On Some Procedures Based on Fisher ’ s Inverse Chi-Square Statistic

We present approximations to the distribution of the weighted combination of independent and dependent values, P     1 2 log . m i i i A P      In case that independence of ’s is not assumed, it is argued that the quantity i P A 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 P -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, i  ’s.


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For independent i 's, the variable P This is Fisher's Inverse Chi-square method.We notice that for the statistic , M all the i '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 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 T and .Zhang [3] showed that by equating the first three cumulants of and  the distribution of can be approximated by ] also proposed a chi-square approximation to the distribution of Others authors have approximated the null distribution of by intensive bootstrap [4][5][6][7][8].
. W W In this article, we concentrate on linear combinations of i (a function of i 's) that have a central chi-square distribution, and involve dependent and independent approach by Makambi [9] when it is assumed that there is homogeneity in correlation coefficients between any pair of the 's.where i has a central S 2  -distribution with 2 degrees of freedom and i  are arbitrary weights.For independent 's, Good [10] provided the following approximation: ,

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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, i  and , j i j,   are equal.To avoid this problem, Bhoj [11] proposed the approximation where IG denotes the incomplete gamma function.This approximation is also for independent probability values.
For an alternative and more general approximation to the distribution of A where independence of i 's is not assumed, it may be argued that P A 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 Therefore, the degrees of freedom can be obtained by solving the above equation for ,  namely, S S , denotes the covariance between i and An estimate of the degrees of freedom, ,  is given by (see [9,14]).
We can now synthesize the probability values m , 1, , with an estimate of the degrees of freedom  given by ˆ.
 Notice that for independent i and S j S and normalized weights, Makambi [9] and Hou [14]
For 2 m  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 4 m  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 P  (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 -P , where m I is identity matrix of order and m m J is a square matrix of order with every element equal to unity.It can be shown that We investigate how well this approximation works compared with the other approximations by simulating data from a variate normal distribution with covariance matrix Just as in Hou [14], we simulated   4-6).
, , ,  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.

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distribution with non-centrality parameter .i Solomon and Stephens [1] approximated the distribution of by a random variable of the form W utilize

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to implement the procedures above for dependent i 's.Let's consider the case of homogeneous nonnegative correlation coefficients, that is, ij P Solving the preceding equation for  yields the ap-Let and define the quadratic form[9] distri statistic for independent and dependent - P values.It has also been shown that, for dependent - P values, the proposed estimate of the constant correlation coefficient 

Table 3 .
 inverse variance weights determined from the data; M/H is Makambi/Hou method using  .M/H is Makambi/Hou method using  .

Table 5 . Simulated estimates of
Results are given in Table6for selected nominal levels.