On Some Procedures Based on Fisher’s Inverse Chi-Square Statistic ()
1. Introduction
Let 
be 
tail probabilities or probability values from continuous distributions. Associate null hypotheses 
to these 
probability values. Using the probability integral transform, we know that 
when 
is true. For 
That is, 
which is the cumulative distribution function of a chi-square variable with 2 degrees of freedom. That is, 
and the decision rule is to reject 
if 
Define a combined statistic by
For independent
’s, the variable 
The overall test procedure is to reject 
if 
This is Fisher’s Inverse Chi-square method. We notice that for the statistic 
all the
’s are weighted equally, which may not be acceptable in some situations and therefore unequal weighting may be necessary. A number of authors have attempted to derive the distribution of a weighted form of 
For instancelet 
where 
has a non-central
distribution with non-centrality parameter 
Solomon and Stephens [1] approximated the distribution of 
by a random variable of the form 
matching the first three moments. The disadvantage with this approximation is that there is no closed-form formula for computing the parameters. Buckley and Eagleson [2] approximation of the distribution of 
involves approximating 
using a variable that takes the form 
and matching the first three cumulants of 
and 
Zhang [3] showed that by equating the first three cumulants of 
and 
the distribution of 
can be approximated by 
Zhang [3] also proposed a chi-square approximation to the distribution of 
Others authors have approximated the null distribution of 
by intensive bootstrap [4-8].
In this article, we concentrate on linear combinations of 
(a function of
’s) that have a central chi-square distribution, and involve dependent and independent
’s and arbitrary weights,
’s. For dependent
’s, we use simulations to investigate the performance of the approach by Makambi [9] when it is assumed that there is homogeneity in correlation coefficients between any pair of the
’s.
2. Distribution of Independent and Dependent Weighted
’s
Let’s focus on the mixture
where 
has a central 
-distribution with 2 degrees of freedom and 
are arbitrary weights. For independent
’s, Good [10] provided the following approximation:
where 
This approximation is usually regarded as the exact distribution of 
The approximation has been criticized because the calculations become ill-conditioned when any two weights, 
and 
are equal. To avoid this problem, Bhoj [11] proposed the approximation
where 
denotes the incomplete gamma function. This approximation is also for independent probability values.
For an alternative and more general approximation to the distribution of 
where independence of
’s is not assumed, it may be argued that 
is a quantity that is implicitly dominated by positive definite quadratic forms that induce a chi-square distribution. Thus by Satterthwaite [12] or Patnaik [13], we have
It follows that
Therefore, the degrees of freedom can be obtained by solving the above equation for 
namely,
Now,
and
where 
denotes the covariance between 
and 
for 
An estimate of the degrees of freedom, 
is given by 
(see [9,14]).
We can now synthesize the 
probability values 
based on the decision rule
For normalized weights, that is, 
the decision rule is:
with an estimate of the degrees of freedom 
given by 
Notice that for independent 
and 
and normalized weights, Makambi [9] and Hou [14] utilize
For 
and 4, Hou [14] presented simulation results indicating that the approximation given above attains probability values close to the nominal level, similar to the Good [10] and Bhoj [11] approximations.
For 
independent 
-values, we use Table 1 in Hou [14] to obtain Table 1, just for purposes of comparing the performance of the approaches. We notice that using 
(column 5, Table 1) yields results that are close to both the exact method by Good [10] and the method by Bhoj [11].
To illustrate the application of the methods for independent probability values, we use data from Canner [15] on four selected multicenter trials involving aspirin and post-myocardial infarction patients carried out in Europe and the United States in the period 1970-1979. Two of these trials, referred to as UK-1 and UK-2 were carried out in the United Kingdom; the Coronary Drug Project Aspirin Study (CDPA); and the Persantine-Aspirin Reinfarction Study (PARIS) (Table 2).
The 
values provided in column 4 of Table 2 are for the log odds ratio as the outcome measure of interest. Using the 
values in Table 2 and the weights from Table 1 of [14], we obtain the values in Tables 3. We have also included results for normalized inverse variance weights determined from the data. The three approximations yield values that are close to each other, and are in good agreement with the exact method by Good [10].
If 
and 
are non-independent, the expression for 
contains a covariance term between 
and 
that has to be estimated. Let 
be the correlation between 
and 
i.e., 
An approximation of the variance of 
is given by [16]
3. A Procedure for Constant Correlation Coefficient
We require estimates of 
to implement the procedures above for dependent
’s. Let’s consider the case of homogeneous nonnegative correlation coefficients, that is, 
for 
Let 
and define the quadratic form [9]
We can write 
where 
is identity matrix of order 
and 
is a square matrix of order 
with every element equal to unity. It can be shown that 
where 
and
is the trace of the matrix 
For homogeneous 
and using results from Brown [16] we have 
We can show that 
Solving the preceding equation for 
yields the approximate admissible solution 
with an estimate for 
given by
(1)
We investigate how well this approximation works compared with the other approximations by simulating data from a 
variate normal distribution with covariance matrix 
with 
and 
Just as in Hou [14], we simulated
Table 2. Data on total mortality in six aspirin trials (Number of Deaths/Number of patients).
10,000 multivariate normal samples and computed the corresponding values of 
For 
and 4, we present values for 
at selected nominal levels and weights (Tables 4-6).
For 
(Table 4) the proposed method attains probability levels that are close to the nominal level, similar to the Makambi/Hou method.
For 
(Table 5) the proposed estimate of the constant correlation coefficient 
leads to attained probability level that are close to the nominal level, 
for 
and 0.9. However, for values of 
close to 0.5, the estimate leads to underestimation of the probability level.
Now, instead of using pre-defined weights, we simulated weights from a beta distribution with parameters 
and 
That is, for 
such that 
Results are given in Table 6 for selected nominal levels.
4. Conclusion
In this article, we have presented chi-square approximations to the distribution of Fisher’s inverse chi-square statistic for independent and dependent 
values. It has also been shown that, for dependent 
values, the proposed estimate of the constant correlation coefficient 
performs well by attaining probability levels close to the nominal level for correlation coefficients close to 0.1 and 0.9. We expect the proposed estimate to underestimate probability levels for relatively large numbers of studies, especially when 
is close to 0.5. However, for values close to 0.1 and 0.9, the proposed estimate works quite well and can be recommended.