Example 9.7: Transformation to Normality

The MI Procedure

Example 9.7: Transformation to Normality

This example applies the MCMC method to the FitMiss data set in which the variable Oxygen is transformed. Assume that Oxygen is skewed and can be transformed to normality with a logarithmic transformation. The following statements invoke the MI procedure and specify the transformation. The TRANSFORM statement specifies the log transformation for Oxygen. Note that the values displayed for Oxygen in all of the results correspond to transformed values.

   proc mi data=FitMiss seed=37921 mu0=50 10 180 out=outmi;
      transform log(Oxygen);
      mcmc chain=multiple displayinit;
      var Oxygen RunTime RunPulse; 
   run;

The following "Missing Data Patterns" table lists distinct missing data patterns with corresponding statistics for the FitMiss data. Note that the values of Oxygen shown in the tables are transformed values.

Output 9.7.1: Missing Data Pattern

The MI Procedure

Missing Data Patterns
Group	Oxygen	RunTime	RunPulse	Freq	Percent	Group Means
Group	Oxygen	RunTime	RunPulse	Freq	Percent	Oxygen	RunTime	RunPulse
1	X	X	X	21	67.74	3.829760	10.809524	171.666667
2	X	X	.	4	12.90	3.851813	10.137500	.
3	X	.	.	3	9.68	3.955298	.	.
4	.	X	X	1	3.23	.	11.950000	176.000000
5	.	X	.	2	6.45	.	9.885000	.
Transformed Variables: Oxygen

The following "Variable Transformations" table lists the variables that have been transformed.

Output 9.7.2: Missing Data Pattern

The MI Procedure

Variable Transformations
Variable	_Transform_
Oxygen	LOG

The following "Initial Parameter Estimates for MCMC" table displays the starting mean and covariance estimates used in the MCMC process.

Output 9.7.3: Initial Parameter Estimates

The MI Procedure

Initial Parameter Estimates for MCMC
_TYPE_	_NAME_	Oxygen	RunTime	RunPulse
MEAN		3.846122	10.557605	171.382949
COV	Oxygen	0.010827	-0.120891	-0.328772
COV	RunTime	-0.120891	1.744580	3.011179
COV	RunPulse	-0.328772	3.011179	82.747608
Transformed Variables: Oxygen

The following table displays variance information from the multiple imputation.

Output 9.7.4: Variance Information

The MI Procedure

Multiple Imputation Variance Information
	Variable	Variance			DF	Relative Increase in Variance	Fraction Missing Information
	Variable	Between	Within	Total	DF	Relative Increase in Variance	Fraction Missing Information
*	Oxygen	0.000004541	0.000398	0.000404	27.766	0.013685	0.013590
	RunTime	0.000814	0.063128	0.064105	27.708	0.015478	0.015356
	RunPulse	0.182700	3.498974	3.718214	25.923	0.062658	0.060595
* Transformed Variables

The following table displays parameter estimates from the multiple imputation. Note that the parameter value of Mu0 has also been transformed using the logarithmic transformation.

Output 9.7.5: Parameter Estimates

The MI Procedure

Multiple Imputation Parameter Estimates
	Variable	Mean	Std Error	95% Confidence Limits		DF	Minimum	Maximum	Mu0	t for H0: Mean=Mu0	Pr > \|t\|
*	Oxygen	3.845991	0.020091	3.8048	3.8872	27.766	3.843860	3.848775	3.912023	-3.29	0.0028
	RunTime	10.586242	0.253190	10.0674	11.1051	27.708	10.547440	10.616746	10.000000	2.32	0.0282
	RunPulse	170.849654	1.928267	166.8855	174.8138	25.923	170.315955	171.324638	180.000000	-4.75	<.0001
* Transformed Variables

The following statements list the first ten observations of the data set outmi. Note that the values for Oxygen are in the original scale.

   proc print data=outmi(obs=10);
      title 'First 10 Observations of the Imputed Data Set';
   run;

Output 9.7.6: Imputed Data Set in Original Scale

First 10 Observations of the Imputed Data Set

Obs	_Imputation_	Oxygen	RunTime	RunPulse
1	1	44.6090	11.3700	178.000
2	1	45.3130	10.0700	185.000
3	1	54.2970	8.6500	156.000
4	1	59.5710	8.4840	155.503
5	1	49.8740	9.2200	166.031
6	1	44.8110	11.6300	176.000
7	1	43.4130	11.9500	176.000
8	1	44.6435	10.8500	173.761
9	1	39.4420	13.0800	174.000
10	1	60.0550	8.6300	170.000

The preceding results can also be produced from the following statements without using a TRANSFORM statement.

   data temp;
      set FitMiss;
      LogOxygen= log(Oxygen);
   run;

   proc mi data=temp seed=37921 mu0=3.91202 10 180 out=outtemp;
      mcmc chain=multiple displayinit;
      var LogOxygen RunTime RunPulse; 
   run;

   data outmi;
      set outtemp;
      Oxygen= exp(LogOxygen);
   run;

Note that a transformed value of log(50)=3.91202 is used in the MU0= option.

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