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Modeling

With a continuous prior on , you can get only a continuous posterior distribution, and this makes the probability of equal to by definition. To consider as a viable solution to the Box-Cox transformation, you need to use a discrete prior that places some probability mass on the point 0 and allows for a meaningful posterior estimate of .

This example uses a simulation study where the data are generated from an exponential likelihood. The simulation implies that the correct transformation should be the logarithm and should be 0. Consider the following exponential model:

where . The transformed data can be fitted with a linear model:

The following statements generate a SAS data set with a gridded x and corresponding y:

   title 'Box-Cox Transformation, Modeling Lambda = 0';
   data boxcox;
      do x = 1 to 8 by 0.025;
         ly = x + normal(7);
         y = exp(ly);
         output;
      end;
   run;

The log-likelihood function, after taking the Jacobian into consideration, is as follows:

where and are two constants.

You can use the function DGENERAL to place a discrete prior on . The function is similar to the function GENERAL, except that it indicates a discrete distribution. For example, you can specify a discrete uniform prior from to using

   prior lda ~ dgeneral(1, lower=-2, upper=2);

This places equal probability mass on five points, , , , , and . This prior might not work well here because the grid is too coarse. To consider smaller values of , you can sample a parameter that takes a wider range of integer values and transform it back to the space. For example, set alpha as your model parameter and give it a discrete uniform prior from to . Then define as alpha/100 so can take values between and but on a finer grid.

The following statements fit a Box-Cox transformation by using a discrete prior on :

   proc mcmc data=boxcox outpost=simout nmc=50000 thin=10 seed=12567
             monitor=(lda);
    
      ods select PostSummaries PostIntervals;
      parms s2 1 alpha 10;
    
      beginnodata;
      prior s2 ~ gamma(shape=3, scale=2);
      if alpha=0 then lp = log(2);
         else lp = log(1);
      prior alpha ~ dgeneral(lp, lower=-200, upper=200);
      lda = alpha * 0.01;
      sd = sqrt(s2);
      endnodata;
    
      if alpha=0 then
         ll = -ly+lpdfnorm(ly, x, sd);
      else do;
         ys = (y**lda - 1)/lda;
         ll = (lda-1)*ly+lpdfnorm(ys, x, sd);
      end;
      model general(ll);
   run;

There are two parameters, s2 and alpha, in the model. They are placed in a single PARMS statement so that they are sampled in the same block.

The parameter s2 takes a gamma distribution, and alpha takes a discrete prior. The IF-ELSE statements state that alpha takes twice as much prior density when it is 0 than otherwise. Note that on the original scale, . Translating that to the log scale, the densities become and , respectively. The lda assignment statement transforms alpha to the parameter of interest: lda takes values between and . You can model lda on a even smaller scale by dividing alpha by a larger constant. However, an increment of 0.01 in the Box-Cox transformation is usually sufficient. The sd assignment statement calculates the square root of the variance term.

The log-likelihood function uses another set of IF-ELSE statements, separating the case of from the others. The formulas are stated previously. The output of the program is shown in Output 52.2.6.

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