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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA). EPP 245 Statistical Analysis of Laboratory Data. 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

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Analysis of Variance (ANOVA)

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  1. Analysis of Variance (ANOVA) EPP 245 Statistical Analysis of Laboratory Data

  2. 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. EPP 245 Statistical Analysis of Laboratory Data

  3. EPP 245 Statistical Analysis of Laboratory Data

  4. 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? EPP 245 Statistical Analysis of Laboratory Data

  5. Data EPP 245 Statistical Analysis of Laboratory Data

  6. Cell Means EPP 245 Statistical Analysis of Laboratory Data

  7. Deviations from Cell Means EPP 245 Statistical Analysis of Laboratory Data

  8. 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. EPP 245 Statistical Analysis of Laboratory Data

  9. insheet using redcell.csv summarize folate tabulate ventilation tabulate ventilation, summarize (folate) graph box folate, over (ventilation) graph export folate1.wmf oneway folate ventilation describe ventilation encode ventilation, generate(dv) Describe dv EPP 245 Statistical Analysis of Laboratory Data

  10. . 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 folate ventilation | 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 EPP 245 Statistical Analysis of Laboratory Data

  11. EPP 245 Statistical Analysis of Laboratory Data

  12. . 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 EPP 245 Statistical Analysis of Laboratory Data

  13. . describe ventilation storage display value variable name type format label variable label ------------------------------------------------------------------------------- ventilation str10 %10s . encode ventilation, generate(dv) . describe dv storage display value variable 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 EPP 245 Statistical Analysis of Laboratory Data

  14. 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 EPP 245 Statistical Analysis of Laboratory Data

  15. 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). EPP 245 Statistical Analysis of Laboratory Data

  16. . 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 EPP 245 Statistical Analysis of Laboratory Data

  17. EPP 245 Statistical Analysis of Laboratory Data

  18. EPP 245 Statistical Analysis of Laboratory Data

  19. . 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 EPP 245 Statistical Analysis of Laboratory Data

  20. EPP 245 Statistical Analysis of Laboratory Data

  21. . 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 EPP 245 Statistical Analysis of Laboratory Data

  22. EPP 245 Statistical Analysis of Laboratory Data

  23. EPP 245 Statistical Analysis of Laboratory Data

  24. EPP 245 Statistical Analysis of Laboratory Data

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