Stata Data Analysis Examples: Discriminant Function Analysis



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Stata Data Analysis Examples
Discriminant Function Analysis

Note: This data analysis example needs Stata 10 or later.

Examples of Discriminant Function Analysis

Example 1. A researcher has collected data on three psychological variables, four academic variables (standardized test scores) and gender for 600 college freshman. She is interested in how the set of psychological variables relates to the academic variables and gender. In particular, the researcher is interested in how many dimensions are necessary to understand the association between the two sets of variables.

Example 2. There is Fisher's (1936) classic example of discriminant analysis involving three varities 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

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.

use http://www.ats.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

Some Strategies You Might Be Tempted To Try

Before we show how you can analyze this with a discriminant function analysis, let's consider some other methods that you might use.

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.
Manova - tests of significance are the same as for discriminant function analysis but manova gives no information on the individual dimensions.
Multinomial logistic repression or multinomial probit are also viable options.

Stata Discriminant Function Analysis

We will run the discriminant analysis using the new candisc procedure in Stata 10. We could also have run the discrim lad to get the same analysis with slightly different output. There is a lot 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

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.

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

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 .32 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 unob served discriminant functions (dimensions). The discriminat functions are a kind of latent variable and the correlations are loadings analgous to factor loadings.

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.

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 don't save the dataset these new lables will not be made permanent.

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

scoreplot, msymbol(i)

As you can see the customer service people tend to be a the more social (negative) end of dimension 1 and dispatchers at the opposite end with mechanics 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 and dispatchers lower.

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

loadingplot

And there is no surprise as 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.

Sample Write-Up of the Analysis

There is a lot of variation in the write-ups of discriminant function analyses. The write-up below is fairly minimal, including only the tests of dimensionality and the standardized coefficients. Typically, one does not include raw coefficients with standard errors and Wald tests of significance.

Table 1: Tests of Discriminant Dimensions
            Canonical  Mult.
Dimension     Corr.      F    df1    df2       p
    1         0.72     52.38    6    478     0.000
    2         0.49     38.46    2    240     0.000

Table 2: Standardized Discriminant Coefficients
                            Dimension
                           1         2 
    outdoor              0.38      0.93 
    social              -0.83      0.21 
    conservative         0.52     -0.29

Tests of dimensionality for the discriminant analysis, as shown in Table 1, indicate that both of the dimensions are statistically significant. The F-tests associated with each dimension are exact. Dimension 1 had a canonical correlation of 0.72 between the response variables and the job classification, while for dimension 2 the canonical correlation was lower at 0.49.

Table 2 presents the standardized canonical coefficients for both dimensions. The first discriminant dimension is positively weighted by outdoor (0.38) and conservation (0.52 and strongly negative on social (-0.83). The second discriminant dimenstion is dominated by the outdoor variable (0.93). These results are interpreted to indicate that the first dimension reflects a bipolar social/non-social dimension while the second is an outdoor/non-outdoor dimension.

Cautions, Flies in the Ointment

Multivatiate normal distribution assumptions holds for the response variables.

Discrtiminant function analysis is not recommended for small samples.

Stata Data Analysis Examples Discriminant Function Analysis