ABSTRACT

We present approximations to the distribution of the weighted combination of independent and dependent values, In case that independence of’s is not assumed, it is argued that the quantity is implicitly dominated by positive definite quadratic forms that induce a chi-square distribution. This gives way to the approximation of the associated degrees of freedom using Satterthwaite (1946) or Patnaik (1949) method. An approximation by Brown (1975) is used to estimate the covariance between the log transformed -values. The performance of the approximations is compared using simulations. For both the independent and dependent cases, the approximations are shown to yield probability values close to the nominal level, even for arbitrary weights,’s.

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

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.

REFERENCES