離散資料中改良p-值之研究

The study of the improved p-value test in discrete distributions.

林哲揚(Che-Yang Lin)

指導教授 ： 楊明宗

國立中央大學/理學院/統計研究所/博士(2006年)

摘要

p-值在統計學上有相當廣泛的應用,
但是對離散資料的統計推論,
虛無假設成立時之模型經常含有干擾參數,
使得p-值的性質較為複雜.
本論文針對一般常用於離散資料分析之檢定提出一改良方法,
若使用p-值進行檢定,
本論文所提出之方法是以該p-值做為檢定統計量,
並使用正確非條件方法以構造改良p-值檢定.
對於兩獨立二項抽樣母體成功機率是否相等之檢定,
當原來的p-值為妥當p-值時,
改良p-值檢定有一致較高的檢定力,
且為一 $alpha$ 水準檢定.
對於兩獨立二項抽樣母體成功機率勝算比之區間估計,
當原先信賴集合(區間)其真實信賴係數不低於名目水準 $1-alpha$ 時,
由改良p-值檢定所建構的改良信賴集合(區間),
可同時改良集合大小(區間長度)與覆蓋機率,
且保證覆蓋機率至少為 $1-alpha$.
特別在中小樣本情況下,
改良p-值檢定與改良信賴集合(區間)之改進成果更是顯著.
另外,
由貝氏學派的觀點,
分別於兩獨立二項抽樣與多項抽樣下,
使用均勻先驗分布與 Jeffreys 先驗分布推導部分後驗預測p-值,
其數值分析顯示:
當虛無假設成立時,
相較於常用的費雪p-值,
部分後驗預測p-值之分布顯然較接近均勻分布.

關鍵字

二項抽樣 ； 正確條件方法 ； 正確非條件方法 ； 改良p-值 ； 信賴區間 ； 貝氏p-值

並列摘要

The p-value is the most commonly used measure of compatibility of the null model in many applied statistics.
For the statistical inference in discrete data,
the null model often involves the nuisance parameters so that the property of the p-value is more complicated.
This paper considers a procedure which can improve tests used in discrete distributions.
If one constructs a test by a p-value,
the procedure takes such a p-value as a test statistic,
and uses the exact unconditional approach to construct an improved p-value test.
For testing the equality of two independent binomial proportions,
the improved p-value test, which is a level $alpha$ test, is at least as powerful as the original one when the original p-value is valid.
For the interval estimation of the odds ratio in two independent binomial samples,
the improved confidence set (interval) constructed by using the improved p-value test has improvement on interval length and coverage probability
if the original one has coverage probability above the nominal level $1-alpha$.
Also the actual confidence coefficient of the improved confidence set (interval) attains at least 1-$alpha$.
Especially, the improved p-value test and the improved confidence set (interval) significantly outperform the original ones
when the sample sizes are small or moderate.
From Bayesian point of view,
we use the uniform prior and Jeffreys prior to derive the partial posterior predictive p-values
as the data is sampling from the two independent binomial sampling scheme and the multinomial sampling scheme, respectively,
and the numerical studies show that the distribution of the partial posterior predictive p-value
is much closer to the uniform distribution than that of Fisher''s p-value under the null model.

並列關鍵字

Binomial sampling ； exact conditional approach ； exact unconditional approach ； improved p-value ； Bayesian p-value ； confidence interval

參考文獻

Agresti, A. (1992). A survey of exact inference for contingency tables (with discussion). Statistical Science 7, 131-177.

Agresti, A. (2002). Categorical Data Analysis, 2nd edition. John Wiley & Sons, Hoboken,

New Jersey.

Agresti, A. (2003). Dealing with discreteness: making `exact'' confidence intervals for proportions, differences of proportions, and odds ratios more exact. Statistical Methods in Medical Research 12, 3-21.

Agresti, A. and Min, Y. (2001). On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57, 963-971.