The following section discusses the statistical tests performed in
the MULTTEST procedure. For continuous data, a t-test for the
mean is available. For discrete variables, available tests are the
Cochran-Armitage (CA) linear trend test, the Freeman-Tukey (FT)
double arcsine test, the Peto mortality-prevalence test, and the
Fisher exact test.
Throughout this section, the discrete and continuous variables are
denoted by Svgsr and Xvgsr, respectively, where v is the
variable, g is the treatment group, s is the stratum, and r is
the replication. A plus sign (+) subscript denotes summation over
Cochran-Armitage Linear Trend Test
The Cochran-Armitage linear trend test (Cochran 1954;
Armitage 1955; Agresti 1990) is implemented using a
Z-score approximation, an exact permutation
distribution, or a combination of both.
Let mvgs denote the sample size for a binary variable v
within group g and stratum s. The pooled probability
estimate for variable v and stratum s is
The expected value (under constant within-stratum treatment
probabilities) for variable v, group g, and stratum s is
pvs = [(Sv+s+)/(mv+s)]
The test statistic for variable v has numerator
Evgs = mvgs pvs
where tg denotes a trend coefficient (specified by the CONTRAST
The binomial variance estimate for this statistic is
The hypergeometric variance estimate (the default) is
For any strata s with , the contribution to the
variance is taken to be zero.
PROC MULTTEST computes the Z-score statistic
The p-value for this statistic comes from the standard normal distribution.
Whenever a 0 is computed for the denominator, the p-value is
set to 1. This p-value approximates the probability obtained
from the exact permutation distribution, discussed in the
The Z-score statistic can be continuity-corrected to better
approximate the permutation distribution. With continuity
correction c, the upper-tailed p-value is computed from
For two-tailed, noncontinuity-corrected tests, PROC MULTTEST
reports the p-value as 2 min(p, 1 - p), where p is the
upper-tailed p-value. The same formula holds for the
continuity-corrected test, with the exception
that when the noncontinuity-corrected Z
and the continuity-corrected Z have opposite signs, the
two-tailed p-value is 1.
When the PERMUTATION= option is specified and no STRATA variable is
specified, PROC MULTTEST uses a continuity correction selected to
optimally approximate the upper-tail probability of permutation
distributions with smaller marginal totals (Westfall and Lin 1988).
Otherwise, the continuity correction is specified using the
CONTINUITY= option in the TEST statement.
The CA Z-score statistic is the Hoel-Walburg (Mantel-Haenszel)
statistic reported by Dinse (1985).
When you use the PERMUTATION= option for CA in the TEST
statement, PROC MULTTEST computes the exact permutation
distribution of the trend score
and then compares the observed value of this trend with the
permutation distribution to obtain the p-value
where X is a random variable from the permutation distribution
and where upper-tailed tests are requested. This probability can be
viewed as a binomial probability, where the within-stratum
probabilities are constant and where the probability is conditional
with respect to the marginal totals Sv+s+. It also can be
considered a rerandomization probability.
Because the computations can be quite time-consuming with large
data sets, specifying the PERMUTATION=number option in
the TEST statement limits the situations where PROC MULTTEST
computes the exact permutation distribution. When marginal
total success or total failure frequencies exceed number
for a particular stratum, the permutation distribution is
approximated using a continuity-corrected normal distribution.
You should be cautious in using the PERMUTATION= option in
conjunction with bootstrap resampling because the permutation
distribution is recomputed for each bootstrap sample. This
recomputation is not necessary with permutation resampling.
The permutation distribution is computed in two steps:
As long as the total success or failure frequency does not
exceed number for any stratum, the computed distributions
are exact. In other words, if number or
number for all s, then the
permutation trend distribution for variable v is computed
- The permutation distributions of the trend scores
are computed within each stratum.
- The distributions are convolved to obtain the distribution of
the total trend.
In step 1, the distribution of the within-stratum trend
is computed using the multivariate hypergeometric distribution
of the Svgs+, provided number is not exceeded.
This distribution can be written as
The distribution of the within-stratum trend is then computed
by summing these probabilities over appropriate configurations.
For further information on this technique, refer to Bickis and
Krewski (1986) and Westfall and Lin (1988). In step 2, the
exact convolution distribution is obtained for the trend
statistic summed over all strata having totals that meet the
threshold criterion. This distribution is obtained by applying
the fast Fourier transform to the exact within-stratum
distributions. A description of this general method can be
found in Pagano and Tritchler (1983) and Good (1987).
The convolution distribution of the overall trend is then computed
by convolving the exact distribution with the distribution of the
continuity-corrected standard normal approximation. To be more
specific, let S1 denote the subset of stratum indices that
satisfy the threshold criterion, and let S2 denote the subset of
indices that do not satisfy the criterion. Let Tv1 denote the
combined trend statistic from the set S1, which has an exact
distribution obtained using Fourier analysis as previously
outlined, and let Tv1 denote the combined trend statistic from
the set S2. Then the distribution of the overall trend Tv =
Tv1 + Tv2 is obtained by convolving the analytic distribution
of Tv1 with the continuity-corrected normal approximation for
Tv2. Using the notation from
the "Z-Score Approximation" section, this convolution can be written as
where Z is a standard normal random variable, and
In this expression, the summation of s in Vv is over
S2, and c is the continuity correction discussed under
the Z-score approximation.
When a two-tailed test is requested, the expected trend
is computed, and the two-tailed p-value is reported as
the permutation tail probability for the observed trend
Tv plus the permutation tail probability for
2Ev - Tv, the reflected trend.
Freeman-Tukey Double Arcsine Test
For this test, the trend scores t1, ... , tG
are centered to the values c1, ... , cG, where
, , and G is the number of groups. The
numerator of this test statistic is
and is weighted by the within-strata sample size (mv+s) to
ensure comparability with the ordinary C-A trend statistic.
The function f(r,n) is the double arcsine
The variance estimate is
and the test statistic is
The Freeman-Tukey transformation and its variance are described by
Freeman and Tukey (1950) and Miller (1978). Since its variance
is not weighted by the pooled probabilities, as is the
CA test, the FT test can be more useful than the CA test for
tests involving only a subset of the groups.
Peto Mortality-Prevalence Test
The Peto test is a modified Cochran-Armitage procedure incorporating
mortality and prevalence information. It represents a special case
in PROC MULTTEST because the data structure requirements are
different, and the resampling methods used for adjusting p-values
are not valid. The TIME= option variable is required
"death" times or, more generally, time of occurrence. In addition,
the test variables must assume one of the following three values.
- 0 = no occurrence
- 1 = incidental occurrence
- 2 = fatal occurrence
Use the TIME= option variable to define the mortality strata, and
use the STRATA statement variable to define the prevalence strata.
The Peto test is computed like two Cochran-Armitage Z-score
approximations, one for prevalence and one for mortality.
In the following notation, the subscript
v represents the variable, g represents the treatment group,
s represents the stratum, and t represents the time.
Recall that a plus sign
(+) in a subscript location denotes summation over that
Let SPvgs be the number of incidental occurrences, and let
mPvgs be the total sample size for variable v in group g,
stratum s, excluding fatal tumors.
Let SFvgt be the number of fatal occurrences in time period
t, and let mFvgt be the number alive at the end of time
The pooled probability estimates are
The expected values are
Define the numerator terms:
where tg denotes a trend coefficient.
Define the denominator variance terms (using the binomial variance)
The hypergeometric variances (the default) are calculated by
weighting the within-strata variances as discussed in
the "Z-Score Approximation" section.
The Peto statistic is computed as
where c is a continuity correction. The p-value is
determined from the standard normal distribution unless
the PERMUTATION=number option is used. When you
use the PERMUTATION= option for PETO in the TEST
statement, PROC MULTTEST computes the ``discrete
approximation'' permutation distribution described by
Mantel (1980) and Soper and Tonkonoh (1993). Specifically, the
permutation distribution of
is computed, assuming that and are independent over all s and t. The
p-values are exact under this independence assumption. However,
the independence assumption is valid only asymptotically, which is
why these p-values are called "approximate."
An exact permutation distribution is available only under the
assumption of equal risk of censoring in all treatment groups;
even then, computing this distribution can be cumbersome. Soper and
Tonkonoh (1993) describe situations where the discrete approximation
distribution closely fits the exact permutation distribution.
Fisher Exact Test
The CONTRAST statement in PROC MULTTEST enables you to compute
Fisher exact tests for two-group comparisons. No stratification
variable is allowed for this test. Note, however, that the FISHER
exact test is a special case of the exact permutation tests
performed by PROC MULTTEST and that these permutation tests allow a
stratification variable. Recall that contrast coefficients can be
-1, 0, or 1 for the Fisher test. The frequencies and sample
sizes of the groups scored as -1 are combined, as are the
frequencies and sample sizes of the groups scored as 1. Groups
scored as 0 are excluded. The -1 group is then compared with
the 1 group using the Fisher exact test.
Letting x and m denote the frequency and sample
size of the 1 group, and y and n denote those of the
-1 group, the p-value is calculated as
where X and Y are independent binomially distributed random
variables with sample sizes m and n and common probability
parameters. The hypergeometric distribution is used to determine
the stated probability; Yates (1984) discusses this
technique. PROC MULTTEST computes the two-tailed p-values by
adding probabilities from both tails of the hypergeometric
distribution. The first tail is from the observed x and y,
and the other tail is chosen so that the resulting probability
is as large as possible without exceeding the probability from
the first tail.
t-Test for the Mean
For continuous variables, PROC MULTTEST automatically centers
the trend coefficients, as in the Freeman-Tukey test. These
centered coefficients cg are then used to form a
t-statistic contrasting the within-group means. Let nvgs
denote the sample
size within group g and stratum s; it
depends on variable v only when there are missing values.
as the sample mean within a group-and-stratum combination, and
as the pooled sample variance. Assume constant variance for
all group-and-stratum combinations. Then the t-statistic
for the mean is
and is weighted by the within-strata sample size (nv+s) to
ensure comparability with the C-A trend and Freeman-Tukey
Let denote the treatment means. Then
under the null hypothesis that
and assuming normality, independence, and homoscedasticity, Mv
follows a t-distribution with degrees of freedom.
Whenever a denominator of 0 is computed, the
p-value is set to 1. When missing data force nvgs = 0,
then the contribution to the denominator of the pooled variance
is 0 and not -1. This is also true for degrees of freedom.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.