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The MULTTEST Procedure |
Statistical Tests |
The following section discusses the statistical tests performed in the MULTTEST procedure. For continuous data, a t test for the mean (MEAN) is available. For discrete variables, available tests are the Cochran-Armitage linear trend test (CA), the Freeman-Tukey double arcsine test (FT), the Peto mortality-prevalence test (PETO), and the Fisher exact test (FISHER).
Throughout this section, the discrete and continuous variables are denoted by and
, respectively, where
is the variable,
is the treatment group,
is the stratum, and
is the replication. Let
denote the sample size for a binary variable
within group
and stratum
. A plus sign (
) subscript denotes summation over an index. Note that the tests are invariant to the location and scale of the contrast coefficients
.
The Cochran-Armitage linear trend test (Cochran; 1954; Armitage; 1955; Agresti; 2002) is implemented by using a -score approximation, an exact permutation distribution, or a combination of both.
The pooled probability estimate for variable and stratum
is
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The expected value (under constant within-stratum treatment probabilities) for variable , group
, and stratum
is
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Letting denote the contrast trend coefficients specified by the CONTRAST statement, the test statistic for variable
has numerator
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The binomial variance estimate for this statistic is
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where
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The hypergeometric variance estimate (the default) is
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For any strata with
, the contribution to the variance is taken to be zero.
PROC MULTTEST computes the -score statistic
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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 following text.
The -score statistic can be continuity-corrected to better approximate the permutation distribution. With continuity correction
, the upper-tailed p-value is computed from
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For two-tailed, noncontinuity-corrected tests, PROC MULTTEST reports the p-value as , where
is the upper-tailed p-value. The same formula holds for the continuity-corrected test, with the exception that when the noncontinuity-corrected
and the continuity-corrected
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 by the CONTINUITY= option in the TEST statement.
The CA -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
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where the contrast trend coefficients must be integer valued. The observed value of this trend is compared to the permutation distribution to obtain the p-value
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where 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
. 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 by a continuity-corrected normal distribution. You should be cautious when 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:
The permutation distributions of the trend scores are computed within each stratum.
The distributions are convolved to obtain the distribution of the total trend.
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
, then the permutation trend distribution for variable
is computed exactly.
In step 1, the distribution of the within-stratum trend
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is computed by using the multivariate hypergeometric distribution of the , provided number is not exceeded. This distribution can be written as
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The distribution of the within-stratum trend is then computed by summing these probabilities over appropriate configurations. For further information about this technique, see 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 denote the subset of stratum indices that satisfy the threshold criterion, and let
denote the subset of indices that do not satisfy the criterion. Let
denote the combined trend statistic from the set
, which has an exact distribution obtained from Fourier analysis as previously outlined, and let
denote the combined trend statistic from the set
. Then the distribution of the overall trend
is obtained by convolving the analytic distribution of
with the continuity-corrected normal approximation for
. Using the notation from the section Z-Score Approximation, this convolution can be written as
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where is a standard normal random variable, and
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In this expression, the summation of in
is over
, and
is the continuity correction discussed under the
-score approximation.
When a two-tailed test is requested, the expected trend is computed
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The two-tailed p-value is reported as the permutation tail probability for the observed trend plus the permutation tail probability for
, the reflected trend.
For this test, the contrast trend coefficients are centered to the values
, where
,
, and
is the number of groups. The numerator of this test statistic is
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where the weights take on three different types of values depending upon your specification of the WEIGHT= option in the STRATA statement. The default value is the within-strata sample size
, ensuring comparability with the ordinary CA trend statistic. WEIGHT=HARMONIC sets
equal to the harmonic mean
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where is the number of nonmissing groups and the summation is over only the nonmissing elements. The harmonic means analysis places more weight on the smaller sample sizes than does the default sample size method, and is similar to a Type 2 analysis in PROC GLM. WEIGHT=EQUAL sets
for all
and
, and is similar to a Type 3 analysis in PROC GLM.
The function is the double arcsine transformation:
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The variance estimate is
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The test statistic is
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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.
The Peto test is a modified Cochran-Armitage procedure incorporating mortality and prevalence information. The Peto test is computed like two Cochran-Armitage -score approximations, one for prevalence and one for mortality (Peto, Pike, and Day; 1980). 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 to specify "death" times or, more generally, times 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.
In the following notation, the subscript represents the variable,
represents the treatment group,
represents the stratum, and
represents the time. Recall that a plus sign
in a subscript location denotes summation over that subscript.
Let be the number of incidental occurrences, and let
be the total sample size for variable
in group
, stratum
, excluding fatal tumors.
Let be the number of fatal occurrences in time period
, and let
be the number of patients alive at the end of time
.
The pooled probability estimates are given by
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The expected values are
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Let denote a contrast trend coefficient, and define the numerator terms as follows:
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Define the denominator variance terms by using the binomial variance:
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The hypergeometric variances (the default) are calculated by weighting the within-strata variances as discussed in the section Z-Score Approximation.
The Peto statistic is computed as
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where 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
and
. Note that the contrast trend coefficients
must be integer valued. 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.
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 ,
, or
for the Fisher test. The frequencies and sample sizes of the groups scored as
are combined, as are the frequencies and sample sizes of the groups scored as
. Groups scored as
are excluded. The
group is then compared with the
group by using the Fisher exact test.
Letting and
denote the frequency and sample size of the
group, and letting
and
denote those of the
group, the p-value is calculated as
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where and
are independent binomially distributed random variables with sample sizes
and
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
and
, and the other tail is chosen so that the resulting probability is as large as possible without exceeding the probability from the first tail.
For continuous variables, PROC MULTTEST automatically centers the contrast trend coefficients, as in the Freeman-Tukey test. These centered coefficients are then used to form a t statistic contrasting the within-group means. Let
denote the sample size within group
and stratum
; it depends on variable
only when there are missing values. Determine the weights
as in the Freeman-Tukey test with
replacing
. Define
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as the sample mean within a group-and-stratum combination, and let denote the treatment means. Write the null hypothesis as
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Also define
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as the pooled sample variance.
Assuming constant variance for all group-and-stratum combinations, the t statistic for the mean is
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Then under the null hypothesis and assuming normality, independence, and homoscedasticity, 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 , the contribution to the denominator of the pooled variance is 0 and not
. This is also true for the degrees of freedom.
If you do not assume constant variance for all group-and-stratum combinations, then the approximate t test is
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Under the null hypothesis and assuming normality and independence, the Satterthwaite (1946) approximation for the degrees of freedom of the t test is given by
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under the restriction .
Whenever a denominator of 0 for is computed, the p-value is set to 1. If the denominator for
is computed as 0, then set
. When missing data force
, that group-and-stratum combination does not contribute to the
computation.
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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.