Multiple Predictor Variables: ANOVA

Transcription

1 Multiple Predictor Variables: ANOVA 1/32 Linear Models with Many Predictors Multiple regression has many predictors BUT - so did 1-way ANOVA if treatments had 2 levels What if there are multiple treatment types and combinations? What if we have spatial gradients in our experiments? 2/32

2 Multiway ANOVA Extends multiple predictor framework Categorical treatments are orthogonal Reflects reality of experiments Stepping-stone to factorial designs 3/32 Blocked Designs 4/32

3 What if you manipulate two factors? Block 1 Block 2 Block 3 Block 4 A B C D B C D A C D A B D A B C Randomized Controlled Blocked Design: Design where each treatment only has 1 replicate of a second treatment 5/32 What if you manipulate two factors? Block 1 Block 2 Block 3 Block 4 A B C D B C D A C D A B D A B C Randomized Controlled Blocked Design: Design where each treatment only has 1 replicate of a second treatment Note: Above is a Latin Squares Design - Every row and column contains one replicate of a treatment. 5/32

4 Effects of Stickleback Density on Zooplankton Units placed across a lake so that 1 set of each treatment was blocked together 6/32 Treatment and Block Effects control high low Treatment Block 7/32

5 Modeling & Evaluating Multiple Factors 8/32 Model for Multiway ANOVA/ANODEV y k = β 0 + β i x i + β j x j + ɛ k ɛ ijk N(0, σ 2 ), x i = 0, 1 9/32

6 Model for Multiway ANOVA/ANODEV y k = β 0 + β i x i + β j x j + ɛ k ɛ ijk N(0, σ 2 ), x i = 0, 1 Or, with matrices... Y = βx + ɛ 9/32 Model for Multiway ANOVA/ANODEV Y = βx + ɛ y1 β i ɛ 1 y2 y3 = β i β j ɛ 2 ɛ 3 y4 β j ɛ 4 10/32

7 Model for Multiway ANOVA/ANODEV Y = βx + ɛ y1 β i ɛ 1 y2 y3 = β i β j ɛ 2 ɛ 3 y4 β j ɛ 4 We can have as many groups as we need, so long as there is sufficient replication of each treatment combination. 10/32 Hypotheses for Multiway ANOVA/ANODEV TreatmentHo: µ i1 = µi2 = µi3 =... Block Ho: µ j1 = µj2 = µj3 =... 11/32

8 Sums of Squares for Multiway ANOVA Factors are Orthogonal and Balanced, so... SST = SSA + SSB + SSR F-Test using Mean Squares as Before Type I and Type II SS will produce the same result 12/32 Before we model it, make sure Block is a factor zoop$block <- factor(zoop$block) 13/32

9 Two-Way ANOVA as a Linear Model zoop_lm <- lm(zooplankton treatment + block, data=zoop) 14/32 Check Diagnostics Residuals vs Fitted Normal Q Q Scale Location Residuals Standardized residuals Standardized residuals Fitted values Theoretical Quantiles Fitted values Cook's distance Constant Leverage: Residuals vs Factor Levels Cook's distance Standardized residuals treatment : control high low 14 Obs. number Factor Level Combinations 15/32

10 Residuals by Groups and No Non-Additivity Pearson residuals control high low treatment block Pearson residuals Pearson residuals Fitted values 16/32 Residuals by Groups and No Non-Additivity Tukey s Test for Non-Additivity library(car) residualplots(zoop_lm, cex.lab=1.4) # Test stat Pr(> t ) # treatment NA NA # block NA NA # Tukey test /32

11 The ANOVA But first, what are the DF for... Treatment (with 3 levels) Block (with 5 blocks) Residuals (with n=15) 18/32 The ANOVA anova(zoop_lm) # Analysis of Variance Table # # Response: zooplankton # Df Sum Sq Mean Sq F value Pr(>F) # treatment # block # Residuals /32

12 Coefficients via Treatment Contrasts summary(zoop_lm)$coef # Estimate Std. Error t value # (Intercept) e e+01 # treatmenthigh e e+00 # treatmentlow e e+00 # block e e-15 # block e e+00 # block e e+00 # block e e-01 # Pr(> t ) # (Intercept) e-06 # treatmenthigh e-04 # treatmentlow e-03 # block e+00 # block e-02 # block e-02 # block e-01 20/32 Unique Effect of Each Treatment crplots(zoop_lm) Component + Residual Plots Component+Residual(zooplankton) Component+Residual(zooplankton) control high low treatment block 21/32

13 Unique Effect of Each Treatment (visreg) zooplankton zooplankton control high low treatment block 22/32 Exercise: Bees! Load the Bee Gene Expresion Data Does bee type or colony matter? How much variation does this experiment explain? 23/32

14 Bee ANOVA anova(bee_lm) # Analysis of Variance Table # # Response: Expression # Df Sum Sq Mean Sq F value Pr(>F) # type # colony # Residuals /32 Bee Effects crplots(bee_lm) Component + Residual Plots Component+Residual(Expression) Component+Residual(Expression) for nurse type colony 25/32

15 What if my data is unbalanced? 26/32 Unbalancing the Zooplankton Data zoop_u <- zoop[-c(1,2),] 27/32

16 An Unbalanced ANOVA zoop_u_lm <- update(zoop_lm, data=zoop_u) anova(zoop_u_lm) # Analysis of Variance Table # # Response: zooplankton # Df Sum Sq Mean Sq F value Pr(>F) # treatment # block # Residuals /32 An Unbalanced ANOVA zoop_u_lm <- update(zoop_lm, data=zoop_u) anova(zoop_u_lm) # Analysis of Variance Table # # Response: zooplankton # Df Sum Sq Mean Sq F value Pr(>F) # treatment # block # Residuals Is this valid? Can we use Type I sequential SS? 28/32

17 Unbalanced Data and Type I SS Missing cells (i.e., treatment-block combinations) mean that order matters in testing SS zoop_u_lm1 <- lm(zooplankton treatment + block, data=zoop_u) zoop_u_lm2 <- lm(zooplankton block + treatment, data=zoop_u) Intercept versus Treatment and Block versus Treatment + Block will not produce different SS 29/32 Unbalanced Data and Type I SS # Analysis of Variance Table # # Response: zooplankton # Df Sum Sq Mean Sq F value Pr(>F) # treatment # block # Residuals # Analysis of Variance Table # # Response: zooplankton # Df Sum Sq Mean Sq F value Pr(>F) # block # treatment # Residuals /32

18 Solution: Marginal, or Type II SS SS of Block: Treatment versus Treatment + Block SS of Treatment: Block versus Block + Treatment Note: Because of marginality, the sum of all SS will no longer equal SST 31/32 Solution: Marginal, or Type II SS Anova(zoop_u_lm1) # Anova Table (Type II tests) # # Response: zooplankton # Sum Sq Df F value Pr(>F) # treatment # block # Residuals Note the capital A - this is a function from the car package. 32/32