If the data are contaminated in the -space, M estimation does not do well. The following example shows how you can use LTS estimation to deal with this situation.

data hbk;
   input index$ x1 x2 x3 y @@;
datalines;
1   10.1  19.6  28.3   9.7     39  2.1  0.0  1.2  -0.7
2    9.5  20.5  28.9  10.1     40  0.5  2.0  1.2  -0.5
3   10.7  20.2  31.0  10.3     41  3.4  1.6  2.9  -0.1
4    9.9  21.5  31.7   9.5     42  0.3  1.0  2.7  -0.7
5   10.3  21.1  31.1  10.0     43  0.1  3.3  0.9   0.6
6   10.8  20.4  29.2  10.0     44  1.8  0.5  3.2  -0.7
7   10.5  20.9  29.1  10.8     45  1.9  0.1  0.6  -0.5
8    9.9  19.6  28.8  10.3     46  1.8  0.5  3.0  -0.4
9    9.7  20.7  31.0   9.6     47  3.0  0.1  0.8  -0.9
10   9.3  19.7  30.3   9.9     48  3.1  1.6  3.0   0.1
11  11.0  24.0  35.0  -0.2     49  3.1  2.5  1.9   0.9
12  12.0  23.0  37.0  -0.4     50  2.1  2.8  2.9  -0.4
13  12.0  26.0  34.0   0.7     51  2.3  1.5  0.4   0.7
14  11.0  34.0  34.0   0.1     52  3.3  0.6  1.2  -0.5
15   3.4   2.9   2.1  -0.4     53  0.3  0.4  3.3   0.7
16   3.1   2.2   0.3   0.6     54  1.1  3.0  0.3   0.7
17   0.0   1.6   0.2  -0.2     55  0.5  2.4  0.9   0.0
18   2.3   1.6   2.0   0.0     56  1.8  3.2  0.9   0.1
19   0.8   2.9   1.6   0.1     57  1.8  0.7  0.7   0.7
20   3.1   3.4   2.2   0.4     58  2.4  3.4  1.5  -0.1
21   2.6   2.2   1.9   0.9     59  1.6  2.1  3.0  -0.3
22   0.4   3.2   1.9   0.3     60  0.3  1.5  3.3  -0.9
23   2.0   2.3   0.8  -0.8     61  0.4  3.4  3.0  -0.3
24   1.3   2.3   0.5   0.7     62  0.9  0.1  0.3   0.6
25   1.0   0.0   0.4  -0.3     63  1.1  2.7  0.2  -0.3
26   0.9   3.3   2.5  -0.8     64  2.8  3.0  2.9  -0.5
27   3.3   2.5   2.9  -0.7     65  2.0  0.7  2.7   0.6
28   1.8   0.8   2.0   0.3     66  0.2  1.8  0.8  -0.9
29   1.2   0.9   0.8   0.3     67  1.6  2.0  1.2  -0.7
30   1.2   0.7   3.4  -0.3     68  0.1  0.0  1.1   0.6
31   3.1   1.4   1.0   0.0     69  2.0  0.6  0.3   0.2
32   0.5   2.4   0.3  -0.4     70  1.0  2.2  2.9   0.7
33   1.5   3.1   1.5  -0.6     71  2.2  2.5  2.3   0.2
34   0.4   0.0   0.7  -0.7     72  0.6  2.0  1.5  -0.2
35   3.1   2.4   3.0   0.3     73  0.3  1.7  2.2   0.4
36   1.1   2.2   2.7  -1.0     74  0.0  2.2  1.6  -0.9
37   0.1   3.0   2.6  -0.6     75  0.3  0.4  2.6   0.2
38   1.5   1.2   0.2   0.9
;

The data set hbk is an artificial data set generated by Hawkins, Bradu, and Kass (1984). Both ordinary least squares (OLS) estimation and M estimation (not shown here) suggest that observations 11 to 14 are outliers. However, these four observations were generated from the underlying model, whereas observations 1 to 10 were contaminated. The reason that OLS estimation and M estimation do not pick up the contaminated observations is that they cannot distinguish good leverage points (observations 11 to 14) from bad leverage points (observations 1 to 10). In such cases, the LTS method identifies the true outliers.

The following statements invoke the ROBUSTREG procedure with the LTS estimation method:

proc robustreg data=hbk fwls method=lts;
   model y = x1 x2 x3 / diagnostics leverage;
   id index;
run;

Figure 75.12 displays the model fitting information and summary statistics for the response variable and independent covariates.

Figure 75.13 displays information about the LTS fit, which includes the breakdown value of the LTS estimate. The breakdown value is a measure of the proportion of contamination that an estimation method can withstand and still maintain its robustness. In this example the LTS estimate minimizes the sum of 57 smallest squares of residuals. It can still estimate the true underlying model if the remaining 18 observations are contaminated. This corresponds to the breakdown value around 0.25, which is set as the default.

Figure 75.14 displays parameter estimates for covariates and scale. Two robust estimates of the scale parameter are displayed. See the section Final Weighted Scale Estimator for how these estimates are computed. The weighted scale estimator (Wscale) is a more efficient estimator of the scale parameter.

Figure 75.15 displays outlier and leverage-point diagnostics. The ID variable index is used to identify the observations. If you do not specify this ID variable, the observation number is used to identify the observations. However, the observation number depends on how the data are read. The first 10 observations are identified as outliers, and observations 11 to 14 are identified as good leverage points.

Figure 75.16 displays the final weighted least squares estimates. These estimates are least squares estimates computed after deleting the detected outliers.