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:

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)

   /* 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);