1 Analysis of Variance (ANOVA) EPP 245/298 Statistical Analysis of Laboratory Data

October 27, 2004EPP 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.

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 3 ABC

October 27, 2004EPP 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?

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 5 > data(red.cell.folate) > help(red.cell.folate) > summary(red.cell.folate) folate ventilation Min. :206.0 N2O+O2,24h:8 1st Qu.:249.5 N2O+O2,op :9 Median :274.0 O2,24h :5 Mean : rd Qu.:305.5 Max. :392.0 > attach(red.cell.folate) > plot(folate ~ ventilation)

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 6 red.cell.folate package:ISwR R Documentation Red cell folate data Description: The 'folate' data frame has 22 rows and 2 columns. It contains data on red cell folate levels in patients receiving three different methods of ventilation during anesthesia. Format: This data frame contains the following columns: folate a numeric vector. Folate concentration ($mu$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.

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 7 > folate.lm <- lm(folate ~ ventilation) > summary(folate.lm) Call: lm(formula = folate ~ ventilation) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) e-14 *** ventilationN2O+O2,op * ventilationO2,24h Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: on 19 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: on 2 and 19 DF, p-value:

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 8 > anova(folate.lm) Analysis of Variance Table Response: folate Df Sum Sq Mean Sq F value Pr(>F) ventilation * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 9 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

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 10 > data(heart.rate) > attach(heart.rate) > heart.rate hr subj time

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 11 > anova(hr.lm) Analysis of Variance Table Response: hr Df Sum Sq Mean Sq F value Pr(>F) subj e-16 *** time * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

October 27, 2004EPP 245 Statistical Analysis of Laboratory Data 12 Assignment for next week Do problems 6.1 and 6.2 in Dalgaard