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LTS Call

CALL LTS( sc, coef, wgt, opt, y <*>, x <*>, sorb ) ;

The LTS subroutine performs Least Trimmed Squares (LTS) robust regression by minimizing the sum of the smallest squared residuals. The subroutine also detects outliers and perform a least squares regression on the remaining observations. The LTS subroutine implements the FAST-LTS algorithm described by Rousseeuw and Van Driessen (1998).

The value of can be specified, but for many applications the default value works just fine and the results seem to be quite stable toward different choices of .

In the following discussion, is the number of observations and is the number of regressors. The input arguments to the LTS subroutine are as follows:

opt

refers to an options vector with the following components (missing values are treated as default values). The options vector can be a null vector.

opt[1]

specifies whether an intercept is used in the model (opt[1]=0) or not (opt[1]). If opt[1]=0, then a column of ones is added as the last column to the input matrix ; that is, you do not need to add this column of ones yourself. The default is opt[1]=0.

opt[2]

specifies the amount of printed output. Higher values request additional output and include the output of lower values.

opt[2]=0

prints no output except error messages.

opt[2]=1

prints all output except (1) arrays of , such as weights, residuals, and diagnostics; (2) the history of the optimization process; and (3) subsets that result in singular linear systems.

opt[2]=2

additionally prints arrays of , such as weights, residuals, and diagnostics; also prints the case numbers of the observations in the best subset and some basic history of the optimization process.

opt[2]=3

additionally prints subsets that result in singular linear systems.

The default is opt[2]=0.

opt[3]

specifies whether only LTS is computed or whether, additionally, least squares (LS) and weighted least squares (WLS) regression are computed:

opt[3]=0

computes only LTS.

opt[3]=1

computes, in addition to LTS, weighted least squares regression on the observations with small LTS residuals (where small is defined by opt[8]).

opt[3]=2

computes, in addition to LTS, unweighted least squares regression.

opt[3]=3

adds both unweighted and weighted least squares regression to LTS regression.

The default is opt[3]=0.

opt[4]

specifies the quantile to be minimized. This is used in the objective function. The default is opt[4], which corresponds to the highest possible breakdown value. This is also the default of the PROGRESS program. The value of should be in the range

opt[5]

specifies the number of generated subsets. Each subset consists of observations , where . The total number of subsets that contain observations out of observations is

     

where is the number of parameters including the intercept.

Due to computer time restrictions, not all subset combinations of observations out of can be inspected for larger values of and . Specifying a value of enables you to save computer time at the expense of computing a suboptimal solution.

When opt[5] is zero or missing:

If , the default FAST-LTS algorithm constructs up to five disjoint random subsets with sizes as equal as possible, but not to exceed 300. Inside each subset, the algorithm chooses subset combinations of observations.

The number of subsets is taken from the following table:

n

1

2

3

4

5

6

7

8

9

10

500

50

22

17

15

14

0

0

0

0

1414

182

71

43

32

27

24

23

22

500

1000

1500

2000

2500

3000

3000

3000

3000

3000

n

11

12

13

14

15

0

0

0

0

0

22

22

22

23

23

3000

3000

3000

3000

3000

If the number of cases (observations) is smaller than , then all possible subsets are used; otherwise, fixed 500 subsets for FAST-LTS or subsets for algorithm before SAS/IML 8.1 are chosen randomly. This means that an exhaustive search is performed for opt[5]=. If is larger than , a note is printed in the log file that indicates how many subsets exist.

opt[6]

is not used.

opt[7]

specifies whether the last argument sorb contains a given parameter vector or a given subset for which the objective function should be evaluated.

opt[7]=0

sorb contains a given subset index.

opt[7]=1

sorb contains a given parameter vector .

The default is opt[7]=0.

opt[8]

is relevant only for LS and WLS regression (opt[3] > 0). It specifies whether the covariance matrix of parameter estimates and approximate standard errors (ASEs) are computed and printed.

opt[8]=0

does not compute covariance matrix and ASEs.

opt[8]=1

computes covariance matrix and ASEs but prints neither of them.

opt[8]=2

computes the covariance matrix and ASEs but prints only the ASEs.

opt[8]=3

computes and prints both the covariance matrix and the ASEs.

The default is opt[8]=0.

opt[9]

is relevant only for LTS. If opt[9]=0, the algorithm FAST-LTS of Rousseeuw and Van Driessen (1998) is used. If opt[9] = 1, the algorithm of Rousseeuw and Leroy (1987) is used. The default is opt[9]=0.

y

refers to an response vector.

x

refers to an matrix of regressors. If opt[1] is zero or missing, an intercept is added by default as the last column of . If the matrix is not specified, is analyzed as a univariate data set.

sorb

refers to an vector that contains either of the following:

  • observation numbers of a subset for which the objective function should be evaluated; this subset can be the start for a pairwise exchange algorithm if opt[7] is specified.

  • given parameters (including the intercept, if necessary) for which the objective function should be evaluated.

Missing values are not permitted in or . Missing values in opt cause the default value to be used.

The LTS subroutine returns the following values:

sc

is a column vector that contains the following scalar information, where rows 1–9 correspond to LTS regression and rows 11–14 correspond to either LS or WLS:

sc[1]

the quantile used in the objective function

sc[2]

number of subsets generated

sc[3]

number of subsets with singular linear systems

sc[4]

number of nonzero weights

sc[5]

lowest value of the objective function attained

sc[6]

preliminary LTS scale estimate

sc[7]

final LTS scale estimate

sc[8]

robust (coefficient of determination)

sc[9]

asymptotic consistency factor

If opt[3] > 0, then the following are also set:

sc[11]

LS or WLS objective function (sum of squared residuals)

sc[12]

LS or WLS scale estimate

sc[13]

value for LS or WLS

sc[14]

value for LS or WLS

For opt[3]=1 or opt[3]=3, these rows correspond to WLS estimates; for opt[3]=2, these rows correspond to LS estimates.

coef

is a matrix with columns that contains the following results in its rows:

coef[1,]

LTS parameter estimates

coef[2,]

indices of observations in the best subset

If opt[3] > 0, then the following are also set:

coef[3]

LS or WLS parameter estimates

coef[4]

approximate standard errors of LS or WLS estimates

coef[5]

-values

coef[6]

-values

coef[7]

lower boundary of Wald confidence intervals

coef[8]

upper boundary of Wald confidence intervals

For opt[3]=1 or opt[3]=3, these rows correspond to WLS estimates; for opt[3]=2, to LS estimates.

wgt

is a matrix with columns that contains the following results in its rows:

wgt[1]

weights (=1 for small, =0 for large residuals)

wgt[2]

residuals

wgt[3]

resistant diagnostic (note that the resistant diagnostic cannot be computed for a perfect fit when the objective function is zero or nearly zero)

Example

Consider Brownlee (1965) stackloss data used in the example for the LMS subroutine.

For and (three explanatory variables including intercept), you obtain a total of 5,985 different subsets of 4 observations out of 21. If you decide not to specify optn[5], the FAST-LTS algorithm chooses random sample subsets, as in the following statements:

   /* 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 };

a = aa[, 2:4]; b = aa[, 5];
optn = j(8, 1, .);
optn[2]= 1;    /* ipri */
optn[3]= 3;    /* ilsq */
optn[8]= 3;    /* icov */

call lts(sc, coef, wgt, optn, b, a);

Figure 23.162 Least Trimmed Squares

LTS: The sum of the 13 smallest squared residuals will be minimized.

Median and Mean
  Median Mean
VAR1 58 60.428571429
VAR2 20 21.095238095
VAR3 87 86.285714286
Intercep 1 1
Response 15 17.523809524


Least Trimmed Squares (LTS) Method


Minimizing Sum of 13 Smallest Squared Residuals.


Highest Possible Breakdown Value = 42.86 %


Random Selection of 517 Subsets


Among 517 subsets 17 is/are singular.


The best half of the entire data set obtained after full iteration consists of the cases:


LTS Objective Function = 0.474940583


Preliminary LTS Scale = 0.9888435617


Robust R Squared = 0.9745520119


Final LTS Scale = 1.0360272594

Dispersion and Standard Deviation
  Dispersion StdDev
VAR1 5.930408874 9.1682682584
VAR2 2.965204437 3.160771455
VAR3 4.4478066555 5.3585712381
Intercep 0 0
Response 5.930408874 10.171622524


Unweighted Least-Squares Estimation

LS Parameter Estimates
Variable Estimate Approx
Std Err
t Value Pr > |t| Lower WCI Upper WCI
VAR1 0.7156402 0.13485819 5.31 <.0001 0.45132301 0.97995739
VAR2 1.29528612 0.36802427 3.52 0.0026 0.57397182 2.01660043
VAR3 -0.1521225 0.15629404 -0.97 0.3440 -0.4584532 0.15420818
Intercep -39.919674 11.8959969 -3.36 0.0038 -63.2354 -16.603949


Weighted Least-Squares Estimation


Weighted Sum of Squares = 10.273044977


Degrees of Freedom = 11


RLS Scale Estimate = 0.9663918355


Weighted R-squared = 0.9622869127


F(3,11) Statistic = 93.558645037


Probability = 4.1136826E-8


There are 15 points with nonzero weight.


Average Weight = 0.7142857143


The run has been executed successfully.


Sum of Squares = 178.8299616


Degrees of Freedom = 17


LS Scale Estimate = 3.2433639182

Cov Matrix of Parameter Estimates
  VAR1 VAR2 VAR3 Intercep
VAR1 0.0181867302 -0.036510675 -0.007143521 0.2875871057
VAR2 -0.036510675 0.1354418598 0.0000104768 -0.651794369
VAR3 -0.007143521 0.0000104768 0.024427828 -1.676320797
Intercep 0.2875871057 -0.651794369 -1.676320797 141.51474107


R-squared = 0.9135769045


F(3,17) Statistic = 59.9022259


Probability = 3.0163272E-9


Least Trimmed Squares (LTS) Method


Least Trimmed Squares (LTS) Method


Minimizing Sum of 13 Smallest Squared Residuals.


Highest Possible Breakdown Value = 42.86 %


Random Selection of 517 Subsets


Among 517 subsets 17 is/are singular.


The best half of the entire data set obtained after full iteration consists of the cases:

5 6 7 8 9 10 11 12 15 16 17 18 19

Estimated Coefficients
VAR1 VAR2 VAR3 Intercep
0.7409210642 0.3915267228 0.0111345398 -37.32332647


LTS Objective Function = 0.474940583


Preliminary LTS Scale = 0.9888435617


Robust R Squared = 0.9745520119


Final LTS Scale = 1.0360272594


Weighted Least-Squares Estimation

RLS Parameter Estimates Based on LTS
Variable Estimate Approx
Std Err
t Value Pr > |t| Lower WCI Upper WCI
VAR1 0.75694055 0.07860766 9.63 <.0001 0.60287236 0.91100874
VAR2 0.45353029 0.13605033 3.33 0.0067 0.18687654 0.72018405
VAR3 -0.05211 0.05463722 -0.95 0.3607 -0.159197 0.054977
Intercep -34.05751 3.82881873 -8.90 <.0001 -41.561857 -26.553163


Weighted Sum of Squares = 10.273044977


Degrees of Freedom = 11


RLS Scale Estimate = 0.9663918355

Cov Matrix of Parameter Estimates
  VAR1 VAR2 VAR3 Intercep
VAR1 0.0061791648 -0.005776855 -0.002300587 -0.034290068
VAR2 -0.005776855 0.0185096933 0.0002582502 -0.069740883
VAR3 -0.002300587 0.0002582502 0.0029852254 -0.131487406
Intercep -0.034290068 -0.069740883 -0.131487406 14.659852903


Weighted R-squared = 0.9622869127


F(3,11) Statistic = 93.558645037


Probability = 4.1136826E-8


There are 15 points with nonzero weight.


Average Weight = 0.7142857143


The run has been executed successfully.

The preceding program produces the following output associated with the LTS analysis. In this analysis, observations, 1, 2, 3, 4, 13, and 21 have scaled residuals larger than 2.5 (table not shown) and are considered outliers.

See the documentation for the LMS subroutine for additional details.

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