1

STATISTICS

Analysis Of Variance

Review Preview ANOVA

F test One-way ANOVA Multiple comparison Two-way ANOVA

2

STATISTICS

),(~ 2Nx

x

Zx

Standard normal distribution Z value:

(Observed - Expected) in terms of UNITS of SD

Review

3

STATISTICS

Central Limit Theorem Review

)/,(~ 2 nNx

),(~ 2Nx

For large n,

X

The beauty of CLT: Easy to calculate V

The ugliness of CLT: Hard to explain p

4

STATISTICS

Sampling Distribution of

)( 21 xx

),(~2

22

1

21

2121nn

Nxx

2

)/,(~ 22222 nNx

21 21 xx

1

)/,(~ 12111 nNx

2

)( 21 xx

)( 21 xx

Review

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STATISTICS

Population & Sampling Distribution

Review

Population parameters known Population parameters unknown

Mean SD Z score Mean SD t score

x N

xi N

xi

2)(

Xz n

xx i

1

)( 2

n

xxS i

S

xxt

x N

xi

x n

SEx

n

xZ i

x )(

n

xi

x n

SSE

x

nSx

t ix

)(

Please add yourself: )( 21 xx

STATISTICS

No of groups

N > 30

ND

1-s t

1-s t

1. TransF for t 2. sign test

N > 30

Independent

N > 30

ND

ND

Equal variance 2-s t

2-s t

2-s t

1. transform for t 2. WRS test

1. TransF for t 2. WRS test

Paired t

Paired t

1. transform for t 2. WSR test

Equal N

1 group

2 group

If Yes, go up; If No, do down

Flowchart of 2G MD testReview

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STATISTICS

ANOVA

Analysis of Variance

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STATISTICS

Analysis of Variance

The logic of ANOVA Partition of sum of squares

F test One way ANOVA

Multiple comparison Two way ANOVA

Interaction and confounding

ANOVA

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STATISTICS

Eyeball test for 3-sample means

ANOVA

A B

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 comparisons

1 / 21 < 0.05 !!

1 2 3 1 2 3

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STATISTICS

Data sheet: k groups MD comparison

Subjects Observed Tx Group

Mean

Grand

Mean

Group

Effect Tx error

Total

Difference

1 X1 X1-Ma X1-M

2 X2 X2-Ma X2-M

3 X3

A Ma Ma-M

X3-Ma X3-M

4 X4

5 X5 B Mb Mb-M

… … … … …

… …

n Xn K Mk

M

Mk-M

ANOVA

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STATISTICS

The Logic of one-way ANOVA

Total Difference divided into two parts (Observed－ group mean)＋ (group mean－ grand mean)

Total sum of squares divided into two parts SS Total = SS Between + SS Within (or Error) SST = SSB + SSE

Partition of TD & TSS Model of one-way ANOVA

j i

j

j i

jijj i

jjijj i

ij XXXXXXXXXX 2.

2.

2..

2 )()()]()[()(

ANOVA

)()( .. XXXXXX jjijij

ijjij eX

A B C

x

x

x

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STATISTICS

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

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STATISTICS

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

ANOVA

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STATISTICS

F test : named after Fisher Characteristics

a sickly, poor-eyesighted child The teacher used no paper/pencil t

o 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

ANOVA

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STATISTICS

One-way ANOVA

ANOVA

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STATISTICS

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:)/(

)1/(

knSSE

kSSB

MESS

MBSS

MS

MSF

E

B

ANOVA

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STATISTICS

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 placeboContrast Coefficients: add to 0

OrthogonalTwo contrasts are orthogonal if they don’t use the same informationEx: (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

ANOVA

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STATISTICS

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 Monthswere in good health

X: grouping by scores for major life events Measurement: Social Readjustment Rating Scale score

Y: immune system functionNK cell activity: lytic units

Example 1: one-way ANOVA

ANOVA

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STATISTICS

Box plot & Error bar plot

0.00

25.00

50.00

75.00

100.00

1 2 3

Box Plot

GROUP

CE

LL

10.0

15.6

21.1

26.7

32.2

37.8

43.3

48.9

54.4

60.0

1 2 3

Error Bar Plot

Printout

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STATISTICS

ANOVA table

Analysis of Variance Table

Source Term DF Sum of Squares Mean Square F-Ratio Prob Power(Alpha=0.05)

A: GROUP 2 4654.156 2327.078 8.35 0.001125* 0.947488

S(A) 34 9479.396 278.8058

Total (Adjusted) 36 14133.55

Total 37

Printout

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STATISTICS

Nonparametric ANOVA

Printout

Kruskal-Wallis One-Way ANOVA on Ranks Test Results

Method DF Chi-Sq (H) Prob. Level Decision (0.05)

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

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STATISTICS

MC: Priori comparisons t test for orthogonal comparisons

t statistic: ; not using SDp but MSE

DF: (n1+n2j); n=n1=n2

Adjusting downward: / (group number) Ex: 4 comparisons, =0.05/4=0.0125

Bonferroni t procedure Applicable for both orthogonal & non-orthogonal t statistic:

Multiplier table: no. of comparisons & DF for MSE Able to find CI for mean difference

nMS

xxt

E

ji

/2

nMSMultiplier E /2

ANOVA

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STATISTICS

MC: Posteriori comparisons

Tukey’s HSD (honestly significant difference) HSD=

Like Bonferroni, HSD multiplier table is needed (P176, table 7-7) Able to find CI for mean difference

Ex:

n

MSMultiplier E

ANOVA

31.2112

82.27842.4 HSD

24.63 22.17

2.46

LOWn=13

MODn=12

HIGHn=12

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STATISTICS

MC: Posteriori comparisons Scheffé’s procedure

S statistic:

j: No. of groups; C: contrast; (alpha, df1, df2)=(0.01, 2, 34)

most versatile (not only pair-wise) & most conservativeEX: Low (Moderate & High) combined; Low Moderate

Note: MD btw L & H not significant Able to find CI for mean difference

j

jEdf n

CMSFjS

2

,)1(

ANOVA

167.012

)1(

12

1;125.0

12

)5.0(

12

)5.0(

12

1 2222222

j

j

j

j

n

C

n

C

24.22167.082.27831.5)13( S

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STATISTICS

MC: Posteriori comparisons

Newman-Keuls procedure NK statistic:

Multiplier table is needed Less conservative than Tukey’s HSD Unable to find CI for mean difference Ex:2 steps ; 3 steps

n

MSmultiplier E

3 Steps

2 Steps 2 Steps

ANOVA

65.1882.487.3 NK 31.2182.442.4 NK

same as HSD

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STATISTICS

MC: Posteriori comparisons

Dunnett’s procedure Dunnett’s statistic:

Only used in several Tx means with single CTL mean Relatively low critical value Ex:

2 units lower than HSD value; 4 units lower than Scheffé value

n

MSmultiplier E2

ANOVA

48.1882.671.2 D

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STATISTICS

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.

ANOVA

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STATISTICS

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 screeningLSD, Duncan’s new multiple-range test are O.K.The above two are not recommended by the authors

ANOVA

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STATISTICS

Multiple comparisonsNewman-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

Printout

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STATISTICS

Two-way ANOVA

ANOVA

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STATISTICS

The 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:

With interaction:

ANOVA

ijjiij eX

ijjijiij eX )(

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STATISTICS

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%

ANOVA

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STATISTICS

Statistical Interaction & confounding

Interaction: 2 lines with different slope

Confounding: 2 parallel lines

T0 T1

C1

C0

Y

C0

C1

1|11ˆˆ: cH

TCCTCTY 321,|

How to test: ANOVA

ANOVA

0ˆ: 31 H

34

STATISTICS

Confounding factors

Mixing effect of X2 with X1 & Y Definition:

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 Associated With the exposure

與危險因子有相關 Not as a result of being exposed.

干擾不能是中介變項： intervening variable Intervening variable: X1X2YExample: S/S of diseases

MI

Obesity

Cholesterol

ANOVA

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STATISTICS

Interaction & confounding

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)

ANOVA

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STATISTICS

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

Interaction SS(RC) (r-1)(c-1) MS(RC) MS(RC)/MSE

Error SSE rc(n-1) MSE

Total SST n-1

ANOVA

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STATISTICS

Example 2: two-way ANOVA

Research problem: Glucose tolerance, insulin secretion, insulin sensitivi

ty and glucose effectiveness in normal and overweight hyperthyroid women. Gonzalo MA. Clin Endocrinol, 1996;45:689-697

X1: BMI; X2: thyroid functionAll categorical variablesBMI: 2 level; thyroid function: 2 level;

Y: Insulin sensitivityContinuous variable

ANOVA

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STATISTICS

Box plot & Error bar plot, ex 2

0.00

0.25

0.50

0.75

1.00

0 1

Means of IS

BMI2

IS

HT

01

0.0

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

1.0

0 1

Error Bar Plot

BMI

IS

HT

0 Normal thyroid1 Hyperthyroid

Printout

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STATISTICS

Descriptive statistics, ex 2Means and Standard Errors of IS

Term Count Mean SE

All 33 0.4647917

A: BMI2

0 19 0.615 5.786324E-02

1 14 0.3145833 6.740864E-02

B: HT

0 19 0.57375 5.786324E-02

1 14 0.3558333 6.740864E-02

AB: BMI2,HT

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

Printout

40

STATISTICS

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 6.361494E-02

Total (Adj.) 32 2.916255

Total 33

Printout

41

STATISTICS

Flowchart of 3G MD test

Indepedent

ND

No. of Factors

One-way ANOVA

Two-way ANOVAor other

ND

RepeatedANOVA

Friedman

1 Factor

2 or more Factors

Kruskal-Wallisfor 1 Factor

3 or more groups

Summary

If Yes, go up; If No, do down

42

STATISTICS

QUIZ

Q: Can I use ANOVA to test 2G MD? A: Yes, you can. Q: What is the relationship btw ANOVA & 2-s t? A: 2-s t test is a special case of ANOVA F, t & Z table:

22/1),1(,2

2

)1(,2/1)1,1(,

,).2(

).1(22

ZFdf

tF nn

43

STATISTICS

Home Work 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

Answering the following questions: Is there a difference in %TBSA (percent of total body surface area affected) related to age at onset? Is there a difference in %TBSA related to type of psoriasis (familial vs. sporadic)? Is the interaction significant? What is your conclusion?