ANOVAs on actual Ratio data?

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    Gavin Revie

    I know it is said you can use ANOVAs on ratio data.  By this they are referring to the properties of the scores (e.g. a meaningful zero).

    But what if your scores are actually ratios of one thing to another?  I’ve read that even where x and y have no relationship, the ratio x/y will often look as if there is one.

    The reason I ask is I’ve been asked to help analyse some data from a chemical analysis.  The quantities of 6 different chemicals have been measured from multiple samples.  However the machine used to collect this information was not properly calibrated between each group of 6 measurements, which means the absolute values produced are not comparable between samples.  However the ratios of one thing to another should be trustworthy.

    One possible way of doing this would be to treat Chemical A as having a value of 1 and then score the other 5 as a ratio to Chemical A.  However I’m concerned this may cause problems when plugged into an ANOVA.  An alternative would be to calculate a mean and SD for each group of 6 chemical scores and turn them into z-scores, although doing so within each group of 6 scores sounds to me like a dangerously small sample size for that.  One advantage of using z-scores is that I wouldn’t lose the scores for Chemical A (which would be 1 in every cell using the ratio method).  However I’m unsure whether it is permissible to convert data to z-scores before conducting an analysis on that data.

    I’d welcome any thoughts or suggestions.

    Dave Collingridge

    There is nothing wrong with running some sort of data transformation on data before using them in ANOVA. Sometimes researchers use data transformations to bring data within acceptable parameters before running parametric tests (although most people probably use a non-parametric equivalent test before transforming data). ANOVA analyzes the variance between and within variables. As long as the variability is preserved in a data transformation then everything is good to go.

    Anyway, if you use x/y to create a ratio variable z, it would be acceptable to then run ANOVA on the z variable as long as the data transformation is the same for all cases entered into the ANOVA. If variable z does not appear to be normally distributed for the groups then you may want to use the Kruskal Wallis test which is the non-parametric equivalent of ANOVA. Remember that descriptive statistics like the mean for the groups will refer to the means of ratios. There is no need to worry about the new variable z not having an true absolute zero point. ANOVA produces rigorous results when run on interval data that do not have a true absolute zero value. 

    Gavin Revie

    Thank you for your reply, and for reminding me to check the normality of the transformed data before using a parametric test.  I’ll bear what you said in mind.

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