此篇論文分為兩部份，第一個部份是Cochran-Armitage test for a linear trend中檢定H_0:π_i=π對於所有的i=1,..,I，想利用迴歸模型中的簡單線性迴歸(Simple Linear Regression: Y=β_0+β_1X+ε)取代，進而取代H_0:β_1=0； 另一個部份是在非線性模型下，對y去做轉換，並且提供Box-Cox的方法，可以得到轉換後的結果。 再來看是否有需要去對x做轉換，利用Box-Tidwell的方法去判別。
最後我們在此論文中嘗試去證明出 Pearson Chi-squared 的檢定統計量與迴歸模型中的簡單線性迴歸有相同的結果。並且提供例子與 SAS 相關程式。

The dissertation is divide into two parts. The first part is in Cochran- Armitage test for a linear trend to test H_0:π_i=π for all i=1,..,I, and wants to use simple linear regression of regression model substitutes. Then to test H_0:β_1=0.
Another part is under the nonlinear model, makes the transformation on y, and provides Box-Cox method.  And obtain the transformation the result.  Using the Box-Tidwell method to check whether has the need to transformation on x.
Finally in this dissertation, we attempt to prove the Pearson Chi-squared test in a view of a simple linear regression model. Also, we have provided some examples and the associatated SAS programs.

Contents
1   Cochran-Armitage test for a linear trend             1
1.1 Introduction                                         1
1.2 Least squares estimation                             2
1.3 Example                                             13

2   Transformation to Correct Model Inadequacies        15
2.1 Variance-Stabiliziry Transformations                15
2.2 Transformations to Linear the model                 22
2.3 Transformation on y : The Box-Cox Method            33
2.4 Transformation on x : The Box-Tidwell Method        41

3   Conclusions                                         45

4   References                                          46

Appendix                                                47
A.1  SAS program for 1.3 example on page 13             47
A.2  SAS program for 2.1 example on page 16             49
A.3  SAS program for 2.2 example on page 23             51
A.4  SAS program for 2.3 example on page 36             53
A.5  SAS program for 2.4 example on page 42             57

Table Contents
Table 1.1.1                                              1
Table 1.3.1                                             13
Table 1.3.2                                             13
Table 1.3.3                                             14
Table 2.1.1 Useful Variance-Stabiliziry Transformations Table      15
Table 2.1.2                                             16
Table 2.1.3                                             19
Table 2.1.4                                             21
Table 2.2.1                                             23
Table 2.2.2                                             26
Table 2.2.3                                             28
Table 2.3.1                                             37
Table 2.4.1                                             43

Graph Contents
Graph 2.1.1                                             18
Graph 2.1.2                                             20
Graph 2.2.1                                             25
Graph 2.2.2                                             26
Graph 2.2.3                                             28
Graph 2.2.4                                             32
Graph 2.3.1                                             40

Alan Agresti, (1990). Categorical Data Analysis, John Wiley ＆ Sons, Chapter 4
Cochran-Armitage, (1955). Cochran-Armitage test for a linear trend.
C.R.Rao, (1993). Linear Statistical Inference and Its Applications, 2nd, c.f. P383~P385.
Douglsd C. Montgomery, Elizabeth A. Peck, and G. Geoffrey Vining, (2001). Introduction to Linear Regression Analysis, third edition, John Wiley ＆ Sons. Chapter 13.
David W. Hosmer, Jr. and Stanley Lemeshow, (1989). Applied Logistic Regression, John Wiley ＆ Sons, Chapter 1- Chapter 2.