REGRESSION MODELS ANOVA

Transcription

1 REGRESSION MODELS ANOVA 141

2 Cotiuous Outcome? NO RECAP: Logistic regressio ad other methods YES Liear Regressio Examie mai effects cosiderig predictors of iterest, ad cofouders Test effect modificatio if scietifically relevat Compute ad plot Residuals Assess ifluece Modify approach NO Do the assumptios appear reasoable? YES REPORT 142

3 COMING UP NEXT: ANOVA a special case of liear regressio What if the idepedet variables of iterest are categorical? I this case, comparig the mea of the cotiuous outcome i the differet categories may be of iterest This is what is called ANalysis Of VAriace We will show that it is just a special case of liear regressio 143

4 ANOVA a special case of liear regressio LINEAR REGRESSION Oe-way Aalysis of Variace Two-way Aalysis of Variace Aalysis of Covariace Oe Categorical POI Two Categorical POIs Oe Categorical POI + Oe cotiuous predictor Uses dummy variables to represet categorical variables! 144

5 Outlie Motivatio: We will cosider some examples of ANOVA ad show that they are special cases of liear regressio ANOVA as a regressio model Dummy variables Oe-way ANOVA models Cotrasts Multiple comparisos Two-way ANOVA models Iteractios ANCOVA models 145

6 ANOVA/ANCOVA: Motivatio Let s ivestigate if geetic factors are associated with cholesterol levels. Ideally, you would have a cofirmatory aalysis of scietific hypotheses formulated prior to data collectio Alteratively, you could cosider a exploratory aalysis hypotheses geeratio for future studies 146

7 ANOVA/ANCOVA: Motivatio Scietific hypotheses of iterest: Assess the effect of rs o cholesterol levels. Assess the effect of rs ad sex o cholesterol levels Does the effect of rs o cholesterol differ betwee males ad females? Assess the effect of rs ad age o cholesterol levels Does the effect of rs o cholesterol differ depedig o subject s age? 147

8 ANOVA: Oe-Way Model Motivatio: Scietific questio: Assess the effect of rs o cholesterol levels. 148

9 Motivatio: Example Here are some descriptive summaries: > tapply(chol, factor(rs174548), mea) > tapply(chol, factor(rs174548), sd)

10 Motivatio: Example Aother way of gettig the same results: > by(chol, factor(rs174548), mea) factor(rs174548): 0 [1] factor(rs174548): 1 [1] factor(rs174548): 2 [1] > by(chol, factor(rs174548), sd) factor(rs174548): 0 [1] factor(rs174548): 1 [1] factor(rs174548): 2 [1]

11 Motivatio: Example Is rs associated with cholesterol? R commad: boxplot(chol ~ factor(rs174548)) 151

12 Motivatio: Example Aother graphical display: mea of chol as.factor(rs174548) R commad: plot.desig(chol ~ factor(rs174548)) Factors 152

13 Motivatio: Example Feature: How do the mea resposes compare across differet groups? Categorical/qualitative predictor 153

14 REGRESSION MODELS Oe-way ANOVA as a regressio model 154

15 ANalysis Of VAriace Models (ANOVA) Compares the meas of several populatios Assumptios for Classical ANOVA Framework: Idepedece Normality Equal variaces 155

16 ANalysis Of VAriace Models (ANOVA) Compares the meas of several populatios

17 ANalysis Of VAriace Models (ANOVA) Compares the meas of several populatios Couter-ituitive ame! 157

18 ANalysis Of VAriace Models (ANOVA) I both data sets, the true populatio meas are: 3 (A), 5 (B), 7(C) Situatio 1 Situatio A B C Low variace withi groups A B C High variace withi groups Where do you expect to detect differece betwee populatio meas? 158

19 ANalysis Of VAriace Models (ANOVA) Compares the meas of several populatios Couter-ituitive ame! Uderlyig cocept: To assess whether the populatio meas are equal, compares: Variatio betwee the sample meas (MSR) to Natural variatio of the observatios withi the samples (MSE). The larger the MSR compared to MSE the more support that there is a differece i the populatio meas! The ratio MSR/MSE is the F-statistic. We ca make these comparisos with multiple liear regressio: the differet groups are represeted with dummy variables 159

20 ANOVA as a multiple regressio model Dummy Variables: Suppose you have a categorical variable C with k categories 0,1, 2,, k-1. To represet that variable we ca costruct k-1 dummy variables of the form The omitted category (here category 0) is the referece group. 160

21 ANOVA as a multiple regressio model Dummy Variables: Back to our motivatig example: Predictor: rs (coded 0=C/C, 1=C/G, 2=G/G) Outcome (Y): cholesterol Let s take C/C as the referece group. x 1 = ì1, í î0, if code1(c/g) otherwise x 2 = ì1, í î0, if code 2 (G/G) otherwise 161

22 ANOVA as a multiple regressio model rs Mea cholesterol X 1 X 2 C/C µ C/G µ G/G µ

23 ANOVA as a multiple regressio model Regressio with Dummy Variables: Example: Model: E[Y x 1, x 2 ] = b 0 + b 1 x 1 + b 2 x 2 Iterpretatio of model parameters? 163

24 ANOVA as a multiple regressio model Mea Regressio Model µ 0 b 0 µ 1 b 0 + b 1 µ 2 b 0 + b 2 164

25 ANOVA as a multiple regressio model Regressio with Dummy Variables: Example: Model: E[Y x 1, x 2 ] = b 0 + b 1 x 1 + b 2 x 2 Iterpretatio of model parameters? µ 0 = b 0 : mea cholesterol whe rs is C/C µ 1 = b 0 +b 1 : mea cholesterol whe rs is C/G µ 2 = b 0 +b 2 : mea cholesterol whe rs is G/G 165

26 ANOVA as a multiple regressio model Regressio with Dummy Variables: Example: Model: E[Y x 1, x 2 ] = b 0 + b 1 x 1 + b 2 x 2 Iterpretatio of model parameters? µ 0 = b 0 : mea cholesterol whe rs is C/C µ 1 = b 0 +b 1 : mea cholesterol whe rs is C/G µ 2 = b 0 +b 2 : mea cholesterol whe rs is G/G Alteratively b 1 : differece i mea cholesterol levels betwee groups with rs equal to C/G ad C/C (µ 1 - µ 0 ). b 2 : differece i mea cholesterol levels betwee groups with rs equal to G/G ad C/C (µ 2 - µ 0 ). 166

27 ANOVA: Oe-Way Model Goal: Compare the meas of K idepedet groups (defied by a categorical predictor) Statistical Hypotheses: (Global) Null Hypothesis: H 0 : µ 0 = µ 1 = = µ K-1 or, equivaletly, H 0 : β 1 = β 2 = = β K-1 =0 Alterative Hypothesis: H 1 : ot all meas are equal If the meas of the groups are ot all equal (i.e. you rejected the above H 0 ), determie which oes are differet (multiple comparisos) 167

28 Estimatio ad Iferece Global Hypotheses µ = µ =... = µ K H 0 : vs. H 1 : ot all meas are equal 1 H 0 : β 1 = β 2 = = β K-1 =0 2 Aalysis of variace table Source df SS MS F 2 Regressio K-1 SSR= (y - y MSR= SSR/(K-1) Residual -K SSE= å 2 (yij - yi) MSE= i, j SSE/-K Total -1 SST= å i å i, j (y i ) 2 ij - y) MSR/ MSE 168

29 ANOVA: Oe-Way Model How to fit a oe-way model as a regressio problem? Need to use dummy variables Create o your ow (ca be tedious!) Most software packages will do this for you R creates dummy variables i the backgroud as log as you state you have a categorical variable (may eed to use: factor) 169

30 ANOVA: Oe-Way Model By had: Creatig dummy variables: > dummy1 = 1*(rs174548==1) > dummy2 = 1*(rs174548==2) Fittig the ANOVA model: > fit0 = lm(chol ~ dummy1 + dummy2) > summary(fit0) Call: lm(formula = chol ~ dummy1 + dummy2) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** dummy ** dummy Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit0) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) dummy ** dummy Residuals Sigif. codes: 0 *** ** 0.01 *

31 ANOVA: Oe-Way Model Better: Let R do it for you! > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) ** factor(rs174548) Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals Sigif. codes: 0 *** ** 0.01 *

32 ANOVA: Oe-Way Model Your tur! Compare model fit results (fit0 & fit1.1) What do you coclude? 172

33 ANOVA: Oe-Way Model > fit0 = lm(chol ~ dummy1 + dummy2) > summary(fit0) Call: lm(formula = chol ~ dummy1 + dummy2) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** dummy ** dummy Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit0) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) dummy ** dummy Residuals > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) ** factor(rs174548) Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals

34 ANOVA: Oe-Way Model > fit0 = lm(chol ~ dummy1 + dummy2) > summary(fit0) Call: lm(formula = chol ~ dummy1 + dummy2) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** dummy ** dummy Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit0) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) dummy ** dummy Residuals > 1-pf(4.4865,2,397) [1] > 1-pf((( )/2)/481,2,397) [1] > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) ** factor(rs174548) Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals

35 ANOVA: Oe-Way Model > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 factor(rs174548) factor(rs174548) Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals Let s iterpret the regressio model results! What is the iterpretatio of the regressio model coefficiets? 175

36 ANOVA: Oe-Way Model > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 factor(rs174548) factor(rs174548) Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals Iterpretatio: Estimated mea cholesterol for C/C group: mg/dl Estimated differece i mea cholesterol levels betwee C/G ad C/C groups: mg/dl Estimated differece i mea cholesterol levels betwee G/G ad C/C groups: mg/dl 176

37 ANOVA: Oe-Way Model > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 factor(rs174548) factor(rs174548) Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals Overall F-test shows a sigificat p-value. We reject the ull hypothesis that the mea cholesterol levels are the same across groups defied by rs (p= ). This does ot tell us which groups are differet! (Need to perform multiple comparisos! More soo ) 177

38 ANOVA: Oe-Way Model Alterative form: (better if you will perform multiple comparisos) > fit1.2 = lm(chol ~ -1 + factor(rs174548)) > summary(fit1.2) Call: lm(formula = chol ~ -1 + factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) factor(rs174548) <2e-16 *** factor(rs174548) <2e-16 *** factor(rs174548) <2e-16 *** --- Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 9383 o 3 ad 397 DF, p-value: < 2.2e-16 > aova(fit1.2) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) < 2.2e-16 *** Residuals Sigif. codes: 0 *** ** 0.01 *

39 ANOVA: Oe-Way Model How about this oe? How is rs beig treated ow? Compare model fit results from (fit1.1 & fit2). > fit2 = lm(chol ~ rs174548) > summary(fit2) Call: lm(formula = chol ~ rs174548) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** rs ** --- Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 398 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 1 ad 398 DF, p-value: > aova(fit2) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) rs ** Residuals Sigif. codes: 0 *** ** 0.01 *

40 ANOVA: Oe-Way Model > fit2 = lm(chol ~ rs174548) > summary(fit2) Model: E[Y x] = b 0 + b 1 x where Y: cholesterol, x: rs Call: lm(formula = chol ~ rs174548) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** rs ** Residual stadard error: o 398 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 1 ad 398 DF, p-value: > aova(fit2) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) rs ** Residuals Iterpretatio of model parameters? b 0 : mea cholesterol i the C/C group [estimate: mg/dl] b 1 : mea cholesterol differece betwee C/G ad C/C or betwee G/G ad C/G groups [estimate: mg/dl] This model presumes differeces betwee cosecutive groups are the same (i this example, liear dose effect of allele) more restrictive tha the ANOVA model! Back to the ANOVA model 180

41 ANOVA: Oe-Way Model > fit1.1 = lm(chol ~ factor(rs174548)) > summary(fit1.1) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 factor(rs174548) factor(rs174548) We rejected the ull hypothesis that the mea cholesterol levels are the same across groups defied by rs (p= ). Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit1.1) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals What are the groups with differeces i meas? MULTIPLE COMPARISONS (comig up) 181

42 Oe-Way ANOVA allowig for uequal variaces We ca also perform oe-way ANOVA allowig for uequal variaces: > oeway.test(chol ~ factor(rs174548)) Oe-way aalysis of meas (ot assumig equal variaces) data: chol ad factor(rs174548) F = , um df = 2.000, deom df = , p-value = We reject the ull hypothesis that the mea cholesterol levels are the same across groups defied by rs (p= ). What are the groups with differeces i meas? MULTIPLE COMPARISONS (comig up) 182

43 Oe-Way ANOVA with robust stadard errors > summary(gee(chol ~ factor(rs174548), id=seq(1,legth(chol)))) Begiig Cgee geeformula.q /01/27 ruig glm to get iitial regressio estimate (Itercept) factor(rs174548)1 factor(rs174548) GEE: GENERALIZED LINEAR MODELS FOR DEPENDENT DATA gee S-fuctio, versio 4.13 modified 98/01/27 (1998) Model: Lik: Idetity Variace to Mea Relatio: Gaussia Correlatio Structure: Idepedet Call: gee(formula = chol ~ factor(rs174548), id = seq(1, legth(chol))) Summary of Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Naive S.E. Naive z Robust S.E. Robust z (Itercept) factor(rs174548) factor(rs174548) Estimated Scale Parameter: Number of Iteratios: 1 183

44 Kruskal-Wallis Test No-parametric aalogue to the oe-way ANOVA Based o raks I our example: > kruskal.test(chol ~ factor(rs174548)) Kruskal-Wallis rak sum test data: chol by factor(rs174548) Kruskal-Wallis chi-squared = , df = 2, p-value = Coclusio: Evidece that the cholesterol distributio is ot the same across all groups. With the global ull rejected, you ca also perform pairwise comparisos [Wilcoxo rak sum], but adjust for multiplicities! 184

45 REGRESSION METHODS MULTIPLE COMPARISONS 185

46 ANOVA: Oe-Way Model What are the groups with differeces i meas? MULTIPLE COMPARISONS: µ 0 = µ 1? µ 0 = µ 2? Pairwise comparisos µ 1 = µ 2? (µ 1 + µ 2 )/2 = µ 0? No-pairwise compariso 186

47 Multiple Comparisos: Family-wise error rates Illustratig the multiple compariso problem Truth: ull hypotheses Tests: pairwise comparisos - each at the 5% level. What is the probability of rejectig at least oe? #groups = K #pairwise comparisos C = K(K-1)/2 P(at least oe sig) =1-(1-0.05) C That is, if you have three groups ad make pairwise comparisos, each at the 5% level, your familywise error rate (probability of makig at least oe false rejectio) is over 14%! Need to address this issue! Several methods!!! 187

48 Multiple Comparisos Several methods: Noe (o adjustmet) Boferroi Holm Hochberg Hommel BH BY FDR Available i R 188

49 Multiple Comparisos Boferroi adjustmet: for C tests performed, use level α/c (or multiply p-values by C). Simple Coservative Must decide o umber of tests beforehad Widely applicable Ca be doe without software! 189

50 Multiple Comparisos FDR (False Discovery Rate) Less coservative procedure for multiple comparisos Amog rejected hypotheses, FDR cotrols the expected proportio of icorrectly rejected ull hypotheses (that is, type I errors). 190

51 Multiple Comparisos This optio cosiders all pairwise comparisos > ## call library for multiple comparisos > library(multcomp) > > ## fit model > fit1 = lm(chol ~ -1 + factor(rs174548)) > > ## all pairwise comparisos > ## -- first, defie matrix of cotrasts > M = cotrmat(table(rs174548), type="tukey") > M Multiple Comparisos of Meas: Tukey Cotrasts > > ## -- secod, obtai estimates for multiple comparisos > mc = glht(fit1, lifct =M) Stads for geeral liear hypothesis testig 191

52 Multiple Comparisos > ## -- third, adjust the p-values (or ot) for multiple comparisos > summary(mc, test=adjusted("oe")) Simultaeous Tests for Geeral Liear Hypotheses Multiple Comparisos of Meas: Tukey Cotrasts Fit: lm(formula = chol ~ -1 + factor(rs174548)) Liear Hypotheses: Estimate Std. Error t value Pr(> t ) 1-0 == ** 2-0 == == Sigif. codes: 0 *** ** 0.01 * (Adjusted p values reported -- oe method) 192

53 Multiple Comparisos > summary(mc, test=adjusted("boferroi")) Simultaeous Tests for Geeral Liear Hypotheses Multiple Comparisos of Meas: Tukey Cotrasts Fit: lm(formula = chol ~ -1 + factor(rs174548)) Liear Hypotheses: Estimate Std. Error t value Pr(> t ) 1-0 == * 2-0 == == Sigif. codes: 0 *** ** 0.01 * (Adjusted p values reported -- boferroi method) 193

54 Multiple Comparisos > summary(mc, test=adjusted("fdr")) Simultaeous Tests for Geeral Liear Hypotheses Multiple Comparisos of Meas: Tukey Cotrasts Fit: lm(formula = chol ~ -1 + factor(rs174548)) Liear Hypotheses: Estimate Std. Error t value Pr(> t ) 1-0 == * 2-0 == == Sigif. codes: 0 *** ** 0.01 * (Adjusted p values reported -- fdr method) 194

55 Multiple Comparisos What about usig other adjustmet methods? For example, we used: > summary(mc, test=adjusted("boferroi")) (all pairwise comparisos, with Boferroi adjustmet) > summary(mc, test=adjusted("fdr")) (all pairwise comparisos, with FDR adjustmet) Other optios are: summary(mc, test=adjusted("holm")) summary(mc, test=adjusted("hochberg")) summary(mc, test=adjusted("hommel")) summary(mc, test=adjusted("bh")) summary(mc, test=adjusted("by")) Results, i this particular example, are basically the same, but they do t eed to be! Differet criteria could lead to differet results! 195

56 Summary: Relatioships: GOAL: Compariso of Meas across K groups Oe-way ANOVA: H 0 :µ 0 = µ 1 = = µ K-1 H 1 : ot all meas are equal Rejected H 0? Multiple Regressio: Model: E[Y groups]= b 0 + b 1 group 2 + +b k-1 group k where group 1 is the referece group H 0 :b 1 = b 2 = = b k-1 =0 H 1 : ot all b i are equal to zero YES Multiple Comparisos (cotrol a overall) e.g. Boferroi: a/#comparisos 196

57 REGRESSION METHODS Two-way ANOVA models 197

58 ANOVA: Two-Way Model Motivatio: Scietific questio: Assess the effect of rs ad sex o cholesterol levels. 198

59 ANOVA: Two-Way Model Factors: A ad B Goals: Test for mai effect of A Test for mai effect of B Test for iteractio effect of A ad B 199

60 ANOVA: Two-Way Model To simplify discussio, assume that factor A has three levels, while factor B has two levels Factor A A 1 A 2 A 3 Factor B B 1 µ 11 µ 21 µ 31 B 2 µ 12 µ 22 µ

61 ANOVA: Two-Way Model Meas B 2 Parallel lies = No iteractio B 1 A 1 A 2 A 3 B 2 Lies are ot parallel = Iteractio B 1 A 1 A 2 A 3 201

62 ANOVA: Two-Way Model Recall: Categorical variables ca be represeted with dummy variables Iteractios are represeted with cross-products 202

63 ANOVA: Two-Way Model Model 1: E[Y A 2, A 3, B 2 ] = b 0 + b 1 A 2 + b 2 A 3 + b 3 B 2. What are the meas i each combiatio-group? A 1 A 2 A 3 B 1 µ 11 =b 0 µ 21 =b 0 + b 1 µ 31 =b 0 + b 2 B 2 µ 12 =b 0 + b 3 µ 22 =b 0 + b 1 + b 3 µ 32 = b 0 + b 2 + b 3 203

64 ANOVA: Two-Way Model Model 1: E[Y A 2, A 3, B 2 ] = b 0 + b 1 A 2 + b 2 A 3 + b 3 B 2. A 1 A 2 A 3 B 1 µ 11 =b 0 µ 21 =b 0 + b 1 µ 31 =b 0 + b 2 B 2 µ 12 =b 0 + b 3 µ 22 =b 0 + b 1 + b 3 µ 32 = b 0 + b 2 + b 3 Model with o iteractio: Differece i meas betwee groups defied by factor B does ot deped o the level of factor A. Differece i meas betwee groups defied by factor A does ot deped o the level of factor B. 204

65 ANOVA: Two-Way Model Model 2: E[Y A 2, A 3, B 2 ] = b 0 + b 1 A 2 + b 2 A 3 + b 3 B 2 + b 4 A 2 B 2 + b 5 A 3 B 2 What are the meas i each combiatio-group? A 1 A 2 A 3 B 1 µ 11 =b 0 µ 21 =b 0 + b 1 µ 31 =b 0 + b 2 B 2 µ 12 =b 0 + b 3 µ 22 =b 0 + b 1 + b 3 + b 4 µ 32 = b 0 + b 2 + b 3 + b 5 205

66 ANOVA: Two-Way Model Three (possible) tests Iteractio of A ad B (may wat to start here) Rejectio would imply that differeces betwee meas of A depeds o the level of B (ad vice-versa) so stop Mai effect of A Test oly if o iteractio Mai effect of B Test oly if o iteractio [ Note: If you have oe observatio per cell, you caot test iteractio! ] 206

67 ANOVA: Two-Way Model Model without iteractio E[Y A 2, A 3, B 2 ] = b 0 + b 1 A 2 + b 2 A 3 + b 3 B 2. How do we test for mai effect of factor A? H 0 : b 1 = b 2 =0 vs. H 1 : b 1 or b 2 ot zero How do we test for mai effect of factor B? H 0 : b 3 =0 vs. H 1 : b 3 ot zero 207

68 ANOVA: Two-Way Model Model with iteractio: E[Y A 2, A 3, B 2 ] = b 0 + b 1 A 2 + b 2 A 3 + b 3 B 2 + b 4 A 2 B 2 + b 5 A 3 B 2 How do we test for iteractios? H 0 : b 4 = b 5 =0 vs. H 1 : b 4 or b 5 ot zero IMPORTANT: If you reject the ull, do ot test mai effects!!! 208

69 ANOVA: Two-Way Model (without iteractio) > fit1 = lm(chol ~ factor(sex) + factor(rs174548)) > summary(fit1) Call: lm(formula = chol ~ factor(sex) + factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(sex) e-07 *** factor(rs174548) ** factor(rs174548) Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 396 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 12.2 o 3 ad 396 DF, p-value: 1.196e-07 > aova(fit0,fit1) Aalysis of Variace Table Model 1: chol ~ factor(sex) Model 2: chol ~ factor(sex) + factor(rs174548) Res.Df RSS Df Sum of Sq F Pr(>F) ** 209

70 ANOVA: Two-Way Model (without iteractio) > fit1 = lm(chol ~ factor(sex) + factor(rs174548)) > summary(fit1) Call: lm(formula = chol ~ factor(sex) + factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(sex) e-07 *** factor(rs174548) ** factor(rs174548) Residual stadard error: o 396 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 12.2 o 3 ad 396 DF, p-value: 1.196e-07 > aova(fit0,fit1) Aalysis of Variace Table Iterpretatio of results: Estimated mea cholesterol for male C/C group: mg/dl Estimated differece i mea cholesterol levels betwee females ad males adjusted by geotype: mg/dl Estimated differece i mea cholesterol levels betwee C/G ad C/C groups adjusted by sex: mg/dl Estimated differece i mea cholesterol levels betwee G/G ad C/C groups adjusted by sex: mg/dl Model 1: chol ~ factor(sex) Model 2: chol ~ factor(sex) + factor(rs174548) Res.Df RSS Df Sum of Sq F Pr(>F) ** There is evidece that cholesterol is associated with sex (p< 0.001). There is evidece that cholesterol is associated with geotype (p=0.005) 210

71 ANOVA: Two-Way Model (without iteractio) I words: Adjustig for sex, the differece i mea cholesterol comparig C/G to C/C is ad comparig G/G to C/C is This differece does ot deped o sex (this is because the model does ot have a iteractio betwee sex ad geotype!) 211

72 ANOVA: Two-Way Model (with iteractio) > fit2 = lm(chol ~ factor(sex) * factor(rs174548)) > summary(fit2) Call: lm(formula = chol ~ factor(sex) * factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(sex) * factor(rs174548) factor(rs174548) factor(sex)1:factor(rs174548) ** factor(sex)1:factor(rs174548) Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 394 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 9.14 o 5 ad 394 DF, p-value: 3.062e

73 ANOVA: Model compariso > aova(fit1,fit2) Aalysis of Variace Table Model 1: chol ~ factor(sex) + factor(rs174548) Model 2: chol ~ factor(sex) * factor(rs174548) Res.Df RSS Df Sum of Sq F Pr(>F) * --- Sigif. codes: 0 *** ** 0.01 *

74 ANOVA: Two-Way Model (with iteractio) > fit2 = lm(chol ~ factor(sex) * factor(rs174548)) > summary(fit2) Call: lm(formula = chol ~ factor(sex) * factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(sex) * factor(rs174548) factor(rs174548) factor(sex)1:factor(rs174548) ** factor(sex)1:factor(rs174548) Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 394 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: 9.14 o 5 ad 394 DF, p-value: 3.062e-08 > aova(fit1,fit2) Aalysis of Variace Table Iterpretatio of results: Estimated mea cholesterol for male C/C group: mg/dl Estimated mea cholesterol for female C/C group? ( ) mg/dl Estimated mea cholesterol for male C/G group: ( ) mg/dl Estimated mea cholesterol for female C/G group: ( ) mg/dl Model 1: chol ~ factor(sex) + factor(rs174548) Model 2: chol ~ factor(sex) * factor(rs174548) Res.Df RSS Df Sum of Sq F Pr(>F) * --- Sigif. codes: 0 *** ** 0.01 * There is evidece for a iteractio betwee sex ad geotype (p= 0.015) 214

75 SUMMARY: Two-Way ANOVA Sigificat Iteractio? NO Iterpret mai effects of factor A ad factor B YES Iterpret the effect of factor A o mea respose for each level of factor B (or effect of factor B o mea respose for each level of factor A) 215

76 REGRESSION METHODS ANCOVA (aka ANACOVA) 216

77 ANalysis of COVAriace Models (ANCOVA) Motivatio: Scietific questio: Assess the effect of rs o cholesterol levels adjustig for age 217

78 ANalysis of COVAriace Models (ANCOVA) ANOVA with oe or more cotiuous variables Equivalet to regressio with dummy variables ad cotiuous variables Primary compariso of iterest is across k groups defied by a categorical variable, but the k groups may differ o some other potetial predictor or cofouder variables [also called covariates]. 218

79 ANalysis of COVAriace Models (ANCOVA) To facilitate discussio assume Y: cotiuous respose (e.g. cholesterol) X: cotiuous variable (e.g. age) Z: dummy variable (e.g. idicator of C/G or G/G versus C/C) Model: Y = b + b X + b Z + b XZ + e Note that: Z = 0 Þ E[ Y X, Z Z = 1Þ E[ Y X, Z = 0] = b = 1] = ( b b X 1 + b ) + 2 ( b + b ) X 1 Iteractio term 3 This model allows for differet itercepts/slopes for each group. 219

80 ANCOVA Testig coicidet lies: H0 : b2 = 0, b3 = Compares overall model with reduced model Y 0 1 = b + b X + e 0 Testig parallelism: H 0 : b3 = Compares overall model with reduced model Y = b + b X + b Z + e 220

81 ANCOVA > fit0 = lm(chol ~ factor(rs174548)) > summary(fit0) Call: lm(formula = chol ~ factor(rs174548)) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) ** factor(rs174548) Sigif. codes: 0 *** ** 0.01 * Residual stadard error: o 397 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 2 ad 397 DF, p-value: > aova(fit0) Aalysis of Variace Table Respose: chol Df Sum Sq Mea Sq F value Pr(>F) factor(rs174548) * Residuals Sigif. codes: 0 *** ** 0.01 *

82 ANCOVA > fit1 = lm(chol ~ factor(rs174548) + age) > summary(fit1) Call: lm(formula = chol ~ factor(rs174548) + age) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) ** factor(rs174548) age e-05 *** Residual stadard error: o 396 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 3 ad 396 DF, p-value: 5.778e-06 > aova(fit0,fit1) Aalysis of Variace Table Model 1: chol ~ factor(rs174548) Model 2: chol ~ factor(rs174548) + age Res.Df RSS Df Sum of Sq F Pr(>F) e-05 *** --- Sigif. codes: 0 *** ** 0.01 *

83 ANCOVA Total cholesterol (mg/dl) C/C C/G G/G Age (years) 223

84 ANCOVA > fit2 = lm(chol ~ factor(rs174548) * age) > summary(fit2) Call: lm(formula = chol ~ factor(rs174548) * age) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** factor(rs174548) factor(rs174548) age ** factor(rs174548)1:age factor(rs174548)2:age Residual stadard error: o 394 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 5 ad 394 DF, p-value: 4.065e

85 ANCOVA > fit0 = lm(chol ~ age) > summary(fit0) Call: lm(formula = chol ~ age) Residuals: Mi 1Q Media 3Q Max Coefficiets: Estimate Std. Error t value Pr(> t ) (Itercept) < 2e-16 *** age e-05 *** --- Sigif. codes: 0 *** ** 0.01 * Test of coicidet lies Residual stadard error: o 398 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: o 1 ad 398 DF, p-value: 4.522e-05 > aova(fit0,fit2) Aalysis of Variace Table Model 1: chol ~ age Model 2: chol ~ factor(rs174548) * age Res.Df RSS Df Sum of Sq F Pr(>F) * --- Sigif. codes: 0 *** ** 0.01 *

86 ANCOVA Test of parallel lies > aova(fit1,fit2) Aalysis of Variace Table Model 1: chol ~ factor(rs174548) + age Model 2: chol ~ factor(rs174548) * age Res.Df RSS Df Sum of Sq F Pr(>F)

87 ANCOVA Total cholesterol (mg/dl) C/C C/G G/G Age (years) 227

88 228 ANCOVA I summary: If the slopes are ot equal, the age is a effect modifier If the slopes are the same, ) ( ) ( ) ( ) ( ], [ GG x CG x GG CG x z x E Y * + * = b b b b b b ) ( ) ( ], [ GG CG x z x Y E b b b b =

89 229 ANCOVA If the slopes are the same, the oe ca obtai adjusted meas for the three geotypes usig the mea age over all groups For example, the adjusted meas for the three groups would be ˆ ) ˆ ˆ ( Y (adj) ˆ ) ˆ ˆ ( Y (adj) ˆ ˆ Y (adj) b b b b b b b b x x x + + = + + = + = ) ( ) ( ], [ GG CG x z x Y E b b b b =

90 ANCOVA > ## mea cholesterol for differet geotypes adjusted by age > predict(fit1, ew=data.frame(age=mea(age),rs174548=0)) > predict(fit1, ew=data.frame(age=mea(age),rs174548=1)) > predict(fit1, ew=data.frame(age=mea(age),rs174548=2))

91 SUMMARY: ANCOVA Sigificat Iteractio? (slopes are differet?) NO Cotrol for potetial cofouder? YES YES Iterpret the differece i meas of the respose for give values of the cotiuous variable Compute adjusted meas at the commo X mea 231

92 Summary We have cosidered: ANOVA ad ANCOVA Iterpretatio Estimatio Iteractio Multiple comparisos 232