Chapter 12 ANOVA of Repeated Measurement Data. §1 Character of Repeated Measurement Data.
-
Upload
vivian-mccarthy -
Category
Documents
-
view
223 -
download
1
Transcript of Chapter 12 ANOVA of Repeated Measurement Data. §1 Character of Repeated Measurement Data.
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.