I have a problem with some data I am currently trying to analyze. I have what could be considered a repeated measures MANOVA with measurements at 2 time points (same animals - 2 different ages), and 3 different between subjects factors (male vs female, prenatal treatment vs. control, and postnatal treatment vs. control). The data violate assumptions of normality pretty badly (kurtosis is around +1/-1) and some of my dvs are also failing levene's test. I have no idea what to use to substitute for the repeated measures MANOVA - help?
Dear Amanda
You can always consider to either 1) function-transform your response-variable so that the MANOVA behaves (remember to afterwards inverse-function-transform your estimates back to the original scale) or 2) use a rank-transform, where the actual response-variable-values are replaced with ranks. Very much like the typical non-parametric methods. In 2) the p-value from the statistical hypothesis test is typically valid, but point estimates (of effects on the response variable) are unusable.
1) is a science in itself. One can start with simple Log-, Sqrt- etc. transformation, and then advance to more sophisticated ones such as Power-transformations in general (Box-Cox in particular)
In 2) remember to handle ties, if any.
Good luck!
One way is to fit a model of change as you only have 2 time points
that is regress outcome at time 2 on outcome at time 1 and include predictors for your factor - the normality assumption is then about the residuals and these may be more normal if you are regressing a skewed variable on a skewed variable
See my longer post at https://www.researchgate.net/post/What_are_the_statistics_for_a_cross_sectional_study
First evaluate the assumptions of the MANOVA, you may be surprised that you do not actually violate them. If you do find that this is a problem, you can try transformations, as suggested above. Alternatively, there are extensions of non-parametrics that can be used as a NMANOVA. See below:
http://d-scholarship.pitt.edu/19411/
http://www.entsoc.org/PDF/MUVE/6_NewMethod_MANOVA1_2.pdf
http://apm.sagepub.com/content/5/4/516.extract
http://www.tandfonline.com/doi/abs/10.1207/s15327906mbr1503_4?journalCode=hmbr20
Hi,
I agree that you should carefully assess the assumptions. Is it the raw data, or the model residuals that are skewed? You should be more concerned with model residuals. Here is one potentially helpful reference
Anderson, M. J. (2006). Distance‐based tests for homogeneity of multivariate dispersions. Biometrics, 62(1), 245-253.
Note that with MANOVA, Pillai's trace can be useful (see refs blow).
You should also consider permutation based approaches if there are still problems. There are several options and software available. One would be Multi Response Blocked Permutation procedures (MRBP). You could address (I think) the repeated measures by treating individuals as blocks. See:
Also, PERMANOVA may work depending on balance in your design/samping. See:
Anderson, M. J., & Walsh, D. C. (2013). PERMANOVA, ANOSIM, and the Mantel test in the face of heterogeneous dispersions: What null hypothesis are you testing?. Ecological Monographs, 83(4), 557-574.
Anderson, M. J. (2005). Permutational multivariate analysis of variance. Department of Statistics, University of Auckland, Auckland.
Several of these procedures are implemented in R in the vegan and Blossom packages.
http://cc.oulu.fi/~jarioksa/softhelp/vegan/html/betadisper.html
http://cc.oulu.fi/~jarioksa/softhelp/vegan/html/adonis.html
http://finzi.psych.upenn.edu/library/Blossom/html/mrbp.html
http://www.inside-r.org/node/97996
-Steve
There are some good suggestions above ... but let me add a couple comments. 1) An alternative to the model suggested by Kelvyn is to do a three way ANOVA on the change scores. If your pretest measure is related to your between group factors then the ANCOVA approach suggested by Kelvyn might be biased. 2) Be very careful with nonparametric tests when you violate the variance homogeneity assumptions. They do not test a null hypothesis regarding only location, but instead regarding any distributional differences. Thus, even if the central tendencies are the same you could find a significant effect but it would not be telling you what you want to know.
Good luck!
Robert's point about distributional differences is illustrated very nicely in the simulation study by Fagerland & Sandvik (2009)--see the link below. HTH.
http://onlinelibrary.wiley.com/doi/10.1002/sim.3561/abstract
Robert
It would be useful to know why they would be biased - I could understand imprecise; and I was not advocating change scores that is the response being YNow - Y past but rather Ynow as a response and Ypast as a predictor because of the arguments given in
Gelman, A., and J. Hill. 2006. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press. www.stat.columbia.edu/~gelman/arm/
I strongly agree about heterogeneity being an issue.
and unfortunately Bruce's link did not work for me.
I could not get Bruce's link to work either, but the paper sounds very interesting. Here is a link: http://onlinelibrary.wiley.com/doi/10.1002/sim.3561/abstract
I should be careful when discussing bias, because as you know the hypothesis tested with ANCOVA (as you propose) differs from that of an ANOVA on difference scores (which I was discussing). If the pretest score (a predictor) is related to other predictors in the ANCOVA model then it can be difficult to explore relationships between the predictors and change. In other words, the effect of pretest on posttest is contaminated by the relationship b/w pretest and the other predictors. There is a paper by Daniel Wright entitled "Comparing groups in a before–after design: When t test and ANCOVA produce different results" that has some nice simulations on this topic.
The Fagerland & Sandvik simulation study is also available via ResearchGate (see link below).
Article The Wilcoxon-Mann-Whitney test under scrutiny
Dear Amanda
First you may do transformation for your data and test transformed data distribution again, in case of not effect of transformation , you can use Friedman nonparametric test.
Good Luck
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