Ludbrook, J. 及 Dudley, H.[7] 檢視了生命科學領域5種常被引用的期刊，發現在252項研究中，只有4%是從母體中隨機抽樣來產生實驗組，這些實驗對象都是動物。其他的研究只是把並非隨機樣本的受試對象隨機分組來產生實驗組及對照組，所以不應假設兩組為獨立之隨機樣本；然而大部份的研究仍然用t檢定來判斷母體平均數是否相等，似乎背後的假設條件是否符合，並沒受到足夠的重視。

    常用的檢定統計量大部分都是根據隨機樣本來做決定的，而隨機樣本符合i.i.d. (independent, identically distributed)之條件。若考慮在 X1,X2,…,Xn 為「可交換 (exchangeable)」，而不符合i.i.d.，即 X1,X2,…,Xn 的分配相同，但並不獨立。此時一些常用的檢定統計量，其分布必然與 X1,X2,…,Xn 為i.i.d.的假設下會有差異，這些差異會有多大？不同的可交換分配之下狀況又如何？本文主要就是探討在 X1,X2,…,Xn 為可交換的條件下，在二項檢定(binomial test)、兩樣本之t檢定、Wilcoxon rank sum檢定以及信賴區間等，利用R模擬出不同分配的可交換隨機變數(多變量常態分配、三種不同的copula)來進行檢定問題的探討。

Ludbrook, J. and Dudley, H.’s survey of 252 prospective comparative studies reported in five frequently cited biomedical journals revealed that experimental groups were constructed by random sampling in only 4% of them, and these experimental objects all are animals. Other researches produce experimental and comparison group by randomization of nonrandom samples, so we should not treat the two groups as independent random samples. Yet most researches are still using T-test to decide if the population means are equal or not, without worrying whether the conditions behind the test are met.

Most of the popular test statistics are based on random samples which are i.i.d. (independent, identically distributed). Now we consider the case when X1,X2,…,Xn are exchangeable, but not i.i.d. ,that means X1,X2,…,Xn are identically distributed, but not independent. In this case, the distribution of these test statistics should be different from the i.i.d. cases, and we want to find the differences.

Let X1,X2,…,Xn be exchangeable variables, we use the R system code to simulate exchangeable variables from different distributions (multivariate normal density and 3 types of copula), then carry out binomial test, two sample T-test and Wilcoxon rank sum test, we also calculate the confidence intervals and discuss the results.

目錄

1  緒論 ........................................................  1
2  文獻回顧 ...................................................  3
3  簡介 ........................................................  5
    3.1可交換隨機變數(exchangeable random variable) ..........  5
    3.2簡介copula .............................................  7

4  可交換隨機變數的檢定問題探討 ...........................  17
    4.1二項檢定(binomial test) .................................  17
    4.2兩樣本之t檢定 ........................................  27
    4.3 Wilcoxon rank sum檢定 ................................  30
4.4信賴區間 ..............................................  33

5  結論 .......................................................  37
6  參考文獻 ..................................................  38


表目錄

表4-2-1(a)multivariate normal density dim=10；t檢定的π(θ) ............... 28
表4-2-1(b)Gumbel normal density dim=10；t檢定的π(θ) ................... 29
表4-2-1(c)Frank normal density dim=10；t檢定的π(θ) ..................... 29
表4-2-2(a)multivariate normal density dim=10；Wilcoxon rank sum檢定的π(θ).. 31
表4-2-2(b)Gumbel normal density dim=10；Wilcoxon rank sum檢定的π(θ)..... 32
表4-2-2(c)Frank normal density dim=10；Wilcoxon rank sum檢定的π(θ)....... 32


圖目錄

圖3-2-1(i)：2-Dim Gumbel Copula with N(0,1) marginals  δ=1.1 ............  12
圖3-2-1(ii)：2-Dim Gumbel Copula with N(0,1) marginals  δ=1.5 ...........  13
圖3-2-1(iii)：2-Dim Gumbel Copula with N(0,1) marginals  δ=2 ............  13
圖3-2-1(iv)：2-Dim Gumbel Copula with N(0,1) marginals  δ=2.5 ...........  13
圖3-2-2(i)：2-Dim Frank Copula with N(0,1) marginals  δ=1×10^-6 ...........  15
圖3-2-2(ii)：2-Dim Frank Copula with N(0,1) marginals  δ=1 ...............  15
圖3-2-2(iii)：2-Dim Frank Copula with N(0,1) marginals  δ=3 ..............  15
圖3-2-2(iv)：2-Dim Frank Copula with N(0,1) marginals  δ=5 ..............  16
圖4-1-1(i) 10-dim Normal Copula with N(0,1) marginals：ρ=0 ............  18
圖4-1-1(ii) 10-dim Normal Copula with N(0,1) marginals：ρ=0.15 ..........  19
圖4-1-1(iii) 10-dim Normal Copula with N(0,1) marginals：ρ=0.25 ..........  20
圖4-1-2(i) 20-dim Normal Copula with N(0,1) marginals：ρ=0 .............  21
圖4-1-2(ii) 20-dim Normal Copula with N(0,1) marginals：ρ=0.15 ..........  22
圖4-1-2(iii) 20-dim Normal Copula with N(0,1) marginals：ρ=0.25 .........  23
圖4-1-3(i) 30-dim Normal Copula with N(0,1) marginals：ρ=0 ............  24
圖4-1-3(ii) 30-dim Normal Copula with N(0,1) marginals：ρ=0.15 ..........  25
圖4-1-3(iii) 30-dim Normal Copula with N(0,1) marginals：ρ=0.25 ..........  26

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[2] Frees, Edward W. and Valdez, Emiliano A.(1998), Understanding relationships using copulas, North American Actuarial Journal, vol. 2, No. 1, pp1-25

[3]Gumbel, E.J.(1960), Bivariate exponential distributions, Journal of the American statistical association, Vol. 55, No. 292, pp698-707

[4]Hollander, M. and Wolfe, D.(1999), Nonparametric statistical methods ,   edi., Wiley-Interscience,pp364-377

[5]Joe H.(1997),Multivariate Models and Dependence Concepts, (Chapman & Hall/CRC).

[6] Langbehn,D.R., Berger,V., Higgins,J.J., Blair,R.C., Mallows,C.L.,  Ludbrook,J. and Dudley,H.(2000), Letter to the editor:”Ludbrook, J. and Dudley, H.(1998), Why permutation tests are superior to t and F tests in biomedical research, The American Statistician, 52, 127-132 “, The American Statistician, vol. 54, Iss. 1, pp85-87


[7]Ludbrook, J. and Dudley, H.(1998), Why permutation tests are superior to t and F tests in biomedical research, The American Statistician, vol. 52, No. 2, pp127-132

[8]Nelsen, Roger B.(1999), An Introduction to Copulas, (Springer).

[9] Alan Genz, Frank Bretz and Torsten Hothorn with contributions by Tetsuhisa Miwa, Xuefei Mi, Friedrich Leisch and Fabian Scheipl(2008),  R’s package：mvtnorm (Multivariate normal and t distributions) ,
http://cran.r-project.org/web/packages/mvtnorm/index.html

[10] Jun Yan and Ivan Kojadinovic(2008), R’s package：copula (Multivariate dependence with Copula) ,
http://cran.r-project.org/web/packages/copula/index.html