Interpret transformed means.

You have done a statistical analysis using a transformation. The means produced by this analysis are on the transformed scale. For example, if your dependent variable had cm units and a log transformation was done, then the means are in log cm.

These are difficult to interpret. Two alternatives are widely used:

Alternative 1: Report the original means, but always use the statistical test results from the transformed analysis. With this strategy, you would report the first set of means, 54 to 68, but use the A, AB, B letter grouping from the transformed analysis (and the ANOVA test results with transformation).

Alternative 2: The reverse transformation can be applied to the transformed means to convert them back to the original scale. This is referred to as back-transformation. %MMAOV does this for you, and reports the results on the page following the transformed means.

For example:

You have a new dataset to be statistically analyzed. You identify the correct design and model, and run the analysis using %MMAOV. Output of the means is:

Looking at the diagnostics, you realize a log transformation is needed. Adding TRANSTYPE=LOG10 to %MMAOV produces new means:

These means are on the log scale, and obviously have no direct meaning. So %MMAOV automatically produces back-transformed means printed immediately after the above table.

Back-transformed, or "bt" means and standard errors are added at the end of the table.
You could verify that 10 raised to the power 1.7957 does give the first back-transformed mean. This is the reverse of the log10 transformation.

The back-transformed means are not equivalent to means of the original variable, due to the mathematical "twisting" that has been done. For example, the log transformation will produce geometric means, not the usual arithmetic means . Geometric means are generally smaller, as they place less weight on large observations (necessary to reduce the influence of skewed data, the original reason for applying the transformation).

Choose the alternative that you are most comfortable with (probably the original means), or provides the fairest measure of treatment means. With either choice, all statistical test results must come from the transformed analysis.

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