Funktionsdauer von Batterien in Abhängigkeit des verwendeten Materials und der Umgebungstemperatur

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Ποδαργη Βενιζέλος
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1 Beispiel: Funktionsdauer von Batterien in Abhängigkeit des verwendeten aterials und der Umgebungstemperatur emp. = 15 emp. = 70 emp. = aterial aterial aterial > battery.df <- read.table(file="c:\\daten\\battery.txt", header =) > attach(battery.df) > battery.df hours > [1] [26] Levels: > [1] [26] Levels: > is.vector() [1] FALSE > is.factor() [1] RUE arcus Hudec

2 Deskriptive Analysen > mean.mat <- tapply(hours, list(, ), mean) > > mean.mat <- cbind(mean.mat, apply(mean.mat, 1, mean)) > mean.mat <- rbind(mean.mat, apply(mean.mat, 2, mean)) > mean.mat par(mfrow=c(1,2)) plot(hours~ +) hours hours plot.design(hours ~ +, ylim=c(60,160)) plot.design(hours ~ +, ylim=c(60,160), fun=median) arcus Hudec

3 mean of hours Factors median of hours Factors arcus Hudec

4 interaction.plot(,, hours, ylim=c(20,160)) win.graph() interaction.plot(,, hours, median, ylim=c(20,160)) win.graph() interaction.plot(,, hours, ylim=c(20,160)) win.graph() interaction.plot(,, hours, median, ylim=c(20,160)) mean of hours arcus Hudec

5 median of hours mean of hours arcus Hudec

6 median of hours arcus Hudec

7 Allgemeine Datenstruktur: Zwei Faktoren A und B mit a und b verschiedenen Kategorien A1 A2 Ai Aa B1 B2 Bj Bb y 111 y 11n11 y.1 y.2 y ij1 y ijnij y. j y.b y 1. y 2. y i. y a. y.. i = 1,, a j = 1,, b Wichtigster Spezialfall: n ij = const.=n y = µ + α + β + ( αβ) + ε = µ + ε ijk i j ij ijk ij ijk E( y ) = µ = µ + α + β + ( αβ) ijk ij i j ij ε α = 0 β = 0 ( αβ) = ( αβ) = 0 i j ij ij i j i j 2 ijk N(0, σ ) B1 B2 Bj Bb A1 A2 y 11 y 1. y 2. Ai y ij y i. Aa y.1 y.2 y ab y. j y.b y a. y.. arcus Hudec

8 Varianzanalytische Zerlegung: Source of Sum of degrees of ean Squares Variation Squares freedom reatment A SSA a-1 SA=SSA/(a-1) reatment B SSB b-1 SB=SSB/(b-1) AB Interaction SSAB (a-1)(b-1) SAB=SSAB/((a-1)(b-1)) Error SSE ab(n-1). SE=SSE/k(n-1) otal SS abn-1 k SSA = bn ( y y ) i= 1 i... k 2 SSAB = n ( yij y.. ) SSA SSB i= 1 SS = ( y y ) i j k ijk k SSB = an ( y y ) i= 1. j.. SSE = ( y y ) i j k ijk 2 ij 2 > contrasts() > contrasts() arcus Hudec

9 > model.matrix(hours~*) (Intercept) :2 3:2 2:3 3: arcus Hudec

10 > fit.treat.add<-aov(hours~+) > > summary(fit.treat.add) Df Sum Sq ean Sq F value Pr(>F) e-06 *** ** Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 > anova(fit.treat.add) Analysis of Variance able Response: hours Df Sum Sq ean Sq F value Pr(>F) e-06 *** ** Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 > coefficients(fit.treat.add) (Intercept) > tapply(fitted(fit.treat.add), list(,), mean) > interaction.plot(,, fitted(fit.treat.add)) arcus Hudec

11 mean of fitted(fit.treat.add) Effekt der Orthogonalität: > aov(hours~+) Call: aov(formula = hours ~ + ) erms: Residuals Sum of Squares Deg. of Freedom Residual standard error: Estimated effects may be unbalanced > aov(hours~+) Call: aov(formula = hours ~ + ) erms: Residuals Sum of Squares Deg. of Freedom arcus Hudec

12 > aov(hours~) Call: aov(formula = hours ~ ) erms: Residuals Sum of Squares Deg. of Freedom 2 33 Residual standard error: Estimated effects may be unbalanced > aov(hours~) Call: aov(formula = hours ~ ) erms: Residuals Sum of Squares Deg. of Freedom 2 33 Residual standard error: Estimated effects may be unbalanced > Schätzung mit Interaktionseffekten > fit.treat<-aov(hours~*) > summary(fit.treat) Df Sum Sq ean Sq F value Pr(>F) e-07 *** ** : * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 > anova(fit.treat) Analysis of Variance able Response: hours Df Sum Sq ean Sq F value Pr(>F) e-07 *** ** : * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 arcus Hudec

13 > coefficients(fit.treat) (Intercept) : :2 2:3 3: > tapply(fitted(fit.treat), list(,), mean) > interaction.plot(,, fitted(fit.treat)) > mean of fitted(fit.treat) > options(contrasts=c("contr.sum", "contr.poly")) > > contrasts() [,1] [,2] > contrasts() [,1] [,2] arcus Hudec

14 > model.matrix(hours~*) (Intercept) :1 2:1 1:2 2: > fit.sum<-aov(hours~*) > summary(fit.sum) Df Sum Sq ean Sq F value Pr(>F) e-07 *** ** : * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 arcus Hudec

15 > anova(fit.sum) Analysis of Variance able Response: hours Df Sum Sq ean Sq F value Pr(>F) e-07 *** ** : * Residuals Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 > coefficients(fit.sum) (Intercept) : :1 1:2 2: > options(contrasts=c("contr.helmert", "contr.poly")) > contrasts() [,1] [,2] > contrasts() [,1] [,2] arcus Hudec

16 > tapply(fitted(fit.treat), list(,), mean) > model.tables(fit.treat) ables of effects rep rep : rep rep rep > model.tables(fit.sum) ables of effects rep rep : rep rep rep > arcus Hudec

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