ANOVA. The Observations y ij

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1 ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2 m k 1

2 The ANOVA table The analyss s usually lad out n a table For a one-way layout (where the response s assumed to vary accordng to groupng on one factor): Source df SS MS F p-val Treatment k-1 Σ(m -m) 2 SST/(k-1) MST/MSE * Error n-k Σ(y j -m ) 2 SSE/(n-k) Total n-1 Σ(y j -m) 2 m = overall mean, m = mean wthn group Sum of Squares The two-sample t-test tests for equalty of the means of two groups. We could express the observatons as: X = μ + E =1, 2 j j Where the E j are assumed to be N(0,σ 2 ) H : μ 1 = μ 2 2

3 Sum of Squares Ths can also be wrtten as: X = μ + α + E =1, 2 j μ could be seen as overall mean α as devaton from μ n group Model s overparameterzed Uses more parameters than necessary Necesstates some constrant, e.g. Σ α =0 j Sum of Squares Goal: test dfference between means of two (or more) groups Between SS measures the dfference The dfference must be measured relatve to the varance wthn the groups Wthn SS F-test: consders the rato of B/W The larger F s, the more sgnfcant the dfference 3

4 The ANOVA Procedure Subdvde observed total sum of squares nto several components Pck approprate sgnfcance pont for a chosen Type I error α from an F table Compare the observed components to test the NULL hypothess Comments Generalzes to any number of groups ANOVAs can be classfed n varous ways, e.g. fxed effects models mxed effects models random effects model For now we consder fxed effect models Parameter α s fxed, but unknown, n group X = μ + α + E j j 4

5 One-Way ANOVA One-Way fxed-effect ANOVA Setup and dervaton Lke two-sample t-test for g number of groups Observatons (n observatons, =1,2,,g) X, X, 1 2 K, X n Usng overparameterzed model for X X = μ + α + E j j j = 1,2, K,n = 1,2, K, g E j assumed N(0,σ 2 ), Σn α = 0, α fxed n group One-Way ANOVA Null Hypothess H 0 s: α 1 = α 2 = = α g = 0 Total sum of squares s Ths s subdvded nto B and W wth B = g = 1 g n = 1 j= 1 ( X j X ) 2 n ( X X ) W = ( X 2 g n = 1 j= 1 j X ) 2 X n = j= 1 X n j X = g n = 1 j= 1 X N j N = g n = 1 5

6 One-Way ANOVA Total degrees of freedom: N 1 Subdvded nto df B = g 1 and df W = N - g Ths gves the test statstc F F = B W N g * g 1 Assumptons Have random samples from each separate populaton The varance s the same n each treatment group The samples are suffcently large that the CLT holds for each sample mean (or the ndvdual populaton dstrbutons are normal) 6

7 What does t mean when we reject H? The null hypothess H s a jont one: that all populaton means are equal When we reject the null, that does NOT mean that the means are all dfferent! It means that at least one s dfferent To fnd out whch s dfferent, can do post hoc testng (parwse t-tests, for example) Addtonal aspects Why not start off dong separate (z or t) tests for each par of samples?... Testng the assumptons Whch mean(s) s/are not equal can do post hoc testng (parwse t-tests, for example) Multple comparsons (multple testng) Data snoopng 7

8 Factoral crossng Compare 2 (or more) sets of condtons n the same experment Desgns wth factoral treatment structure allow you to measure nteracton between two (or more) sets of condtons that nfluence the response you wll look at ths n more detal durng the exercses today Factoral desgns may be ether observatonal or expermental 3 types of 2-factor factoral desgns 2 expermental factors you randomze treatments to each unt 2 observatonal factors you cross-classfy your populatons nto groups and get a sample from each populaton 1 expermental and 1 observatonal factor you get a sample of unts from each populaton, then use randomzaton to assgn levels of the expermental factor (treatments), separately wthn each sample 8

9 Interacton Interacton s very common (and very mportant) n scence Interacton s a dfference of dfferences Interacton s present f the effect of one factor s dfferent for dfferent levels of the other factor Man effects can be dffcult to nterpret n the presence of nteracton, because the effect of one factor depends on the level of the other factor Interacton plot no nteracton nteracton 9

10 Two-Way ANOVA Two-Way Fxed Effects ANOVA More complcated setup; example: Expresson levels of one gene n lung cancer patents a dfferent rsk classes E.g.: ultrahgh, very hgh, ntermedate, low b dfferent age groups n ndvduals for each rsk/age combnaton Two-Way ANOVA Observatons: X jk s the rsk class ( = 1, 2,, a) j ndcates the age group k corresponds to the ndvdual n each group (k = 1,, n) Each group s a possble rsk/age combnaton The number of ndvduals n each group s the same, n Ths s a balanced desgn (equal numbers n each group Theory for unbalanced desgns s a lttle more complcated 10

11 Two-Way ANOVA The model for each X jk s X = μ + α + β + δ + E jk j = 1,2, K,a j = 1,2, K, b k =1,2, K, n j jk E jk are N(0, σ 2 ) The mean of X jk s μ + α + β + δ j α addtve for rsk class β addtve for age group δ j rsk/age nteracton parameter Should be added f a possble group/group nteracton exsts Two-Way ANOVA Constrants: Σ α = Σ β = 0 Σ δ j = 0 for all j Σ j δ j = 0 for all The total sum of squares s then subdvded nto four groups: Rsk class SS Age group SS Interacton SS Wthn cells ( resdual or error ) SS 11

12 Two-Way ANOVA Assocated wth each sum of squares Correspondng degrees of freedom (df) Correspondng mean square (MS) Sum of squares dvded by degrees of freedom The mean squares are compared usng F ratos to test varous effects Frst test for a sgnfcant rsk/age nteracton If there s an nteracton, t may not be reasonable to test for sgnfcant rsk or age dfferences Mult-Way ANOVA One-way and two-way fxed effects ANOVA can be extended to mult-way ANOVA Example: four-way ANOVA (saturated) model: X jkl = μ + α + β + γ j k + ε + l ( αβ ) j + ( αγ ) k + ( αε ) l + ( βγ ) jk + ( βε ) jl + ( γε ) kl + ( αβγ ) jk + ( αβε ) jl + ( βγε ) jkl + ( αβγε ) jkl One observaton per cell In general, nterested n unsaturated models 12

13 Model formulas n R A smple model formula n R looks somethng lke: yvar ~ xvar1 + xvar2 + xvar3 We could wrte ths model (algebracally) as Y = a + b 1 *x 1 + b 2 *x 2 + b 3 *x 3 By default, an ntercept s ncluded n the model - you don t have to nclude a term n the model formula If you want to leave the ntercept out: yvar ~ -1 + xvar1 + xvar2 + xvar3 More on model formulas We can also nclude nteracton terms n a model formula: yvar ~ xvar1 + xvar2 + xvar3 Examples yvar ~ xvar1 + xvar2 + xvar3 + xvar1:xvar2 yvar ~ (xvar1 + xvar2 + xvar3)^2 yvar ~ (xvar1 * xvar2 * xvar3) 13

14 More on model formulas The generc form s response ~ predctors The predctors can be numerc or factor Other symbols to create formulas wth combnatons of varables (e.g. nteractons) + to add more varables - to leave out varables : to ntroduce nteractons between two terms * to nclude both nteractons and the terms (a*b s the same as a+b+a:b) ^n adds all terms ncludng nteractons up to order n I() treats what s n () as a mathematcal expresson Interpretng R output > chcks.aov <- aov(weght ~ House + Proten*LP*LS) > summary(chcks.aov) Df Sum Sq Mean Sq F value Pr(>F) House ** Proten * LP * LS *** Proten:LP ** Proten:LS LP:LS Proten:LP:LS Resduals Sgnf. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 14

15 Numercal and graphcal analyss Tables of group means: Numercal and graphcal analyss Desgn plot Boxplots of outcome for each factor Interacton plots Wrte out model, assumptons, defne all parameters anova table Plots for assumpton checkng/model assessment 15

16 Model assessment: Normalty Boxplots of observatons (or resduals) should be symmetrc Plot of sample means vs sample varances should not show a pattern QQ normal plots of observatons (or resduals) should be a straght lne Check for outlers QQ-Plot Quantle-quantle plot Used to assess whether a sample follows a partcular (e.g. normal) dstrbuton (or to compare two samples) A method for lookng for outlers when data are mostly normal Sample Sample quantle s Theoretcal Value from Normal dstrbuton whch yelds a quantle of (= -1.15) 16

17 Typcal devatons from straght lne patterns Outlers Curvature at both ends (long or short tals) Convex/concave curvature (asymmetry) Horzontal segments, plateaus, gaps Outlers 17

18 Long Tals Short Tals 18

19 Plateaus/Gaps Model assessment: Varance homogenety Boxplots of observatons should have smlar spread Spread of resduals should be smlar when plotted aganst group means There are also formal tests (e.g., Bartlett, Levene) but these are not so useful for dagnoss 19

20 Dagnostc plots Model assessment: Independence Plot resduals aganst group means, mght ndcate e.g. autocorrelaton Usually need to deal wth the ndependence ssue at the desgn stage, through randomzaton or other means 20

21 chcks.dat Demo 21

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