Transcript Interpretation of Study Results

腦研所
資料分析三次上課大綱:
 I. Biostatistics: basic concepts
 基本觀念
 課堂練習：描述統計
 II. Probability Density Function





1-sample t test
2-sample t test
Non-parametric
Multiple comparison
課堂練習：1-s t test; 2-s t test
 III. Analysis of Variance





One-way ANOVA
Two-way ANOVA
Interaction
Some study designs
課堂練習：2-way ANOVA
1
I. Biostatistics: basic concepts
Institute of Brain Science, 2008
2
腦研所
The study of statistics
 Origin of Statistics
 Telling a story with numbers:
 Exploring the collection, organization, analysis, and interpretation of
numerical data.
 Applied field:
 Business, marketing, economics, politics, agriculture, education,
psychology, sociology, anthropology, biology, Public Health….
 Biostatistics:
 focusing on biological and health science
 Objectives:
 以隨機抽樣樣本估計母群體
 樣本平均值等於母群體平均值；
 樣本標準差等於母群體標準差；
3
1. 統計學速成的幾個重要觀念:
4
腦研所
『有統計差異』是什麼意思? ND
常態分佈下的觀察值與期望值
 Normal distribution parameter: mean, variance
x ~ N (  , 2 )
Z
x

x
5
2-s t test
腦研所
Eyeball test for 2-sample means
 Using 95% Confidence
Limits
 A: Significant
 B: non-significant
 C: need to do a statistical
test
A
B
C
6
ANOVA
腦研所
Eyeball test for 3-sample means
1
2
3
1
2
3
 Using 95% Confidence
Limits
 A: Non-Significant
 B: Significant
 Why?
 Between group variation
 Within group variation
 Why not do 2-s test 3 times?
 Alpha error inflated
 Ex: 7 groups MD
A
B
comparisons
1 / 21 < 0.05 !!
7
ANOVA
腦研所
One-way ANOVA table
Source of variation
SS
DF
Mean SS
F ratio
Between k groups
SSB
k-1
MSB
MSB/MSE
Error(within groups)
SSE
n-k
MSE
Total
SST
n-1
F test:
MS B MBSS SSB/(k  1)
F


MS E MESS SSE /(n  k )
8
腦研所
Statistical Interaction & confounding
Interaction: 2 lines with different slope
Y |T ,C    1T   2C   3TC
H1 : ˆ3  0
C0
Confounding: 2 parallel lines
C1
C1
Y
H1 : ˆ1|c  ˆ1
C0
T0
T1
9
腦研所
Univariate analysis: 1X1Y
X
Y
Comparisons
Methods
Num._normal
Num._normal
Num._non-normal
Num._non-normal
Num._normal
Num._normal
Num._non-normal
Categorical
Categorical
Categorical_Binary
Categorical_Binary Categorical_Binary
Categorical_Binary Categorical_Binary
2 indep. means
>= 2 indep. means
2 indep. medians
>= 2 indep. medians
Two-sample t test*
One-way ANOVA*
Wilcoxon rank sum
Kruskal-Wallis
Regression*
Paired t
Wilcoxon signed rank
Pearson's Chi-sq
McNemar Chi-sq
Pearson's Chi-sq
2-Z
 說明：有*的分析方法需要有以下假
設:
 名詞縮寫
Binary
Categorical
Binary
Categorical
num._normal
 normality
 Independence..
2 related means
2 related medians
X related to Y
2 related prop.
2 indep. Prop.
2 indep. Prop.
 Cat.: categorical; Num.: numerical
10
腦研所
Multivariate analysis: Xs1Y
Xs
Y
Methods
Categorical
Cat.
Log-linear
Cat.+Num.
Cat.(binary)
Logistic regression
 Multivariate normality
Cat.+Num.
Cat.(>=3)
Logistic regression
 Independence..
Dicriminant analysis*
Cluster analysis
Propensity scores
 說明：有*的分析方法需要有以下假設:
 名詞縮寫
 Cat.: categorical; Num.: numerical
 CART: classification and
CART
Cat.
Num.
ANOVA*

MANOVA*


Num.
Num.
Multiple regression*
Cat.+Num.
Num.(censored)
Cox Propotional hazard model
Confounding factors
Num.
ANCOVA*

regression tree
ANOVA: analysis of variance
ANCOVA: analysis of covariance
MANOVA: multivariate analysis of
variance
GEE: generalized estimating
equations
MANOVA*
GEE*
Confounding factors
Num.
Cat.
Mantel-Haenszel
Factor analysis
11
2.研究的出發點: 研究問題的提出
12
腦研所
Conceiving the Research Q.
 Origins of a RQ
 Mastering the literature
 Being alert to new ideas and techniques
 Keeping the imagination roaming
 Characteristics of a good RQ: FINER
 Feasible
 Adequate NO. of subjects; adequate technical expertise; affordable in time and
money; manageable in scope;
 Interesting
 Novel
 Confirms or refutes previous findings; extends previous findings; provides new
findings;
 Ethical
 Relevant
 To scientific knowledge; to clinical and health policy; to further research
directions..
13
Steven R. Cummings et al: Designing Clinical Research: An Epidemiologic Approach. 2001
腦研所
重要的研究: X?→Y?
14
腦研所
幾個有趣的研究
 尿療法
 喝尿前後三個月，所有慢性病指標明顯改善?!
 特異功能




I can see your letters through the envelop.
When I was young, I can tell animal talking.
When I am in a good mood, I can tell the future.
李嗣涔: 演講題目「人體特異功能」，提及高橋小妹的史丹福申請resume
 會議：「身心靈科學」學術研討會
 時間：1999/12/11,陽明大學國際會議廳第三會議室
 單位：中華生命電磁科學學會主辦、中華民國預防醫學學會協辦
 其他：崔玖(人體身心靈的健康測試) 、陳國鎮(穴道電檢法探測到的生命相，東吳大學物理
系教授)、王唯工(穴診儀之測量與血液循環之關係，中研院物理所研究員)、周碧瑟(另類醫
學在評估上的困難，陽明大學教授)
 書籍：
 難以置信II尋訪諸神的網站，心靈拓展系列118(張老師2004)、
 難以置信，心靈拓展系列64(張老師2000)、
 人身極機密－－人體X檔案(時報出版1998)
15
腦研所
重要的問題: 台灣愛滋問題與drug use
2008/10/31止
愛滋病問題是道德淪喪問題?
16
腦研所
HIV, drug abuse and Brain Science
17
3. 疾病自然史的演進：
18
腦研所
疾病自然史：醫學觀點
5
Risk Factors
分子: 新病人
分母: 沒病的母群體
I (1,2)
Prognostic Factors
(Surrogate Indicators?)
分子: 病重或死亡者
分母: 新病人
II (3)
III (4,5)
三段五級防治架構
臨床醫學
預防醫學
（社會醫學）
臨床水平
可感受期
0
Biological onset
症狀前期
臨床期
殘障期
19
腦研所
三段五級疾病防治策略
 第一段預防：針對可感受期
 第一級：健康促進
 運動、飲食、衛生教育、環境衛生
 第二級：特殊防護
 疫苗注射、預防意外事件、消除致癌物質、安全性行為
 第二段預防：針對症狀前期
 第三級：早期診斷、早期治療
 篩檢
 第三段預防：針對臨床期以後
 第四級：限制殘障
 第五級：復健
 策略運用：
 每個疾病都有其最有效率的防治策略，但其他工作也很重要
20
腦研所
疾病防治的其他觀點：社會學
 英國每百萬兒童的肺結核死亡率
 從1838的4000，到了1960已低於200/M,
Mckeown, 1976
 請依以下歷史事件的年代，選出正確的死亡率趨勢圖
 1880年發現TB菌、1945發病治療藥物、1955發明BCG
 A B C D 哪條線對呢？
21
腦研所
疾病自然史的連續演進
1. Exp.
A
B
C
D
E
2. Exp.
A
B
C
D
E
 時間點說明：






Exp: 危險因子暴露
A: 生物學上的疾病起始(biologic onset)
B: 可被診斷技術偵測出
C: 症狀出現後被診斷出
D: 病情變嚴重
E: 死於該疾病
 自然史型態
 1. : 一般疾病自然史
 2. : 疾病診斷後，卻死於其他疾病
22
腦研所
介入治療與疾病自然史的改變
3. Exp.
A
B
C
D
E
4. Exp.
A
B
C
D
E
5. Exp.
A
B
C
6. Exp.
A
B
C
D
D
E
E
自然史型態
3
4
5
6
: ‘pseudodisease’, 臨床期未出現，死於其他疾病
: 早期治療無療效，疾病進程一樣
: 早期治療有療效，延緩病重D與死亡E 的發生
: 早期治療可治癒，死於其他疾病
23
腦研所
篩檢、治療與疾病死亡率
5. 治療有效，延緩發病後的Disease-specific mortality rate
screening
6. 治癒疾病，避免死亡後的Disease-specific mortality rate
screening
24
腦研所
與疾病自然史有關的幾個時間名詞
Induction time誘發期
 暴露於致病因到發生疾病後果disease onset之間
 每個疾病的誘發期都有其最大值與最小值範圍
 無法因診斷技術進步而縮短
Latent period潛伏期：
 發生疾病後果到可檢測出之間的時間
 可因診斷技術進步而所短
Apparent induction time: (AIT=IT+LP)
 暴露於致病因到可檢測出之間的時間
 可因診斷技術進步而所短
At risk of exposure暴露危險期
 即流行病學上的觀察人年
25
腦研所
Log hazard ratio
舉例：時間軸與Reverse causality bias
老問題：抽菸與肺癌
弔詭處：剛戒菸者易得肺癌
關鍵點：戒菸從何時算起?
Time since cessation(years)
Modeling Smoking history: A comparison of different approaches, Leffondre, 2002, AJE
26
腦研所
因的過程：自然史與致病因時間軸
暴露開始
B’
Exp(因)
A(果)
累積足夠暴露
疾病發生biologic onset 篩檢工具極限 篩檢查出
Detectable
Detected
Induction time
Latent period
B
Delay time
Apparent induction time
C
症狀出現查出
Lead time
Sojourn time (=DPCP*)
DPCP*: Detectable pre-clinical period
?=0
Time at risk of exposure
lifelong?
Definition of exposure
Exposure onset
=0 ? In study of delayed effects in survivors of the atomic bombs in Japan
27
=> Dilute the effect estimate unless the AIT had only 1 day (Ms to Ys)
腦研所
暴露病因與疾病後果的幾種關係
A：血壓與症狀
B：輻射與血癌
C：眼壓與青光眼
 有一個明顯的閾值關係
D：膽固醇與CHD
 大多數慢性病屬於此類
B.
C.
D.
Incidence
A.
Exposure
28
腦研所
慢性疾病防治的根本定理
25
25
17%
8%
20
4%
15
20
22%
15
19%
10
10
13%
9% 8%
5
0
5
Prevalence %
CHD deaths/1000/6 year
The strategy of preventive medicine, Oxford press, 1992
0
0-4
4.04.5
4.5- 5.0- 5.5- 6.0- 6.55.0 5.5 6.0 6.5 7.0
serum cholesterol(mmol/L)
7.07.5
7.58.0
 為數眾多的人暴露在小量的危險因子當中，比起少數人暴露在高量危險因子，會產生出更多的病例數

Martin MJ et al: Serum cholesterol, blood pressure and mortality. Lancet, II 933-6,1986.
 微觀層次的研究也有相同現象，QTL
29
腦研所
慢性病防治的弔詭
Prevalence %
 為數眾多的人暴露在小量的危險因子當中，比起少數人暴露在高量
危險因子，會產生更多病例數
 對社區整體有很大利益的預防性措施，卻對每個參與的個人沒有太
大的好處：
30
25
20
15
10
5
0
Truncation of the ditribution
60
80
100
120
140
160
180
200
SBP(mmHg)
30
4. 實證科學的思考：
假說提出：
因果推論：
科學哲學：
31
腦研所
假說的提出：您贊成哪一個？
血管老化導致痔瘡
 某學生申請到美國中西部一家教學醫院實習。當地有許多牛仔，
主要以畜牧業為生。他發現，當地痔瘡病患年紀偏大，病況也
較嚴重
牧場工人常得到的牛痘對天花有免疫作用
 E Jenner觀察到，牧場女工都沒得天花
電燈開關是造成電燈亮的原因
 五歲的小明發現，每次打開開關電燈就會亮
公雞啼叫是太陽出來的原因
 怪叔叔發現，每次他養的公雞啼叫不久天就亮了
生命科學與電燈開關
32
腦研所
事實真相與科學哲學
 科學哲學：
 歸納法：學
 缺點一：英雄所見略有不同
 缺點二：沒有邏輯：「豬長翅膀」式思考的危險
 演繹法：思
 缺點：型式邏輯的陷阱
 舉例：肥胖易導致心臟病 → 中國肥胖婦女易導致心臟病：熊?!
 猜測與否證：Conjecture & refutation, Karl Popper
基本觀點
正面「證實」看法
反面「否證」看法
科學基於證實假說
科學基於推翻錯誤的假說，存留較可能
為真的假說
思考出發點
觀察顯示真理（先有觀察才有假說） 觀察涉及解釋，所以觀察者乃是心中先
有假說，才觀察到他所要觀察的
對定理的看法 好的定理是被多次證實的
好的定理是被否證多次而未被擊倒的
基本假設
歸納法是合邏輯的
只有演繹法合乎邏輯
定理的科學性 定理被觀察所證實，也就越具科學性 一個定理經歷越多否證而不被推翻，也
就越合乎科學
33
腦研所
豬長翅膀：簡單的邏輯謬誤
Bertrand Russell (1945):
 假設：如果豬有翅膀，所以有些長翅膀的動物的肉很好吃
 推論：有些長翅膀的動物的肉很好吃，所以豬有長翅膀
臨床實例：愛滋病門診
 如果你不好好吃藥，你的病毒量就會上升。
 你現在的病毒量上升，可見你沒有好好吃藥。
34
腦研所
致病因果的邏輯判斷
充分致因(sufficient cause)：
 只要有此致因，一定會發病
必要致因(necessary cause)：
 一定要有此致因存在，才會發病
促成致因(contributing cause)：
 一種致因若不是充分致因，則是一個促成致因
B
A
E
D
C
A
H
G
B
F
C
A
J
I
F
35
腦研所
因果推論Causation & Causal Inference
因果與強度：
 三個疾病的充分致因
 四個成分致因A, B, E, U
Exposures
假設A沒有測量
Response
Frequency of exposure pattern
族群 1
族群 2
A
B
E
(outcome)
1
1
1
1
100
900
1
1
0
1
100
900
1
1
0
0
1
0
1
0
900
900
100
100
0
0
1
1
1
0
1
0
900
900
100
100
0
0
1
0
100
900
0
0
0
0
100
900
Note: (A or B) in 1 is common, 90%, 3600/4000; (A or B) in 2 is uncommon, 10%, 400/4000
36
腦研所
因果推論, cont.
族群 1 的 Incidence propotions for combinations of B & E，假設 A 沒有測量
B=1, E=1
B=1, E=0
B=0, E=1
B=0, E=0
Cases
1,000
100
900
0
Total
1,000
1,000
1,000
1,000
Proportion
1
0.1
0.9
0
族群 2 的 Incidence propotions for combinations of B & E，假設 A 沒有測量
B=1, E=1
B=1, E=0
B=0, E=1
B=0, E=0
Cases
1,000
900
100
0
Total
1,000
1,000
1,000
1,000
1
0.9
0.1
0
Proportion
 結果：
Modern Epidemiology, 1998
 E 在族群 1 對疾病發生率有較強的作用
 B 在族群 2 對疾病發生率有較強的作用
 因果互補性(Causal complement, CC)： E的CC為(A & U) or (B & U)
 某因子的作用強度，取決於其CC的相對普遍性( 暴露盛行率)
 作用強度：在預防醫學上有重要意義，在生物學機轉上卻沒有太大意義
37
腦研所
因果關係判定的基準
必要基準：
 時序性(temporality)
次必要基準：
 一致性(consistency)
 機會無法解釋, P<0.05?
 沒有其他干擾性的解釋因素
 合理性(coherence)
 相關強度(strength of association)
 相關的特定程度(specificity of association)
 生物性漸增趨勢(biological gradient)或劑量反應關係
 生物學上的贊同性(biological plausibility)
38
腦研所
這種研究是新年笑話嗎?!
 研究題目：
 Effects of remote, retroactive intercessory prayer on outcomes in patients
with bloodstream infection: randomised controlled trial BMJ 2001;323:14501451 ( 22-29 December )
 研究設計：
 Objective: To determine whether intercessory prayer has an effect
 Design: Double blind, parallel group, randomised controlled trial
 Subjects: All 3393 adult patients whose bloodstream infection was detected




at the hospital in 1990-6
Intervention: In July 2000 patients were randomised to a control group and an
intervention group.
Main outcome measures: Mortality in hospital, length of stay in hospital, and
duration of fever.
Results: Mortality was 28.1% (475/1691) in the intervention group and 30.2%
(514/1702) in the control group (P for difference=0.4). Length of stay in
hospital and duration of fever were significantly shorter in the intervention
group than in the control group (P=0.01 and P=0.04, respectively).
Conclusions: Remote, retroactive intercessory prayer said for a group is
associated with a shorter stay in hospital and shorter duration of fever in
patients with a bloodstream infection and should be considered for use in
39
clinical practice.
腦研所
研究問題爭議與因果推論
 施打HBV疫苗到底有沒有副作用?
 A shadow falls on hepatitis B vaccination effort. Science, 1998.
 歐美陸續傳出施打B型肝炎疫苗後遺症的病例，包括許多自體免疫疾病，如
多發性硬化症(multiple sclerosis, MS)、視神經炎等
 巴黎有反對疫苗聯盟的活躍份子於1998/07/17控告法國政府低估疫苗的危險
性，該團體聲稱代表了15000人的立場。法國一位醫師收集了曾施打過B型
肝炎疫苗而出現類似MS的600多個病患資料。美國休士頓一位分子生物學家
(本身就是避孕疫苗的研發者)堅信，她的哥哥就是因為施打B型肝炎疫苗後
才出現類似MS的症狀。美國FDA的VAERS(vaccine adverse event
reporting system)收集了111個施打疫苗後才併發MS的案例，但FDA宣稱，
回顧這些病人的病歷資料，並沒有得到任何證據顯示，這些病例是因為施打
B型肝炎疫苗而引起的。
 中研院院士陳定信發表意見指出，台灣已超過四百萬人施打過疫苗，從沒有
類似問題，MS是西方人叫常見的疾病，應與施打疫苗無關…
 您的意見如何?
 Hepatitis B Vaccination and the Risk of Multiple Sclerosis, NEJM2001
40
腦研所
癌末不死: 醫學奇蹟?
台大數學系退休教授黃武雄
資料來源: 我的數學生涯: http://www.math.ntu.edu.tw/library/history/article_03_09_16.htm
樞機主教單國璽
為什麼會有醫學奇蹟?
41
5. 研究偏差與研究設計
42
腦研所
Bias: a systematic error
偏差與影響研究結果的可比性：(王榮德，流行病學方法論)
 族群可比性Comparability：年齡、性別、種族、生活習慣等
 資料或測量方法的可比性：如臨床試驗的遮盲(Blinding)設計
 作用可比較性：干擾因子
Results of Bias
 Bias away from the null hypothesis, BAN
overestimating RR
 Bias toward to the null hypothesis, BTN:
underestimating RR
H0
1
BTN
2
BAN
RR
43
腦研所
Types of Bias:
三種影響研究結果的偏差 Bias:
 Selection bias: 選樣偏差
 Information bias: 訊息偏差
 Confounding: 干擾因子
44
腦研所
Selection bias
 Selection & Maintenance of subjects in a study (Fig 10-6, p146)
 目標族群、抽樣族群、合格族群、參與族群、全程參與族群
 Clinical trial or Cohort studies: loss to follow-up
 Case-control studies:
 對照組：Willing to participation
 個案組：Incident vs. Prevalent cases
 Risk factors also a prognostic factor: MI vs. cholesterol
 Hospital-based subjects: admission rate bias (Berkson, 1946)
 Detection bias (Feinsten, 1979): 很混亂、但很重要的名詞。
45
腦研所
Information bias (misclassification)
Random or systematic inaccuracy in measurement
 Non-differential MxC(random):
MxC of E or D are independent of other variables.
Social unacceptability bias
 Differential MxC(systemic):
MxC of one variable depends on the other.
Examples:
 Recall bias, Interviewer bias
減少MxC的方法:
 以biological markers 取代pt. self report
46
腦研所
Differential misclassification
Recall bias in Case-control studies
 個案會較努力去回想危險因子，對照組則不會
Interviewer bias
 訪員的問法、問卷題目的設計都會影響結果，例如：
您是否贊成「開放三通」以促進經濟發展？
您是否贊成「戒急用忍」以防止產業空洞化？
47
腦研所
Non-differential misclassification:
social unacceptability bias
 Case-Control Study: Myocardial infarction v.s Fat
 危險因子：高脂肪食物攝取量
D
D
E
a
 最後結果：
 可能是Bias toward the Null, BTN (Int J Epi, 2005)
E
c
 Large sample size; independence of errors;
 No confounding, selection bias, and mismeasurement of covariates
b
 可能情境：無論是否有病，受訪者都會低報脂肪攝取量
d
 Clinical trial: Intention to treat principle
 某些實驗組病人沒到吃新藥、對照組卻吃了新藥：該怎麼辦？
 依照原來的分組，作資料分析
 依照實際吃藥情況，作資料分析
 將這種病人的資料刪掉
48
腦研所
干擾因子的定義
Mixing effect of X2 with X1 & Y
Definition:
Obesity
 Associated With the disease of
interest in the absence of exposure
本身單獨與疾病有相關；本身是危險因子
Cholesterol
MI
 Associated With the exposure
與危險因子有相關：分布不平均
 Not as a result of being exposed.
干擾不能是中介變項：intervening variable
Intervening variable: X1X2Y
Example: S/S of diseases
49
腦研所
干擾因子的控制
控制干擾因子
 研究設計階段：
Randomization、Restriction(Sex:male)、Match(只在世代研究才可以)
 資料分析階段：
分層分析：Stratification
多變數迴歸：multiple regression
50
腦研所
三種主要研究設計與控制干擾因子
Case-control study
 Matched case-control study
Cohort study
Randomization clinical trial
 Complete matched cohort study
Causality and correlation
 Y=a+b1X1+b2X2+b3X3+b4X4+b5X5…
51
腦研所
Early Case-control Study in AIDS
Onset
of study
Large number of
sexual partners
Time
Cases
Small number of
sexual partners
Large number of
sexual partners
Controls
Small number of
sexual partners
Direction of inquiry
52
腦研所
Cohort Study in AIDS
Onset
of study
Time
HIV-1 +
Receptive
anal
intercourse
HIV-1 -
HIV-1 +
HIV-1
negative
homosexual
men
No receptive
anal intercourse
HIV-1 Direction of inquiry
53
腦研所
AIDS Prognostic Factors Study
Onset
of study
Age
CD4+
Hemoglobin
Time
No progression
Prognostic
factor present
Progression
All patients
with HIV
infection
Prognostic
factor absent
No progression
Progression
Direction of inquiry
54
腦研所
Clinical Trial for AIDS Treatment
Onset
of study
Time
AIDS or death
Eligible
subjects
PI
+
RTIs
No progression
Randomize
RTIs
only
AIDS or death
No progression
Direction of inquiry
55
腦研所
Case-control study: 如何選對照組是關鍵
E
E
D
D
D
D
E
9
400
E
1
600
OR=13.5
Over-Matched CTL:
結果是對照組越來越像 OR=13.5
個案組, 結果是低估OR Good CTL
OR<13.5
OM CTL
過度配對
OR>13.5
UM CTL
56
腦研所
各種Study Designs之間的關係
Case-control study
 Matched case-control study
Cohort study
E
E
 Matched cohort study
Randomization clinical trial
 =Complete matched cohort study
Causality and correlation
 Y=a+b1X1+b2X2+b3X3+b4X4+b5X5…
越好的研究設計越不需複雜的統計分析
covariate, confounder
57
腦研所
6.變項的定義與測量
All science is measurement
 Helmholts, German physiologist
Good measurement
 Great science breakthrough is always about
measurement improvement
 To do everything well is to measure it well
Good study
 A good story about the relations btw variables
Definition
 Definition of variables
 Definition of Denominator and Numerator
58
腦研所
Variables
Variables
 A characteristic of interest in a study
 With different VALUE for different measurement
units
Gender: male, female
Blood pressure: 130 mHg
59
腦研所
Variables and study question
Outcome (dependent) variable: Y
 Systolic blood pressure
 Angina pectoris:
0:no; 1:mild; 2:severe
Explanatory (independent) variable: X,
 Treatment
1:new drug; 0: standard drug
 BMI
Causality and correlation
 Y=a+b1X1+b2X2+b3X3+b4X4+b5X5…
covariate, confounder
60
腦研所
Types of data measurements
Nominal scale : ( =, ≠ )
 Race, gender, etc
 Also called categorical data, qualitative scale
Ordinal scale : ( =, ≠, >, < )
 Your satisfaction score about YMU
Numerical scale :
 Interval scale (=, ≠, >, <, +,  )
Temperature: zero is NOT NOTHING in quantity in measuring
 Ratio scale (
=, ≠, >, <, +, , ,  )
Weight: zero is nothing in quantity in measuring
61
腦研所
Between measurement scales
Definition of mountain
 A prominence of 1000
feet, by Royal
Geographical Society
An interesting movie
 山丘上的情人
 The Englishman who
went up a hill but came
down a mountain
62
7. Descriptive statistics
63
腦研所
描述統計的內容
Data Presentation
 Types of numerical data
 Tables
 Graphs
Numerical Summary Measures
 Measures of central tendency
 Measures of dispersion
 Grouped data
64
腦研所
Numerical Summary Measures
Measures of Central Tendency
PDF機率密度函數
 Mean

1
   xi     xi  Pi    xf ( x)dx

母體:
N
樣本: x 
1
xi

n
 Median: 50th percentile of a set of measurement,
n=
奇數:
偶數:
n 1
th;
2
n
n

th

(

1
)
th
/2
2

2


For ordinal and discrete/continuous data
65
腦研所
Measure of central tendency, cont.
Mode
A
unimodal
C
R. skewed
B
bimodal
D
L. skewed
E
2 distributions with
Identical means, medians
and modes
66
腦研所
Measure of Dispersion
 Range: (Max-Min)
 Interquartile Range
75th – 25th : the middle 50% of the observations
 Variance
母體:
樣本:
 2  E( x   ) 2  E( x 2 )   2 
S 
2
2
(
x

x
)
 i
n 1

2
n 1

2
(
x


)

2
(
x

x
)
 i
2
n

2
n 1
  (2n1)
n-1: degree of freedom
 Coefficient of Variation: SD/mean
When to use C.V.?
67
腦研所
研究文獻舉例: 描述統計表
Effects of remote, retroactive intercessory prayer on outcomes in patients with bloodstream infection:
randomised controlled trial BMJ 2001;323:1450-1451 ( 22-29 December )
68
腦研所
研究文獻舉例: 分析統計表
BMJ 2001;323:1450-1451 ( 22-29 December )
69
腦研所
資料變異性的例子: Ab structure
variability and regularity Wu TT, Kabat EA
 An analysis of the sequences of the
variable regions of Bence Jones proteins
and myeloma light chains and their
implications for antibody complementarity.
J Exp Med. 1970 Aug 1;132(2):211-50.
 Accurate prediction of Ab structure more
than 10 years before the emergence of
relevant technology
 Definition of variability:
(No. of a.a. found) / freq of the most common a.a.
70
腦研所
Data problem 1:
Changes in patient characteristics, number of admissions, duration of hospital stay, and mortality rate of 309 HIV-infected patients
Without hemophiliacs from June 24, 1994 to June 23, 1999
Clinical Spectrum, Morbidity, and Mortality of Acquired Immunodeficiency Syndrome
71
in Taiwan: A 5-Year Prospective Study. JAIDS, 24:378–385, 2000
課堂練習作業 1：
72
II. Probability Density Function
Normal Distribution
1 sample t test
2 sample t test
73
腦研所
1.『有統計差異』是什麼意思? ND
常態分佈下的觀察值與期望值
 Normal distribution parameter: mean, variance
x ~ N (  , 2 )
Z
x

x
74
腦研所
Standard normal distribution
Z Transformation and Z value:
 (Observed - Expected) in terms of UNITS of SD
舉例
x ~ N (  , 2 )
 考試分數
 兒童身高智商
Z
x

x
75
腦研所
Probability Density Function
Area under probability distribution
 For continuous random variables:
b
f ( a  x  b)   f ( x ) d ( x )
f(x)
a



f ( x)d ( x)  1
f ( x )  0, for all x
a
b
76
腦研所
Normal density function
Point of inflection
x ~ N (  , 2 )
σ
1
1 2
f ( x) 
exp( Z )
2
2
σ
Z
μ
x

x
77
腦研所
Characteristics of N Distribution
Area under curve=1
Symmetric about the mean
mean=median=mode
Points of inflection:
 μσ
78
腦研所
One tailed & two tailed in Hypothesis testing
Two tailed test:
One tailed test:
 H0: μ1 = μ2 ; H1: μ1 ≠ μ2
 H0: μ1 = μ2 ; H1: μ1 > μ2
 較保守
 較寬鬆, 因多了專業訊息
**Z1-α/2 = 1.96, α=0.05
**Z1-α = 1.645, α=0.05
68.27%
95.45%
  2
 

 
  2
79
腦研所
Statistical Power & α, β
 Type One Error: α error
 the probability of rejecting the true null hypothesis
 無中生有的機率
 Type Two Error: ß error
 the probability of accepting the false null hypothesis
 視而不見的機率
Fact
H0 is ture
H1 is true
Accept H0
1
 error
Reject H0
 error
1
Decision
80
腦研所
Power的計算
 Power: 1-ß，正確推翻虛無假說的機率
 Ex: one sample Z test
 如何算出研究所需的樣本數
 一般定Alfa=0.05, beta=0.2, 粗估σ值並訂出有臨床意義的絕對值差，即可算出
H1
H0
N ( 1 ,  2 / n)
[ Z   (  0   1 ) n /  ]
if
 0  1
[ Z   (  1   0 ) n /  ]
if
 0  1
N (  0 ,  2 / n)
Area  1  
 [ Z 1 ]  1  
 Z    Z 1 ,
1
Area  
0
Area  
 0  Z   / n
 Z 1 
1   0

 Z 1

n
81
腦研所
Decision Principle
Avoid  error v.s. Avoid  error principle
 Panacea:
Doctors & terminal patients
 SARS epidemic:
Doctors & Director of DOH
 God:
Onlooker & members of Holy UFO from Taiwan
 教主陳恆明預言1998/03/25上帝會降臨美國德州
 挺貪反貪?
Making a wrong decision
 Probability: objective
 Risk: highly subjective
82
Review
腦研所
影響統計檢力的因素
Type I error: α
Mean difference: |μ1-μ2|
Standard deviation: σ
Sample size: n
 You can’t decrease α, β error simultaneously unless
you get more information, i.e., increasing the sample
size.
83
腦研所
2. 瞭解真相哪一種方法比較好?
研究整個母全體或是抽樣?
研究樣本
 內在校度：因果推論
該研究結果是否有所偏差?
 外在校度：統計推論
該研究結果能否外推?
 內在效度是外在效度的前提
84
腦研所
Sampling methods
Simple random sampling
Stratified random sampling
 CATI (computer-aided telephone interview)
Cluster sampling
Systemic sampling
Multi-stage sampling
Conventional sampling
85
腦研所
Sampling distribution
 Sampling distribution
 Occurring in REPEATED SAMPLING
 Distribution of values of
over all possible samples
X
 Using sample Statistics to inference population Parameters
 Quiz:
 If we want to select 6 students from the class with total 49
students, how many possible samples would we have?
 作業:
 買一張電腦選號的大樂透，算算樣本的平均值與標準差，並與母群體的平
均值標準差做比較。
86
腦研所
Central Limit Theorem
For large n,
x ~ N (  ,  / n)
2
The beauty of CLT:
 Easy to calculate V
The ugliness of CLT:
 Hard to explain p
Standard Error:
 SE of the mean
 SD of the means

n
X N ( ,
x~
)
2
87
腦研所
Statistical Inference
Parameter symbol Statistical symbol
Mean
μ
x
Standard Deviation
σ
SD
Standard Error (CTL)
x 

n
Sx 
S
n
88
腦研所
95% Confidence Interval
 In repeated sampling with same sample size n,
 95% of the all CI will contain the true population mean
μ
89
腦研所
Estimation
點估計
 樣本平均值＝母群體平均值
區間估計
 Confidence Interval: level of confidence
信賴區間太大會失去意義
 Typical values are 95%CI
 100*(1 - α)% CI for μ
95% CI for the population meanμwhen σis unknown
x  t(1 / 2),df
S
S
   x  t(1 / 2),df
n
n
90
腦研所
Parameters and Statistics
Parameter
symbol
Statistical
symbol
Mean
μ
X
Standard Deviation
σ
SD
Variance
σ2
S2
Correlation
ρ
r
Proportion
π
p
91
Review
腦研所
Population & Sampling Distribution
Population parameters known
SD
Mean
x
x
 i
N
x
xi
x  
N

 (x
Z score
i
 )2
z
N
SEx 
Population parameters unknown

n
Zx 
X 

( xi   )

n
Mean
x
x i
n
xi
x  
n
SD
S
t score
2
(
x

x
)
 i
SEx 
n 1
S
n
t
tx 
xx
S
( xi   )
S
n
92
腦研所
3. 一組平均值差的統計檢定
 某年齡群正常成年男子平均收縮壓為125mmHg, 標準差為10, 從眾
多糖尿病患隨機抽100人, 其收縮壓為130, 請問糖尿病人血壓是否
異於常人?
解
 假說檢定
 虛無假說：糖尿病人與正常人血壓相同 μ1 = μ0
 對立假說：糖尿病人與正常人血壓不同 μ1 ≠μ0
 結論
 P<001, 推翻虛無假說, 糖尿病人血壓與常人不同
 x  1   x  0
1
0
 x  10 / 100  1
Z
x  x
x
130  125

5
1
93
腦研所
4. 兩組平均值差的檢定與假設
Assumptions for 2-s t distribution
 Random samples
 Normal distribution
 Equal Standard deviations
Homogeneous variances, if not:
t test is Robust when the sample sizes are equal
 Independence observation between 2 groups
94
2-s t test
腦研所
兩組平均值差的抽樣分布
x1 ~ N (1, 12 / n1 )
x2 ~ N(2 , 22 / n2 )
1
2
x 1  x 2 ~ N ( 1   2 ,
( x x
1
 12
n1

 22
n2
)
 (2x  x
2)
1
1   2 x1  x 2
2)
95
腦研所
Population & Sampling Distribution
Population parameters known
Mean
x
x

xi
N
x  
xi
N
Population parameters unknown
SD

 (x
Z score
i
 )2
z
N
SEx 

n
Zx 
x1  x2 ( x1 x2 )  1  2 SE ( x1  x2 )   1 / n1  1 / n2 Z ( x1  x2 ) 
X 

( x  )

n
( x1  x 2 )  0
 1 / n1  1 / n2
Mean
x
SD
xi
n
x  
S
xi
n
 (x
t score
i
 x) 2
t
n 1
SEx 
S
n
tx 
SE ( x  x )  S p 1 / n1  1 / n2 t( x1  x2 ) 
1
2
xx
S
(x  )
S
n
( x1  x 2 )  0
S p 1 / n1  1 / n2
96
2-s t test
腦研所
Eyeball test for 2-sample means
 Using 95% Confidence
Limits
 A: Significant
 B: non-significant
 C: need to do a statistical
test
A
B
C
97
2-s t test
腦研所
實例 1: Paracervical block(X) and cramping score(Y): Harper, 1997
computer printout
Descriptive Statistics Section
Variable
Count Mean
SD
SE
95%LCL
95%UCL
BLOCK_TY Group=1 No Block 39
51.41026
28.11135
4.50142
42.29761
60.5229
BLOCK_TY Group=2 Block
45
35.6
28.45123
4.24126
27.0523
44.1477
Note: T-alpha (BLOCK_TY Group=1 No Block) = 2.0244,
T-alpha (BLOCK_TY Group=2 Block) = 2.0154
Confidence-Limits of Difference Section
Variance Assumption
DF
Mean D
SD
Equal
82
15.81026
28.29424
Unequal
80.57 15.81026
39.99651
Note: T-alpha (Equal) = 1.9893,
T-alpha (Unequal) = 1.9898
SE
95%LCL
95%UCL
6.19012
6.184744
3.496136
3.503572
28.12438
28.11694
Power
(Alpha=.05)
0.713673
0.000015
0.812755
Power
(Alpha=.01)
0.470767
0.000001
0.573611
Equal-Variance T-Test Section
Prob
Decision
Level
(5%)
Difference <> 0
2.5541
0.012496
Reject Ho
Difference < 0
2.5541
0.993752
Accept Ho
Difference > 0
2.5541
0.006248
Reject Ho
Difference: (BLOCK_TY Group=1 No Block)-(BLOCK_TY Group=2 Block)
Alternative Hypothesis
T-Value
98
腦研所
實例 1: Box plot & Error Bar Plot
VASALLCR T otal Cramping Score
Box Plot
Error Bar Plot
100.00
65.0
60.6
56.1
75.00
51.7
47.2
50.00
42.8
38.3
25.00
33.9
29.4
0.00
G1
G2
Groups
25.0
1 No Blo
2 Block
99
腦研所
實例 1: Assumption test
Tests of Assumptions Section
Assumption
Value
Probability Decision(5%)
Skewness Normality (No Block)
-0.7312
0.464671
Cannot reject normality
Kurtosis Normality(No Block)
-1.8119
0.070002
Cannot reject normality
Omnibus Normality (No Block)
3.8176
0.148258
Cannot reject normality
Skewness Normality (Block)
0.7487
0.454054
Cannot reject normality
Kurtosis Normality (Block)
-3.2609
0.001111
Reject normality
Omnibus Normality (Block)
11.1937
0.003709
Reject normality
Variance-Ratio Equal-Variance Test
1.0603
0.852504
Cannot reject equal variances
Modified-Levene Equal-Variance Test
0.1999
0.655980
Cannot reject equal variances
100
腦研所
實例 1: Distribution curve
Histogram of BLOCK_TY Group=2 Block
10.0
10.0
7.5
7.5
Count
Count
Histogram of BLOCK_TY Group=1 No Block
5.0
2.5
2.5
0.0
0.0
5.0
25.0
50.0
75.0
BLOCK_TY Group=1 No Block
100.0
0.0
0.0
25.0
50.0
75.0
100.0
BLOCK_TY Group=2 Block
101
課堂練習作業 2：
Dennison,
五歲小朋友的熱量攝取與全國平均是否有所不同:
 1-s-t test
兩歲與五歲小朋友的熱量攝取是否有所不同:
2-s-t test
102
腦研所
兩組平均值差的檢定假設出問題1
Not Normal distribution
Not Equal variance
Independent groups
103
2-s t test
腦研所
2-Sample t test, unequal variance
σ1≠σ2 , n≥30
 Test for equal variance
F test: If variance unequal, SDs can not be pooled.
 Very sensitive if data are not normal distribution
Levene test:
 A modified t test: average deviations (O-E) are the same in 2
group
 Solution:
Transform the data to make variance equal
Satterthwaite correction: reducing DF, to be more conservative
Non-parametric approach
104
NP test
腦研所
Distribution issue
Outliers violate t test assumption
2-s t test
P>0.05!
But Sir, it’s
normal and
healthy!
EXP
You kill that
God damned
abnormal
mouse!
CTL
105
NP test
腦研所
Nonparametric analysis
History
 Wilcoxson’s Lab data in a pharmaceutical company (1940)
 You don’t need to have the parameters (mean, variance)
to test the distribution!
Measure and test Median, not mean
 Avoid outlier problem
Outlier issue by using nonparametric test
 Problem solved
If the numbers of outlier are small
Due to random error
 Problem worse
Due to systemic error
106
NP test
腦研所
Nonparametric test
Wilcoxon Rank Sum Test
 Null hypothesis (logic of the test):
1
Means of the ranks are equal in 2
groups
 Procedure:
2
3
4
5
.
.
EXP
Rank all the scores from low to high
If same score, assign the average
rank
7
8
9
 Test:
The ranks are tested as 2-s t test
CTL
107
腦研所
Example 1: Nonparametric result 1
Median Statistics
Variable
Count
Median
95% LCL
95% UCL
No Block
39
50
37
65
Block
45
32
19
50
Mann-Whitney U or Wilcoxon Rank-Sum Test for Difference in Medians
Variable
MWU
W Sum Ranks Mean of W
SD of W
No Block
1139.5
1919.5
1657.5
111.4458
Block
615.5
1650.5
1912.5
111.4458
Number Sets of Ties = 18,
Multiplicity Factor = 528
Exact Probability
Alternative H
Prob
Approximation W/O Correction
Decision(5%) Z-Value
Approximation With Correction
Prob
Decision(5%) Z-Value
Prob
Decision(5%)
Diff<>0
2.3509
0.018
Reject Ho
2.3464
0.018
Reject Ho
Diff<0
2.3509
0.990
Accept Ho
2.3554
0.990
Accept Ho
Diff>0
2.3509
0.009
Reject Ho
2.3464
0.009
Reject Ho
108
腦研所
Example 1: Nonparametric result 2
t test on RANK of total cramping score
Variable
Count Mean
SD
SE
95% LCL
95% UCL
No Block
39
49.21795
23.3246
3.734925
41.65699
56.77891
Block
45
36.67778
24.01802
3.580395
29.46197
43.89359
Note: T-alpha (BLOCK_TY Group=1 No Block) = 2.0244,
T-alpha (BLOCK_TY Group=2 Block) = 2.0154
Equal-Variance T-Test Section
Alternative
T-Value
Prob .Level
Decision (5%) Power (0.05)
Power(0.01)
Difference <> 0
2.4186
0.017794
Reject Ho
0.666503
0.418374
Difference < 0
2.4186
0.991103
Accept Ho
0.000026
0.000001
Difference > 0
2.4186
0.008897
Reject Ho
0.774497
0.520876
Hypothesis
Difference: (BLOCK_TY Group=1 No Block)-(BLOCK_TY Group=2 Block)
109
腦研所
5. 兩組平均值差的檢定假設出問題2
Normal distribution
Equal variance
Not Independent groups
110
Paired t test
腦研所
Paired t test for matched groups
Repeated measure studies
 Very Powerful in detecting differences
 Common used in clinical study
One sample t test for mean difference
 Null hypothesis (logic of the test): Mean difference = 0
 Test statistic:
t
d  d
Sd

d 0
SD d / n
,  SD d 
2
(
d

d
)

n 1
 More statistical efficient than 2-s t test
Less samples needed to reach significant level
Pre-test post-test covariance v.s. independence btw 2 groups
111
ANOVA
腦研所
Data sheet: paired t test
Subjects
Pre-T
Post-T
Difference
1
X11
X12
X1d
2
X21
X22
X2d
3
X32
X32
X3d
4
X42
X42
5
X52
X52
…
…
…
…
…
…
n
Xn2
Xn2
Xnd
112
腦研所
paired t test
 paired (repeated measures) designs:
 非獨立樣本平均數的t檢定
 舉例: 降血壓藥物服用後，病人血壓是否有所不同？
t
d  d
Sd

d 0
SDd / n
,  SDd 
2
(
d

d
)

n 1
113
腦研所
Paired t test 實例
Sauter GH的研究
 Bowel habits and bile acid malabsorption in the
months after cholecystectomy. Am. J.
Gastroenterol. 2002
 資料見Table 5-8: difference bwt baseline and 1
month measures of 7-alpha-HCO
114
課堂練習作業 3：
Sauter
膽囊切除手術病人7-alfa HCO前後數值是
否有不同
Paired t test
115
腦研所
Problem 4: Pre/post Tx BP Difference
Means difference in 2 groups:
Not Normal distribution
Not Equal variance
Not Independent groups
116
NP test
腦研所
Nonparametric test for matched groups (1)
 Time to use:
 Not normally distributed and n < 30
 Sign test:
 Null hypothesis (logic of the test):
 The probability of above and below the median score is equal to 0.5
 Procedure:
 Assign ( ) sign for the score which is above the median
 Assign ( ) sign for the score which is below the median
 Test:
 The probability follow binomial distribution,
 Test statistic
z
X  n   0 .5
n (1   )
117
NP test
腦研所
Nonparametric test for matched groups (2)
Wilcoxon Signed Rank test:
 Null hypothesis (logic of the test):
The mean of the signed rank before and after intervention = 0
 Procedure:
Assign ( ) sign for the score if pre-post difference is > o
Assign ( ) sign for the score if pre-post difference is < o
Rank all the scores from low to high
 Test:
The ranks follow t distribution
Test statistic
t
( Mean_ rank)  0
SDrk / n
118
N
腦研所
6. Sample size determination
1-s mean study
n
 2 ( Z 1   Z 1 ) 2
( 1   0 ) 2
1-s proportion study
 Z  0 (1   0 )  Z   1 (1   1 ) 
n

 0  1


2-s means study
 2 ( Z 1   Z 1 / 2 ) 2
n  2
( 1   0 ) 2
 Very Important for getting enough power & research fund
 PASS is a very good software to deal with this problem
119
2
Summary
Flowchart of 2G MD test
腦研所
STATISTICS
1-s t
1 g ro u p
N > 30
1-s t
ND
1. Tra n s F fo r t
2. s ig n te s t
No o f g ro u p s
2-s t
2-s t
N > 30
Eq u a l va ria n c e
2-s t
Eq u a l N
ND
In d e p e n d e n t
2 g ro u p
1. tra n s fo rm fo r t
2. WRS te s t
1. Tra n s F fo r t
2. WRS te s t
P a ire d t
N > 30
P a ire d t
ND
If Ye s , g o u p ; If No , d o d o wn
1. tra n s fo rm120
fo r t
2. WS R te s t
Summary
腦研所
Summary of t tests
Always check the raw data before any test
 Distribution, outlier, definition of zero and missing, etc
 Descriptive analysis is very important
Always check test assumptions
 Sample size
 Normal distribution
 Equal variance
 Independence
A simple & powerful test for mean differences
 Study Design plays the key role:
randomized controlled trial and other designs
121
腦研所
資料分析練習
Dennison,
 五歲小朋友的熱量攝取與全國平均是否有所不同
1-s-t test
 兩歲與五歲小朋友的熱量攝取是否有所不同
2-s-t test
Sauter
 膽囊切除手術病人7-alfa HCO前後數值是否有所不同
Paired t test
Gonzalo
 不同甲狀腺功能、體重狀態的病人，其胰島素敏感性是否有所
不同
2-way ANOVA
122
腦研所
期中考例題1
123
腦研所
期中考例題2
 花名在外的陳先生被告有一名非婚生子，去找親子鑑定專家高中同學王醫師請教問題，王醫師指出，懷
孕期的長短是接近平均值為270天，標準差為10的常態分布。陳先生能證明在小孩出生前的270天到290
天之間他人是在國外的。若被告事實上是小孩的父親，請問母親有很長(大於290天)或很短(小於240天)
的懷孕期的機率是多少？
(A). 0.001 (B).0.012 (C).0.0241
(D).0.05
124
腦研所
期中考例題3



某一服藥前後的臨床試驗研究結果，用的是2-s t test，結果有統
計差異。該研究所使用的統計方法是否正確?研究結論是否可信?
某綜合大學採入學申請制，結果男生的錄取率為50%，女生的錄
取率則僅為20%，結果引來女性主義者的嚴重抗議。如果您是該
校校長，如何為自己辯護?
美國專欄作家Ann Landers有一次問她的讀者：「如果可以重來
一次，你要孩子嗎？」結果她接到將近一萬份的回函，其中將近
70%的父母說：「不要！」。每日新聞Newsday針對同樣的問題
作了一次全美隨機抽樣調查，訪問了1300多位父母，其中91%表
示還是會要孩子。請問哪一個調查結果比較可信，為什麼?


(A).第一個，因為研究樣本數較大；
(B).第二個，因為研究對象是隨機樣本
125
ANOVA
III. Analysis of Variance
Analysis of Variance
三組以上平均值檢定
The logic of ANOVA
Partition of sum of squares
F test
One way ANOVA
Multiple comparison
Two way ANOVA
Interaction and confounding
126
ANOVA
腦研所
1. Logic of ANOVA : Eyeball test
1
2
3
1
2
3
 Using 95% Confidence
Limits
 A: Non-Significant
 B: Significant
 Why?
 Between group variation
 Within group variation
 Why not do 2-s test 3 times?
 Alpha error inflated
 Ex: 7 groups MD
A
B
comparisons
1 / 21 < 0.05 !!
127
ANOVA
腦研所
Data sheet: k groups MD comparison
Subjects
Observed
1
X1
2
X2
3
X3
4
X4
5
X5
…
…
…
…
n
Xn
Tx
A
Group
Mean
Grand
Mean
Ma
Group
Effect
Ma-M
M
Total
Tx error
Difference
X1-Ma
X1-M
X2-Ma
X2-M
X3-Ma
X3-M
Mb-M
B
Mb
…
…
…
K
Mk
Mk-M
128
ANOVA
腦研所
Logic of ANOVA: SS Partition
Total Difference divided into two parts
 (Observed－group mean)＋(group mean－grand mean)
  X ij  X  ( X ij  X . j )  ( X . j  X )
Total sum of squares divided into two parts
 SS Total = SS Between + SS Within (or Error)  SST = SSB
+ SSE
   ( X ij  X ) 2   [( X ij  X . j )  ( X . j  X )] 2  ( X ij  X . j ) 2   ( X . j  X ) 2
j
i
j
i
Partition of TD & TSS
Model of one-way ANOVA
  X ij     j  eij
x
j
i
j
i
x
x
129
A
B
C
腦研所
Assumptions in ANOVA
Normal Distribution: Y values in each group
 Not very important, esp. for large n
 If not ND and small n: Kruskal-Wallis
nonparametric
Equal variance: homogeneity
 If not: data transformation or ask for help
Random & independent sample
130
ANOVA
腦研所
F test: variance ratio test
Review:
 F test for equal variance in 2-s t test
F test: F=V1/V2
 The larger V is divided by the smaller V
 If two variances are about equal, the ratio is about 1
 The critical value of F distribution depends on DFs
ANOVA for mean difference, k groups
 Null hypothesis: 1= 2 = 3=…= k
 Variance Between / Variance within
 If F is about to 1, it’s meaningless for grouping
131
ANOVA
腦研所
F test : named after Fisher
 Characteristics
 a sickly, poor-eyesighted child
 The teacher used no paper/pencil
to teach him
 Very strong instinct on geometry
 Mathematicians take years to prove
his formulas
 Persistence
 Calculation of ANOVA tables takes
Fisher 8 months, 8h/D to finish!!
 Reference:
 The lady tasting tea, Salsburg, 2001
 「統計，改變了世界」天下，2001
Sir Ronald Aylmer Fisher 1890-1962
132
ANOVA
腦研所
One-way ANOVA table
Source of variation
SS
DF
Mean SS
F ratio
Between k groups
SSB
k-1
MSB
MSB/MSE
Error(within groups)
SSE
n-k
MSE
Total
SST
n-1
F test:
MS B MBSS SSB/(k  1)
F


MS E MESS SSE /(n  k )
133
ANOVA
腦研所
2. Multiple Comparison
 Definition:
 Contrast btw 2 means: 1 2
 More than 2 means is OK: [(1  2 )/2] c
 Compare the overall effect of the drug with that of placebo
 Contrast Coefficients: add to 0
 Orthogonal
 Two contrasts are orthogonal if they don’t use the same information
 Ex: (1 2) and (3 4), i.e. the questions asked are INDEPENDENT
 Types of MC: before or after ANOVA
 Priori(planned) comparisons
 post hoc(posteriori) comparisons
134
ANOVA
腦研所
Example 1: one-way ANOVA
Research problem:
 Life events, depressive symptoms, and immune function.
Irwin M. Am J Psychiatry, 1987; 144:437-441
 Subjects: women whose husbands
treated for lung Ca.
died of lung Ca. in the preceding 1-6 Months
were in good health
 X: grouping by scores for major life events
Measurement: Social Readjustment Rating Scale score
 Y: immune system function
NK cell activity: lytic units
135
Printout
腦研所
Box plot & Error bar plot
Error Bar Plot
Box Plot
60.0
100.00
54.4
48.9
CELL
75.00
43.3
37.8
50.00
32.2
26.7
25.00
21.1
15.6
10.0
0.00
1
2
3
1
2
3
GROUP
136
Printout
腦研所
ANOVA table
Analysis of Variance Table
Source Term
DF
Sum of Squares Mean Square F-Ratio
A: GROUP
2
4654.156
2327.078
S(A)
34
9479.396
278.8058
Total (Adjusted)
36
14133.55
Total
37
8.35
Prob
Power(Alpha=0.05)
0.001125* 0.947488
137
Printout
腦研所
Nonparametric ANOVA
Kruskal-Wallis One-Way ANOVA on Ranks Test Results
Method
Prob. Level Decision (0.05)
DF
Chi-Sq (H)
Not Corrected for Ties
2
11.16963
0.003754
Reject Ho
Corrected for Ties
2
11.17095
0.003752
Reject Ho
Group Detail
Group
Count
Sum of Ranks
Mean Rank
Z-Value
Median
1
13
351.00
27.00
3.3087
37
2
12
163.50
13.63
-2.0927
14.5
3
12
188.50
15.71
-1.2815
14.05
138
ANOVA
腦研所
MC: Priori comparisons
t test for orthogonal comparisons
 t statistic: t 
xi  x j
2MS E / n
 DF: (n1+n2j); n=n1=n2
; not using SDp but MSE
Adjusting  downward:  / (group number)
 Ex: 4 comparisons, =0.05/4=0.0125
Bonferroni t procedure
 Applicable for both orthogonal & non-orthogonal
 t statistic:
Multiplier 2MSE / n
 Multiplier table: no. of comparisons & DF for MSE
 Able to find CI for mean difference
139
ANOVA
腦研所
MC: Posteriori comparisons
Tukey’s HSD (honestly significant difference)
 HSD= Multiplier
MSE
n
 Like Bonferroni, HSD multiplier table is needed (P176,
table 7-7)
 Able to find CI for mean difference
278.82
HSD

4
.
42

 21.31
 Ex:
12
24.63
22.17
2.46
LOW
n=13
MOD
n=12
HIGH
n=12
140
ANOVA
腦研所
MC: Posteriori comparisons
Scheffé’s procedure
 S statistic:
S  ( j  1) F ,df  MS E 
C 2j
nj
 j: No. of groups; C: contrast; (alpha, df1, df2)=(0.01, 2, 34)
 most versatile (not only pair-wise) & most conservative
 EX: Low (Moderate & High) combined; Low Moderate

C 2j 12 (1) 2
12 (0.5) 2 (0.5) 2
 

 0.125; 
 
 0.167
n j 12
12
12
n j 12 12
C 2j
S  (3 1)  5.31 278.82 0.167  22.24
 Note: MD btw L & H not significant
 Able to find CI for mean difference
141
ANOVA
腦研所
MC: Posteriori comparisons
Newman-Keuls procedure
MS E
n
 Multiplier table is needed
 NK statistic: m ultiplier
2 Steps
2 Steps
3 Steps
 Less conservative than Tukey’s HSD
 Unable to find CI for mean difference
 Ex:2 steps NK  3.87  4.82  18.65 ; 3 steps NK  4.42 4.82  21.31
same as HSD
142
ANOVA
腦研所
MC: Posteriori comparisons
Dunnett’s procedure
2MS E
 Dunnett’s statistic: m ultiplier
n
 Only used in several Tx means with single CTL mean
 Relatively low critical value
 Ex:
D  2.71  6.82  18.48
2 units lower than HSD value;
4 units lower than Scheffé value
143
ANOVA
腦研所
Other posteriori comparisons
Duncan’s new multiple-range test
 Same principle as NK test; but with smaller multiplier
Least significant difference, LSD
 Use t distribution corresponding to the No. of DF for MSE
  levels are inflated.
 Proposed by Fisher
The above two procedures are NOT recommended
by statisticians for medical research.
144
ANOVA
腦研所
Summary of Multiple Comparisons
Don’t care about the formulas
Which procedure is better? depends on you!
 Pairwise comparisons:
Tukey’s test: the first choice; Newman-Keuls test: second choice
 Several Txs with single CTL:
Dunnett’s is the best
 Non-pairwise comparisons:
Scheffé is the best
 When  larger than 0.05 is OK to you: e.x., drug screening
LSD, Duncan’s new multiple-range test are O.K.
The above two are not recommended by the authors
145
Printout
腦研所
Multiple comparisons
Newman-Keuls Multiple-Comparison Test
Group
Count
Mean
Different From Groups
2
12
15.60000
1
3
12
18.05833
1
1
13
40.23077
2, 3
Response: CELL; Term A: GROUP; DF=34; MSE=278.8058
Scheffe's Multiple-Comparison Test
Group
Count
Mean
Different From Groups
2
12
15.60000
1
3
12
18.05833
1
1
13
40.23077
2, 3
Critical Value=2.5596
146
腦研所
3. Two-way ANOVA
Logic of two-way ANOVA
 SST divided into 3 or 4 parts
SST = SSR + SSC + SSE
SST = SSR + SSC + SS(RC) +SSE
Models of two-way ANOVA
 Without interaction:
X ij     i   j  eij
 With interaction:
X ij    i   j  (i  j )  eij
147
腦研所
4. Confounding & Interaction交互作用
Interaction:
 The effect of X1 varies with the level of X2
 A phenomenon you have to present
 Main effects of X1, X2: not meaningful anymore
 Ex: X1(Sex), X2(teaching method) & Y (language score)
Confounding:
 Given condition: no interaction
 A condition you have to control (or adjust)
148
腦研所
干擾與交互作用
研究設計
有干擾、有交互作用
無干擾、無交互作用
有干擾、無交互作用
強烈交互作用，
干擾已無意義
填鴨
Adjusted vs. Crude
第二層估計值
粗率估計值
(Effect Measures)
第一層估計值
F-U(RR)
1.02
1.86
4.00
F-U(RR)
1.74
3.00
1.00
C-C(OR)
0.96
0.45
1.83
F-U(RR)
4.00
4.00
4.00
F-U(RR)
1.00
1.00
1.00
C-C(OR)
1.83
1.83
1.83
F-U(RR)
1.01
1.03
4.00
F-U(RR)
3.00
3.00
1.00
C-C(OR)
0.83
0.83
1.83
F-U(RR)
1.07
9.40
4.00
F-U(RR)
3.00
0.33
1.00
C-C(OR)
0.36
6.00
1.83
啟發
F
F
D
D
E
110
390
500
E
380
2620
3000
D
D
E
90
1410
1500
E
20
980
1000
RR=3.00
RR=1.74
SEX
D
D
E
200
1800
2000
E
400
3600
4000
RR=1.00
Epidemiologic research, 1982
 變項：性別(X1)、教學方法(X2: 啟發式教育、填鴨式教育)、語言成績(Y)
 干擾：
 女生整體的語言成績較男生好，但受到教學方法的干擾：填鴨式較低、啟發式較高
 交互作用：
 女生以啟發式教育法的語言成績較高，以填鴨式教育的成績較低
 男生以啟發式教育法的語言成績較低，以填鴨式教育的成績較高
149
腦研所
Simpson’s Paradox: 陳小姐買帽子
第一天
第二天
第一櫃(大人)
第二櫃(小孩)
兩櫃一起
紅色
黑色
紅色
黑色
紅色
黑色
合適
9
17
3
1
12
18
不合適
1
3
17
9
18
12
Total
10
20
20
10
30
30
90%
85%
15%
10%
40%
60%
150
腦研所
Statistical Interaction & confounding
Interaction: 2 lines with different slope
Y |T ,C    1T   2C   3TC
H1 : ˆ3  0
C0
Confounding: 2 parallel lines
C1
C1
Y
H1 : ˆ1|c  ˆ1
C0
T0
T1
151
ANOVA
腦研所
Two-way ANOVA table
Source of variation
SS
DF
Mean SS
F ratio
Among rows
SSR
r-1
MSR
MSR/MSE
Among columns
SSC
c-1
MSC
MSC/MSE
SS(RC)
(r-1)(c-1)
MS(RC)
MS(RC)/MSE
Error
SSE
rc(n-1)
MSE
Total
SST
n-1
Interaction
152
ANOVA
腦研所
Example 2: two-way ANOVA
Research problem:
 Glucose tolerance, insulin secretion, insulin
sensitivity and glucose effectiveness in normal
and overweight hyperthyroid women. Gonzalo MA.
Clin Endocrinol, 1996;45:689-697
 X1: BMI; X2: thyroid function
All categorical variables
BMI: 2 level; thyroid function: 2 level;
 Y: Insulin sensitivity
Continuous variable
153
Printout
腦研所
Box plot & Error bar plot, ex 2
Means of IS
1.00
Error Bar Plot
1.0
HT
HT
0.9
0
1
0 Normal thyroid
1 Hyperthyroid
0.8
0.75
IS
IS
0.7
0.50
0.6
0.4
0.3
0.25
0.2
0.1
0.00
0.0
0
1
BMI2
0
1
BMI
154
Printout
腦研所
Descriptive statistics, ex 2
Means and Standard Errors of IS
Term
All
Count
Mean
SE
33
0.4647917
0
19
0.615
5.786324E-02
1
14
0.3145833
6.740864E-02
0
19
0.57375
5.786324E-02
1
14
0.3558333
6.740864E-02
0,0
11
0.68
0.0760472
0,1
8
0.55
8.917324E-02
1,0
8
0.4675
8.917324E-02
1,1
6
0.1616667
0.1029684
A: BMI2
B: HT
AB: BMI2,HT
155
Printout
腦研所
2-way ANOVA table, ex 2
Analysis of Variance Table for IS (alpha = 0.05)
Source
DF
SS
MSS
F-Ratio
Prob.
Power
A: BMI2
1
0.7112253
0.7112253
11.18
0.002293*
0.898154
B: HT
1
0.3742312
0.3742312
5.88
0.021745*
0.649738
AB
1
6.091182E-02 6.091182E-02
0.96
0.335909
0.157220
S
29
1.844833
Total (Adj.)
32
2.916255
Total
33
6.361494E-02
156
課堂練習作業 3
Gonzalo
不同甲狀腺功能、體重病人，其胰島素敏感性是否有不同
2-way ANOVA
157
腦研所
Example 3: two-way ANOVA
 Chapter 7, exercise 7, (table 7-20, p187)
 Analysis of phenotypic variation in psoriasis as a function of age at onset and family
history. Arch. Dermatol. Res. 2002;294:207-213
 請回答下列問題:
 不同發作年齡病人的%TBSA (percent of total body surface area affected)是否有所差異?
 不同類型病人(familial vs. sporadic)的%TBSA是否有所差異?
 是否有interaction ?
158
5. 其它研究設計與ANOVA
CRD: Complete Randomized Design
RCBD: Randomized Complete Block Design
LDS: Latin Square Design
Cross-over Design
Factorial Experiment
159
腦研所
Design 1: repeated measure design 1
Science 2007
160
腦研所
Design 1: repeated measure design 2
 Tau reduction prevented water maze deficits in hAPP mice
n=7-11 mice per genotype
 Cued platform learning curves: Fig. 1A
6 genotype total

Day 0 indicates performance on the first trial, and subsequent
points represent average of all daily trials.
 Performance differed by genotype. RMANOVA(repeated
measures analysis of variance): P <0.001;
 hAPP by tau interaction,P = 0.058).
 In post-hoc comparisons, only hAPP/Tau+/+ differed from
groups without hAPP (P < 0.001).
 Hidden platform learning curves : Fig. 1B



n = 7 to 11 mice per genotype, age 4 to 7 months
Differed by genotype RMANOVA: P < 0.001;
hAPP by Tau interaction, P < 0.02
In post-hoc comparisons,
 hAPP/Tau+/+ differed from all groups without hAPP (P < 0.001);
 hAPP/Tau+/– differed from hAPP/Tau+/+ (P < 0.02) and groups
without hAPP (P < 0.01);
 hAPP/Tau–/– differed from hAPP/Tau+/+ (P < 0.001) but not from
any group without hAPP.
161
腦研所
Design 1: repeated measure design 3
 Number of target platform crossings versus
crossings of the equivalent area in the three other
quadrants: Fig. 1D



differed by genotype, p<?
target crossing by genotype interaction, P < 0.001
In post-hoc comparisons, all genotypes except
hAPP/Tau+/+ and hAPP/Tau+/– exhibited a
preference for the target location over equivalent
areas in the other three quadrants (*P < 0.05; **P <
0.01; ***P < 0.001).
 Probe trial 72 hours after completion of 5 days of
hidden-platform training: Fig. 1E




Target platform preference differed by genotype
target crossing by genotype interaction, P < 0.001;
target crossing by hAPP by tau interaction, P<0.05.
In posthoc comparisons, all genotypes except
hAPP/Tau+/+ exhibited a preference for the target
location (**P < 0.01; ***P < 0.001).
162
腦研所
Design 1: repeated measure design 4
 Tau reduction prevented
 behavioral abnormalities
 A. total arm entries: n=49-58 mice per genotype
 ANOVA: genotype effect, p<0.0001
 hAPP by tau interaction, p<0.0001
 P<0.0001 versus groups without hAPP
 B. % time spent active: n=7-14 mice per genotype
 ANOVA: genotype effect, p<0.01
 hAPP by tau interaction, p<0.05
 P<0.05 versus groups without hAPP
 C. total distance: open and closed arm: n=49-59 mice per genotype
 ANOVA: genotype effect, p<0.0001
 hAPP by tau interaction, p<0.05
 P<0.001 versus groups without hAPP
 D. total distance: elevated plus maze: n=6-13 mice per genotype
 ANOVA: genotype effect, p<0.01
 hAPP by tau interaction, p=0.079
 P<0.05 versus groups without hAPP
 premature mortality in hAPP mice.
 Kaplan-Meier survival curve: n=887
 Logrank test, p<0.005
 only hAPP/Tau+/+ differed from all other groups,
163
腦研所
Design 1: repeated measure design 5
164
腦研所
Design 1: repeated measure design 5
 Tau reduction increased resistance
 excitotoxin PTZ-induced seizures
(single injection, n=10-11 per genotype)
 A. Seizure severity
 B. & C. Seizure latency
 Kainate induced Tonic-clonic seizure
ANOVA: tau effect, P < 0.0001
RMNOVA , P < 0.001
RMNOVA , P < 0.01
Logistic regression, P < 0.05
n=10 to 11 mice per genotype
165
腦研所
Design 2: Latin Square 1
166
腦研所
Design 2: Latin Square 2
167
腦研所
Experiment principles
Fundamental principles in Experiment
 Replication
 Randomization
 Error control: Blocking
Conditions:
 Experimental material: homogeneity
 Experimental Space: same environment
 Experimental Time: same time
168
腦研所
Complete Randomized Design
Principles
 Exp. Material: homogeneous or unknown
 Random arrangement for treatment
Advantages
 No. of replication: unlimited, even missing occurs
 Simple to analyze
 DF max. ; MSE min.
Disadvantage:
 heterogeneous exp. material
169
腦研所
CRD: example, one-way ANOVA
Tumor Growth:
 3 Tumor Cells Treatment, 4 mice each, n=12
Tx.
Observations
sum
mean Tx Effect
X ij
X i.
X i.
ti  X i.  X ..
A
14
19
20
15
68
17
-4
B
20
24
18
22
84
21
0
C
26
28
25
21
100
25
+4
170
腦研所
Randomized Complete Block Design
Homogeneity:
 Homogeneous material: CRD
 Heterogeneous material: RCBD
Block:
 Homogeneous within block
 Heterogeneous between blocks
Examples:
 Clinical trial: heterogeneous patients
171
腦研所
Block example:
分層
年齡
性別
抽煙與否
實驗分配
1
2
3
4
5
6
7
8
9
10
11
12
40-49
40-49
40-49
40-49
40-49
40-49
50-59
50-59
50-59
50-59
50-59
50-59
男
男
男
女
女
女
男
男
男
女
女
女
抽煙
已戒菸
不抽煙
抽煙
已戒菸
不抽煙
抽煙
已戒菸
不抽煙
抽煙
已戒菸
不抽煙
ABBA BABA…
BABA BBAA…
etc.
172
腦研所
Statistical Model: two-way ANOVA
Model: yij  μ  Ti  b j  eij
 I = 1,2,…m, treatment
 j = 1,2,… N, block
ANOVA table
Source
Treat
Block
Error
TSS
DF
m-1
n-1
(m-1)(n-1)
mn-1
SS
SStx
SSB
SSE
SST
MS
MStx
MSB
MSE
F
MSt/MSE
MSB/MSE
173
腦研所
RCBD: example,
Tumor Growth: 4 treatment, 4 species mice, n=16
Treatment
Block I
II
III
IV
Sum
Mean
A
10
15
16
15
56
14.00
B
16
20
25
22
83
20.75
C
14
18
20
16
68
17.00
D
12
16
18
15
61
15.25
16.75
SS
DF
MS
F
0.05
0.01
SSTx
93.5
3
31.17
28.05
3.86
6.99
SSB
103.5
3
34.50
31.05
3.86
6.99
SSE
10
9
1.11
TSS
207
15
174
腦研所
When to Use Blocking
MSB=MSE, F=1:
 No effect
MSB<MSE, F<1:
 Negative effect: wrong blocking
MSB>MSE, F>1:
 Positive effect: right blocking
175
腦研所
Latin Square Design: LDS, 3 way-ANOVA
2 dimensional Block
 Row Block & Column Block
Requirement: Equal No. of Blocks & Treatment
Advantage:
 需要的研究樣本數最少
Disadvantage:
 Equal No. of Treatment and Blocks
 Small Treatment(<5): DF of SSE is small
 Large Treatment(>8): Blocking is too large
176
腦研所
LSD assumptions:
Main effects only design
 interaction ignored
Independent effects
 treatments, row factor, and column factor effect
177
腦研所
LSD的設計規則
 三個x變項
 x1為treatment; x2為row block; x3為column block
LSD
Std. Sq. No.
排列數
M!×(M-1)!
全部不同可能排列數
2×2
1
2
2
3×3
1
12
12
4×4
4
144
576
5×5
56
2880
161280
6×6
9408
86,400
812,851,200
 Equal No. of Blocks & Treatment
 行列block內的treatment必須隨機排列，且僅能出現1次
 標準方(std square): 如上圖
 標準方: 第1行第1列照順序排列
 循環標準方: 其餘行列也照順序排列，如L1
 步驟: 以4×4為例
L1
1
2
3
4
L2
1
2
3
4
1
A
B
C
D
3
C
D
A
B
2
B
C
D
A
1
A
B
C
D
B
4
D
A
B
C
C
2
B
C
D
A
L3
4
3
1
2
3
B
A
C
D
1
D
C
A
B
4
C
B
D
A
2
A
D
B
C
 先隨機抽出1個標準方
3
C
D
A
 隨機抽出亂數22，22/4=5 餘2
4
D
A
B
 選第2個標準方L1，並將各行列編號
 再隨機抽出各行各列的排列
 列號改排: 隨機號碼除以4, 若餘數分別為3,1,4,2，則將L1改為L2
 行號改排: 隨機號碼除以4, 若餘數分別為4,3,1,2，則將L2改為L3
 最後的L3為應用方Applied square
178
腦研所
Statistical Model: 3-way ANOVA
Model:
yijk  μ  Ri  C j  Tk  eijk
 i = 1,2,… m, R block; j=1,2,…m, C block;
 k = 1,2,…m, treatment;
ANOVA table
Source
DF
SS
MS
F
R Block
m-1
SSR
MSR
MSR/MSE
C Block
m-1
SSB
MSC
MSC/MSE
Treat
m-1
SStx
MStx
MSt/MSE
Error
(m-1)(m-2)
SSE
MSE
TSS
m2-1
SST
179
腦研所
Latin Square: blocking efficiency
Latin Square
RCBD
Source
SS
DF
MS
Source
SS
DF
MS
R
SSR
m-1
MSR
B
SSR
m-1
MSB
C
SSC
m-1
MSC
E
SSE1
(m-1)2
MSE1
E(E2)
SSE2
m(m-1)
MSE2
T
SST
m2-1
T
SST
m2-1
SSC  SSE1
SSE2
MSC  (m  1) MSE1
MSE2= MSE2  (m  1)  (m  1) 2  m(m  1) 
m
[(m  1)(m  2)  1][(m  1) 2  3]MSE2
(6  1)(9  3)  2.0225

100

 132.93%
[(m  1)(m  2)  3][(m  1) 2  1]MSE1
(6  3)(9  1) 1.42
RE(LSD/RCBD) =
180
腦研所
LSD: example
4 species, 4 age group, 4 treatment, n=16
Species
Age
new born
young
adult
sum
old
W
B
16
A
10
C
14
D
12
52
X
D
16
C
18
A
15
B
20
69
Y
C
20
B
25
D
18
A
16
79
Z
A
14
D
20
B
22
C
16
79
Sum
66
73
69
64
272
181
腦研所
Results: LSD
Source
SS
DF
MS
F
Falfa
0.05 0.01
R block (Species)
98.5
3
32.8
23.12**
4.76 9.78
C block (Age)
11.5
3
3.83
2.697
4.76 9.78
Treatment
99.5
3
33.17
23.357**
4.76 9.78
Error
8.5
6
1.42
Total
218
15
182
腦研所
Repeated LSD: RLSD
Used when treatment no. is small
To increase experiment efficiency
Model: X ijkl    Li  R j  Ck  Tl  eijkl
列
區
集
實驗1(L1)
實驗1(L2)
行區集
行區集
1(青)
2(壯)
3(老)
1(青)
2(壯)
3(老)
X
C
B
A
B
A
C
Y
A
C
B
C
B
A
Z
B
A
C
A
C
B
183
腦研所
RLSD: 4-way ANOVA table
Source
Exp. (L)
R
C
Treatment
Error
SS
SSL
SSR
SSC
SSt
SSE
DF
r-1
r(m-1)
r(m-1)
m-1
Total
SST
rm2-1
MS
MSL
MSR
MSC
MSt
MSE
F
MSL/MSE
MSR/MSE
MSC/MSE
MSt/MSE
Combine total experiment results
184
腦研所
RLSD: example
Exp.1 (L1)
Exp.2 (L2)
O(青)
P(壯)
Q(老)
Sum
O(青)
P(壯)
Q(老)
Sum
X
C8
B10
A 5
23
B15
A12
C10
37
Y
A10
C12
B18
40
C12
B20
A16
48
Z
B16
A 8
C12
36
A11
C 9
B14
34
sum
34
30
35
99
38
41
40
119
185
腦研所
RLSD: results
Source
SS
DF
MS
F
Falfa
0.05 0.01
L
22.22
1
22.22
4.940
5.99 13.74
R
88.99
4
22.223
4.940*
4.53 9.15
C
6.23
4
1.5575
0.3462
4.53 9.15
Treatment
103.45
2
51.725
11.4988**
5.14 10.92
Error
26.99
6
4.4983
Total
247.78
17
R block significant
C block, Repeated L not significant
186
腦研所
Experimental Designs
Cross-over Design
Factorial Experiment
187
腦研所
Cross-over (change-over) design
 Infeasible conditions
 Irreversible changes in patient’s health
 Long periods of time for intervention
 Suitable conditions
 Rapid relief of symptoms in chronic diseases
 Commonly used in
 phase I & II trial,
 equivalence trial: to show that treatments are equally effective
 Human right issue
 Separation of periods
 Wash-out period is needed: carry-over effect
 Run-in period is optional: pt settle down for baseline observation
188
腦研所
Example of Cross-over trial: Heart Rate
Patient
1
2
3
4
5
6
…24
Period 1
a
p
c
a
c
p
…
Period 2
p
c
a
c
p
a
…
Period 3
c
a
p
p
a
C
…
Sequence
A
B
C
D
E
F
p=placebo c=control
a=test drug
y: heart rate hr1h
Mixed Models Analyses Using the SAS System
189
腦研所
Data example: cross-over trial
Treatment
Difference
Patient
A
B
AB
Patient
mean
1
20
12
8
16.0
2
26
24
5
25.0
3
16
17
-1
16.5
4
29
21
8
25.0
5
22
21
1
21.5
6
24
17
7
20.5
Mean
22.83
18.67
4.17
20.75
Applied Mixed Models in Medicine, p3. Brown H, 1999
190
腦研所
Data example: cross-over trial
Patient
Y_P1
Y_P2
Sequence
T1
T2
1
20
12
1
A
B
2
26
24
1
A
B
3
16
17
1
A
B
4
21
29
2
B
A
5
21
22
2
B
A
6
17
24
2
B
A
191
腦研所
NCSS result: ANOVA
DF
SS
MS
F
P
A: DRUG
1
52.08333
52.08333
6.61
0.05
B: ID1
5
154.75
30.95
3.93
0.08
AB
5
39.41667
7.883333
S
0
Total (Adjusted)
11
246.25
192
腦研所
Interaction and Carry-over
Period 1
Period 2
Group I
(AB)
μ+σA+β1
μ+σB +β2+γAB
Group II
(BA)
μ+σB+β1
μ+σA +β2+ γBA
yijkm    i   j  pm  eijkm
Possible reasons for an Interaction
 wash-out period is too short
 wash-out period is long enough for a period 1 drug
elimination
 Strong period-effect
193
腦研所
Factorial Design
Two-way ANOVA
 RCBD: Randomized Complete Block Design
 Factorial Design: both R & C are treatment
Advantage:
 Less observation needed
 Detect Interaction effect
194
腦研所
ANOVA of 3-factor design
SS
DF
A
SA
I-1
B
SB
J-1
C
SC
K-1
AB
SAB
(I-1)(J-1)
AC
SAC
(I-1)(K-1)
BC
SBC
(J-1)(K-1)
SABC
(I-1)(J-1)(K-1)
MS
F=VR
Main Effects
2-factor interaction
3-factor interaction
ABC
Residual
Total
SR
IJK(n-1)
N-1
195