Jian WANG, PhD. Room A115 College of Fishery and Life Science Shanghai Ocean University

Transcription

1 Jian WANG, PhD Room A115 College of Fishery and Life Science Shanghai Ocean University

2 Contents 1. Introduction to R 2. Data sets 3. Introductory Statistical Principles 4. Sampling and experimental design with R 5. Graphical data presentation 6. Simple hypothesis testing 7. Introduction to Linear models 8. Correlation and simple linear regression 9. Single factor classification (ANOVA) 10. Nested ANOVA 11. Factorial ANOVA 12. Simple Frequency Analysis

3 Factorial ANOVA A two-way ANOVA design Factor A & Factor B are independent

4 Factorial ANOVA temperature (high vs low) fertilizer (w/ vs w/o) A two-way ANOVA design Factor A & Factor B are independent

5 Null hypotheses associated with each of the main effects : Factor A Factor B A:B Interaction

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7 Null hypotheses associated with each of the main effects : Factor A Factor B A:B Interaction

8 Linear models Model 1 - fixed effects H0(A): µ1 = µ2 = = µi = µ H0(B): µ1 = µ2 = = µi = µ H0(AB): µij = µi + µj µ Null hypotheses of three models Model 2 - random effects H0(A): σα 2 = 0 H0(B): σβ 2 = 0 H0(AB): σαβ 2 = 0 Model 3 - mixed effects H0(A): µ1 = µ2 = = µi = µ H0(B): σβ 2 = 0 H0(AB): σαβ 2 = 0 (Fixed) (Random) (Random)

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10 Two factor factorial designs

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12 Small sample size Yates continuity correction (df<=1) 2 c k ( Oi Ei 0.5) E i 1 i 2

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14 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 1 - Import data set > quinn <- read.table("quinn.csv", header = T, sep = ",") #Step 2 Redefine class of the density vector (variable) as factor. > class(quinn$density) > quinn$density <- factor(quinn$density) > class(quinn$density)

15 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 3&4 - Assess assumptions of normality and homogeneity of variance > boxplot(eggs ~ DENSITY, quinn) > boxplot(eggs ~ SEASON, quinn) > boxplot(eggs ~ DENSITY * SEASON, quinn) Normal distribution & equal variance

16 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 5 - Determine whether or not the design is missing any factor combinations (cells) or is unbalanced > replications(eggs ~ DENSITY * SEASON, quinn) each Density: each season: each combination: 6 replicates 12 replicates 3 replicates Balanced

17 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 6- Define polynomial contrasts >contrasts(quinn$density) <- contr.poly(4, scores = c(8, 15, 30,45)) Step 7 (Key) - Fit the factorial linear model ## Option 1 > quinn.aov <- aov(eggs ~ DENSITY + SEASON + DENSITY:SEASON, data = quinn) ## Option 2- common > quinn.aov <- aov(eggs ~ DENSITY * SEASON, data = quinn)

18 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 8 - Examine the fitted model diagnostics > plot(quinn.aov, which = 1) Change the No. Figure out the difference

19 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 8 - Examine the fitted model diagnostics > plot(quinn.aov, which = 2) Change the No. Figure out the difference

20 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 9 - Examine the balanced model I ANOVA table, including the set of defined planned polynomial contrasts. >summary(quinn.aov) Impacts to eggs were found for density and season in a linear model But no evidence of an interaction between density and season

21 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 9 - Examine the balanced model I ANOVA table, (linear & planned polynomial contrasts) #option 2 >summary(quinn.aov, split = list(density = list(linear = 1, Quadratic = 2))) Linear model is better appropriate Impacts to eggs were found for density and season in a linear model But no evidence of an interaction between density and season

22 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 10 - Summarize the trends in a interaction plot ##option 1 >with(quinn, interaction.plot(density, SEASON, EGGS,type="b")) Conclusions-No evidence of an interaction between density and season

23 Example A: Adult density and season on egg mass production by intertidal limpets. #Step 11 - Summarize the trends in a interaction plot ##option 2 (try make the following work) >library(gmodels) >quinn.means <- tapply(quinn$eggs, list(quinn$density,quinn$season), mean) >quinn.se <- tapply(quinn$eggs, list(quinn$density, quinn$season),function(x) ci(x)[4]) >quinn$dens <- as.numeric(as.character(quinn$density)) >plot(eggs ~ DENS, quinn, type = "n", axes = F, xlab = "",ylab = "") >points(quinn.means[, 1] ~ unique(quinn$dens), pch = 16,type = "b", lwd = 2) >arrows(unique(quinn$dens), quinn.means[, 1] - quinn.se[, 1],unique(quinn$DENS), quinn.means[, 1] + quinn.se[, 1],code = 3, angle = 90, len = 0.1) >points(quinn.means[, 2] ~ unique(quinn$dens), pch = 16, type = "b", lwd = 2, lty = 2) >arrows(unique(quinn$dens), quinn.means[, 2] - quinn.se[, 2], unique(quinn$dens), quinn.means[, 2] + quinn.se[, 2], code = 3, angle = 90, len = 0.1) >axis(1, cex.axis = 0.8) >mtext(text = "Adult Density", 1, line = 3) >axis(2, cex.axis = 0.8, las = 1) >mtext(text = "Egg production", side = 2, line = 3) >legend("topright", leg = c("winter-spring", "Summer-autumn"), lwd = 2, lty = c(1, 2), bty = "n") >box(bty = "l")

24 Example B: Adult density and season on egg mass production by intertidal limpets (another dataset). #Step Import data set > quinn1 <- read.table("quinn1.csv", header = T, sep = ",") > quinn1$density <- factor(quinn1$density) > class(quinn1$density) >boxplot(eggs ~ DENSITY * SEASON, quinn1) >replications(eggs ~ DENSITY * SEASON, quinn1) > contrasts(quinn1$density) <- cbind(c(1, -0.5, -0.5)) Normality, Balanced #Step Fit the factorial linear mode > quinn1.aov <- aov(eggs ~ DENSITY * SEASON, data = quinn1) > plot(quinn1.aov, which = 1)

25 Example B: Adult density and season on egg mass production by intertidal limpets (another dataset). #Step 9 - Examine the model #option - 1 > summary(quinn1.aov) ## simple model

26 Example B: Adult density and season on egg mass production by intertidal limpets (another dataset). interaction #Step 9 - Examine the model (linear & polynomial) #option -2 >summary(quinn1.aov, split = list(density = list(linear = 1, Quadratic = 2))) Linear model is better appropriate Strong evidence of a interaction between density and season

27 Example B: Adult density and season on egg mass production by intertidal limpets (another dataset). # Step 11 - Summarize the trends in a interaction plot >library(gmodels) >quinn1.means <- tapply(quinn1$eggs, list(quinn1$density,quinn1$season), mean) >quinn1.se <- tapply(quinn1$eggs, list(quinn1$density, quinn1$season), function(x) ci(x)[4]) >quinn1$dens <- as.numeric(as.character(quinn1$density)) >plot(eggs ~ DENS, quinn1, type = "n", axes = F, xlab = "", ylab = "") >points(quinn1.means[, 1] ~ unique(quinn1$dens), pch = 16, type = "b", lwd = 2) >arrows(unique(quinn1$dens), quinn1.means[, 1] - quinn1.se[, 1], unique(quinn1$dens), quinn1.means[, 1] + quinn1.se[, 1], code = 3, angle = 90, len = 0.1) >points(quinn1.means[, 2] ~ unique(quinn1$dens), pch = 16, type = "b", lwd = 2, lty = 2) >arrows(unique(quinn1$dens), quinn1.means[, 2] - quinn1.se[, 2], unique(quinn1$dens), quinn1.means[, 2] + quinn1.se[, 2], code = 3, angle = 90, len = 0.1) >axis(1, cex.axis = 0.8) >mtext(text = "Adult Density", 1, line = 3) >axis(2, cex.axis = 0.8, las = 1) >mtext(text = "Egg production", side = 2, line = 3) >legend("topright", leg = c("winter-spring", "Summer-autumn"), lwd = 2, lty = c(1, 2), bty = "n") >box(bty = "l")