used Quade's Test. However, its statistical power is weaker than ANCOVA - its parametric test. Can anyone suggest alternative to Quade's Analysis? Thanks in adavnce.
You could just use a robust regression version of the current ancova model to
fix any nonnormal.situations in your data.see the screenshot for details. Best wishes, David Booth
Read this text too. It is interesting.
https://www.graphpad.com/support/faq/can-any-graphpad-software-do-analysis-of-covariance-ancova/
Nonparametric tests are less powerful than parametric tests (generally about 4.5% less power, for example, Mood 1954) when the data is normally distributed. However, when the data is not normally distributed, nonparametric tests often have more power than parametric tests. The ANOVA is particularly sensitive to failures of its several critical assumptions including that of normality.
A commonly cited paper in defense of the ANOVA is Blanca et al. (2017) who ran massive Monte Carlo simulations. “In order to examine the effect of non normality on F-test robustness, a one-way design with 3 groups and homogeneity of variance was considered." Like most other studies they focused on Type I error; they did not consider Type II error (the power aspect). However, they used Bradley’s (1978) liberal criterion that a statistical test is considered robust if the empirical Type I error rate is between 0.025 and 0.075 for a nominal alpha level of 0.05 . Thus they went from accepting one false positive in twenty tests to accepting one false positive in thirteen tests. Considering that data is almost never normal, without Bradley’s, extravagant adaption of the meaning of alpha =5%, the ANOVA is not robust. Blanca et al. (2018) then continued their excellent Monte Carlos and revealed additional apparent weaknesses of the ANOVA with respect to other assumptions, and I think they have continued pursuing further understanding of the ANOVA via more Monte Carlos.
Blanca, María J, Rafael Alarcón, Jaume Arnau, Roser Bono, Rebecca Bendayan 2017.Non-normal data: Is ANOVA still a valid option? Psicothema 2017, Vol. 29, No. 4, 552-557.
Blanca, María J, Rafael Alarcón, Jaume Arnau, Roser Bono & Rebecca Bendayan 2018, Effect of variance ratio on ANOVA robustness: Might 1.5 be the limit? Behavior Research Methods volume 50, pages937–962
Mood, A M 1954. On the asymptotic efficiency of certain nonparametric two-sample tests. The Annals of Mathematical Statistics Vol. 25, No. 3 (Sep.), pp. 514-522. P.J.
Bradley, J. V. 1978. Robustness? British Journal of Mathematical and Statistical Psychology, 31, 144-152.
Oops! I just realized that the term "cognitive performance" suggests a Likert scale. That is the different values may not be equidistant; a difference of one unit on one part of the scale might not have the same meaning on another part of the scale. However such scales preserve order and are fine for order based nonparametric statistics. If cognitive performance is not equal value (eg. Likert), statistics using the assumption of normality such as ANOVA type statistics cannot work. There is no 'fixing the non-normality' in such a case. Robustness to non-normality will not help fix some ANCOVA alternative.
An example of this is IQ; a person with an IQ=65 cannot be said to he half as smart as a person with IQ=130. However, folks with IQs equal 65, 90, and 145 can still be ordered as first, second, and third most capable of whatever IQ is measuring (scholarly ability?). Maybe you need to stick to Quade's Test.
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