Topic: Advanced Design of Experiment

Author: Zongwei Zhou | 周纵苇
Weibo: @MrGiovanni
Email: zongweiz@asu.edu

选错课系列笔记...
Textbook: Design and Analysis of Experiments, edited by Dean, Morris, Stufken and Bingham, CRC Press, 2015

Lecture #1：2018.08.20

Basic Terminology

Treatment：试验的变量
Response Variable
Experimental Unit：试验的对象
Observational Unit

Basic Concept

Replication：多次重复试验，统计上的差异
Randomization: Not full randomization but restricted randomization.
Blocking [?] (随机区组设计): can increase the precision of inferences. To group some experiments together. blocking is the grouping of similar experimental units.

Lecture #2: 2018.08.22

``8'' estimate freedom [?]

CRD: Completely Randomized Designs
unrestricted randomization.

Goal: assess the effect of increasing acidity of the soil on scab disease among potatoes (scab disease does not thrive in acid soil)

The acidity is to be increased by adding sulphur to the soil at some time prior to the potatoes being planted

Does it matter when the sulphur is added? Does it matter how much sulphur is added?

Two times of application are considered: Fall or Spring
Four amounts: 0, 3, 6, 12

0	F3	S3	F6	S6	F12	S12
1	2	3	4	5	6	7

统计学根据显著性检验方法所得到的P 值，一般以P < 0.05 为显著， P <0.01 为非常显著，其含义是样本间的差异由抽样误差所致的概率小于0.05 或0.01。

degree of freedom 自由度

在统计学中，自由度(degree of freedom, df)指的是计算某一统计量时，取值不受限制的变量个数。通常df=n-k。其中n为样本数量，k为被限制的条件数或变量个数，或计算某一统计量时用到其它独立统计量的个数。自由度通常用于抽样分布中。

Treatment Contrasts

When treatments have a factorial treatment structure, we may be interested in contracts for main-effects, two-factor interactions, and so on.
In general, a treatment contrast is any function $p'\tau$ with $p'1=0$ [?]
A point estimator is $p'\hat\tau$ , which can also be used to construct a confidence interval.
One contrast of possible interest is control vs the average of the active treatments, i.e., $\tau_1-\frac{1}{6}(\tau_2+...+\tau_7)$

Each contrast corresponds to 1 df (degree freedom) and its SS (sum of square):
$\frac{(p'\hat\tau)^2}{Var(p'\hat\tau)/\sigma^2}$

To partition a Treatment SS we use orthogonal treatment contrasts. Two treatments contrasts $p'\tau$ and $q'\tau$ are orthogonal if $Cov(p'\hat\tau, q'\hat\tau)=0$
For a CRD, the treatment contrasts $\sum_id_i\tau_i$ and $\sum_if_i\tau_i$ are orthogonal if $\sum_id_if_i/r_i=0$
We can always find $t-1$ pairwise orthogonal treatment contrasts, but want to use treatment contrasts that are meaningful.