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I am confused about when I can and/or cannot run an ANOVA on percentage data. MY situation is as follows:

I have two separate treatment groups (T1 and T2). Each of these treatment groups receives 1 of 3 drugs (D0, D3, D30) only once. My DV is muscle activity (continuous data). Only one muscle response is recorded. 3 consistent pre-drug baseline measurements and 3 consistent post-drug measurements are taken.

I take the mean of the pre and post to give single values, and then calculate the post-drug as a % of pre drug for each subject, and then compare mean % for each treatment/drug group

The raw data are generally not normally distributed (though one group out of six comes out as normal but only just)

The problem I have is that pre-drug values are quite variable and so I think it's better to look at the proportion change (if indeed that it the correct thing to do?). Previously I have just expressed the post-drug values as a percentage of the pre-drug (not percentage difference), and then compared the mean percentages of the various different groups with a two-way ANOVA. (nb. the percentage data are normally distributed according to shapiro-wilks test)

So for example, pre-drug value = 500, post drug = 100, thus the % baseline is 0.2 (20%).

Is this a valid comparison?? I am less interested in absolute size of pre or post-drug values. It seems quite logical to me but my stats knowledge is pretty basic.

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    $\begingroup$ Can you say more about your situation & your data? Do you have repeated measures? Are the data normally distributed within each group x time combination? $\endgroup$ – gung Sep 13 '15 at 13:57
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    $\begingroup$ Do you have a before & after measure for each participant? How many muscles are measured on each participant? Etc. $\endgroup$ – gung Sep 13 '15 at 14:30
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There is a more and more strongly emerging consensus that you cannot analyze percentage data with ANOVA. Some of the two latest references are Jaeger (2008) and Dixon (2008) the abstract of which I post below.

Jaeger, T. F. (2008). Categorical data analysis: Away from ANOVAs (transformation or not) and towards logit mixed models. Journal of Memory and Language, 59(4), 434–446. http://doi.org/10.1016/j.jml.2007.11.007

This paper identifies several serious problems with the widespread use of ANOVAs for the analysis of categorical outcome variables such as forced-choice variables, question-answer accuracy, choice in production (e.g. in syntactic priming research), et cetera. I show that even after applying the arcsine-square-root transformation to proportional data, ANOVA can yield spurious results. I discuss conceptual issues underlying these problems and alternatives provided by modern statistics. Specifically, I introduce ordinary logit models (i.e. logistic regression), which are well-suited to analyze categorical data and offer many advantages over ANOVA. Unfortunately, ordinary logit models do not include random effect modeling. To address this issue, I describe mixed logit models (Generalized Linear Mixed Models for binomially distributed outcomes, Breslow and Clayton [Breslow, N. E. & Clayton, D. G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Society 88(421), 9–25]), which combine the advantages of ordinary logit models with the ability to account for random subject and item effects in one step of analysis. Throughout the paper, I use a psycholinguistic data set to compare the different statistical methods.

Dixon, P. (2008). Models of accuracy in repeated-measures designs. Journal of Memory and Language, 59(4), 447–456. http://doi.org/10.1016/j.jml.2007.11.004

Accuracy is often analyzed using analysis of variance techniques in which the data are assumed to be normally distributed. However, accuracy data are discrete rather than continuous, and proportion correct are constrained to the range 0–1. Monte Carlo simulations are presented illustrating how this can lead to distortions in the pattern of means. An alternative is to analyze accuracy using logistic regression. In this technique, the log odds (or logit) of proportion correct is modeled as a linear function of the factors in the design. In effect, accuracy is rescaled in terms of a logit “response-strength” measure. Because the logit scale is unbounded, it is not susceptible to the same scaling artifacts as proportion correct. However, repeated-measures designs are not readily handled in standard logistic regression. I consider two approaches to analyzing such designs: conditional logistic regression, in which a Rasch model is assumed for the data, and generalized linear mixed-effect analysis, in which quasi-maximum likelihood techniques are used to estimate model parameters. Monte Carlo simulations demonstrate that the latter is superior when effect size varies over subjects.

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    $\begingroup$ These papers both seem to be about summarizing categorical data as percentages. The question here is about continuous data tabulated as percent change from a control measurement. $\endgroup$ – Harvey Motulsky Feb 28 '16 at 19:33

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