7.1. Single classification analysis of variance (ANOVA) Why not use multiple 2-sample 2. When to use ANOVA

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1 Sngle classfcaton analyss of varance (ANOVA) When to use ANOVA ANOVA models and parttonng sums of squares ANOVA: hypothess testng ANOVA: assumptons A non-parametrc alternatve: Kruskal-Walls ANOVA Power analyss n sngle classfcaton ANOVA When to use ANOVA Tests for effect of dscrete ndependent varables. Each ndependent varable s called a factor, and each factor may have two or more levels or treatments (e.g. crop yelds wth ntrogen (N) or ntrogen and phosphorous (N + P) added). ANOVA tests whether all group means are the same. Use when number of levels (groups) s greater than two. Control Expermental (N+P) Why not use multple -sample tests? For k comparsons, the probablty of acceptng a true H 0 for all k s ( - α) k. For 4 means, ( - α) k = (0.95) 6 =.735. So α (for all comparsons) = So, when comparng the means of four samples from the same populaton, we would expect to detect sgnfcant dfferences among at least one par 7% of the tme. µ C : µ N+P µ c :µ N µ N :µ N+P Control Expermental (N+P) 3 7.

2 What ANOVA does/doesn t t do Tells us whether all group means are equal (at a specfed α level)......but f we reject H 0, the ANOVA does not tell us whch pars of means are dfferent from one another. Control Expermental (N+ P) µ N+P µ C µ N 4 Model I ANOVA: effects of temperature on trout growth 3 treatments determned (set) by nvestgator. Dependent varable s growth rate (λ), factor (X) s temperature. Snce X s controlled, we can estmate the effect of a unt ncrease n X (temperature) on λ (the effect sze)... and can predct λ at other temperatures. Growth rate λ (cm/day) Water temperature ( C) 5 Model II ANOVA: geographcal varaton n body sze of black bears 3 locatons (groups) sampled from set of possble locatons. Dependent varable s body sze, factor (X) s locaton. Even f locatons dffer, we have no dea what factors are controllng ths varablty... so we cannot predct body sze at other locatons. Body sze (kg) Rdng Kluane Mountan Algonqun 6 7.

3 Model dfferences In Model I, the putatve causal factor(s) can be manpulated by the expermenter, whereas n Model II they cannot. In Model I, we can estmate the magntude of treatment effects and make predctons, whereas n Model II we can do nether. In one-way (but NOT mult-way!) ANOVA, calculatons are dentcal for both models. 7 How s t done? And why call t ANOVA? In ANOVA, the total varance n the dependent varable s parttoned nto two components: among-groups: varance of means of dfferent groups (treatments) wthn-groups (error): varance of ndvdual observatons wthn groups around the mean of the group 8 Statstcal analyss as model buldng All statstcal analyses begn wth a mathematcal model that supposedly descrbes the data, e.g., regresson, ANOVA. Model fttng s then the process by whch model parameters are estmated. Y = α + β + ε Y X Lnear regresson ANOVA Y µ Group Group Group 3 X ε 4 µ α Y j = µ + α + ε j 9 7.3

4 Least squares estmaton (LSE) An ordnary least squares (OLS) estmate of a model parameter Θ s that whch mnmzes the sum of squared dfferences between OLS Θ observed and predcted values: Predcted values are derved from some model whose N parameters we wsh to SSR = ( y yˆ) estmate = SS R yˆ = f ( x, Θ) 0 Example: LSE of model parameters n smple lnear regresson Data conssts of a set of n pared observatons (x, y ),, (x n y n ) The model for the I th observaton s: Y = α + βx+ ε What s the LSE of the model parameters α and β? Y Y $Y ε X = Y Y$ Resdual: ε The general ANOVA model The general model s: Y j = µ + α + ε j ANOVA algorthms ft the above model (by ordnary least squares) to estmate the α s. Y µ H 0 : all α s = 0 Group Group Group 3 Group ε 4 µ α Y µ µ =µ = µ = µ 3 α =α =α 3 = 0 7.4

5 Parttonng the total sums of squares µ Y µ µ 3 µ Total SS Model (Groups) SS Error SS Group Group Group 3 3 The ANOVA table Source of Varaton Total Error Sum of Squares k n ( Y ) j Y = j= k ( ) = k n ( Y ) j Y Degrees of freedom (df) n - n - k Mean Square SS/df Groups n Y Y k - SS/df = j= SS/df F MS groups MS error 4 Varance components and group means MS groups measures average squared dfference among group means. MS error s a measure of precson. t = X s C X MS T groups F = X C XT MS error Control Expermental (N+ P) µ N+P µ C µ N 5 7.5

6 ANOVA: the null hypothess H 0 : all group means are the same, or... H 0 : all group effects (α ) are zero, or... H 0 : F = MS groups / MS error = For k groups and N observatons, compare wth F dstrbuton at desred α level wth k - and N - k degrees of freedom. Control Expermental (N+ P) µ N+P µ C µ N 6 Lab example: temporal varaton n sze of sturgeon (Model II ANOVA) Predcton: dam constructon resulted n loss of large sturgeon Test: compare sturgeon sze before and after dam constructon H 0 : mean sze s the same for all years (?) Fork Length/Talle Dam constructon Year/Année 7 Temporal varaton n sze of sturgeon (ANOVA results) *** Analyss of Varance Model *** Short Output: Call: aov(formula = FKLNGTH ~ YEAR, data = Dam0dat, na.acton = na.exclude) Terms: YEAR Resduals Sum of Squares Deg. of Freedom 3 4 Resdual standard error: Estmated effects may be unbalanced Type III Sum of Squares Df Sum of Sq Mean Sq F Value Pr(F) YEAR Resduals Concluson: reject H

7 ANOVA assumptons Resduals are ndependent of one another. Resduals are normally dstrbuted. Varance of resduals wthn groups s the same for all groups (homoscedastcty). Note: all assumptons apply to the resduals, not the raw data. Snce all assumptons apply to the resduals, not the raw data all tests of assumptons are done after the analyss s completed (and resduals have been generated). 9 The general ANOVA model The general model s: Y j = µ + α + ε j so the predcted value of all observatons n the th group s µ α Y µ Group Group Group 3 ε 4 µ =µ = µ = µ 3 Y = µ + α ˆ Y µ The dfference between the predcted value for an observaton and the observed value s ts resdual. α =α =α 3 = 0 Group 0 Why does observatons need to be ndependent? If observatons are not ndependent, then the true degrees of freedom s less (sometmes much less) than the calculated degrees of freedom the dstrbuton used to calculate p wll be wrong and p wll be smaller than t ought to be. Probablty true df t calcuated df Calculated t 7.7

8 Checkng ndependence of observatons (resduals) Does the expermental desgn suggest that samplng unts may not be ndependent (e.g. spatotemporal correlaton?) Do autocorrelaton plots to check for seral autocorrelaton. Testng normalty of resduals Generate normal probablty plot of resduals and check for lnearty. If warranted, run Lllefors test, keepng n mnd the power ssue! Resduals Outlers? Quantles of Standard Normal 3 Testng homoscedastcty I: plottng resduals aganst estmates Does spread of resduals appear the same for each group? Outler? 59 Resduals Ftted : YEAR 4 7.8

9 Testng homoscedastcty II: Levene s test Calculate mean absolute resdual for each group. Does ths value vary among groups? Abs (Res) Year/Année 5 Testng homoscedastcty II: Levene s test (cont d) *** Analyss of Varance Model *** Short Output: Call: aov(formula = absres ~ YEAR, data = Dam0dat, na.acton = na.exclude) Terms: YEAR Resduals Sum of Squares Deg. of Freedom 3 4 Resdual standard error: Estmated effects may be unbalanced Df Sum of Sq Mean Sq F Value Pr(F) YEAR Resduals Effects of volatons of assumptons Calculaton of p assumes p(f) = p(f*) but as resduals conform less to requred assumptons, the devaton between the two ncreases. Therefore, calculated p values are ncorrect. Probablty F F, low conformty F, hgh conformty True F (F*) 7 7.9

10 Robustness of ANOVA wth respect to volaton of assumptons Assumpton Robustness Remark Normalty Hgher Only f sample szes are reasonably large (>0) Independence Low But depends on strength of correlaton Homoscedastcty Lower Especally wth smaller sample szes 8 Resdual analyss: questons Whch assumptons are not met, and how robust s ANOVA to ther volaton? What s the sample sze? Is the volaton of assumptons due to a couple of outlers? How close s p to α? Elmnate outlers and rerun analyss. Transform data. Try a non-parametrc alternatve (generally recommended f sample szes are small,.e. < 0 per group) such as Kruskal-Walls ANOVA. 9 Calculate rank sum (R g ) for each group. H 0 : R C = R = R Calculate K-W H statstc: k R H = 3( N + ) N ( N + ) = n whch s dstrbuted as χ wth k- df f N for each group s not too small, otherwse use crtcal values for H. A non-parametrc alternatve: Kruskal-Walls ANOVA Control Treatment Treatment Plot Rank Rank Rank Rank Sum

11 Power and sample sze n sngle- classfcaton ANOVA If H 0 s true, then varance rato MS groups /MS error follows central F dstrbuton. But, f H 0 s false, then MS groups /MS error follows non-central F, defned by ν, ν and noncentralty parameter φ. So, power calculatons depend on non-central F. Control Expermental (N+P) 3 Power and sample sze n sngle- classfcaton ANOVA Power of a test nvolvng k groups wth n replcates per group at specfed α when () group means are known; () mnmal detectable dstance s specfed. estmaton of mnmum sample sze and mnmal detectable dfference among groups Control Expermental (N+P) 3 Power and sample sze n ANOVA wth k groups wth sngle-classfcaton n replcates per group at ANOVA specfed α. If we have an estmate of the wthn-group varablty s (MS error ), we can calculate φ: φ = k n = ( µ µ ) ks Control Expermental (N+P) 33 7.

12 Calculatng power gven φ Decreasng ν ν = Gven ν,ν, α and φ, we can read -β from sutable tables or curves (e.g. Zar (996), Appendx Fgure B.). -β α =.05 α =.0 φ(α =.0) φ(α =.05) Model I ANOVA: mnmal detectable dfference Suppose we want to detect a dfference between the two most dfferent sample means of at least δ. To test at the α sgnfcance level wth - β power, we can calculate the mnmal sample sze n mn requred to detect δ, gven a sample group varance s by solvng teratvely. δ ks n > φ mn δ 35 Model I ANOVA: power of the test If H 0 s accepted, t s good practce to calculate power! Knowng MS groups, s (= MS error ), and k, we can calculate φ. Source SS df MS Total SS T N- Among SS groups k- MS groups groups Error SS error N-k MS error φ = ( k )( MS s ) ks groups 36 7.

13 Power of the test: an example Effect of temperature on nsect development tme 4 eggs each at two temperatures, 5 at the thrd (k = 3, n = n = 4, n 3 = 5) Source SS df MS F Total 6.9 Among groups Error ( k )( MSgroups s ) φ = ks 59 (. 66. ) = = (. ) βν ( =, ν = 0, φ= 9. ) =. 33 So, there s a 67% chance of commttng a Type II error. 37 Factors determnng power n sngle classfcaton ANOVA Power ncreases wth ncreasng φ. Therefore, power ncreases wth () ncreasng sample sze n; () ncreasng dfferences among group means (MS groups ); (3) decreasng number of groups; (4) decreasng wthn-group varablty s (MS error ). φ = k n = φ = ( µ µ ) ks nδ ks 38 Power n sngle- classfcaton Model II ANOVA F In ths case, we can calculate - β from central F: ( β), ν, ν = ν F Knowng ν, ν, α and MS groups, we can estmate - β. Body sze (kg) α(), ν, ν MS groups ( ν ) Rdng Kluane Mountan Algonqun

14 Power n non-parametrc sngle- classfcaton ANOVA If assumptons of parametrc ANOVA are met, then non-parametrc ANOVA s 3/π = 95% as powerful. If non-parametrc ANOVA s used, calculate power for parametrc ANOVA to get a rough estmate of power of non-parametrc test. Control Treatment Treatment Plot Rank Rank Rank Rank Sum Power wth G*Power f = φ n φ f n δ k n ( µ µ ) ( k )( MS ) nδ groups s = = = = k ( µ µ ) ( )( ) δ k MSgroups s = = = = ks ks nks φ ks f nks mn > = = ks f = δ ks ks ks δ ks φ n 4 Concepts map Contnuous dependent varable Total SS= Model SS + Error SS One categorcal ndependent varable Statstc k-, n-k df k- df Kruskall-Walls 95% as powerful as ANOVA when ANOVA assumptons are met One-way ANOVA Null: all means are equal Independence Autocorrelaton plot Vsual test Normalty Lllefors Assumptons Vsual test Homoscedastcty Levene's Most crtcal Model Scheffé Tukey GT OK multple comparsons Solutons to the problem of experment wse error rate Planned vs unplanned comparsons a' = a/k Bonferonn a' = -(-a)/k Sdak Conservatve multple comparsons Multple comparsons Two strateges Use smaller crtcal p Use a dfferetn statstc than t H0 rejecton wth conservatve s sold H0 acceptance wth lberal s sold Strategy: use one lb and one conservatve to bracket conclusons Lberal multple comparsons Unadjusted alpha SNK 4 7.4