Reporting results from transformed analyses

Objective:  Transformed data, for example log(y), is analyzed to correct normality or equal variance requirements.  But we want to report means and standard errors in the original units.

SAS example:
data one;
 do treat=1 to 3;
 do rep=1 to 5;
   y=10 + treat+ exp(rannor(111));
   logy=log(y);
   output;
 end;end;
run;
proc mixed plots=all;
  class treat;
  model y=treat;
  lsmeans treat/pdiff;
run;
proc mixed plots=all;
  class treat;
  model logy=treat;
  lsmeans treat/pdiff;
run;

The original data, variable y, might have units of pounds.  If a transformation is needed, we simply calculate a new variable by applying a mathematical function known to improve normality or equal variance, and run the same analysis on the new variable.  Commonly used choices are listed in the second table below.
However, looking at the results for both analyses we see

treat	Mean Y	SE Y	Mean logY	SE logY	BT Mean	BT SE
1	12.55	0.771	2.52	0.054	12.43	0.67

The mean and standard error for logY are completely "wrong", do not match the data, because they have units of "log pounds".  Not very useful for scientific interpretation.  There are two common remedies:

1) report means and standard errors from the untransformed analysis, but use statistical test p-values from the transformed analysis (normality and equal variance are primarily needed to make p-values accurate).

2) back-transform the transformed results to give them the original units.  We need to calculate the BT values in the above table.

Both choices are in common use, both are statistically acceptable, but always clearly state which you used as it does make a difference in the results.

Back-transformation of the mean is fairly logical, we simply apply the opposite function.  For example exp(log(5)=5, so the exp function will "undo" the log transformation.  In the above table, BT Mean=exp(Mean logY)=exp(2.52)=12.43, reasonably close to the untransformed mean.  Back-transformed means from a log transformation will always be smaller than the original means, because they are correcting for the positive skew, giving a mean that is more like the median.

Back-transformation for standard errors is less logical, as the variance of a function of Y equals the variance of Y times the second derivative of the function evaluated at the mean of Y (see Wikipedia or other statistical theory sources).  The following table lists common transformations, their purpose, and back transformation formulas.  Using the table, we calculate the example BT SE as 0.054*12.43 = 0.67, again believably close to the untransformed SE.

Transform	Formula	Good for...	BT Mean	BT SE
Sqrt	sqrt(Y + tv)	Slight +Skew	mean**2 - tv	2*SE*mean
Log	ln( Y + tv)	Strong +Skew	exp(mean) - tv	SE*(Btmean+tv)
Log10	log10(Y + tv)	Strong +Skew	10**mean - tv	SE*log(10)*(Btmean+tv)
Arcsinsqrt	arcsine(sqrt(Y/tv))	Percentage Data	tv*sine(mean)**2	SE*sqrt(1-Btmean/tv)*sqrt(Btmean/tv)*2*tv
Power	Y**tv	Anything else	mean**(1/tv)	SE*(1/abs(tv)) * mean**[(1-tv)/tv]
Rank	rank(Y)	Last resort	NA	NA

Table notes:  a) tv is a constant used in the first 4 transformations to avoid illegal mathematical operations, such as log or sqrt of a negative number. b) tv for Power transformation can be any number, but usually is between -3 and 3, larger values considered to produce too drastic transformations of the data.  Note that sqrt is a special case of power tv=0.5, and log transformation is approximately tv=0.25.  Negative values might work for negative skew, but there are no guarantees. c) For some BT SE's, the BT mean is calculated first, then used in the BT SE formula.  Otherwise the mean and SE from the transformed analysis are used. d) Log and Log10 have identical transformation properties, either can be used. e) Linear transformations of the form a*Y+b do not help normality or equal variance, but if used can be back-transformed by Btmean=(mean-b)/a, and BT SE=SE/a.  f)  Rank transformations can not be back-transformed, so report means and SE from the untransformed analysis.  Rank is used if no other transformation can be found that corrects normality or equal variance issues.

DANDA.sas:
If using the macro collection, simply choose transtype=[name from first column] and transvalue=tv from second column in the above table.  For the SAS example code above, running
%include 'd:\danda.sas';
%mmaov(one, y, class=treat, fixed=treat, transtype=log, transvalue=0);
will produce all untransformed, transformed, and back-transformed results.