Language Reference

Consider results for Brownlee (1965) stackloss data. The three explanatory variables correspond to measurements for a plant that oxidizes ammonia to nitric acid on 21 consecutive days:

The response variable contains the permillage of ammonia lost (stackloss). These data are also given by Rousseeuw and Leroy (1987).

        /* X1  X2  X3   Y    Stackloss data */
 aa = { 1  80  27  89  42,
        1  80  27  88  37,
        1  75  25  90  37,
        1  62  24  87  28,
        1  62  22  87  18,
        1  62  23  87  18,
        1  62  24  93  19,
        1  62  24  93  20,
        1  58  23  87  15,
        1  58  18  80  14,
        1  58  18  89  14,
        1  58  17  88  13,
        1  58  18  82  11,
        1  58  19  93  12,
        1  50  18  89   8,
        1  50  18  86   7,
        1  50  19  72   8,
        1  50  19  79   8,
        1  50  20  80   9,
        1  56  20  82  15,
        1  70  20  91  15 };

Rousseeuw and Leroy (1987) cite a large number of papers where this data set was analyzed and state that most researchers "concluded that observations 1, 3, 4, and 21 were outliers"; some people also reported observation 2 as an outlier.

By default, subroutine MVE chooses only 2,000 randomly selected subsets in its search. There are in total 5,985 subsets of 4 cases out of 21 cases, as shown in Figure 261, which is produced by the following statements:

a = aa[, 2:4];
opt = j(8, 1, .);
opt[1] = 2;              /* ipri */
opt[2] = 1;              /* pcov: print COV */
opt[3] = 1;              /* pcor: print CORR */
opt[5] = -1;             /* nrep: use all subsets */

call mve(sc, xmve, dist, opt, a);

The first part of the output (Figure 261) shows the classical scatter and correlation matrix, along with the means of each variable.

The second part of the output (Figure 262) shows the results of the optimization (complete subset sampling):

The third part of the output (Figure 263) shows the optimization results after local improvement:

The final output (Figure 264) presents a table that contains the classical Mahalanobis distances, the robust distances, and the weights that identify the outlying observations (that is leverage points when explaining y with these three regressor variables):