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One Way ANOVA Repeated Measures

花蓮慈濟醫院

葉日弌

每個個體都接受多種處理或多次測量的實驗設計

比較兩組樣本之平均數

1.獨立樣本(Independent samples): t-test.

2.相關樣本(Dependent/Related samples): paired t-test.

兩組獨立與相關樣本之檢定

•兩組樣本之 t-test oPaired t-test

oIndependent samples

•Equal variance

•Unequal variance

獨立與相關樣本

兩個不同的醫師針對10個腫瘤分別利用相同的工具測量其體積，結果如下表所示

腫瘤 A醫師 B醫師

1 17.2 15.8

2 22.3 20.3

3 14.5 14.2

4 18.5 15.7

5 28.0 26.8

6 24.8 24.0

7 21.8 20.3

8 25.4 23.0

9 29.3 27.5

10 20.5 19.7

配對(非獨立)樣本 腫瘤 A醫師 B醫師 差異

1 17.2 15.8 1.4

2 22.3 20.3 2.0

3 14.5 14.2 0.3

4 18.5 15.7 2.8

5 28.0 26.8 1.2

6 24.8 24.0 0.8

7 21.8 20.3 1.5

8 25.4 23.0 2.4

9 29.3 27.5 1.8

10 20.5 19.7 0.8

t-test for the differences

General definition of t-statistic

Standard deviation of the paired differences

Standard error of the differences (n=sample size)

t-statistic for differences where δ is the null hypothesis

t-statistic for the null hypothesis that δ = 0

差異 • A醫師之結果

o Population mean:

o Sample mean: 22.23

• B醫師之結果 o Population mean:

o Sample mean: 20.73

• 差之標準差 6.61 • 差異

o Population mean:

o Sample mean:

1x

2x

n

d

d

n

i

i 1

21

2

1

差異之分布 • 假設di 是常態分布, 以下統計量是自由度為n-1的 t-distribution

• 標準差

ns

dt

d

1

1

2

n

dd

s

n

i

i

d

t-test 1) Null hypothesis:

1) 獨立樣本; alpha=0.05

2) p = 0.48, df = 18

0:0 drBdrAdrBdrAH

72.0

1061.6

5.1

21

21

ns

xxt

xx

Confidence interval • 計算方式和one-sample t-test一樣

• 結果為 (-2.9, 5.9)

n

std

n

std da

da 2/12/1 ,

Non-paired vs. paired

The paired t test

15

paired t-test 使用時機 1. 同一個體接受多次測量.

2. 表現隨著時間增強或減弱.

3. 配對比較

16

Before-After

前測 後測 T

reat

ment

17

Matched Subject Design 根據已知影響結果的因子將受試配對分組比較，如

1. 年齡

2. 性別

3. 用藥記錄

4. 是否吸煙

5. 其他共病

Paired t-test

• 個體在接受處置前後均接受測量

• 依據個體前後測的差值而非組平均的差值做比較

Paired t-test 1) Null hypothesis:

1) 非獨立樣本; alpha=0.05

2) p = 0.00016

0:0 drBdrAdrBdrAH

16.61077.0

5.1

ns

dt

d

Analysis of variance

(ANOVA)

Analysis of Variance

比較三組以上樣本之平均數

1. 獨立樣本 : Analysis of Variance (ANOVA)

2. 相關樣本: Repeated Measures ANOVA

一因子變異數分析

一因子變異數分析

母體參數的推估

母體參數的推估

withinbetweentotal SSSSSS

• dfbetween=k-1; k=# of treatment (level)

• dfwithin =N-k; N=total sample size

• dftotal =N-1

withinbetweentotal dfdfdf

一般情形之完全隨機化設計的ANOVA表

F = MSbetween / MSwithin

= (SSbetween/dfbetween)/(SSwithin/dfwithin)

=MSB/MSE

完全隨機化設計的F檢定

reject H0

KH 210 :

),1( knkFMSE

MSBF

Repeated measures

ANOVA

comprehensive

double blind

study

ANOVA approach: different subjects in the treatment groups

Null hypothesis: the 4 group means are equal

F-distribution

Sums of squares are additive

F-statistic is used to determine significance

SSt are the squared differences of the treatment means from the mean of all the treatments

MSt = SSt / (number of groups-1)

MSwithin or s2

wit

Squared differences of individuals from their group’s mean divided by the number of people in the group minus 1

全部變異TSS

dftotal= mn-1

組間變異BSS

dfbetween subject = n-1

組內變異WSS

dfwithin = n(m-1)

隨機變異

個體間差異之變異

dftreatment = m-1 隨機化之殘差變異

dfresidual = (n-1)(m-1)

o group means:

• diet 1 = 4.96

• diet 2 = 5.23

• diet 3 = 4.93

• diet 4 = 4.8

o Grand mean:

= 1/4 ( + + + )= 4.98

= 4.45

1x

2x

3x

4x

2x 4x3x1xx

1.x

• SS total (df = 27) = (4.6-4.98)^2+….+ (5.6-4.98)^2 = 4.50

• SS between subjects (df = 6) = (4. 6454.98)^2 …..+ (5.55-4.98)^2 +…= 3.49

• SS between treatments (df = 3) =

7*(4.96-4.98)^2 +……..+7*(5.55-4.98)^2 = 0.681

• SS residual (df = 18) = SS total - SS between subjects - SS between treatments = 0.681-0.3 = 0.338

• F = MSE between treatments / MSE residual

= (SS between treatments /df between treatments ) /

(SS residual /df residual)

Repeated Measures Designs

• 優點

• 以自身當對照組，減少變異 • 檢定力較高

• 缺點 • 可能有前次處理殘存的效果(carryover effect) • 重複測量會有練習效果 • 有的狀況無法重複

Post hoc testing

• F 值達到顯著水準

o 至少有一組平均值和其他組不同

• 到底是哪一組?

• 使用事後檢定 (Post hoc tests)

o Bonferroni

o Tukey

Sphericity檢定

• Mauchly’s Test

• 若 p > .05 接受變異數相等

• 若 p < .05 採取以下措施

• 使用校正因子例如Greenhouse-Geisser • 或者使用不同的自由度(比較保守)

Post Hoc Analysis • 如果拒絕H0 就不用作了…

對應的無母數方法

Marijuana during pregnancy and behavioral characteristics at age 10

Kruskal-Wallis Statistic

Rank observations without regard to treatment group

P < 0.05

Chi-square calculation

The Friedman test

Friedman chi-square statistic

Repeated Measures ANOVA in Stata

person drug score

1 1 30

1 2 28

1 3 16

1 4 34

2 1 14

2 2 18

2 3 10

2 4 22

3 1 24

3 2 20

3 3 18

3 4 30

4 1 38

4 2 34

4 3 20

4 4 44

5 1 26

5 2 28

5 3 14

5 4 30

Repeated Measures ANOVA in SPSS

Data

Run repeated measures

Descriptive statistics

Mauchly’s test of Sphericity

Result of F test

Confidence interval of the means

Post Hoc

Plot

The End