If the data are contaminated in the X space, M estimation might yield improper results. It is better to use the high breakdown
value method. This example shows how you can use LTS estimation to deal with X-space contaminated data. The following data
set, hbk
, is an artificial data set that was generated by Hawkins, Bradu, and Kass (1984).
data hbk; input index $ x1 x2 x3 y @@; datalines; 1 10.1 19.6 28.3 9.7 2 9.5 20.5 28.9 10.1 3 10.7 20.2 31.0 10.3 4 9.9 21.5 31.7 9.5 5 10.3 21.1 31.1 10.0 6 10.8 20.4 29.2 10.0 7 10.5 20.9 29.1 10.8 8 9.9 19.6 28.8 10.3 9 9.7 20.7 31.0 9.6 10 9.3 19.7 30.3 9.9 11 11.0 24.0 35.0 -0.2 12 12.0 23.0 37.0 -0.4 13 12.0 26.0 34.0 0.7 14 11.0 34.0 34.0 0.1 15 3.4 2.9 2.1 -0.4 16 3.1 2.2 0.3 0.6 17 0.0 1.6 0.2 -0.2 18 2.3 1.6 2.0 0.0 19 0.8 2.9 1.6 0.1 20 3.1 3.4 2.2 0.4 21 2.6 2.2 1.9 0.9 22 0.4 3.2 1.9 0.3 23 2.0 2.3 0.8 -0.8 24 1.3 2.3 0.5 0.7 25 1.0 0.0 0.4 -0.3 26 0.9 3.3 2.5 -0.8 27 3.3 2.5 2.9 -0.7 28 1.8 0.8 2.0 0.3 29 1.2 0.9 0.8 0.3 30 1.2 0.7 3.4 -0.3 31 3.1 1.4 1.0 0.0 32 0.5 2.4 0.3 -0.4 33 1.5 3.1 1.5 -0.6 34 0.4 0.0 0.7 -0.7 35 3.1 2.4 3.0 0.3 36 1.1 2.2 2.7 -1.0 37 0.1 3.0 2.6 -0.6 38 1.5 1.2 0.2 0.9 39 2.1 0.0 1.2 -0.7 40 0.5 2.0 1.2 -0.5 41 3.4 1.6 2.9 -0.1 42 0.3 1.0 2.7 -0.7 43 0.1 3.3 0.9 0.6 44 1.8 0.5 3.2 -0.7 45 1.9 0.1 0.6 -0.5 46 1.8 0.5 3.0 -0.4 47 3.0 0.1 0.8 -0.9 48 3.1 1.6 3.0 0.1 49 3.1 2.5 1.9 0.9 50 2.1 2.8 2.9 -0.4 51 2.3 1.5 0.4 0.7 52 3.3 0.6 1.2 -0.5 53 0.3 0.4 3.3 0.7 54 1.1 3.0 0.3 0.7 55 0.5 2.4 0.9 0.0 56 1.8 3.2 0.9 0.1 57 1.8 0.7 0.7 0.7 58 2.4 3.4 1.5 -0.1 59 1.6 2.1 3.0 -0.3 60 0.3 1.5 3.3 -0.9 61 0.4 3.4 3.0 -0.3 62 0.9 0.1 0.3 0.6 63 1.1 2.7 0.2 -0.3 64 2.8 3.0 2.9 -0.5 65 2.0 0.7 2.7 0.6 66 0.2 1.8 0.8 -0.9 67 1.6 2.0 1.2 -0.7 68 0.1 0.0 1.1 0.6 69 2.0 0.6 0.3 0.2 70 1.0 2.2 2.9 0.7 71 2.2 2.5 2.3 0.2 72 0.6 2.0 1.5 -0.2 73 0.3 1.7 2.2 0.4 74 0.0 2.2 1.6 -0.9 75 0.3 0.4 2.6 0.2 ;
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 and use the LTS estimation method:
proc robustreg data=hbk fwls method=lts; model y = x1 x2 x3 / diagnostics leverage; id index; run;
Figure 84.12 displays the model-fitting information and summary statistics for the response variable and independent covariates.
Figure 84.12: Model-Fitting Information and Summary Statistics
Model Information | |
---|---|
Data Set | WORK.HBK |
Dependent Variable | y |
Number of Independent Variables | 3 |
Number of Observations | 75 |
Method | LTS Estimation |
Summary Statistics | ||||||
---|---|---|---|---|---|---|
Variable | Q1 | Median | Q3 | Mean | Standard Deviation |
MAD |
x1 | 0.8000 | 1.8000 | 3.1000 | 3.2067 | 3.6526 | 1.9274 |
x2 | 1.0000 | 2.2000 | 3.3000 | 5.5973 | 8.2391 | 1.6309 |
x3 | 0.9000 | 2.1000 | 3.0000 | 7.2307 | 11.7403 | 1.7791 |
y | -0.5000 | 0.1000 | 0.7000 | 1.2787 | 3.4928 | 0.8896 |
Figure 84.13 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 a breakdown value around 0.25, which is set as the default.
Figure 84.13: LTS Profile
LTS Profile | |
---|---|
Total Number of Observations | 75 |
Number of Squares Minimized | 57 |
Number of Coefficients | 4 |
Highest Possible Breakdown Value | 0.2533 |
Figure 84.14 displays parameter estimates for covariates and scale. Two robust estimates of the scale parameter are displayed. For information about computing these estimates, see the section Final Weighted Scale Estimator. The weighted scale estimator (Wscale) is a more efficient estimator of the scale parameter.
Figure 84.14: LTS Parameter Estimates
LTS Parameter Estimates | ||
---|---|---|
Parameter | DF | Estimate |
Intercept | 1 | -0.3431 |
x1 | 1 | 0.0901 |
x2 | 1 | 0.0703 |
x3 | 1 | -0.0731 |
Scale (sLTS) | 0 | 0.7451 |
Scale (Wscale) | 0 | 0.5749 |
Figure 84.15 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 84.15: Diagnostics
Diagnostics | ||||||
---|---|---|---|---|---|---|
Obs | index | Mahalanobis Distance | Robust MCD Distance | Leverage | Standardized Robust Residual |
Outlier |
1 | 1 | 1.9168 | 29.4424 | * | 17.0868 | * |
2 | 2 | 1.8558 | 30.2054 | * | 17.8428 | * |
3 | 3 | 2.3137 | 31.8909 | * | 18.3063 | * |
4 | 4 | 2.2297 | 32.8621 | * | 16.9702 | * |
5 | 5 | 2.1001 | 32.2778 | * | 17.7498 | * |
6 | 6 | 2.1462 | 30.5892 | * | 17.5155 | * |
7 | 7 | 2.0105 | 30.6807 | * | 18.8801 | * |
8 | 8 | 1.9193 | 29.7994 | * | 18.2253 | * |
9 | 9 | 2.2212 | 31.9537 | * | 17.1843 | * |
10 | 10 | 2.3335 | 30.9429 | * | 17.8021 | * |
11 | 11 | 2.4465 | 36.6384 | * | 0.0406 | |
12 | 12 | 3.1083 | 37.9552 | * | -0.0874 | |
13 | 13 | 2.6624 | 36.9175 | * | 1.0776 | |
14 | 14 | 6.3816 | 41.0914 | * | -0.7875 |
Figure 84.16 displays the final weighted least squares estimates. These estimates are least squares estimates that are computed after the detected outliers are deleted.
Figure 84.16: Final Weighted LS Estimates
Parameter Estimates for Final Weighted Least Squares Fit | |||||||
---|---|---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error | 95% Confidence Limits | Chi-Square | Pr > ChiSq | |
Intercept | 1 | -0.1805 | 0.1044 | -0.3852 | 0.0242 | 2.99 | 0.0840 |
x1 | 1 | 0.0814 | 0.0667 | -0.0493 | 0.2120 | 1.49 | 0.2222 |
x2 | 1 | 0.0399 | 0.0405 | -0.0394 | 0.1192 | 0.97 | 0.3242 |
x3 | 1 | -0.0517 | 0.0354 | -0.1210 | 0.0177 | 2.13 | 0.1441 |
Scale | 0 | 0.5572 |