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F6 Karl B Christensen http://biostat.ku.dk/ ~ kach/CSS1 Karl B Christensenhttp://biostat.ku.dk/ ~ kach/CSS1 F6
F6
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Kirkwood & Sterne kapitel 9. Comparison of means fromseveral groups - analysis of variance
t-test
Sammenligner to grupper.
Beregner differensen melle de to middelværdier
Variansanalyse (ANOVA)
Sammenligner tre eller flere grupper.
Beregner variansen indenfor grupperne
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA
t-test
To middelværdier µ1 og µ0. Tester nulhypotesen
H0 : µ1 = µ0
Variansanalyse (ANOVA)
k middelværdier µ1, µ2, . . . , µk . Tester nulhypotesen
H0 : µ1 = µ2 = · · · = µk
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
Er der signifikant forskel p̊a de tre grupper mht. alder ?The SAS System 09:37 Tuesday, September 4, 2018 1
The MEANS Procedure
The SAS System 09:37 Tuesday, September 4, 2018 1
The MEANS Procedure
Analysis Variable : age
group N MeanLower 95%
CL for MeanUpper 95%
CL for Mean
1: eksp. > 10 år 12 49.75 43.96 55.54
2: eksp. < 10 år 28 37.79 34.22 41.35
3: Ikke eksponeret 44 39.80 36.15 43.45
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
09:37 Tuesday, September 4, 2018 209:37 Tuesday, September 4, 2018 2
3
2
1
grou
p
35 40 45 50 55
age (Mean)
age (Mean), 95% Confidence Limits
(kahoot.it)
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
Er der signifikant forskel p̊a de tre grupper mht. vital capacity ?The SAS System 09:37 Tuesday, September 4, 2018 1
The MEANS Procedure
The SAS System 09:37 Tuesday, September 4, 2018 1
The MEANS Procedure
Analysis Variable : vitcap
group N MeanLower 95%
CL for MeanUpper 95%
CL for Mean
1: eksp. > 10 år 12 3.95 3.29 4.61
2: eksp. < 10 år 28 4.47 4.21 4.74
3: Ikke eksponeret 44 4.46 4.25 4.67
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
09:37 Tuesday, September 4, 2018 209:37 Tuesday, September 4, 2018 2
3
2
1
grou
p
3.25 3.50 3.75 4.00 4.25 4.50 4.75
vitcap (Mean)
vitcap (Mean), 95% Confidence Limits
(kahoot.it)
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA
Parvise t-tests ikke en god løsning
Kan regne p̊a det ved at se p̊a varianser
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA. FEV1. Tre aldersgrupper (s = SD, s2 varians)
Lungefunktionsdata sex=1, heightgroup=1
1 N = 57, x̄1 = 1.441, s1 = 0.235,∑
(x − x̄1)2 = 3.083, s21 = 0.055
2 N = 34, x̄2 = 1.477, s2 = 0.208,∑
(x − x̄2)2 = 1.425, s22 = 0.043
3 N = 8, x̄3 = 1.467, s3 = 0.164,∑
(x − x̄3)2 = 0.188, s23 = 0.027
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA. FEV1. Tre aldersgrupper (s = SD, s2 varians)
Lungefunktionsdata sex=1, heightgroup=1
1 N = 57, x̄1 = 1.441, s1 = 0.235,∑
(x − x̄1)2 = 3.083, s21 = 0.055
2 N = 34, x̄2 = 1.477, s2 = 0.208,∑
(x − x̄2)2 = 1.425, s22 = 0.043
3 N = 8, x̄3 = 1.467, s3 = 0.164,∑
(x − x̄3)2 = 0.188, s23 = 0.027
N = 99, x̄ = 1.456, s = 0.220,∑
(x − x̄)2 = 4.726, s2 = 0.048
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA. FEV1. Tre aldersgrupper (s = SD, s2 varians)
Lungefunktionsdata sex=1, heightgroup=1
SSbetween =∑
ni (x̄i − x̄)2 = 57 · (1.441− 1.456)2 + . . . ' 0.029
SSwithin =∑
(ni − 1)s2i = (57− 1) · 0.055 + . . . ' 4.688
SStotal =∑
(x − x̄)2 ' 4.726
Kan bevise matematisk at
SStotal = SSbetween + SSwithin
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA. FEV1. Tre aldersgrupper (s = SD, s2 varians)
Lungefunktionsdata sex=1, heightgroup=1
SS d.f. MS=SS/d.f. Fbetween 0.029 2 0.029/2 =0.0145within 4.688 96 4.688/96=0.0488 F = 0.0145
0.0488= 0.30
total 4.726 98
F skal være omkring 1 hvis der er samme middelværdi i de tregrupper. (Computer: F -fordeling med (2,96) frihedsgrader)
http://www.statdistributions.com/f?p=0.05&df1=2&df2=96
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Variation within groups and variation between groups
The model (grouping) explains part of the variation
Variation︷ ︸︸ ︷between gr. within gr.
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Variation within groups and variation between groups
Small variation within groups
..
..........
..
....
-
A B C
Large variation within groups
..
..
.
.
.
..
.
..
.
.
.
.
..
-
A B C
The evidence of a true difference between groups is strongerwhen the variation between groups is large compared to thevariation within groups.
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA
Antagelser
Normalfordelte data1
Samme SD i grupperne [vigtigt]
ANOVA med to grupper: præcis det samme som t-test.
1egentlig nok at middelværdierne er normalfordelteKarl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
ANOVA p̊a computer
Data p̊a hjemmeside
http://biostat.ku.dk/~kach/CSS1/F6.txt
http://biostat.ku.dk/~kach/CSS1/F6.csv
http://biostat.ku.dk/~kach/CSS1/F6.sav
http://biostat.ku.dk/~kach/CSS1/F6.xlsx
computereksempler
http://biostat.ku.dk/~kach/CSS1/R_ANOVA.pdf
http://biostat.ku.dk/~kach/CSS1/SAS_ANOVA.pdf
http://biostat.ku.dk/~kach/CSS1/SPSS_ANOVA.pdf
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
The SAS System 09:37 Tuesday, September 4, 2018 4
The ANOVA Procedure
Dependent Variable: age
The SAS System 09:37 Tuesday, September 4, 2018 4
The ANOVA Procedure
Dependent Variable: age
Source DFSum of
Squares Mean Square F Value Pr > F
Model 2 1254.68615 627.34307 5.41 0.0062
Error 81 9392.12338 115.95214
Corrected Total 83 10646.80952
R-Square Coeff Var Root MSE age Mean
0.117846 26.55670 10.76811 40.54762
Source DF Anova SS Mean Square F Value Pr > F
group 2 1254.686147 627.343074 5.41 0.0062
20
30
40
50
60
age
1 2 3
group
Distribution of age
20
30
40
50
60
age
1 2 3
group
0.0062Prob > F5.41F
Distribution of age
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
Vital capacity data
The SAS System 09:37 Tuesday, September 4, 2018 4
The ANOVA Procedure
Dependent Variable: vitcap
The SAS System 09:37 Tuesday, September 4, 2018 4
The ANOVA Procedure
Dependent Variable: vitcap
Source DFSum of
Squares Mean Square F Value Pr > F
Model 2 2.74733766 1.37366883 2.48 0.0902
Error 81 44.89361829 0.55424220
Corrected Total 83 47.64095595
R-Square Coeff Var Root MSE vitcap Mean
0.057668 16.95060 0.744474 4.392024
Source DF Anova SS Mean Square F Value Pr > F
group 2 2.74733766 1.37366883 2.48 0.0902
3
4
5
6
vitc
ap
1 2 3
group
Distribution of vitcap
3
4
5
6
vitc
ap
1 2 3
group
0.0902Prob > F2.48F
Distribution of vitcap
Karl B Christensenhttp://biostat.ku.dk/~kach/CSS1 F6
