含蓄什么意思Power Comparison of Exact Unconditional Tests for Comparing Two Binomial Proportions
及物和不及物动词Roger L.Berger
Department of Statistics
North Carolina State University
Raleigh,NC27695-8203
June29,1994
Institute of Statistics Mimeo Series No.2266
Abstract
The powers of six exact,unconditional tests for comparing two binomial proportions are studied.Total sample sizes range from20to100with balanced and imbalanced designs included.Some previously propod tests are found to have poor power for imbalanced designs.Two new tests,confidence interval modifications of Boschloo’s and Suissa and Shuster’s tests,are found to have the best powers.Overall,the modified Boschloo test is recommended.
pasteurizedKEY WORDS:Confidence interval,Contingency table,Homogeneity test,Independence, value,22table.
1INTRODUCTION
In this article,we study the power functions of six exact unconditional tests for com-paring two binomial proportions.Wefind that two new tests are superior to four tests that have been studied previously.Although no one test is uniformly better than all the rest in all situations,wefind that Boschloo’s(1970)test,with the confidence in-terval modification of Berger and Boos(1994),generally has the best power properties. The Suissa and Shuster(1985)test,using the pooled variance estimate and the confi-dence interval modification of Berger and Boos(1994),also has generally good power properties.
This article differs from most previous ones in focusing on the powers of the tests.Many previous articles have studied the sizes of various tests.For example, Upton(1982)compared the sizes of twenty-two tests but did no power comparisons. Storer and Kim(1990)thoroughly compared the sizes of ven tests but only briefly compared their powers at a few alternative parameter points.Haber(1986)prented a small power comparison.Our purpo is to provide a more thorough comparison of the powers of six tests.
The analysis in this article is unconditional.That is,the size and power com-parisons we make are bad on the binomial distributions of the model.There is continuing debate as to whether conditional or unconditional calculations are more relevant for the problems.Little(1989)and Greenland(1991)provide good recent summaries of the issues in this debate.The purpo of this paper is not to continue this debate.Rather,we agree with Greenland that in some(we believe most)situations the unconditional analysis is appropriate.This article is relevant to tho situations.
All the tests we compare are exact tests.The sizes of the tests are computed using the exact binomial distributions,not normal or chi-squared approximations.We follow the standard Neyman-Pearson paradigm of restricting consideration to level-tests and then comparing the powers of the tests.For a specified error probability, all six tests we consider are level-tests.Tests that are liberal,that sometimes have type-I error probabilities that are greater than,are not considered.However,the tests do not have sizes exactly equal to the specified.Becau of the discrete nature
1
of this data,equality can(usually)be achieved only with a randomized test.We do not think randomize
d tests are of any practical interest.So all the tests we consider are level-;their sizes are not greater than the specified.
2MODEL AND TESTS TO BE COMPARED
Let and be independent binomial random variables.The sample size for is and the success probability is1.The sample size for is and the success probability is2.We will u
;11110
to denote the binomial probability mass function of,where!!!is the binomial coefficient.Similarly,;2will denote the binomial probability mass function of.The sample space of will be denoted by00.
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This kind of data is often displayed in a22contingency table as follows.
ud to doyes no
1.–Fisher’s(1935)Exact Test.
min
that this modification of the usual definition of a value yields a valid value.
That is,the test that rejects if and only if is an unconditional level-test.In our comparison,we u001(as suggested by Berger and Boos)and the binomial confidence interval in Calla and Berger(1990,Exerci
9.21).The confidence interval is bad on,a binomial()random
variable if12.
4.–Suissa and Shuster’s(1985)-pooled Test.Haber(1986)also propod this
小学英语论文集test.Define the-pooled statistic(score statistic)as
21
1
where1,2and,the pooled estimate of12.
Then
sup 01impaction
sup
jealous怎么读01
;;
where:and.Thus,is analogous to ,except using,rather than,as the test statistic.
5.–Confidence Interval Modified Test.This test is a modification of in the
same way as is a modification of.The supremum is taken over a confidence interval,and is added to the error probability.In our comparison,we u the same001and confidence interval,as we ud for.
6.–Suissa and Shuster’s(1985)-unpooled Test.This test is defined in the same
way as the test,but using the-unpooled statistic(Wald statistic)division
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111
mobySuissa and Shuster showed that,if,is the same test as.But the tests typically differ for unequal sample sizes.
7.–Confidence Interval Modified Test.A modification of using a confidence
interval,like and,could be defined.But becau has such poor power properties,as we shall e,is not included in our comparison of tests.
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