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Repeated measures analysis in R

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richardchandler

The document provides an overview of a repeated measures design and analysis using data from a plant growth experiment. It introduces univariate and multivariate approaches for analyzing the data, including the use of adjusted p-values and MANOVA. Additionally, it discusses handling sphericity assumptions and comparing results from ANOVA and MANOVA.

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Lab 10 – Repeated Measures October 22 & 23, 2018 FANR 6750 Richard Chandler and Bob Cooper
Overview Design • We randomly assign each “subject” to a treatment • We record the response to the treatment over time Intro Univariate Split-plot Approach Multivariate 2 / 18
Overview Design • We randomly assign each “subject” to a treatment • We record the response to the treatment over time Sources of variation • Treatment • Time • Treatment-time interaction • Random variation among subjects • Random variation within subjects Intro Univariate Split-plot Approach Multivariate 2 / 18
Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values Intro Univariate Split-plot Approach Multivariate 3 / 18
Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values MANOVA • Testing based on Wilks’ lambda or Pillai’s trace • This is usually followed by a profile analysis Intro Univariate Split-plot Approach Multivariate 3 / 18
Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values MANOVA • Testing based on Wilks’ lambda or Pillai’s trace • This is usually followed by a profile analysis Mixed effects model with (ARMA) correlation structure • This can be done using lme • We might cover this later Intro Univariate Split-plot Approach Multivariate 3 / 18
The plant data plantData <- read.csv("plantData.csv") plantData$plant <- factor(plantData$plant) plantData$week <- factor(plantData$week) str(plantData) ## 'data.frame': 50 obs. of 4 variables: ## $ plant : Factor w/ 10 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 . ## $ fertilizer: Factor w/ 2 levels "H","L": 2 2 2 2 2 2 2 2 2 2 ... ## $ week : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 .. ## $ leaves : int 4 5 6 8 10 3 4 6 6 9 ... Intro Univariate Split-plot Approach Multivariate 4 / 18
The plant data plantData <- read.csv("plantData.csv") plantData$plant <- factor(plantData$plant) plantData$week <- factor(plantData$week) str(plantData) ## 'data.frame': 50 obs. of 4 variables: ## $ plant : Factor w/ 10 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 . ## $ fertilizer: Factor w/ 2 levels "H","L": 2 2 2 2 2 2 2 2 2 2 ... ## $ week : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 .. ## $ leaves : int 4 5 6 8 10 3 4 6 6 9 ... head(plantData, n=8) ## plant fertilizer week leaves ## 1 1 L 1 4 ## 2 1 L 2 5 ## 3 1 L 3 6 ## 4 1 L 4 8 ## 5 1 L 5 10 ## 6 2 L 1 3 ## 7 2 L 2 4 ## 8 2 L 3 6 Intro Univariate Split-plot Approach Multivariate 4 / 18
Leaf Growth 1 2 3 4 5 468101214 Week Leaves High fertilizer Low fertilizer Intro Univariate Split-plot Approach Multivariate 5 / 18
Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) Intro Univariate Split-plot Approach Multivariate 6 / 18
Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 6 / 18
Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 We need to adjust the p-values for the time and interaction effects Intro Univariate Split-plot Approach Multivariate 6 / 18
Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 We need to adjust the p-values for the time and interaction effects In R, this requires reformatting the data and running a MANOVA Intro Univariate Split-plot Approach Multivariate 6 / 18
Format data for MANOVA plantData2 <- reshape(plantData, idvar="plant", timevar="week", v.names="leaves", direction="wide") Intro Univariate Split-plot Approach Multivariate 7 / 18
Format data for MANOVA plantData2 <- reshape(plantData, idvar="plant", timevar="week", v.names="leaves", direction="wide") plantData2 ## plant fertilizer leaves.1 leaves.2 leaves.3 leaves.4 leaves.5 ## 1 1 L 4 5 6 8 10 ## 6 2 L 3 4 6 6 9 ## 11 3 L 6 7 9 10 12 ## 16 4 L 5 7 8 10 12 ## 21 5 L 5 6 7 8 10 ## 26 6 H 4 6 9 9 11 ## 31 7 H 3 5 7 10 12 ## 36 8 H 6 8 11 10 14 ## 41 9 H 5 7 9 10 12 ## 46 10 H 5 8 9 11 11 Intro Univariate Split-plot Approach Multivariate 7 / 18
MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) Intro Univariate Split-plot Approach Multivariate 8 / 18
MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 Intro Univariate Split-plot Approach Multivariate 8 / 18
MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 The last 3 columns are p-values corresponding to the effects of time (Intercept) and interaction (fertilizer). No adjustment is necessary for the main effect so you can use the p-value from aov . Intro Univariate Split-plot Approach Multivariate 8 / 18
An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Intro Univariate Split-plot Approach Multivariate 9 / 18
An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Intro Univariate Split-plot Approach Multivariate 9 / 18
An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Here is how you test the assumption: mauchly.test(manova1, X=~1) ## ## Mauchly's test of sphericity ## Contrasts orthogonal to ## ~1 ## ## ## data: SSD matrix from manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, ## W = 0.099297, p-value = 0.1062 Intro Univariate Split-plot Approach Multivariate 9 / 18
An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Here is how you test the assumption: mauchly.test(manova1, X=~1) ## ## Mauchly's test of sphericity ## Contrasts orthogonal to ## ~1 ## ## ## data: SSD matrix from manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, ## W = 0.099297, p-value = 0.1062 We fail to reject the null hypothesis, so sphericity can be assumed. Intro Univariate Split-plot Approach Multivariate 9 / 18
Compare aov and manova results summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 Intro Univariate Split-plot Approach Multivariate 10 / 18
Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. Intro Univariate Split-plot Approach Multivariate 11 / 18
Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. anova(manova1, X=~1, test="Wilks") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Wilks approx F num Df den Df Pr(>F) ## (Intercept) 1 0.008487 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.144772 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 11 / 18
Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. anova(manova1, X=~1, test="Wilks") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Wilks approx F num Df den Df Pr(>F) ## (Intercept) 1 0.008487 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.144772 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 As before, we conclude that the effect of fertilizer changes over time Intro Univariate Split-plot Approach Multivariate 11 / 18
Multivariate tests – Pillai’s trace Pillai’s trace is an alternative to Wilks’ lambda. In this case, it returns the same p-values as Wilks’ test. anova(manova1, X=~1, test="Pillai") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Pillai approx F num Df den Df Pr(>F) ## (Intercept) 1 0.99151 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.85523 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 12 / 18
Profile analysis Profile analysis requires calculating the differences (ie, the number of leaves grown each week). manova2 <- manova( cbind(leaves.2-leaves.1, leaves.3-leaves.2, leaves.4-leaves.3, leaves.5-leaves.4) ~ fertilizer, data=plantData2) Intro Univariate Split-plot Approach Multivariate 13 / 18
Profile Analysis During which intervals do the growth rates differ? Intro Univariate Split-plot Approach Multivariate 14 / 18
Profile Analysis During which intervals do the growth rates differ? summary.aov(manova2) ## Response 1 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 2.5 2.5 12.5 0.00767 ** ## Residuals 8 1.6 0.2 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Response 2 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 1.6 1.6 3.2 0.1114 ## Residuals 8 4.0 0.5 ## ## Response 3 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 0.1 0.1 0.0625 0.8089 ## Residuals 8 12.8 1.6 ## ## Response 4 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 0.1 0.1 0.0909 0.7707 ## Residuals 8 8.8 1.1 Intro Univariate Split-plot Approach Multivariate 14 / 18
Plot the growth rates Calculate mean growth rate for each time interval Intro Univariate Split-plot Approach Multivariate 15 / 18
Plot the growth rates Calculate mean growth rate for each time interval leavesMat <- plantData2[,3:7] growthMat <- leavesMat[,2:5] - leavesMat[,1:4] colnames(growthMat) <- paste("interval", 1:4, sep=".") (lowFertilizer <- colMeans(growthMat[1:5,])) ## interval.1 interval.2 interval.3 interval.4 ## 1.2 1.4 1.2 2.2 (highFertilizer <- colMeans(growthMat[6:10,])) ## interval.1 interval.2 interval.3 interval.4 ## 2.2 2.2 1.0 2.0 Intro Univariate Split-plot Approach Multivariate 15 / 18
Plot the growth rates Calculate mean growth rate for each time interval leavesMat <- plantData2[,3:7] growthMat <- leavesMat[,2:5] - leavesMat[,1:4] colnames(growthMat) <- paste("interval", 1:4, sep=".") (lowFertilizer <- colMeans(growthMat[1:5,])) ## interval.1 interval.2 interval.3 interval.4 ## 1.2 1.4 1.2 2.2 (highFertilizer <- colMeans(growthMat[6:10,])) ## interval.1 interval.2 interval.3 interval.4 ## 2.2 2.2 1.0 2.0 Calculate the standard errors for these growth rates SE <- sqrt(diag(stats:::vcov.mlm(manova2))) SE <- SE[names(SE)==":(Intercept)"] # Only use "intercept" SEs unname(SE) ## Ignore the names ## [1] 0.2000000 0.3162278 0.5656854 0.4690416 Intro Univariate Split-plot Approach Multivariate 15 / 18
Plot the growth rates plot(1:4-0.05, lowFertilizer, type="b", xlim=c(0.9, 4.1), ylim=c(-1, 4), xaxp=c(1,4,3), cex.lab=1.5, xlab="Time interval", ylab="Growth rate (leaves/week)") abline(h=0, lty=3) arrows(1:4-.05, lowFertilizer-SE, 1:4-.05, lowFertilizer+SE, angle=90, code=3, length=0.05) lines(1:4+0.05, highFertilizer, type="b", pch=17, col=4) arrows(1:4+0.05, highFertilizer-SE, 1:4+0.05, highFertilizer+SE, angle=90, code=3, length=0.05) legend(1, 4, c("Low fertilizer", "High fertilizer"), col=c("black", "blue"), pch=c(1,17)) Intro Univariate Split-plot Approach Multivariate 16 / 18
Plot the growth rates q q q q 1 2 3 4 −101234 Time interval Growthrate(leaves/week) q Low fertilizer High fertilizer Intro Univariate Split-plot Approach Multivariate 17 / 18
Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. Intro Univariate Split-plot Approach Multivariate 18 / 18
Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. (1) Conduct the univariate repeated measures ANOVA using aov . Calculate the adjusted p-values using the Huynh-Feldt method. Does the effect of density change over time? (2) Conduct a multivariate repeated measures ANOVA and use Wilks’ lambda to test if the effect of density changes over time. What is your conclusion? (3) Conduct a profile analysis. In which time intervals is the effect of density on growth rate significant? Intro Univariate Split-plot Approach Multivariate 18 / 18
Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. (1) Conduct the univariate repeated measures ANOVA using aov . Calculate the adjusted p-values using the Huynh-Feldt method. Does the effect of density change over time? (2) Conduct a multivariate repeated measures ANOVA and use Wilks’ lambda to test if the effect of density changes over time. What is your conclusion? (3) Conduct a profile analysis. In which time intervals is the effect of density on growth rate significant? Upload your self-contained R-script to ELC at least one day before your next lab Intro Univariate Split-plot Approach Multivariate 18 / 18

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Repeated measures analysis in R

  • 1. Lab 10 – Repeated Measures October 22 & 23, 2018 FANR 6750 Richard Chandler and Bob Cooper
  • 2. Overview Design • We randomly assign each “subject” to a treatment • We record the response to the treatment over time Intro Univariate Split-plot Approach Multivariate 2 / 18
  • 3. Overview Design • We randomly assign each “subject” to a treatment • We record the response to the treatment over time Sources of variation • Treatment • Time • Treatment-time interaction • Random variation among subjects • Random variation within subjects Intro Univariate Split-plot Approach Multivariate 2 / 18
  • 4. Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values Intro Univariate Split-plot Approach Multivariate 3 / 18
  • 5. Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values MANOVA • Testing based on Wilks’ lambda or Pillai’s trace • This is usually followed by a profile analysis Intro Univariate Split-plot Approach Multivariate 3 / 18
  • 6. Approaches Univariate • This is just a split-plot analysis with adjusted p-values • Adjustments: Greenhouse-Geisser or Huynh-Feldt methods • In R, you must do a MANOVA to obtain these adjusted P-values MANOVA • Testing based on Wilks’ lambda or Pillai’s trace • This is usually followed by a profile analysis Mixed effects model with (ARMA) correlation structure • This can be done using lme • We might cover this later Intro Univariate Split-plot Approach Multivariate 3 / 18
  • 7. The plant data plantData <- read.csv("plantData.csv") plantData$plant <- factor(plantData$plant) plantData$week <- factor(plantData$week) str(plantData) ## 'data.frame': 50 obs. of 4 variables: ## $ plant : Factor w/ 10 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 . ## $ fertilizer: Factor w/ 2 levels "H","L": 2 2 2 2 2 2 2 2 2 2 ... ## $ week : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 .. ## $ leaves : int 4 5 6 8 10 3 4 6 6 9 ... Intro Univariate Split-plot Approach Multivariate 4 / 18
  • 8. The plant data plantData <- read.csv("plantData.csv") plantData$plant <- factor(plantData$plant) plantData$week <- factor(plantData$week) str(plantData) ## 'data.frame': 50 obs. of 4 variables: ## $ plant : Factor w/ 10 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 . ## $ fertilizer: Factor w/ 2 levels "H","L": 2 2 2 2 2 2 2 2 2 2 ... ## $ week : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 .. ## $ leaves : int 4 5 6 8 10 3 4 6 6 9 ... head(plantData, n=8) ## plant fertilizer week leaves ## 1 1 L 1 4 ## 2 1 L 2 5 ## 3 1 L 3 6 ## 4 1 L 4 8 ## 5 1 L 5 10 ## 6 2 L 1 3 ## 7 2 L 2 4 ## 8 2 L 3 6 Intro Univariate Split-plot Approach Multivariate 4 / 18
  • 9. Leaf Growth 1 2 3 4 5 468101214 Week Leaves High fertilizer Low fertilizer Intro Univariate Split-plot Approach Multivariate 5 / 18
  • 10. Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) Intro Univariate Split-plot Approach Multivariate 6 / 18
  • 11. Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 6 / 18
  • 12. Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 We need to adjust the p-values for the time and interaction effects Intro Univariate Split-plot Approach Multivariate 6 / 18
  • 13. Univariate split-plot approach aov1 <- aov(leaves ~ fertilizer*week + Error(plant), data=plantData) summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 We need to adjust the p-values for the time and interaction effects In R, this requires reformatting the data and running a MANOVA Intro Univariate Split-plot Approach Multivariate 6 / 18
  • 14. Format data for MANOVA plantData2 <- reshape(plantData, idvar="plant", timevar="week", v.names="leaves", direction="wide") Intro Univariate Split-plot Approach Multivariate 7 / 18
  • 15. Format data for MANOVA plantData2 <- reshape(plantData, idvar="plant", timevar="week", v.names="leaves", direction="wide") plantData2 ## plant fertilizer leaves.1 leaves.2 leaves.3 leaves.4 leaves.5 ## 1 1 L 4 5 6 8 10 ## 6 2 L 3 4 6 6 9 ## 11 3 L 6 7 9 10 12 ## 16 4 L 5 7 8 10 12 ## 21 5 L 5 6 7 8 10 ## 26 6 H 4 6 9 9 11 ## 31 7 H 3 5 7 10 12 ## 36 8 H 6 8 11 10 14 ## 41 9 H 5 7 9 10 12 ## 46 10 H 5 8 9 11 11 Intro Univariate Split-plot Approach Multivariate 7 / 18
  • 16. MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) Intro Univariate Split-plot Approach Multivariate 8 / 18
  • 17. MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 Intro Univariate Split-plot Approach Multivariate 8 / 18
  • 18. MANOVA and adjusted P-values manova1 <- manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, leaves.5) ~ fertilizer, data=plantData2) anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 The last 3 columns are p-values corresponding to the effects of time (Intercept) and interaction (fertilizer). No adjustment is necessary for the main effect so you can use the p-value from aov . Intro Univariate Split-plot Approach Multivariate 8 / 18
  • 19. An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Intro Univariate Split-plot Approach Multivariate 9 / 18
  • 20. An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Intro Univariate Split-plot Approach Multivariate 9 / 18
  • 21. An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Here is how you test the assumption: mauchly.test(manova1, X=~1) ## ## Mauchly's test of sphericity ## Contrasts orthogonal to ## ~1 ## ## ## data: SSD matrix from manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, ## W = 0.099297, p-value = 0.1062 Intro Univariate Split-plot Approach Multivariate 9 / 18
  • 22. An aside – sphericity Technically, adjusted p-values and MANOVA aren’t necessary if the assumption of sphericity holds. However, we recommend doing the adjustments (or the MANOVA) anyway because the test of sphericity has low power. Sphericity is the multivariate analogue of the homogeneity of variance assumption of ANOVA. Here is how you test the assumption: mauchly.test(manova1, X=~1) ## ## Mauchly's test of sphericity ## Contrasts orthogonal to ## ~1 ## ## ## data: SSD matrix from manova(cbind(leaves.1, leaves.2, leaves.3, leaves.4, ## W = 0.099297, p-value = 0.1062 We fail to reject the null hypothesis, so sphericity can be assumed. Intro Univariate Split-plot Approach Multivariate 9 / 18
  • 23. Compare aov and manova results summary(aov1) ## ## Error: plant ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 16.82 16.82 2.604 0.145 ## Residuals 8 51.68 6.46 ## ## Error: Within ## Df Sum Sq Mean Sq F value Pr(>F) ## week 4 267.40 66.85 158.225 <2e-16 *** ## fertilizer:week 4 5.08 1.27 3.006 0.0326 * ## Residuals 32 13.52 0.42 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 anova(manova1, X=~1, test="Spherical") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Greenhouse-Geisser epsilon: 0.5882 ## Huynh-Feldt epsilon: 0.8490 ## ## Df F num Df den Df Pr(>F) G-G Pr H-F Pr ## (Intercept) 1 158.2249 4 32 0.000000 0.000000 0.00000 ## fertilizer 1 3.0059 4 32 0.032613 0.066622 0.04224 ## Residuals 8 Intro Univariate Split-plot Approach Multivariate 10 / 18
  • 24. Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. Intro Univariate Split-plot Approach Multivariate 11 / 18
  • 25. Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. anova(manova1, X=~1, test="Wilks") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Wilks approx F num Df den Df Pr(>F) ## (Intercept) 1 0.008487 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.144772 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 11 / 18
  • 26. Multivariate tests – Wilks’ lambda An alternative to the adjusted p-value approach is to do a multivariate analysis relaxing the assumptions about the structure of the variance-covariance matrix. We’re already most of the way there. anova(manova1, X=~1, test="Wilks") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Wilks approx F num Df den Df Pr(>F) ## (Intercept) 1 0.008487 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.144772 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 As before, we conclude that the effect of fertilizer changes over time Intro Univariate Split-plot Approach Multivariate 11 / 18
  • 27. Multivariate tests – Pillai’s trace Pillai’s trace is an alternative to Wilks’ lambda. In this case, it returns the same p-values as Wilks’ test. anova(manova1, X=~1, test="Pillai") ## Analysis of Variance Table ## ## ## Contrasts orthogonal to ## ~1 ## ## Df Pillai approx F num Df den Df Pr(>F) ## (Intercept) 1 0.99151 146.042 4 5 2.308e-05 *** ## fertilizer 1 0.85523 7.384 4 5 0.02503 * ## Residuals 8 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Intro Univariate Split-plot Approach Multivariate 12 / 18
  • 28. Profile analysis Profile analysis requires calculating the differences (ie, the number of leaves grown each week). manova2 <- manova( cbind(leaves.2-leaves.1, leaves.3-leaves.2, leaves.4-leaves.3, leaves.5-leaves.4) ~ fertilizer, data=plantData2) Intro Univariate Split-plot Approach Multivariate 13 / 18
  • 29. Profile Analysis During which intervals do the growth rates differ? Intro Univariate Split-plot Approach Multivariate 14 / 18
  • 30. Profile Analysis During which intervals do the growth rates differ? summary.aov(manova2) ## Response 1 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 2.5 2.5 12.5 0.00767 ** ## Residuals 8 1.6 0.2 ## --- ## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 ## ## Response 2 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 1.6 1.6 3.2 0.1114 ## Residuals 8 4.0 0.5 ## ## Response 3 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 0.1 0.1 0.0625 0.8089 ## Residuals 8 12.8 1.6 ## ## Response 4 : ## Df Sum Sq Mean Sq F value Pr(>F) ## fertilizer 1 0.1 0.1 0.0909 0.7707 ## Residuals 8 8.8 1.1 Intro Univariate Split-plot Approach Multivariate 14 / 18
  • 31. Plot the growth rates Calculate mean growth rate for each time interval Intro Univariate Split-plot Approach Multivariate 15 / 18
  • 32. Plot the growth rates Calculate mean growth rate for each time interval leavesMat <- plantData2[,3:7] growthMat <- leavesMat[,2:5] - leavesMat[,1:4] colnames(growthMat) <- paste("interval", 1:4, sep=".") (lowFertilizer <- colMeans(growthMat[1:5,])) ## interval.1 interval.2 interval.3 interval.4 ## 1.2 1.4 1.2 2.2 (highFertilizer <- colMeans(growthMat[6:10,])) ## interval.1 interval.2 interval.3 interval.4 ## 2.2 2.2 1.0 2.0 Intro Univariate Split-plot Approach Multivariate 15 / 18
  • 33. Plot the growth rates Calculate mean growth rate for each time interval leavesMat <- plantData2[,3:7] growthMat <- leavesMat[,2:5] - leavesMat[,1:4] colnames(growthMat) <- paste("interval", 1:4, sep=".") (lowFertilizer <- colMeans(growthMat[1:5,])) ## interval.1 interval.2 interval.3 interval.4 ## 1.2 1.4 1.2 2.2 (highFertilizer <- colMeans(growthMat[6:10,])) ## interval.1 interval.2 interval.3 interval.4 ## 2.2 2.2 1.0 2.0 Calculate the standard errors for these growth rates SE <- sqrt(diag(stats:::vcov.mlm(manova2))) SE <- SE[names(SE)==":(Intercept)"] # Only use "intercept" SEs unname(SE) ## Ignore the names ## [1] 0.2000000 0.3162278 0.5656854 0.4690416 Intro Univariate Split-plot Approach Multivariate 15 / 18
  • 34. Plot the growth rates plot(1:4-0.05, lowFertilizer, type="b", xlim=c(0.9, 4.1), ylim=c(-1, 4), xaxp=c(1,4,3), cex.lab=1.5, xlab="Time interval", ylab="Growth rate (leaves/week)") abline(h=0, lty=3) arrows(1:4-.05, lowFertilizer-SE, 1:4-.05, lowFertilizer+SE, angle=90, code=3, length=0.05) lines(1:4+0.05, highFertilizer, type="b", pch=17, col=4) arrows(1:4+0.05, highFertilizer-SE, 1:4+0.05, highFertilizer+SE, angle=90, code=3, length=0.05) legend(1, 4, c("Low fertilizer", "High fertilizer"), col=c("black", "blue"), pch=c(1,17)) Intro Univariate Split-plot Approach Multivariate 16 / 18
  • 35. Plot the growth rates q q q q 1 2 3 4 −101234 Time interval Growthrate(leaves/week) q Low fertilizer High fertilizer Intro Univariate Split-plot Approach Multivariate 17 / 18
  • 36. Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. Intro Univariate Split-plot Approach Multivariate 18 / 18
  • 37. Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. (1) Conduct the univariate repeated measures ANOVA using aov . Calculate the adjusted p-values using the Huynh-Feldt method. Does the effect of density change over time? (2) Conduct a multivariate repeated measures ANOVA and use Wilks’ lambda to test if the effect of density changes over time. What is your conclusion? (3) Conduct a profile analysis. In which time intervals is the effect of density on growth rate significant? Intro Univariate Split-plot Approach Multivariate 18 / 18
  • 38. Assignment A researcher wants to assess the effects of crowding on the growth of the dark toadfish (Neophrynichthys latus). 15 fish tanks are stocked with three densities of conspecifics. Five tanks have low density (1 fish), 5 tanks have medium density (5 fish), and 5 tanks have high density (10 fish). In each tank, the weight of one “focal fish” is recorded on 6 consecutive weeks. The data are in the file fishData.csv. (1) Conduct the univariate repeated measures ANOVA using aov . Calculate the adjusted p-values using the Huynh-Feldt method. Does the effect of density change over time? (2) Conduct a multivariate repeated measures ANOVA and use Wilks’ lambda to test if the effect of density changes over time. What is your conclusion? (3) Conduct a profile analysis. In which time intervals is the effect of density on growth rate significant? Upload your self-contained R-script to ELC at least one day before your next lab Intro Univariate Split-plot Approach Multivariate 18 / 18