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Macro to test multivariate normality

PURPOSE:

The %MULTNORM macro provides tests and plots of multivariate normality. A test of univariate normality is also given for each of the variables. A chi-square quantile-quantile plot of the observations' squared Mahalanobis distances can be obtained allowing a visual assessment of multivariate normality. Univariate histograms with overlaid normal curves are also available.

HISTORY:

Version	Update Notes
1.4	SAS/IML is no longer required if SAS/ETS PROC MODEL is not found. SAS/STAT PROC PRINCOMP is required instead. Univariate plots, if requested, and tests are now presented first. High resolution plotting is done by Base SAS PROC SGPLOT if available, or by SAS/GRAPH PROC GPLOT if not. Checks that the specified data set and variables exist.
1.3	Added message showing whether MODEL or IML is selected. Added check for error status after MODEL or IML. Errors will terminate macro. Added automatic check for newer version. Documented difference between tests in PROC MODEL and PROC UNIVARIATE.
1.2	Use SAS/ETS PROC MODEL if available to get all tests, then SAS/IML, then univariate only. Use ODS SELECT to obtain only normal table from MODEL (requires SAS 8 or later). Provide univariate histograms with overlaid normal curves and tests controlled by expanded PLOT= parameter.
1.1	Use PVALUE format. Prefix notes from macro with MULTNORM: instead of NOTE:.
1.0	Initial coding.

REQUIREMENTS:

Multivariate normality tests require SAS/IML or SAS/STAT Software in SAS 8 or later. If neither is found, only univariate normality tests and plots can be provided.

The high resolution multivariate plot requires SAS/STAT in SAS 8 or later. SAS/GRAPH Software in SAS 8 or later is required if Base SAS PROC SGPLOT is not found.

USAGE:

Follow the instructions in the Downloads tab of this sample to save the %MULTNORM macro definition. Replace the text within quotes in the following statement with the location of the %MULTNORM macro definition file on your system. In your SAS program or in the SAS editor window, specify this statement to define the %MULTNORM macro and make it available for use:

   %inc "<location of your file containing the MULTNORM macro>";

Following this statement, you may call the %MULTNORM macro. See the Results tab for an example.

The options and allowable values are:

DATA=data-set-name

SAS data set to be analyzed. If the DATA= option is not supplied, the most recently created SAS data set is used.

VAR=variable-list

REQUIRED. The list of variables to use when testing for univariate and multivariate normality. Individual variable names, separated by blanks, must be specified. Special variable lists may NOT be used (such as VAR1-VAR10, ABC--XYZ, _NUMERIC_).

PLOT=BOTH | MULT | UNI | NONE

PLOT=MULT requests a chi-square quantile-quantile (Q-Q) plot of the squared Mahalanobis distances of the observations from the mean vector. PLOT=UNI requests univariate histograms of each variable with overlaid normal curves and additional univariate tests of normality. PLOT=BOTH (the default) requests both of the above. PLOT=NONE suppresses all plots and the univariate normality tests.

HIRES=YES | NO

Ignored if PLOT=NONE. HIRES=YES (the default) requests that high-resolution graphics be used when creating plots. PROC SGPLOT is used by default. If SAS/GRAPH PROC GPLOT is used, you must specify any needed graphics-related options before invoking the macro. HIRES=NO requests that the multivariate plot be drawn with low-resolution using PROC PLOT. The univariate plots are not available with HIRES=NO.

DETAILS:

If a set of variables is distributed as multivariate normal, then each variable must be normally distributed. However, when all individual variables are normally distributed, the set of variables may not be distributed as multivariate normal. Hence, testing each variable only for univariate normality is not sufficient. Mardia (1974) proposed tests of multivariate normality based on sample measures of multivariate skewness and kurtosis.

For multivariate normal data, Mardia (1974) shows that the expected value of the multivariate skewness statistic is

            p(p+2)[(n+1)(p+1)-6] / (n+1)(n+3)

and the expected value of the multivariate kurtosis statistic is

            p(p+2)(n-1)/(n+1)  .

As discussed in Testing for Normality in the PROC MODEL documentation, the Henze-Zirkler test of multivariate normality is based on a nonnegative function that measures the distance between two distribution functions and is used to assess distance between the distribution function of the data and the Multivariate Normal distribution function.

Univariate normality is tested using the Shapiro-Wilk W test or the Kolmogorov-Smirnov test. Additional tests are provided if univariate plots are requested. For details on the univariate tests of normality, see Goodness Of Fit Tests in the PROC UNIVARIATE documentation. There may be differences in the Shapiro-Wilk test statistics and p-values produced by PROC MODEL and PROC UNIVARIATE as documented in this SAS Note. The algorithm in PROC UNIVARIATE uses the updated method of Royston (1992). PROC MODEL uses the older method of Royston (1982). While the PROC MODEL results are correct, the Shapiro-Wilk tests provided by PROC UNIVARIATE are preferred.

If the p-value of any of the tests is small, then multivariate normality can be rejected. However, it is important to note that the univariate Shapiro-Wilk W test is a very powerful test and is capable of detecting small departures from univariate normality with relatively small sample sizes. This may cause you to reject univariate, and therefore multivariate, normality unnecessarily if the tests are being done to validate the use of methods that are robust to small departures from normality. For such situations, the plots provide a visual assessment of approximate normality.

When the covariance matrix for the data is singular, the macro quits and issues the following message:

        ERROR: Covariance matrix is singular.

The PRINCOMP procedure in SAS/STAT Software is required for this check. If it is not found, a message is printed, nonsingularity is assumed, and the macro attempts to perform the multivariate test and plot.

For p variables and a large sample size, the squared Mahalanobis distances of the observations to the mean vector are distributed as chi-square with p degrees of freedom. However, the sample size must be quite large for the chi-square distribution to obtain unless p is very small. Also, this plot is sensitive to the presence of outliers. So, this plot should be cautiously used as a rough indicator of multivariate normality.

LIMITATIONS:

Memory usage for the multivariate tests is quadratically related to the number of observations. Moderately large data sets (1000 is big) may require excessive running time or result in Unable to allocate sufficient memory and macro termination errors.

MISSING VALUES:

Observations with missing values are omitted from the analysis and plot.

REFERENCES:

Henze, N. and Zirkler, B. (1990), "A Class of Invariant Consistent tests for Multivariate Normality," Communications in Statistics, Part A - Theory and Methods., 19(10), 3595-3617.

Mardia (1974), "Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies," Sankhya B, 36, 115-128.

Mardia (1975), "Assessment of Multinormality and the Robustness of Hotelling's T-squared Test," Applied Statistics, 1975, 24(2).

Mardia, K.V., Kent, J.T., and Bibby, J.M. (1979), "Multivariate Analysis," New York: Academic Press.

Mardia (1980), "Measures of Multivariate Skewness and Kurtosis with Applications," Biometrika, 57(3).

Royston, J.P. (1982), "An Extension of Shapiro and Wilk's W Test for Normality to Large Samples," Applied Statistics, 31.

Royston, J.P. (1992), "Approximating the Shapiro-Wilk W-Test for Non-normality," Statistics and Computing, 2, 117 - 119.

Shapiro, S.S. and Wilk, M.B. (1965), "An Analysis of Variance Test for Normality (complete samples)," Biometrika, 52.

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These sample files and code examples are provided by SAS Institute Inc. "as is" without warranty of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Recipients acknowledge and agree that SAS Institute shall not be liable for any damages whatsoever arising out of their use of this material. In addition, SAS Institute will provide no support for the materials contained herein.

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These sample files and code examples are provided by SAS Institute Inc. "as is" without warranty of any kind, either express or implied, including but not limited to the implied warranties of merchantability and fitness for a particular purpose. Recipients acknowledge and agree that SAS Institute shall not be liable for any damages whatsoever arising out of their use of this material. In addition, SAS Institute will provide no support for the materials contained herein.

EXAMPLE:

The following example is from Mardia, Kent, Bibby (1979). The multivariate skewness test statistic differs due to the use in %MULTNORM of an alternative approximation to the chi-square distribution given in Mardia (1974).

         data cork;
           input n e s w @@;
           datalines;
         72 66 76 77   91 79 100 75
         60 53 66 63   56 68 47 50
         56 57 64 58   79 65 70 61
         41 29 36 38   81 80 68 58
         32 32 35 36   78 55 67 60
         30 35 34 26   46 38 37 38
         39 39 31 27   39 35 34 37
         42 43 31 25   32 30 30 32
         37 40 31 25   60 50 67 54
         33 29 27 36   35 37 48 39
         32 30 34 28   39 36 39 31
         63 45 74 63   50 34 37 40
         54 46 60 52   43 37 39 50
         47 51 52 43   48 54 57 43
         ;

         %inc "<location of your file containing the MULTNORM macro>";
         %multnorm(data=cork, var=n e s w, plot=mult)

For p variables listed in the VAR= option, the first p rows of the Normality Test table contain univariate tests of normality, including the name of the test (Shapiro-Wilk or Kolmogorov-Smirnov), the values of the test statistic, and the corresponding p-values. The next two lines of the table contain Mardia's tests of multivariate normality, including the name of the test (skewness or kurtosis), the values of the multivariate skewness and kurtosis statistics (when the PRINCOMP procedure is used), the values of the test statistics, and their p-values. When the MODEL procedure is used, the Henze-Zirkler test of multivariate normality is also provided.

If PLOT=MULT or BOTH is specified, a chi-square quantile-quantile plot is produced which plots the squared Mahalanobis distances against corresponding quantiles of the limiting chi-square distribution. If the data are distributed as multivariate normal, then the points should fall on a straight line with slope one and intercept zero. See DETAILS for more information.

Following are the results from the above example. The Mardia tests do not reject multivariate normality, but the Henze-Zirkler multivariate test does. Also, the univariate tests reject univariate normality, and therefore multivariate normality. The multivariate plot seems to indicate approximate normality, but the sample is quite small.

MULTNORM macro: Univariate and Multivariate Normality Tests The MODEL Procedure Normality Test Equation Test Statistic Value Prob n Shapiro-Wilk W 0.91 0.0167 e Shapiro-Wilk W 0.90 0.0116 s Shapiro-Wilk W 0.89 0.0067 w Shapiro-Wilk W 0.94 0.1093 System Mardia Skewness 24.13 0.2369   Mardia Kurtosis -0.40 0.6904   Henze-Zirkler T 2.29 0.0222

Right-click on the link below and select Save to save the %MULTNORM macro definition to a file. It is recommended that you name the file multnorm.sas.

Download and save multnorm.sas

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