**Version info:** Code for this page was tested in SAS 9.3.

Linear discriminant function analysis (i.e., discriminant analysis) performs a multivariate test of differences between groups. In addition, discriminant analysis is used to determine the minimum number of dimensions needed to describe these differences. A distinction is sometimes made between descriptive discriminant analysis and predictive discriminant analysis. We will be illustrating predictive discriminant analysis on this page.

**Please note:** The purpose of this page is to show how to use various data
analysis commands. It does not cover all aspects of the research process which
researchers are expected to do. In particular, it does not cover data
cleaning and checking, verification of assumptions, model diagnostics or
potential follow-up analyses.

## Examples of discriminant function analysis

Example 1. A large international air carrier has collected data on employees in three different job classifications; 1) customer service personnel, 2) mechanics and 3) dispatchers. The director of Human Resources wants to know if these three job classifications appeal to different personality types. Each employee is administered a battery of psychological test which include measures of interest in outdoor activity, sociability and conservativeness.

Example 2. There is Fisher’s (1936) classic example of discriminant analysis involving three varieties of iris and four predictor variables (petal width, petal length, sepal width, and sepal length). Fisher not only wanted to determine if the varieties differed significantly on the four continuous variables but he was also interested in predicting variety classification for unknown individual plants.

## Description of the data

Let’s pursue Example 1 from above.

We have a data file, discrim.sas7bdat, with 244 observations on four variables.
The psychological variables are **outdoor interests**, **social** and
**conservative**. The categorical variable is **job type** with three
levels; 1) customer service, 2) mechanic, and 3) dispatcher.

Let’s look at the data.

proc means data=mylib.discrim n mean std min max; var outdoor social conservative; run;The MEANS Procedure Variable N Mean Std Dev Minimum Maximum ----------------------------------------------------------------------------------- OUTDOOR 244 15.6393443 4.8399326 0 28.0000000 SOCIAL 244 20.6762295 5.4792621 7.0000000 35.0000000 CONSERVATIVE 244 10.5901639 3.7267890 0 20.0000000 -----------------------------------------------------------------------------------proc means data=mylib.discrim n mean std; class job; var outdoor social conservative; run;The MEANS Procedure N JOB Obs Variable N Mean Std Dev -------------------------------------------------------------------------- 1 85 OUTDOOR 85 12.5176471 4.6486346 SOCIAL 85 24.2235294 4.3352829 CONSERVATIVE 85 9.0235294 3.1433091 2 93 OUTDOOR 93 18.5376344 3.5648012 SOCIAL 93 21.1397849 4.5506602 CONSERVATIVE 93 10.1397849 3.2423535 3 66 OUTDOOR 66 15.5757576 4.1102521 SOCIAL 66 15.4545455 3.7669895 CONSERVATIVE 66 13.2424242 3.6922397 --------------------------------------------------------------------------proc corr data=mylib.discrim; var outdoor social conservative; run;<**SOME OUTPUT OMITTED**> Pearson Correlation Coefficients, N = 244 Prob > |r| under H0: Rho=0 OUTDOOR SOCIAL CONSERVATIVE OUTDOOR 1.00000 -0.07130 0.07938 0.2672 0.2166 SOCIAL -0.07130 1.00000 -0.23586 0.2672 0.0002 CONSERVATIVE 0.07938 -0.23586 1.00000 0.2166 0.0002proc freq data=mylib.discrim; tables job; run;The FREQ Procedure Cumulative Cumulative JOB Frequency Percent Frequency Percent -------------------------------------------------------- 1 85 34.84 85 34.84 2 93 38.11 178 72.95 3 66 27.05 244 100.00

## Analysis methods you might consider

Below is a list of some analysis methods you may have encountered. Some of the methods listed are quite reasonable, while others have either fallen out of favor or have limitations.

- Discriminant function analysis – The focus of this page. This procedure is multivariate and also provides information on the individual dimensions.
- Multinomial logistic regression or multinomial probit – These are also viable options.
- MANOVA – The tests of significance are the same as for discriminant function analysis, but MANOVA gives no information on the individual dimensions. However, the psychological variables will be the dependent variables and job type the independent variable.
- Separate one-way ANOVAs – You could analyze these data using separate one-way ANOVAs for each psychological variable. The separate ANOVAs will not produce multivariate results and do not report information concerning dimensionality. Again, the designation of independent and dependent variables is reversed as in MANOVA.

## Discriminant function analysis

We will run the discriminant analysis using **proc discrim **with the
canonical option in the proc discrim statement to output the canonical
coefficients and canonical structure.
We could also have used **proc candisc** with essentially the same syntax to
obtain the same results but with slightly different output.
Please note that we will not be using all of the output that SAS
provides nor will the output be presented in the same order as it appears.
There is still a lot of output remaining so we will comment at various places
along the way.

proc candisc data=mylib.discrim out=discrim_out ; class job; var outdoor social conservative; run;The DISCRIM Procedure Canonical Discriminant Analysis Adjusted Approximate Squared Canonical Canonical Standard Canonical Correlation Correlation Error Correlation 1 0.720661 0.716099 0.030834 0.519353 2 0.492659 . 0.048580 0.242713 Test of H0: The canonical correlations in Eigenvalues of Inv(E)*H the current row and all that follow are zero = CanRsq/(1-CanRsq) Likelihood Approximate Eigenvalue Difference Proportion Cumulative Ratio F Value Num DF Den DF Pr > F 1 1.0805 0.7600 0.7712 0.7712 0.36398797 52.38 6 478 <.0001 2 0.3205 0.2288 1.0000 0.75728681 38.46 2 240 <.0001

There are two discriminant dimensions both of which are statistically significant. The canonical correlations for the dimensions one and two are 0.72 and 0.49 respectively.

Pooled Within-Class Standardized Canonical Coefficients Variable Can1 Can2 OUTDOOR -.3785725108 0.9261103825 SOCIAL 0.8306986150 0.2128592590 CONSERVATIVE -.5171682475 -.2914406390 Pooled Within Canonical Structure Variable Can1 Can2 OUTDOOR -0.323098 0.937215 SOCIAL 0.765391 0.266030 CONSERVATIVE -0.467691 -0.258743

The standardized discriminant coefficients function in a manner analogous to standardized
regression coefficients in OLS regression. For example, a one standard deviation increase
on the **outdoor** variable will result in a .378 standard deviation decrease in the predicted
values on discriminant function 1. The canonical structure, also known as canonical loading or
discriminant loadings, represent correlations between observed variables and the unobserved
discriminant functions (dimensions). The discriminant functions are a kind of latent variable
and the correlations are loadings analogous to factor loadings.

Class Means on Canonical Variables JOB Can1 Can2 1 1.219100186 -0.389003864 2 -0.106724637 0.714570441 3 -1.419668555 -0.505904888 Number of Observations and Percent Classified into JOB From JOB 1 2 3 Total 1 70 11 4 85 82.35 12.94 4.71 100.00 2 16 62 15 93 17.20 66.67 16.13 100.00 3 3 12 51 66 4.55 18.18 77.27 100.00 Total 89 85 70 244 36.48 34.84 28.69 100.00 Priors 0.33333 0.33333 0.33333

The output includes the means on the discriminant functions for each of the three groups and a classification table. Values in the diagonal of the classification table reflect the correct classification of individuals into groups based on their scores on the discriminant dimensions.

Next, we will plot a graph of individuals on the discriminant dimensions.
Specifically, we are going to overlay a scatter plot of the individuals’ scores
on each dimension on top of a contour plot indicating predicted job
classification as a function of the two discriminant dimensions. To create the
contour plot, we first create a fake dataset containing a range of plausible
values. Then we will use these fake data as a test dataset to calculate
predicted job classifications using coefficient estimates derived from the real
dataset. The predicted job classifications are saved in the output dataset
under the variable name _into_. We will then merge the actual job
classifications of the real data with the predicted job classifications of the
fake data into a single dataset for plotting. Finally we will use **proc**
**sgrender** to plot the scatter plot and contour plot in one graph. To
use **proc** **sgrender**, we first need to define a graph template with
**proc** **template**, in which we provide all of our graphing
specifications. Here, we are requesting a contour plot, where x and y are
the discriminant dimensions and the coloring indicates job classification, as
well as a scatter plot, where x and y are the two discriminant dimensions and
the job classification of each data point is indicated by a number (1,2,3).

data fakedata; do outdoor = 0 to 30 by 1; do social = 5 to 40 by 1; do conservative = 0 to 25 by 1; output; end; end; end; run; proc discrim data=mylib.discrim testdata=fakedata testout=fake_out out=discrim_out canonical; class job; var outdoor social conservative; run; data plotclass; merge fake_out discrim_out; run; proc template; define statgraph classify; begingraph; layout overlay; contourplotparm x=Can1 y=Can2 z=_into_ / contourtype=fill nhint = 30 gridded = false; scatterplot x=Can1 y=Can2 / group=job includemissinggroup=false markercharactergroup = job; endlayout; endgraph; end; run; proc sgrender data = plotclass template = classify; run;

Here we see that the two discriminant dimensions do a reasonable job of separating the job classifications: most of the actual job classifications fall within the boundaries of the matching predicted job classification, indicated by the coloring of the underlying contour map. Moreover, it seems job levels 1 and 3 are rather different, while job level 2 is less separable from the other two.

## Things to consider

- Multivariate normal distribution assumptions holds for the response variables. This means that each of the dependent variables is normally distributed within groups, that any linear combination of the dependent variables is normally distributed, and that all subsets of the variables must be multivariate normal.
- Each group must have a sufficiently large number of cases.
- Different classification methods may be used depending on whether the variance-covariance matrices are equal (or very similar) across groups.
- Non-parametric discriminant function analysis, called k
^{th}nearest neighbor, can also be performed.

## See Also

- SAS Online Manual

## References

- Afifi, A, Clark, V and May, S. 2004.
*Computer-Aided Multivariate Analysis.*4th ed. Boca Raton, Fl: Chapman & Hall/CRC. - Grimm, L. G. and Yarnold, P. R. (editors). (1995).
*Reading and Understanding Multivariate Statistics*. Washington, D.C.: American Psychological Association. - Huberty, C. J. and Olejnik, S. (2006). Applied MANOVA and Discriminant Analysis, Second Edition. Hoboken, New Jersey: John Wiley and Sons, Inc.
- Stevens, J. P. (2002).
*Applied Multivariate Statistics for the Social Sciences, Fourth Edition*. Mahwah, New Jersey: Lawrence Erlbaum Associates, Inc. - Tatsuoka, M. M. (1971). Multivariate Analysis:
*Techniques for Educational and Psychological Research*. New York: John Wiley and Sons.