Version info: Code for this page was tested in Stata 12.

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.dta, 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.  It is always a good idea to start with descriptive
statistics.

use https://stats.idre.ucla.edu/stat/stata/dae/discrim, clear

summarize outdoor social conservative

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
     outdoor |       244    15.63934    4.839933          0         28
      social |       244    20.67623    5.479262          7         35
conservative |       244    10.59016    3.726789          0         20

tabstat outdoor social conservative, by(job) stat(n mean sd min max) col(stat)

Summary for variables: outdoor social conservative
     by categories of: job 

             job |         N      mean        sd       min       max
-----------------+--------------------------------------------------
customer service |        85  12.51765  4.648635         0        22
                 |        85  24.22353  4.335283        12        35
                 |        85  9.023529  3.143309         2        17
-----------------+--------------------------------------------------
        mechanic |        93  18.53763  3.564801        11        28
                 |        93  21.13978   4.55066         9        29
                 |        93  10.13978  3.242354         0        17
-----------------+--------------------------------------------------
        dispatch |        66  15.57576  4.110252         4        25
                 |        66  15.45455  3.766989         7        26
                 |        66  13.24242   3.69224         4        20
-----------------+--------------------------------------------------
           Total |       244  15.63934  4.839933         0        28
                 |       244  20.67623  5.479262         7        35
                 |       244  10.59016  3.726789         0        20
--------------------------------------------------------------------

correlate outdoor social conservative
(obs=244)

             |  outdoor   social conser~e
-------------+---------------------------
     outdoor |   1.0000
      social |  -0.0713   1.0000
conservative |   0.0794  -0.2359   1.0000

tabulate job

             job |      Freq.     Percent        Cum.
-----------------+-----------------------------------
customer service |         85       34.84       34.84
        mechanic |         93       38.11       72.95
        dispatch |         66       27.05      100.00
-----------------+-----------------------------------
           Total |        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

We will run the discriminant analysis using the candisc procedure.  We could also have run the discrim lda command to get the same analysis with slightly
different output.  There is a great deal of output, so we will comment at various places
along the way.

candisc outdoor social conservative, group(job)

Canonical linear discriminant analysis

      |                                 | Like- 
      | Canon.   Eigen-     Variance    | lihood
  Fcn | Corr.    value   Prop.   Cumul. | Ratio     F      df1    df2  Prob>F
  ----+---------------------------------+------------------------------------
    1 | 0.7207  1.08053  0.7712  0.7712 | 0.3640  52.382     6    478  0.0000 e
    2 | 0.4927  .320504  0.2288  1.0000 | 0.7573   38.46     2    240  0.0000 e
  ---------------------------------------------------------------------------
  Ho: this and smaller canon. corr. are zero;                     e = exact F

Standardized canonical discriminant function coefficients

                 | function1  function2 
    -------------+----------------------
         outdoor |  .3785725   .9261104 
          social | -.8306986   .2128593 
    conservative |  .5171682  -.2914406 

Canonical structure

                 | function1  function2 
    -------------+----------------------
         outdoor |  .3230982   .9372155 
          social | -.7653907   .2660298 
    conservative |   .467691  -.2587426

Group means on canonical variables

            | job              
    --------+------------------
     group1 | customer service 
     group2 | mechanic         
     group3 | dispatch         

                 | function1  function2 
    -------------+----------------------
          group1 |   -1.2191  -.3890039 
          group2 |  .1067246   .7145704 
          group3 |  1.419669  -.5059049 

Resubstitution classification summary

    +---------+
    | Key     |
    |---------|
    | Number  |
    | Percent |
    +---------+
                 | Classified                     
    True         | group1  group2  group3 |  Total
    -------------+------------------------+-------
          group1 |     70      11       4 |     85
                 |  82.35   12.94    4.71 | 100.00
                 |                        |       
          group2 |     16      62      15 |     93
                 |  17.20   66.67   16.13 | 100.00
                 |                        |       
          group3 |      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 | 0.3333  0.3333  0.3333 |

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.

By default, Stata assumes a priori an equal number of people in each
job.  This is represented by the 0.3333 Priors in the table above.  If
you have different expected proportions in mind, you may specify them with the
priors option.

Next, we will plot a graph of individuals on the discriminant dimensions.  Due to the
large number of subjects we will shorten the labels for the job groups to make the
graph more legible.  As long as we do not save the dataset, these new labels will not be
made permanent.

label define job 1 "c" 2 "m" 3 "d", modify

scoreplot, msymbol(i)

The discrimant functions are:

discriminant_score_1 = 0.517*conservative + 0.379*outdoor – 0.831*social.

discriminant_score_2 = 0.926*outdoor + 0.213*social – 0.291*conservative.

As you can see, the customer service employees tend to be at the more social (negative) end
of dimension 1; the dispatchers are at the opposite end; the mechanics are in the middle.  On
dimension 2 the results are not as clear; however, the mechanics tend to be higher on the
outdoor dimension and customer service employees and dispatchers are lower.

We can also plot the discriminant loadings for the variables onto the discriminant
dimensions.

loadingplot

There is no surprise that the variable social is strong on the social dimension, i.e.,
it has a high negative loading, and the outdoor variable is high on the outdoor dimension.

Things to consider

See also

References