Lecture 19 Topic 13: Analysis of Covariance (ANCOVA), Part I [ST&D, chapter 17]

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1 Lecture 19 Topc 13: Analyss of Covarance (ANCOVA), Part I [ST&D, chapter 17] Suppose that the response varable (Y) s lnearly related to some other contnuous varable (X) that the expermenter cannot control but can observe, along wth Y: Examples: Intal weght of anmals n a feedng tral. Natve sol fertlty n a yeld tral. Overall level of DNA transcrpton n a gene expresson study. Maturty at harvest n a vegetable qualty study. In such studes: X s a covarable (or covarate or concomtant varable) ANCOVA uses X essentally as a contnuous blockng varable to mprove the precson of an experment. ANCOVA can provde great nsght nto the nature of treatment effects. ANCOVA s a combnaton of ANOVA and Lnear Regresson Let us regress (a revew of regresson concepts) The equaton of a straght lne s Y = a + bx a = the ntercept b = the slope 1

2 Example: Body weght (X) vs. ndvdual food consumpton (Y) for 10 anmals. Body weght Food consumpton (X) (Y) Food Consumpton (Y) Body Weght (X) The Prncple of Least Squares The lne of best ft s the one whch mnmzes the sum of squared devatons. 2

3 Calculatng a and b Equatons for the ntercept a and the slope b that mnmze the SSE: ( X X )( Y Y) S( XY ) b = 2 ( X X ) SS( X ) and a = Y b X For the sample dataset above: b = [( )( ) ( )( ) ] ( ) ( ) = 2 2 a = (4.98) = The equaton of the best ft lne: Y = X 7.69 S(XY) s called the corrected sum of cross products S( XY ) n 1 s called the sample covarance In R: X <- c(4.6, 4.7, 5.1, 5.1, 4.8, 5.2, 4.4, 4.9, 5.9, 5.1) Y <- c(87.1, 92.1, 93.1, 95.5, 89.8, 99.3, 91.4, 93.4, 99.5, 94.4) regresson <- lm(y ~ X) anova(regresson) summary(regresson) 3

4 Output Analyss of Varance Table Response: Y X ** Resduals Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) *** X ** Multple R-squared: The model accounts for 67% of the varaton n the experment. Y = X Analyss of adjusted Y s The SSE (44.77) represents the varaton n food consumpton (Y) that would have been observed f all the anmals used n the experment had had the same ntal body weght (X): X Y Adjusted Y = Y b( X X ) X = 4.98 SS = SS =

5 The results of a regresson on the adjusted Y's adjy <- Y * (X - mean(x)) adj_regresson <- lm(adjy ~ X) anova(adj_regresson) summary(adj_regresson) Output Analyss of Varance Table Response: adjy X Resduals Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) e e e-06 *** X e e Multple R-squared: e-10 Wth all anmals adjusted to the same ntal weght: 1. Body weght (X) no longer explans any varaton n the study (SSX = 0, slope ~ 0). 2. The SSE (44.77) s exactly the same as we saw before! Adjustng each Y to a common X by the best-ft equaton s equvalent, n terms of accountng for varaton, to a lnear regresson. 5

6 Oysters! The objectves of a plot experment to study oyster growth: 1. To determne f exposure to artfcally-heated water affects growth 2. To determne f poston n the water column (surface vs. bottom) affects growth Twenty bags of ten oysters each were placed across 5 locatons near a rversde power-generaton plant (.e. 4 bags / locaton): TRT1: cool-bottom TRT2: cool-surface TRT3: hot-bottom TRT4: hot-surface TRT5: control (md-depth, md-temperature) SURFACE 2 BOTTOM COOL INTERMEDIATE HOT The bags were weghed at the begnnng and the end of the experment. The data: Trtmt Rep Intal Fnal

7 The code: # I. Smple overall regresson oyster_reg.mod<-lm(fnal ~ Intal, oyster.dat) anova(oyster_reg.mod) summary(oyster_reg.mod) Analyss of Varance Table Response: Fnal Intal e-13 *** Resduals Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) * Intal e-13 *** Multple R-squared: # II. Usng a loop to perform regressons at each treatment level Trtmt_levels<-c(1:5) for ( n Trtmt_levels) { wth(subset(oyster.dat, Trtmt == Trtmt_levels[]), { prnt(trtmt_levels[]) prnt(summary(lm(fnal ~ Intal))) }) } Parameter estmates wthn each treatment group: Coeffcents: Estmate Std. Error t value Pr(> t ) Slope(Trt1) * Slope(Trt2) Slope(Trt3) * Slope(Trt4) ** Slope(Trt5) *** 7

8 TRT1: Y = X TRT3: Y = X Fnal Weght Fnal Weght Intal Weght Intal Weght # III. The one-way ANOVA oyster_anova.mod<-lm(fnal ~ Trtmt, oyster.dat) anova(oyster_anova.mod) The ANOVA "Are there dfferences n fnal weght across locatons?" Response: Fnal Trtmt * Resduals Ths model explans roughly 55% of the observed varaton. 8

9 The ANCOVA "Are there dfferences n fnal weght across locatons, adjustng for dfferences n ntal weght?" # IV. The ANCOVA #lbrary(car) oyster_ancova.mod<-lm(fnal ~ Trtmt + Intal, oyster.dat) anova(oyster_ancova.mod) Anova(oyster_ancova.mod, type = 2) The ANCOVA (Type I SS) Analyss of Varance Table Response: Fnal Trtmt e-11 *** Intal e-12 *** Resduals The ANCOVA (Type II SS) Anova Table (Type II tests) Response: Fnal Sum Sq Df F value Pr(>F) Trtmt *** Intal e-12 *** Resduals The Type I SS for TRT (198.4) s the unadjusted treatment SS. The Type II SS for TRT (12.1) s the adjusted treatment SS and allows us to test the treatment effects, adjustng for all other factors ncluded n the model. Type II SS produces the approprate results n ANCOVA. (not all levels of Trtmt are present n all levels of contnuous X) The power of the test for ncreases when the covarate s ncluded because most of the error n the smple ANOVA s due to varaton among INITIAL values. 9

10 The take-home vsualzaton of ANCOVA The data for Treatments 2 (whte squares) and 3 (whte crcles) from the oyster example Fnal Weght Adjusted Dfference Observed Dfference Intal Weght Comparng unadjusted (observed) means: Y 3 = < = Y 2 Comparng adjusted means: Y 3 = > = Y 2 10

11 Least squares adjusted means For vald comparsons, treatment means should be adjusted to what ther values would have been f all oysters had had the same ntal weght. Fnal_means <- aggregate(oyster.dat$fnal, lst(oyster.dat$trtmt), mean) oyster.lsm <- lsmeans(oyster_ancova.mod, "Trtmt") TRT Intal Unadjusted Adjusted Calculaton [ Yadj Y ( X X ) Means Means LS Means = β ] ( ) ( ) ( ) ( ) ( ) The coeffcent β = represents a "best" sngle slope value that descrbes the relatonshp between X and Y, accountng for all other classfcaton varables: > summary(oyster_ancova.mod) Call: lm(formula = Fnal ~ Trtmt + Intal, data = oyster.dat) Resduals: Mn 1Q Medan 3Q Max Coeffcents: Estmate Std. Error t value Pr(> t ) (Intercept) Trtmt Trtmt *** Trtmt ** Trtmt Intal e-12 *** Ths "best" slope can be used to create a new adjusted response varable: Z = Y β ( X X ) 11

12 Contrasts The adjusted means can be analyzed further wth orthogonal contrasts: #Comparng LSMeans, usng the "lsmeans" package (functon contrast()) oyster.lsm <- lsmeans(oyster_ancova.mod, "Trtmt") #Contrasts contrast(oyster.lsm, lst("control vs. trtmt"=c(-1,-1,-1,-1,4), "bottom vs. surface"=c(-1,1,-1,1,0), "cool vs. hot"=c(-1,-1,1,1,0), "depth*temp"=c(1,-1,-1,1,0))) The output: contrast estmate SE df t.rato p.value control.vs..trtmt bottom.vs..surface cool.vs..hot depth.temp If the covarable s not ncluded n the model, these exact same contrasts produce completely dfferent results: Trtmt * Trtmt: Cont v. Trt ** Trtmt: Bot vs. Surf Trtmt: Cool vs. Hot Trtmt: Depth*Temp Resduals

13 Why not use (Fnal Intal) as the response varable? ANOVA (no covarable) Dependent Varable: Fnal Weght R 2 = 0.55 Response: Fnal Trtmt * Resduals Dependent Varable: Dfference (Fnal Intal) R 2 = 0.70 Response: Dff Trtmt *** Resduals ANCOVA (Intal Weght as covarable) Dependent Varable: Fnal Weght R 2 = 0.99 Anova Table (Type II tests) Response: Fnal Sum Sq Df F value Pr(>F) Trtmt *** Intal e-12 *** Resduals Dependent Varable: Dfference (Fnal Intal) R 2 = 0.75 Anova Table (Type II tests) Response: Dff Sum Sq Df F value Pr(>F) Trtmt *** Intal Resduals

14 Comparson between ANCOVA and ANOVA of ratos In a study of the effect of stress on the presence of enzyme A n the lver, researchers measured the total actvty of enzyme A from lver homogenates of 10 control and 10 shocked anmals and the total amount of N as an ndcator of total enzyme actvty n the lver. A/N = the actvty of enzyme A per unt proten. Control anmals Shocked anmals N A A/N N A A/N ANOVA of the varable (A/N) R 2 = 0.16 Response: A/N Group NS Resduals ANCOVA of the varable A, usng N as a covarable R 2 = 0.99 Response: A Sum Sq Df F value Pr(>F) Group * N Resduals The use of ANOVA to analyze ratos Z = Y/X s not correct. Both X and Y, beng random varables, exhbt random varaton Varaton n Y affects Z n a lnear way, but varaton n X affects Z n a hyperbolc way The error n Z depends not only on the error n X but also on the absolute value of X. 14

15 ANCOVA model The lnear model for ANOVA of a CRD: Yj = µ + τ + ε j The lnear model for lnear regresson: Y = µ + β ) + ε ( X X. ANCOVA s a combnaton of ANOVA and regresson: Y j = µ + τ + β ) + ε ( X j X.. j A smple rearrangement: Y j β ) = µ + τ + ε ( X j X.. j An ANCOVA on the unadjusted values of Y s equvalent to a regular ANOVA on the adjusted values of Y Assumptons of the ANCOVA model OLD 1. The resduals are normally and ndependently dstrbuted wth zero mean and a common varance. NEW 2. The X s are fxed, measured wthout error, and ndependent of treatments. 3. The regresson of Y on X s lnear and ndependent of treatments. 15