Two problems often occur regarding clinical trials: 1. Often, clinical studies do not encompass large enough samples of patients to reach definitive conclusions. 2. Two or more studies may have conflicting results (possibly because of type I and type II errors).

**META-ANALYSIS**

Two problems often occur regarding clinical trials:

1. Often, clinical studies do not
encompass large enough samples of patients to reach definitive conclusions.

2. Two or more studies may have
conflicting results (possibly because of type I and type II errors).

A technique that is being used more and more
frequently to address these problems is meta-analysis. Meta-analyses are
statistical techniques for combining data, sum-mary statistics, or *p*-values from various similar tests to
reach stronger and more consistent conclusions about the results from clinical
trials and other empirical studies than is possible with a single study.

Care is required in the selection of the trials to
avoid potential biases in the process of combining results. Several excellent
books address these issues, for ex-ample, Hedges and Olkin (1985). The volume
edited by Stangl and Berry (2000) presents several illustrations that use the
Bayesian hierarchical modeling approach. The hierarchical approach puts a Bayesian
prior distribution on the unknown para-meters. This prior distribution will
depend on other unknown parameters called hy-perparameters. Additional prior
distributions are specified for the hyperparameters, thus establishing a
hierarchy of prior distributions. It is not important for you to un-derstand
the Bayesian hierarchical approach, but if you are interested in the details,
see Stangl and Berry (2000). We will define prior and posterior distributions
and Bayes rule in the next section. Bayesian hierarchical models are also used
in an in-ferential approach called the empirical Bayes method. You might
encounter this ter-minology if you study some of the literature.

In this section, we will show you two real-life
examples in which Chernick used a particular method, Fisher’s test, which R. A.
Fisher (1932) and K. Pearson (1933) developed for combining *p*-values in a meta-analysis. These
illustrations will give you some appreciation of the value of meta-analysis and
will provide you with a simple tool that you could use, given an appropriate
selection of studies.

The rationale for Fisher’s test is as follows: The
distribution theory for a test sta-tistic proposed that under the null
hypothesis each study would have a *p*-value
that comes from a uniform distribution on the interval [0, 1]. Denote a
particular *p*-value by the random
variable *U*. Let *L* also refer to a random variable. Now consider the transformation *L* = –2 ln(*U*) where ln is the logarithm to the base *e*. It can be shown mathematically that the random variable *L* has a chi-square distribution with 2
de-grees of freedom. (You will encounter a more general discussion of the
chi-square distribution in Chapter 11.)

Suppose we have *k*
independent trials to be combined and *U*_{1},
*U*_{2}, *U*_{3}, . . . , *U _{k}*
are the random variables denoting the

The chi-square with 2*k* degrees of freedom is, therefore, the reference distribu-tion
that holds under the null hypothesis of no effect. We will see in the upcoming
examples that the alternative of a significant difference should produce *p*-values that are concentrated closer to
zero rather than being uniformly distributed. Lower values of the *U*’s lead to higher values of *L _{k}*. So we select a cutoff
based on the up-per tail of the chi-square with 2

In the first example, one of us (Chernick) was
consulting for a medical device company that manufactured an instrument called a
cutting balloon for use in angio-plasty procedures. The company conducted a
controlled clinical trial in Europe and in the United States to show a
reduction in restenosis rate for the cutting balloon an-gioplasty procedure
over conventional balloon angioplasty. Other studies indicated that
conventional angioplasty had a restenosis rate near 40%.

The manufacturer had seen that procedures with the
cutting balloon were achiev-ing rates in the 20%–25% range. They powered the
trial to detect at least a 10% im-provement (i.e., reduction in restenosis).
However, results were somewhat mixed, possibly due to physicians’differing
angioplasty practices and differing patient selection criteria in the various
countries.

Example 8.5.2 in Chernick (1999) presents the clinical
trial results using the bootstrap for a comparative country analysis. The
results of the meta-analysis, not reported there, are given in Table 9.5.
Countries A, B, C, and D are European coun-tries, and country E is the United
States.

The difficulty for the manufacturer was that although the rate of 22% in the United States was statistically significantly lower than the 40% that is known for conventional balloon angioplasty, the values in
countries *A* and *B* were not lower, and the combined results for all countries were
not statistically significantly lower than 40%. Some additional statistical
analyses gave indications about variables that ex-plained the differences. These
explanations led to hypotheses about the criteria for selection of patients.

However, these data were not convincing enough for
the regulatory authorities to approve the procedure without some labeling
restrictions on the types of patients eligible for it. The procedure did not
create any safety issues relative to convention-al angioplasty. The company was
aware of several other studies that could be com-bined with this trial to
provide a meta-analysis that might be more definitive. Cher-nick and associates
conducted the meta-analysis using Fisher’s method for combining *p*-values.

In the analysis, Chernick considered six
peer-reviewed studies of the cutting bal-loon along with the combined results
for the clinical trial already mentioned (re-ferred to as GRT). In the latter
study, sensitivity analyses also were conducted re-garding the choice of
studies to include with the GRT. The other six studies are referred to by the
name of the first listed author of each study. (Refer to Table 9.6.)

The variable CB ratio refers to the restenosis rate
for the cutting balloon, where-as PTCA ratio is the corresponding restenosis
rate for conventional balloon-angio-plasty-treated patients. Table 9.6 shows
the results for these studies and the com-bined Fisher test. Here *k* = 7 (the number of independent
trials), so the reference chi-square distribution has 14 (2*k*) degrees of freedom.

The table provides the individual *p*-values (the *U*’s for the Fisher chi-square test) that are based on a procedure
called Fisher’s exact test for comparing two propor-tions (see Chapter 11).
Note that we have two test procedures here; both are called Fisher’s test
because they were devised by the same famous statistician, R. A. Fish-er.
However, there is no need for confusion. Fisher’s exact test is applied in each
study to compare the restenosis rates and calculate the individual *p*-values. Then we use these seven *p*-values to compute Fisher’s chi-square
statistic in order to deter-mine their combined *p*-value. Note that the most significant test was Suzuki with a *p*-value of 0.001, and the least
significant was the GRT itself with a* p*-value
equal to* *0.7455. However, the
combined *p*-value is a convincing
0.000107.

In the next example, we look at animal studies of
blood loss in pigs when com-paring the use of Novo Nordisk’s clotting agent
NovoSeven® with conventional treatment. Three investigators performed five
studies; the results of the individual tests for mean differences and Fisher’s
chi-square test are given in Table 9.7.

It is interesting to note here that although in all
studies we used the Wilcoxon test for differences, it does not matter what
tests are used to obtain the individual *p*-val-ues.
All we need is that the individual *p*-values
have a uniform distribution under the null hypothesis and be independent of the
other tests. Generally, these condi-tions are met for a large variety of
parametric and nonparametric tests. We could have mixed *t* tests with Wilcoxon or with any other test of the null
hypotheses.

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