Developing Two Heterogeneous Test Statistics for the Fixed-Effect Unbalanced Nested Design
Jiin-Huarng Guo
Department of applied Mathematics National Pingtung University of Education
Taiwan, R. O. C.
Abstract
To deal heterogeneous data, the present study adopts Alexander-Govern test and Welch test as well to develop two heterogeneous test statistics to test Factor A and B, respectively, for the fixed-effect unbalanced two-stage nested design. The results of simulation experiments show that the proposed procedures can control Type I error rates on the nominal level in various conditions.
Keywords: Alexander-Govern test; Computer simulation; Robustness, two-stage, Welch test
1. Introduction
The simplest types of factorial designs involve only two factors, in which with one nested (hierarchical) factor is particularly useful for behavioral science and medical research because constraints prevent the experimenter from crossing every level of one factor with every level of the other factor. That is, the experimenter has specifically chosen the a levels of factor A, the b levels of factor B nested under each level of A, and n replicates (Montgomery, 1984, p. 358). Both factors of A and B are so-called fixed because the inferences drawn from the analysis of variance are applicable only to the levels of A and B actually used in the design. If there are an equal number of levels of B within each level of A, and an equal number of replicates, this is a balanced nested design. Otherwise, this is an unbalanced nested design. For instance, patients in each hospital have to apply a particular treatment. So each
hospital (B) is within only one level of treatment (A). This is denoted by B(A). Number of hospitals for each level of treatment might be unequal, or number of patients in each hospital is unequal too. The use of nested design enables a researcher to isolate the nuisance variable or to reduce the variability due to sites (Kirk, 1995; p. 477).
It should be also noted that nested designs are constructed from completely randomized designs, and the assumptions of independence, homogeneity, and
normality are also applied. However, it is well known that the F test for one-way fixed effect ANOVA is no longer robust when the data are heterogeneous (Kim & Cohen, 1998; Lix, Keselman, & Keselman, 1996; Scheffé, 1970; Wang, 1971). Therefore, the
F test statistic in the nested design is problematic. Furthermore, the analysis for
(Clinch & Keselman, 1982; Harwell, Rubinstein, Hayes, & Olds, 1992; Hsiung & Olejnik, 1996; Keselman, Carriere, & Lix, 1995; Keselman, Wilcox, Taylor, & Kowalchuk, 2000; Oshima & Algina, 1992; Schneider & Penfield, 1997). Therefore, in this study, we apply the Welch test (1951) and the Alexander-Govern test (1994) to develop two heterogeneous test statistics, respectively, to test the effects of A and B(A) for the fixed-effect unbalanced two-stage nested design.
2. Definitions of Test Statistics The linear statistical model for the two-stage nested design is
( ) ( ) ijk j k j i jk
X ,
for j1,...,J, k 1,...,Kj, and i1,...,njk.
For testing the effect of factorA
For testing the effect of factorA, the null hypothesis pertains to the equality of population treatment effect:
0: 1 ... J 0 H . We denote ( ) ( ) 1 , j K j k j k R X
2 ( ) ( ) , k j k j jk S d n and
2 ( ) 1 ( ) 2 ( ) 1 ˆ , /( 1) j j K k j k j K k j jk k d d n
Moreover, ( ) ( ) ( ) 1 1 / J J j j j j j R r R r
, where ( ) ( ) 1 1 , j j K k j k r d
2.1. Welch Test Statistic Let 1 2 1 ( ) ( ) 1 ( 1) ( ) , J j j j A J r R R
2 1 ( ) 1 ( ) ( ) 1 ˆ 1 / / J J j j j j j B r r
, and then the Welch test statistic for testing factor A is
2
1 1/ 1 2( 2) 1/( 1) .
W A J B J (1)
quantile of an F distribution with degrees of freedom v1 andJ 1 2 2 ( 1) /(3 ).1 v J B 7 5 3 ( )j 4 ( )j 33 ( )j 240 ( )j 855 ( )j , D C C C C 2 4 ( )j 10 ( )j 8 ( )j ( )j 1000 ( )j , E a a C a
3
( )j ( )j ( )j 3 ( )j / ( )j ( )j / ( )j Z C C C a D E . Then Alexander-Govern’s (1994) test statistic for testing factor A is2 1 ( ) 1 . J j j AG Z
(2)If AG exceeds the 1-αquantile of a chi-square distribution with J-1 degrees of1
freedom, then reject the null hypothesis of equal means for factor A. 3. The Design of the Simulation Experiment
A B D-1 σi 1 1 1 1 1 1 1 1 ni 9 9 9 9 9 9 9 9 D-2 σi 1 2 3 4 1 2 3 4 ni 9 18 27 36 9 18 27 36 D-3 σi 1 2 3 4 1 2 3 4 ni 36 27 18 9 36 27 18 9 D-4 σi 1 1 1 1 1 2 3 4 ni 9 9 9 9 9 18 27 36 D-5 σi 1 1 1 1 1 1 2 3 ni 36 36 24 12 36 36 24 12
Table 1. The resulting Type I error rates (%) for the 2 levels of factor A and 4 levels of factor B nested in each level A
Design F W AG PF PW PAG Factor A D-1 4.76 4.70 4.58 98.82 98.77 98.72 D-2 4.96 4.90 4.86 55.68 75.92 75.85 D-3 5.41 4.50 4.29 85.23 40.93 40.27 D-4 0.52 4.91 4.88 71.98 88.51 88.39 D-5 0.87 4.92 4.86 40.92 60.46 60.31 Factor B D-1 5.05 5.30 4.90 100 99.96 99.96 D-2 1.16 4.75 4.73 61.23 97.41 96.94 D-3 28.10 5.62 5.10 98.11 98.82 99.20 D-4 1.23 4.87 4.72 66.62 99.02 98.35 D-5 15.06 5.49 5.03 74.97 96.92 93.82 Note:
a. Underline entries indicate values are less than the criterion 2.5. b. Bold-faced entries indicate values exceeding the criterion 7.5. D-1: =(1,1,1,1 ; 1,1,1,1), n = (9, 9, 9, 9; 9, 9, 9, 9);
赴國外研究心得報告
計畫編號 NSC 96-2118-M-153-002 計畫名稱巢狀實驗設計的異質性檢定統計量及樣本數
出國人員姓名 服務機關及職稱 郭錦煌 國立屏東教育大學 應用數學系教授 出國時間地點 七月二十二日至八月八日 美國喬治亞州雅典城 國外研究機構 喬治亞大學統計系 工作記要：本次訪問美國喬治亞大學統計系, 主要與 Dr. Billard 討論國科會專題研究計畫案 的一些相關問題. 由之前合作的議題: 區間值線性迴歸的估計檢定與預測開始, 再討論目前 執行的巢狀實驗設計檢定問題, 以及未來可以繼續探討的問題. 當初探討線性迴歸在區間值的性質, 主要是區間值是 symbolic data 的一種現象. 過去幾 年 Dr. Billard 一直在研究 symbolic data 的一些相關性質, 最近與法國學者 Diday 共同寫 一本討論 symbolic data 的書. 由於 Dr. Billard 在這方面已有很多經驗與資料, 給本人很 多寶貴的建議. 諸如提供更多的 data 來驗證模型預測, 協助編修以便投稿發表. 另外繼續拓 展更一般化 linear model 的情況, 這是未來可以研究的課題.巢狀設計(Nested design)以往都探討每組的變異數相等, 樣本數也相等的 balance design, 比較少討論 unbalance 的情形. 本人目前研究主要討論在 變異數不相等, 樣本數也不相等 時, 如何檢定 two-stage nested design. Dr. Billard 目前教授實驗設計, 提供不少建議 都已列入研究範疇. 另外在 unbalance nested design 時, 如何求檢定所需的樣本數, 我們 也有進一步討論, 未來研究時會繼續再探討.
