1 Analysis of Variance (ANOVA) EPP 245 Statistical Analysis of Laboratory Data.
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Transcript of 1 Analysis of Variance (ANOVA) EPP 245 Statistical Analysis of Laboratory Data.
1
Analysis of Variance (ANOVA)
EPP 245
Statistical Analysis of
Laboratory Data
November 2, 2006 EPP 245 Statistical Analysis of Laboratory Data
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The Basic Idea
• The analysis of variance is a way of testing whether observed differences between groups are too large to be explained by chance variation
• One-way ANOVA is used when there are k ≥ 2 groups for one factor, and no other quantitative variable or classification factor.
November 2, 2006 EPP 245 Statistical Analysis of Laboratory Data
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A B C
9 10 12
7 9 14
7 8 14
9 9 12
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Data = Grand Mean + Row Deviations from grand mean +Cell Deviations from row mean
Are the row deviations from the grand mean too big to be accounted for by the cell deviations from the row means?
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A B C
9 10 12
7 9 14
7 8 14
9 9 12
Data
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A B C
8 9 13
8 9 13
8 9 13
8 9 13
Cell Means
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A B C
1 1 -1
-1 0 1
-1 -1 1
1 0 -1
Deviations from Cell Means
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Red cell folate data
Description:
22 rows and 2 columns. data on red cell folate levels in patients receiving three different methods of ventilation during anesthesia.
Format:
folate a numeric vector. Folate concentration (g/l).
ventilation a factor with levels 'N2O+O2,24h': 50% nitrous oxide and 50% oxygen, continuously for 24 hours; 'N2O+O2,op': 50% nitrous oxide and 50% oxygen, only during operation; 'O2,24h': no nitrous oxide, but 35-50% oxygen for 24 hours.
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insheet using redcell.csvsummarize folatetabulate ventilationtabulate ventilation, summarize (folate)graph box folate, over (ventilation)graph export folate1.wmfoneway folate ventilationdescribe ventilationencode ventilation, generate(dv)Describe dv
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. summarize folate
Variable | Obs Mean Std. Dev. Min Max-------------+-------------------------------------------------------- folate | 22 283.2273 51.28439 206 392
. tabulate ventilation
ventilation | Freq. Percent Cum.------------+----------------------------------- N2O+O2,24h | 8 36.36 36.36 N2O+O2,op | 9 40.91 77.27 O2,24h | 5 22.73 100.00------------+----------------------------------- Total | 22 100.00
. tabulate ventilation, summarize (folate)
| Summary of folateventilation | Mean Std. Dev. Freq.------------+------------------------------------ N2O+O2,24h | 316.625 58.717088 8 N2O+O2,op | 256.44444 37.121797 9 O2,24h | 278 33.756481 5------------+------------------------------------ Total | 283.22727 51.284391 22
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200
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N2O+O2,24h N2O+O2,op O2,24h
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. oneway folate ventilation
Analysis of Variance Source SS df MS F Prob > F------------------------------------------------------------------------Between groups 15515.7664 2 7757.88321 3.71 0.0436 Within groups 39716.0972 19 2090.32091------------------------------------------------------------------------ Total 55231.8636 21 2630.08874
Bartlett's test for equal variances: chi2(2) = 2.0951 Prob>chi2 = 0.351
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. describe ventilation
storage display valuevariable name type format label variable label-------------------------------------------------------------------------------ventilation str10 %10s. encode ventilation, generate(dv). describe dv
storage display valuevariable name type format label variable label-------------------------------------------------------------------------------dv long %10.0g dv
. anova folate dv
Number of obs = 22 R-squared = 0.2809 Root MSE = 45.72 Adj R-squared = 0.2052
Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 15515.7664 2 7757.88321 3.71 0.0436 | dv | 15515.7664 2 7757.88321 3.71 0.0436 | Residual | 39716.0972 19 2090.32091 -----------+---------------------------------------------------- Total | 55231.8636 21 2630.08874
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Two- and Multi-way ANOVA
• If there is more than one factor, the sum of squares can be decomposed according to each factor, and possibly according to interactions
• One can also have factors and quantitative variables in the same model (cf. analysis of covariance)
• All have similar interpretations
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Heart rates after enalaprilat
Description:
36 rows and 3 columns. data for nine patients with congestive heart failure before and shortly after administration of enalaprilat, in a balanced two-way layout.
Format:
hr a numeric vector. Heart rate in beats per minute.
subj a factor with levels '1' to '9'.
time a factor with levels '0' (before), '30', '60', and '120' (minutes after administration).
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. drop _all
. insheet using heart.rate.csv(4 vars, 36 obs)
. anova hr subj time
Number of obs = 36 R-squared = 0.9685 Root MSE = 3.5165 Adj R-squared = 0.9540
Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 9117.52778 11 828.866162 67.03 0.0000 | subj | 8966.55556 8 1120.81944 90.64 0.0000 time | 150.972222 3 50.3240741 4.07 0.0180 | Residual | 296.777778 24 12.3657407 -----------+---------------------------------------------------- Total | 9414.30556 35 268.980159
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6080
100
120
140
1 2 3 4 5 6 7 8 9
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6080
100
120
140
0 30 60 120
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. anova hr subj
Number of obs = 36 R-squared = 0.9524 Root MSE = 4.07226 Adj R-squared = 0.9383
Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 8966.55556 8 1120.81944 67.59 0.0000 | subj | 8966.55556 8 1120.81944 67.59 0.0000 | Residual | 447.75 27 16.5833333 -----------+---------------------------------------------------- Total | 9414.30556 35 268.980159
. predict hrhat(option xb assumed; fitted values)
. generate hrres = hr - hrhat
. graph box hrres, over (time)
. graph export hrresxtime.wmf
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-50
510
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0 30 60 120
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. anova hr subj time
Number of obs = 36 R-squared = 0.9685 Root MSE = 3.5165 Adj R-squared = 0.9540
Source | Partial SS df MS F Prob > F -----------+---------------------------------------------------- Model | 9117.52778 11 828.866162 67.03 0.0000 | subj | 8966.55556 8 1120.81944 90.64 0.0000 time | 150.972222 3 50.3240741 4.07 0.0180 | Residual | 296.777778 24 12.3657407 -----------+---------------------------------------------------- Total | 9414.30556 35 268.980159
. rvfplot
. graph export hrrvf.wmf
. rvpplot subj
. graph export hrrvpsubj.wmf
. rvpplot time
. graph export hrrvptime.wmf
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-10
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idua
ls
60 80 100 120 140Fitted values
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idua
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0 2 4 6 8 10subj
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0 50 100 150time