MCYang

列聯表之假設估計簡介

傳統的檢定假設建立在Neyman-Pearson的方法論上，除了隨機化檢定實務上使用有困擾，此一方法導致0-策，即接受或拒絕。雖然它是最佳檢定法(在Neyman-Pearson的架構下)，但仍然遭受來自各方面之批評。例如，貝氏學家指出Neyman-Pearson方法只能做0 (接受)或1 (拒絕)的決策行動，並且檢定的評估通常是（觀測資料前）以型I與型II誤差來做評估。Kiefer (1977)考慮使用觀測到資料的p-值做為虛無假設的評估，如此假設檢定可導到觀測資料後的證據力評估。

對於2×2列聯表，我們可將某一檢定所導得的p-值視為檢定假設(例如獨立模式)成立與否指標之估計，最近此一處理方法（假設估計）變得很熱門。當考慮的情況沒有干擾參數時，此一方法已被Hwang, Casella, Robert, Wells 與 Farrell (1992) 用來評估p-值。對於2×2列聯表的獨立性檢定（含有干擾參數），在平方誤差損失之下，可導得最佳程序，稱為期望p-值。在單尾檢定，它正好是中間p-值。而雙尾檢定，此一期望p-值是一新的程序並可由數值方法來取得。各種不同p-值的風險數值比較在Hwang與Yang (2001)中有圖形顯示。同時，期望p-值的真正顯著水準會最靠近指定的水準，並且比傳統使用的卡方p-值更靠近。

Hypothesis estimates for contingency tables

The contingency table arises in nearly every application of statistics. However, even the basic problem of testing independence is not totally resolved. More than thirty- five years ago, Lancaster(1961) proposed using the mid p-value for testing independence in a contingency table. The mid p-value is defined as half the condition probability of the observed statistic plus the conditional probability of more extreme values, given the marginal totals. Recently there seems to be recognition that the mid p-value is quite an attractive procedure. It tends to be less conservative than the p-value derived from Fisher's exact test. However, the procedure is considered to be some what ad-hoc. We provide theory to justify mid p-values and apply the Neyman-Pearson fundamental lemma and the estimated truth approach, to derive optimal procedures, named expected p-values. The estimated truth approach views p-values as estimators of the truth function which is one or zero depending on whether the null hypothesis holds or not. A decision theory approach is taken to compare the traditional p-value with the mid p-value. For the two-sided case, the expected p-value is a new procedure that can be constructed numerically. In a contingency table of two independent binomial samplings with balanced sample sizes, the expected p-value reduces to a two-sided mid p-value . Further, numerical evidence shows that the expected p-values lead to tests which have type one error very close to the nominal level. Our theory provides strong support for mid p-value.

Selected Publications:

1. An empirical investigation of some effects of spareness in constingency tables, Computational Statistics and Data Analysis（with Alan Agresti）5, 9-21 (1987).(SCI) .

2. Simultaneous estimation of Poisson means under entropy loss (with Malay Ghosh), Ann. Stat. 16, 278-291 (1988).(SCI) .

3. Ridge estimation of independent Poisson means under entropy loss, Statistics & Decisions, 10, 1-23 (1992).

4. Simultaneous estimation of Poisson means under relative squared error loss (with Y. H. Lin), Statistics & Decisions, 11, 357-375 (1993).

5. Bounded risk conditions in simultaneous estimation of independent Poisson means (with Y. H. Liu) , Journal of Statistical Planning and Inference, 47, 319-331 (1995).(SCI) .

6. Noninformative priors for the two sample normal problem (with M. Ghosh), TEST, V5, #1, 145-157 (1996).(SCI) .

7. Posterior robustness in simulataneous estimation problem with exchangeable contaminated priors (with Y. H. Liu), Journal of Statistical Planning and Inference, 65, 129-143 (1997).(SCI) .

8. An optimality theory for mid p-values in 2×2 contingency tables (with J. T. Gene Hwang), Statistica Sinica, 11, 807-826 (2001).(SCI) .

9. The Equivalence of the mid p-value and the expected p-value for testing equality of two balanced binomial proportions (with D. W. Lee J. T. Gene Hwang), Journal of statistical planning and Inference, 126, 273-280 (2004).(SCI) .

10. Improved exact confidence intervals for the odds ratio in two independent binomial samples (with Lin, C. Y.), Biometrical Journal, 48, 1008-1019 (2006).(SCI) .

11. Improved p-values for testing marginal homogeneity in 2x2 contingency tables (with Lee-Shen Chen), Communications in Statistics-Theory and Methods, 38, 1649-1663 (2009).(SCI).

12. Improved p-value tests for comparing two independent binomial proportions (with Che-Yang Lin), Communications in Statistics-Simulation and Computation, 38, 78-91(2009).(SCI).