The LMS subroutine performs least median of squares (LMS) robust regression (sometimes called resistant regression) by minimizing the th-ordered squared residual. The subroutine is able to detect outliers and perform a least squares regression on the remaining observations.

The algorithm used in the LMS subroutine is based on the PROGRESS program of Rousseeuw and Hubert (1996), which is an updated version of Rousseeuw and Leroy (1987). In the special case of regression through the origin with a single regressor, Barreto and Maharry (2006) show that the PROGRESS algorithm does not, in general, find the slope that yields the least median of squares. Starting with SAS/IML 9.2, the LMS subroutine uses the algorithm of Barreto and Maharry (2006) to obtain the correct LMS slope in the case of regression through the origin with a single regressor. In this case, input arguments that are specific to the PROGRESS algorithm are ignored and output specific to the PROGRESS algorithm is suppressed.

The value of can be specified, but in most applications the default value works well 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 LMS 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.

0

prints no output except error messages.

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.

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.

3

additionally prints subsets that result in singular linear systems.

The default is opt[2]=0.

opt[3]

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

0

computes only LMS.

1

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

2

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

3

adds both unweighted and weighted least squares regression to LMS 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.

If opt[5] is zero or missing, the default 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, subsets 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.

0

sorb contains a given subset index.

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.

0

does not compute covariance matrix and ASEs.

1

computes covariance matrix and ASEs but prints neither of them.

2

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

3

computes and prints both the covariance matrix and the ASEs.

The default is opt[8]=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 LMS subroutine returns the following values:

sc

is a column vector that contains the following scalar information, where rows 1–9 correspond to LMS 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 LMS scale estimate

sc[7]

final LMS scale estimate

sc[8]

robust R square (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]

R square 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,]

LMS 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, these rows correspond to LS estimates.

wgt

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

wgt[1,]

weights (1 for small residuals; 0 for large residuals)

wgt[2,]

residuals

wgt[3,]

resistant diagnostic (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];
opt = j(8, 1, .);
opt[2]= 2;    /* ipri */
opt[3]= 3;    /* ilsq */
opt[8]= 3;    /* icov */

call lms(sc, coef, wgt, opt, b, a);