Chapter 12

ANOVA of Repeated Measurement Data

§1 Character of Repeated Measurement Data

Content

• Data characteristic

• Analysis of two factors and two levels

• Analysis of two factors and several levels

• Notices

Objective: Inference the effects of treatment, time, and t

reatment*time on experimental objects

Character of data:

treatment factor: g （≥ 1 ） levels and n experimental

objectives in each level, add up to gn experimental ob

jectives.

time factor: m valves at m time in each experimental

objective, add up to gnm valves.

Method: ANOVA

I premeasure-postmeasure design

It is particular case in repeated measurem

ent data and is also called single group pr

emeasure-postmeasure design. g=1, m=2.

number pretreatment posttreatment margin 1 2 3 4 5 6 7 8 9

10

130 124 136 128 122 118 116 138 126 124

114 110 126 116 102 100 98 122 108 106

16 14 10 12 20 18 18 16 18 18

X 126.2 110.2 16.0 S 7.08 9.31 3.13

table12-1 BHP patient’s diastolic pressure in pretreatment and post-treatment （ mmHg ）

number geteli－luozi Fatty acid hydrolyze Margin d

1 0.840 0.580 0.260 2 0.591 0.509 0.082 3 0.674 0.500 0.174 4 0.632 0.316 0.316 5 0.687 0.337 0.350 6 0.978 0.517 0.461 7 0.750 0.454 0.296 8 0.730 0.512 0.218 9 1.200 0.997 0.203

10 0.870 0.506 0.364

Table 3-3 measurement result of fat content in lactic acid (%)

Compare:

Differences with paired test:

1.Two experimental units in the same pair in

the paired test may be managed at random.

Two experimental units can be observed at the same tim

e. We can compare the difference of the group.

The results of premeasure-postmeasure design can’t be

observed at the same time

although it can be arranged at pre-post experiment. But i

n substance it is compared with the difference of pre-po

st experiment. We infer the treatment to be effective on

condition that we assume the time don’t affect the result

s.

2. paired/matched t-test requires the results of two experi

mental units in the same pair and difference to be fit for

independent. The difference obeys to normal distributio

n.

Two results of premeasure-postmeasure

design are common not to independent with differences.

The first time result is negative correlation to the differ

ence in most cases. Table 12-1 as follows, the correlati

on of diastolic pressure before treatment with

the difference is 0.602.

3. It is inferred the effective of treatment by average differe

nce in paired design. While we can analyze average diffe

rence and analyze correlation and regression in premeas

ure-postmeasure design.

Calculate table 12-1 as follows, the correlation coefficient

of diastolic pressure

in pre-post treatment is 0.963, P<0.01. We

can infer the diastolic pressure after treatment by the dias

tolic pressure before treatment.

test intercept P=0.014 ,regression coefficient P<0.01.

ˆ 49.534 1.266Y X

II premeasure-postmeasure design with contrast

The diastolic pressure of HBP patients after treatment dro

ps 16 mmHg .Although using paired test,

,it can still not be proved effective .Because the differen

ces of rest in hospital ,surroundings, emotion can resume

the diastolic pressure. So in order to prove effective, pre

measure-postmeasure design should be set parallel com

parison.

We assign 20 light HBP patients to disposal group

and comparison group at random.

16.18, 0.01t P

Table 12-2 BHP patient’s diastolic pressure in pre-post treatment （ mmHg ） Treatment group Comparison group

number before post difference( )d

number before post differdnce ( )d

1 2 3 4 5 6 7 8 9 10

130 124 136 128 122 118 116 138 126 124

114 110 126 116 102 100 98 122 108

106

11 12 13 14 15 16 17 18 19 20

118 132 134 114 118 128 118 132 120 134

124 122 132 96 124 118 116 122 124

128

sum 1262 1102 sum 1248 1206

mean 126.2 110.2 16.0 mean 124.8 120.6 4.2

S.D 7.08 9.31 3.13 S.D 7.90 9.75 8.02

Though analysis of homogeneity of variance of the differences of

treatment group and comparison group （ 2 21 2/ 6.58F S S ，

0.01P ），it is not fit for two-sample t-test.

III repeated measurement design

When the times of repeated measurement design are over three, it is call

ed repeated measurement design or repeated measurement data.

time（min） number 0 45 90 135

1 5.32 5.32 4.98 4.65

2 5.32 5.26 4.93 4.70

3 5.94 5.88 5.43 5.04

4 5.49 5.43 5.32 5.04

5 5.71 5.49 5.43 4.93

6 6.27 6.27 5.66 5.26

7 5.88 5.77 5.43 4.93

8 5.32 5.15 5.04 4.48

table12-3 Density of experimenter‘s blood glucose （ mmol/L ） (g=1)

Test of sphericity: 2 15.44, 5, 0.01P

The difference with randomized block design

• It is to assign among the the granule (experimenters )s at

random to measure in the repeated measurement design,

every time point in the granule is fixed , can't assign at

random , such as form 12-5, A,, B distribute behind the

each patient at random, each time that patient measures

the same. Randomized block design require the

experiment units are separate in each granule with

random granule, can only assign in the granule at random

to deal with, the treatment that each experiment unit

accepts is different, such as the form 4-9.

Table 12-4 The form 12-3 randomized block design data variance analysis table

It is very similar to repeated measurement design data (form 4-

9 ) that measurements and chapter four introduce, such as form

12-3 , and can calculate the variance analytical table (form 12-4

) that the randomized block design data too.

Table 12-5 　 The pre-post symptom of patient's operation grades (g=2)

post group before

10days 2months 4months 6months 9months

A 0.60 0.67 2.84 2.10 2.00 1.60 A 1.42 3.40 4.10 2.92 2.65 3.40 A 0.90 2.30 2.70 1.70 1.10 1.30 A 1.10 1.40 1.00 2.60 0.90 2.10 A 2.30 2.20 3.80 3.50 2.50 1.80 A 0.81 1.20 1.12 1.61 1.49 1.61 B 1.20 1.10 1.13 3.49 1.57 1.54 B 2.71 2.04 2.61 2.17 2.15 1.81

B 1.80 1.40 1.00 1.30 2.40 2.40

Block drugA drugB drugC

1 0.82 0.65 0.51

2 0.73 0.54 0.23

3 0.43 0.34 0.28

4 0.41 0.21 0.31

5 0.68 0.43 0.24

Table 4-9 Little white mouse's sarcoma weight after different medi

cine function （ g ）

Variance

source df SS MS F P

Total variance 14 0.5328

Between group 2 0.2280 0.1140 11.88 <0.01

Between block 4 0.2284 0.0571 5.95 <0.05

error 8 0.0764 0.0096

Table 4-10 Variance analysis table of the example 4-4

2. It is independent each other of experiment unit in the

repeated measurement design, such as form 12-3, i.e. t

he same experimenter's blood specimen measures are

highly relevant, its correlation coefficient is seen the list

12-6.

Repeated measurement design like relatively dealing wi

th the difference among the groups by the random gran

ule analysis of variance of chapter four, the preconditio

n is satisfied it is supposed ( examines ) that

"sphericity".

2

time time（min）

（min） 0 45 90 135

0 1 0.978** 0.936** 0.860**

45 1 0.879** 0.876**

90 1 0.896**

135 1

Coefficient correlation that each puts the density of blood glucose of time point of the form 12-6 form 12-3

**P<0.01

table12-7 table12-3 the result of “sphericity”

2

df P Greenhouse -Geisser

Huynh -Feldt

Lower -bound

15.44 5 0.010 0.536 0.671 0.333

If the "sphericity" assumptions is met, AN

OVA for randomized block design data

can be used; If not, ANOVA for

randomized block design data can also be

used, but it requires to correct the freedo

m of degree of F value.

§2 repeated measurement design data of two factors and two levels analysis

§3 repeated measurement design data of two factors and many levels analysis

One experiment designTreatment---factor A g levels n experiments in each levelTime---factor B m time Experiment data: Xijk i=1,2, … ,g j=1,2, … ,m k=1,2, … ,n

Experiment data: gmn

gaaa ,, 21

mbbb ,, 21

kiX 1 kiX 2 imkX

Variation and degree of freedom decomposition

*theory：

SS SS SS total bg wg1 、 total bg wg

)1()1(1

)()(

mgngngmn

XXXXXX ikijkikijk

* theory ：

bg A

A

SS SS SS

bgerror

bg bgerror

2 、

)1()1(1

)()(

ngggn

XXXXXX iikiik

*theory：

3 、

)1)(1()1)(1()1()1(

)(

)()(

mngmgmmgn

XXXX

XXXXXXXX

iikijijk

jiijjikijk

B AB

B AB wgerror

SS SS SS SS

wg wgerror

wg

Attention: , when reject "sphericity"

the degree of freedom must use " sphericity " the coefficient to adjust.

2m

attention Factorial design: A variance analytical table: The a

nalysis processes the main effect, the correlation.

Repeated measurement design: Two variances an

alytical tables, processing effect 1, time effect, tim

e and processing correlation 1.

*theory ：factorial design ：

repeated measurement design ：

A B AB

A B AB

SS SS SS SS SS

total error

total error

A B AB

A B AB

SS SS SS SS SS SS

total bg wg

total bg wg

example :12-2 According to table 1

2-2 ， To treatment group and comparison

group, the treatment the diastolic pressure

difference carries on the statistical analysis

variance resource df SS MS F P

Total of between group（患者间） 19 2517.90

group(A) 1 202.50 202.50 1.57 0.05

Error of between group 18 2315.40 128.63

table 12-13 Comparison between treatment group and comparison group

variance resource df SS MS F P

totle of within group 20 1702.0

Pre-post of treatment (B) 1 1020.1 1020.10 55.0 <0.01

AB 1 348.1 348.10 18.8 <0.01

error of within group 18 333.8 18.54

table 12-12 Around survey comparison and correlation variance analytical table

Attention: Although processing does not have the main

effect, it has the correlation with the time, therefore it has

the auxiliary effect.

4.Conclusions ?

Example 12-2:According to data 12-2,we

have analysis of repeated measurement

in three methods at different five time

points.

Time methods

0T 1T 2T 3T 4T sum

（ iA ）

A 605 562 592 629 604 2992

B 606 599 590 641 676 3112

C 631 615 593 713 653 3205

totle（iB ） 1842 1776 1775 1983 1933 9309

table 12-17 different anesthesia induction, at the same time the patient does not contract press fits estimates the value ( ) ( 5)ijT n

Decompose ASS ， BSS ， ABSS the grouping computation different anaesthesia induction, at the same time patient's systolic pressure does not equal the value（ ijT ），to see table

12-17。

Table 12-18 Different induction method patient systolic pressure comparison variance analytical table

variance resource df SS MS F P

Sum of patients 14 1858.72

mthods（A） 2 912.24 456.12 5.78 <0.05

Error of between patients 12 946.48 78.87

variance analytical table according to table 12-14 and the table 12-15 。

Table 12-19 anesthesia induction and its with induction method correlation variance analytical table

variance

resource

df df(adjust) SS MS F P(adjust)

Sum of patients 60 3437.20

Time （B） 4 1 2336.45 584.11 106.59 0.01

AB 8 2 837.63 104.70 19.11 0.01

Error of between

patients

48 48 263.12 5.48

critical value ’adjust of FB、FAB According to the way of adjust degree of freedom of table 12-15，FB ’s adjust degree of freedom

1 1 、 2 ( 1)( 1)g n m 3 (5 1) (5 1) 48 ，look up F critical value table ， FB ’s adjust critical value

0.01(1,48)F =7.19(pre-adjust) 0.01(4,48)F =3.74)。FAB ’s adjust degree of freedom 度

1 1 2g 、 2 ( 1)( 1)g n m 3 (5 1) (5 1) 48 ，

FB ’s adjust critical value 0.01(2,48)F = 5.08(pre-adjust 0.01(8,48)F =2.90)。degree

of freedom of Post-asjust to see table 12-19。

Conclusion: different anesthesia induction

method existence group difference (table 1

2-18), patient's systolic pressure when diff

erent induction methods different induction

changes tendency different (table 12-19),

when A group of different inductions ,the s

ystolic pressure is stable (table 12-20).

Table 12-20 Different anesthesia induction, different time patient's systolic pressure (mmHg ）

Time Induction method

T0 T1 T2 T3 T4

A X 121.00 112.40 118.40 125.80 120.80

S 3.54 5.13 5.64 4.71 3.70

B X 121.20 119.80 118.00 128.20 135.20

S 4.32 5.97 5.43 5.22 4.38

C X 126.20 123.00 118.60 142.60 130.60

S 3.63 3.39 1.95 4.83 3.71

III Notices

1. It is request that number of objects in each group to be equal.

2. examine "sphericity".

3. repeated measurement design data without a parallel comparison

§4 The situations of the repeated measurement design data statistical analysis commonly misuses.

1. repetitions carry on t tests for various time.

2. Neglects the individual curve change characteristic.

3. Lack of validity when compare the differences.