Workshop 9.3a: Randomized block designs

Transcription

1 -1- Workshop 93a: Randomized block designs Murray Logan November 23, 16 Table of contents 1 Randomized Block (RCB) designs 1 2 Worked Examples 12 1 Randomized Block (RCB) designs 11 RCB design Simple Randomized block design 12 RCB design y ij = µ + β i + α j + ε ij µ - the mean of the first treatment group

2 -2- β - the random (Block) effect α - the main within Block effect eg abundance = base + block + treatment 13 Repeated measures designs 0 0 T T T T Assumptions Normality, homogeneity of variance No Block by within-block interaction

3 -3- Q4 Q1 Q4 Q1 Q3 Q3 Q4 Q3 Q2 Q2 Q1 Q T0 T Q4 Q3 Q1 Q2 Q3 Q4 Q1 Q2 Q4 Q2 Q1 Q T0 T Assumptions Normality, homogeneity of variance No Block by within-block interaction Independence (variance-covariance structure) 16 Var-cov structure T1 T2 T3 T4 T T T T T1 T2 T3 T4 Block B Block A T1 T2 T3 T4 Block D Block C T1 T2 T3 T4 T T T T Subject C Subject B Subject A T1 T2 T3 T4 T T T T Time (mins)

4 -4-17 Assumptions Normality, homogeneity of variance No Block by within-block interaction Independence (variance-covariance structure) RCB - usually met Repeated measures - rarely met 18 Example > datarcb1 <- readcsv('/data/datarcb1csv', stripwhite=true) > head(datarcb1) y A Block A B B B C B A B B B C B2 19 Exploratory data analysis Normality and homogeneity of variance > boxplot(y~a, datarcb1) A B C 110 Exploratory data analysis No block by within-block interaction

5 -5- > library(ggplot2) > ggplot(datarcb1, aes(y=y, x=a, group=block,color=block)) + + geom_line() + + guides(color=guide_legend(ncol=3)) 100 Block B1 B B31 B10 B21 B32 80 B11 B12 B22 B23 B33 B34 B13 B24 B35 y B14 B25 B4 60 B15 B16 B26 B27 B5 B6 B17 B28 B7 B18 B29 B8 40 B19 B2 B3 B30 B9 A B C A 111 Exploratory data analysis No block by within-block interaction > library(car) > residualplots(lm(y~block+a, datarcb1)) Pearson residuals Pearson residuals B1 B14 B19 B23 B28 B32 B5 B9 Block A B C A Pearson residuals Fitted values Test stat Pr(> t ) Block NA NA A NA NA Tukey test

6 Sphericity > library(nlme) > datarcb1lme <- lme(y~a, random=~1 Block, + data=datarcb1) > acf(resid(datarcb1lme)) Series resid(datarcb1lme) ACF Lag 113 Model fitting > #Assuming sphericity > datarcb1lme <- lme(y~a, random=~1 Block, data=datarcb1, + method='reml') > datarcb1lme1 <- lme(y~a, random=~a Block, data=datarcb1, + method='reml') > AIC(datarcb1lme, datarcb1lme1) df AIC datarcb1lme datarcb1lme > anova(datarcb1lme, datarcb1lme1) Model df AIC BIC loglik Test LRatio p-value datarcb1lme datarcb1lme vs Model fitting > #Assuming sphericity > datarcb1lmear1 <- lme(y~a, random=~1 Block, data=datarcb1, + correlation=corar1(),method='reml') > datarcb1lme1ar1 <- lme(y~a, random=~a Block, data=datarcb1, + correlation=corar1(),method='reml') > AIC(datarcb1lme, datarcb1lme1,datarcb1lmear1, datarcb1lme1ar1)

7 -7- df AIC datarcb1lme datarcb1lme datarcb1lmear datarcb1lme1ar > anova(datarcb1lme, datarcb1lme1,datarcb1lmear1, datarcb1lme1ar1) Model df AIC BIC loglik Test LRatio p-value datarcb1lme datarcb1lme vs datarcb1lmear vs datarcb1lme1ar vs Model validation > plot(datarcb1lme) 2 Standardized residuals Fitted values > plot(resid(datarcb1lme, type='normalized') ~ + datarcb1$a) resid(datarcb1lme, type = "normalized") A B C datarcb1$a 116 Effects plots > library(effects) > plot(effect('a',datarcb1lme))

8 -8- A effect plot y A B C A 117 Parameter estimates > summary(datarcb1lme) Linear mixed-effects model fit by REML Data: datarcb1 AIC BIC loglik Random effects: Formula: ~1 Block (Intercept) Residual StdDev: Fixed effects: y ~ A Value StdError DF t-value p-value (Intercept) AB AC Correlation: (Intr) AB AB AC Standardized Within-Group Residuals: Min Q1 Med Q3 Max Number of Observations: 105 Number of Groups: 35

9 Parameter estimates > intervals(datarcb1lme) Approximate 95% confidence intervals Fixed effects: lower est upper (Intercept) AB AC attr(,"label") [1] "Fixed effects:" Random Effects: Level: Block lower est upper sd((intercept)) Within-group standard error: lower est upper Parameter estimates > VarCorr(datarcb1lme) Block = pdlogchol(1) Variance StdDev (Intercept) Residual ANOVA table > anova(datarcb1lme) numdf dendf F-value p-value (Intercept) <0001 A < R 2 [1] [1] [1] [1]

10 What about lmer? > library(lme4) > datarcb1lmer <- lmer(y~a+(1 Block), data=datarcb1, REML=TRUE, + control=lmercontrol(checknobsvsnre="ignore")) > datarcb1lmer1 <- lmer(y~a+(a Block), data=datarcb1, REML=TRUE, + control=lmercontrol(checknobsvsnre="ignore")) > AIC(datarcb1lmer, datarcb1lmer1) df AIC datarcb1lmer datarcb1lmer > anova(datarcb1lmer, datarcb1lmer1) Data: datarcb1 Models: datarcb1lmer: y ~ A + (1 Block) datarcb1lmer1: y ~ A + (A Block) Df AIC BIC loglik deviance Chisq Chi Df Pr(>Chisq) datarcb1lmer datarcb1lmer What about lmer? > summary(datarcb1lmer) Linear mixed model fit by REML ['lmermod'] Formula: y ~ A + (1 Block) Data: datarcb1 Control: lmercontrol(checknobsvsnre = "ignore") REML criterion at convergence: 7121 Scaled residuals: Min 1Q Median 3Q Max Random effects: Groups Name Variance StdDev Block (Intercept) Residual Number of obs: 105, groups: Block, 35 Fixed effects: Estimate Std Error t value (Intercept) AB AC Correlation of Fixed Effects: (Intr) AB AB AC

11 What about lmer? > anova(datarcb1lmer) Analysis of Variance Table Df Sum Sq Mean Sq F value A What about lmer? > # Perform SAS-like p-value calculations (requires the lmertest and pbkrtest package) > library(lmertest) > datarcb1lmer <- update(datarcb1lmer) > summary(datarcb1lmer) Linear mixed model fit by REML t-tests use Satterthwaite approximations to degrees of freedom [ lmermod] Formula: y ~ A + (1 Block) Data: datarcb1 Control: lmercontrol(checknobsvsnre = "ignore") REML criterion at convergence: 7121 Scaled residuals: Min 1Q Median 3Q Max Random effects: Groups Name Variance StdDev Block (Intercept) Residual Number of obs: 105, groups: Block, 35 Fixed effects: Estimate Std Error df t value Pr(> t ) (Intercept) <2e-16 *** AB <2e-16 *** AC <2e-16 *** --- Signif codes: 0 '***' 0001 '**' 001 '*' 005 '' 01 ' ' 1 Correlation of Fixed Effects: (Intr) AB AB AC What about lmer? > anova(datarcb1lmer) # Satterthwaite denominator df method Analysis of Variance Table of type III with Satterthwaite approximation for degrees of freedom Sum Sq Mean Sq NumDF DenDF Fvalue Pr(>F) A < 22e-16 *** --- Signif codes: 0 '***' 0001 '**' 001 '*' 005 '' 01 ' ' 1

12 -12- > anova(datarcb1lmer,ddf="kenward-roger") # Kenward-Roger denominator df method Analysis of Variance Table Df Sum Sq Mean Sq F value A Worked Examples Worked Examples