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Fisher [1] proposed a simple method to combine
*p*-values from independent investigations without using detailed information of the original data. In recent years, likelihood-based asymptotic methods have been developed to produce highly accurate
*p*-values. These likelihood-based methods generally required the likelihood function and the standardized maximum likelihood estimates departure calculated in the canonical parameter scale. In this paper, a method is proposed to obtain a
*p*-value by combining the likelihood functions and the standardized maximum likelihood estimates departure of independent investigations for testing a scalar parameter of interest. Examples are presented to illustrate the application of the proposed method and simulation studies are performed to compare the accuracy of the proposed method with Fisher’s method.

Supposed that

1) If

2) If

Since

For illustration, Fisher [

which gives moderate evidence against the null hypothesis. Fisher [

As an illustrative example is the study of rate of arrival. It is common to use a Poisson model to model the number of arrivals over a specific time interval. Let

An alternate way of investigating the rate of arrival over a period of time is by modeling the time to first arrival, T with the exponential model with rate

By Fisher’s way of combining the p-values, we have

which gives strong evidence that

In recent years, many likelihood-based asymptotic methods have been developed to produce highly accurate p-values. In particular, both the Lugannani and Rice’s [

In Section 2, a brief review of the third-order likelihood-based method for a scalar parameter of interest is presented. In Section 3, the relationship between the score variable and the locally defined canonical parameter is determined. Using the results in Section 3, a new way of combining likelihood information is proposed in Section 4. Examples and simulation results are presented in Section 5 and some concluding remarks are recorded in Section 6.

Fraser [

where

and

or the Barndorff-Nielsen [

where

is the observed information evaluated at

Fraser and Reid [

where

is the rate of change of

with

and the standardized maximum likelihood departure

Since

where

Applications of the general method discussed above can be found is Reid and Fraser [

Note that

study of the sensitivity analysis of the third-order method. And

variable with respect to the change of

In Bayesian analysis, Jeffreys [

yields an information function

might be difficult to obtain, we can approximate it by the observed information evaluated at the maximum like- lihood estimate

Hence,

Fraser et al. [

is a pivotal quantity to the second-order. A change of variable from the maximum likelihood estimate of locally defined canonical parameter

which relates the score varaible to the locally defined canonical parameter. Taking the total derivative of (12), and evaluate at the observed data point, we have

Moreover, at

Therefore, the rate of change of the score variable with respect to the change of the locally defined canonical parameter at the observed data point is

This describes how the locally defined canonical parameter

Assume we have

and hence the maximum likelihood estimate of

From (13), the rate of change of the score variable from the

where

Hence, the combined canonical parameter is

The standardized maximum likelihood departure based on the combined canonical parameter can be cal- culated from (5). Thus, a new p-value can be obtained from the combined log-likelihood function and the com- bined canonical parameter using the Lugannani and Rice formula or the Barndorff-Nielsen formula.

In this section, we first revisit the rate of arrival problem discussed in Section 1 and show that the proposed method gives results that is quite different from the results obtained by the Fisher’s way of combining p-values. Then simulation studies are performed to compare the accuracy of the proposed method with the Fisher’s method for the rate of arrival problem. Moreover, two well-known models: scalar canonical exponential family model and normal mean model, are examined. It is shown that, theoretically, the proposed method gives the same results as obtained by the third-order method that was discussed in Fraser and Reid [

From the first investigation discussed in Section 1, the log-likelihood function for the Poisson model is

where

Moreover, from the second investigation discussed in Section 1, the log-likelihood function for the exponen- tial model is

where

The combined log-likelihood function is

and we have

Therefore,

and from (17) we have

Hence,

Simulation studies are performed to compare the three methods discussed in this paper. We examine the rate of arrival problem that was discussed in Section 1. For each combination of

1) generate

2) calculate p-values obtained by the three methods discussed in this paper;

3) record if the p-value is less than a preset value

4) repeat this process

Finally, report the proportion of p-values that is less than

of this process is

Consider

where

From the above model, we have

where

Fisher | LR | BN | Fisher | LR | BN | Fisher | LR | BN | ||

5 | 0.1 | 0.2459 | 0.1084 | 0.1073 | 0.1231 | 0.0525 | 0.0521 | 0.0225 | 0.0099 | 0.0097 |

1.0 | 0.3252 | 0.0992 | 0.0992 | 0.1908 | 0.0496 | 0.0496 | 0.0515 | 0.0123 | 0.0123 | |

2.0 | 0.3256 | 0.1025 | 0.1025 | 0.1961 | 0.0513 | 0.0513 | 0.0547 | 0.0112 | 0.0112 | |

10 | 0.5 | 0.3318 | 0.1014 | 0.1014 | 0.1942 | 0.0490 | 0.0490 | 0.0513 | 0.0128 | 0.0128 |

1.0 | 0.3325 | 0.1005 | 0.1005 | 0.1965 | 0.0530 | 0.0530 | 0.0574 | 0.0105 | 0.0105 | |

2.0 | 0.3269 | 0.1006 | 0.1006 | 0.1975 | 0.0513 | 0.0513 | 0.0562 | 0.0107 | 0.0107 | |

20 | 1.0 | 0.3365 | 0.1000 | 0.1000 | 0.2018 | 0.0526 | 0.0526 | 0.0546 | 0.0098 | 0.0096 |

2.0 | 0.3387 | 0.1064 | 0.1064 | 0.2027 | 0.0528 | 0.0528 | 0.0578 | 0.0109 | 0.0109 | |

5.0 | 0.3356 | 0.1048 | 0.1048 | 0.2037 | 0.0528 | 0.0528 | 0.0582 | 0.0111 | 0.0111 |

and the log-likelihood ratio statistic obtained from the combined log-likelihood function can be obtained from (12). Moreover, from (17), we have

and hence the combined canonical parameter is

The maximum likelihood departure in the combined canonical parameter space is

with the observed information evaluated at

and thus,

which is the same as directly applying the third-order method to the canonical exponential family model with

Consider

where

with

with

and

and the standardized maximum likelihood departure calculated in the locally defined canonical parameter scale can be obtained from Equation (8) and is

These are exactly the same as those obtained in DiCiccio et al. [

In this paper, a method is proposed to obtain a p-value by combining the likelihood functions and the standardized maximum likelihood estimates departure calculated in the canonical parameter space of independent investigations for testing a scalar parameter of interest. It is shown that for the canonical exponential model and the normal mean model, the proposed method gives exactly the same results as using the joint likelihood function. Moreover, for the rate of arrival problem, the proposed method gives very different results from the results obtained by the Fisher’s way of combining p-values. And simulation studies illustrate that the proposed method is extremely accurate.

This research was supported in part by and the National Sciences and Engineering Research Council of Canada.