LSMEANS
fixedeffects </ options> ;
The LSMEANS statement computes least squares means (LSmeans) of fixed effects. As in the GLM procedure, LSmeans are predicted population margins—that is, they estimate the marginal means over a balanced population. In a sense, LSmeans are to unbalanced designs as classification
and subclassification arithmetic means are to balanced designs. The matrix constructed to compute them is the same as the matrix formed in PROC GLM; however, the standard errors are adjusted for the covariance parameters in the model.
Each LSmean is computed as , where is the coefficient matrix associated with the least squares mean and is the estimate of the fixedeffects parameter vector. The approximate standard errors for the LSmean is computed as the
square root of .
LSmeans can be computed for any effect in the MODEL statement that involves CLASS variables. You can specify multiple effects in one LSMEANS statement or in multiple LSMEANS
statements, and all LSMEANS statements must appear after the MODEL statement. As in the ESTIMATE statement, the matrix is tested for estimability, and if this test fails, PROC HPMIXED displays “Nonest” for the LSmeans entries.
Assuming the LSmean is estimable, PROC HPMIXED constructs an approximate t test to test the null hypothesis that the associated population quantity equals zero. By default, the denominator degrees
of freedom for this test are the same as those displayed for the effect in the “Type III Tests of Fixed Effects” table (see the section TEST Statement).
Table 46.6 summarizes the options available in the LSMEANS statement.
Table 46.6: LSMEANS Statement Options
Option

Description

ALPHA=

Specifies the confidence level

CL

Constructs ttype confidence limits

CORR

Displays the estimated correlation matrix

COV

Displays the estimated covariance matrix

DF=

Specifies the degrees of freedom

DIFF or PDIFF

Displays the differences of the LSmeans

E

Displays the matrix coefficients for LSMEANS effects

SINGULAR=

Tunes the estimability checking

SLICE=

Partitions interaction LSMEANS effects

You can specify the following options in the LSMEANS statement after a slash (/).

ALPHA=number

requests that a ttype confidence interval be constructed for each of the LSmeans with confidence level . The value of number must be between 0 and 1; the default is 0.05.

CL

requests that ttype confidence limits be constructed for each of the LSmeans. If DDFM=NONE, then PROC HPMIXED uses infinite degrees of freedom for this test, essentially computing a z interval. The confidence level is 0.95 by default; this can be changed with the ALPHA= option.

CORR

displays the estimated correlation matrix of the least squares means as part of the “Least Squares Means” table.

COV

displays the estimated covariance matrix of the least squares means as part of the “Least Squares Means” table.

DF=number

specifies the degrees of freedom for the t test and confidence limits. The default is the denominator degrees of freedom taken from the “Type III Tests of Fixed Effects” table corresponding to the LSmeans effect. For these DDFM= methods, degrees of freedom are determined separately for each
test; see the DDFM= option for more information.

DIFF<=difftype>
PDIFF<=difftype>

requests that differences of the LSmeans be displayed. You can specify the following values for the optional difftype.
 DIFF=ALL

requests all pairwise differences; it is the default.
 DIFF=ANOM

requests differences between each LSmean and the average LSmean, as in the analysis of means (Ott, 1967). The average is computed as a weighted mean of the LSmeans, with the weights being inversely proportional to the diagonal
entries of the matrix. When a WEIGHT statement is specified, then the preceding matrix is replaced with where is the diagonal matrix that contains the weights. If LSmeans are nonestimable, this designbased weighted mean is replaced
with an equally weighted mean. Note that the ANOM procedure in SAS/QC software implements both tables and graphics for the
analysis of means with a variety of response types. For oneway designs and normally distributed data, the DIFF=ANOM computations
are equivalent to the results of PROC ANOM.
 DIFF=CONTROL

requests differences with a control; by default, the control is the first level of each of the specified LSMEANS effects.
To specify which levels of the effects are the controls, list the quoted formatted values in parentheses after the CONTROL
keyword. For example, if the effects A
, B
, and C
are classification variables, each having two levels, 1 and 2, the following LSMEANS statement specifies the (1,2) level
of A
*B
and the (2,1) level of B
*C
as controls:
lsmeans A*B B*C / diff=control('1' '2' '2' '1');
For multiple effects, the results depend upon the order of the list, and so you should check the output to make sure that
the controls are correct.
CONTROL produces twotailed tests and confidence limits.
 DIFF=CONTROLL

requests onetailed results and tests whether the noncontrol levels are significantly smaller than the control. The upper
confidence limits for the control minus the noncontrol levels are considered to be infinity and are displayed as missing.
 DIFF=CONTROLU

requests onetailed results and tests whether the noncontrol levels are significantly larger than the control. The upper confidence
limits for the noncontrol levels minus the control are considered to be infinity and are displayed as missing.
The differences of the LSmeans are displayed in a table titled “Differences of Least Squares Means.” The table name is “Diffs.”

E

requests that the matrix coefficients for all LSMEANS effects be displayed. The name of this “Matrix Coefficients” table is “Coef.”

PDIFF

is the same as the DIFF option.

SINGULAR=number

tunes the estimability checking as documented for the SINGULAR= in the CONTRAST statement.

SLICE=fixedeffect
SLICE=(fixedeffects)

specifies effects by which to partition interaction LSMEANS effects. This can produce what are known as tests of simple effects
(Winer, 1971). For example, suppose that A
*B
is significant, and you want to test the effect of A
for each level of B
. The appropriate LSMEANS statement is
lsmeans A*B / slice=B;
This statement tests for the simple main effects of A
for B
, which are calculated by extracting the appropriate rows from the coefficient matrix for the A
*B
LSmeans and by using them to form an F test.
The SLICE= option produces F tests that test the simultaneous equality of cell means at a fixed level of the slice effect (Schabenberger, Gregoire, and
Kong, 2000).
The SLICE= option produces a table titled “Tests of Effect Slices.” The table name is “Slices.”