The Analysis of Variance

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1 The Analyss of Varance Introducton Researchers often perform experments to compare two treatments, for example, two dfferent fertlzers, machnes, methods or materals. The objectves are to determne whether there s any real dfference between these treatments, to estmate the dfference, and to measure the precson of the estmate. In the Descrptve Statstcs course we have dscussed comparsons of two means. It s often mportant to compare more than two means. For example, we may be nterested n determnng f there s any evdence for real dfferences among the mean values assocated wth varous dfferent treatments that have been randomly allocated to the expermental unts. Ths corresponds to a hypothess of the form where µ H 0 : µ 1 = µ 2 = K = µ I H : at least two of the µ 's are dfferent A s the mean of the th treatment or populaton. Analyss of varance (ANOVA) provdes the framework to test hypotheses lke the one above, on the supposton that the data can be treated as random samples from I normal populatons havng the same varance σ and possbly only dfferng n ther means. The sample szes for the treatment groups are possbly dfferent, say. The analyss resultng from these assumptons may be approxmately J justfed by randomsaton, to guarantee nferental valdty. It s normally the case n performng an ANOVA that the data come from an experment rather than an observatonal study, snce the expermental condtons mply that balance has been acheved through randomsaton. The calculatons to explore these hypotheses are set out n an analyss of varance table. Essentally, ths calculaton determnes whether the dscrepances between the treatment averages are greater than could reasonably be expected from the varaton that occurs wthn the treatment classfcatons. 2 One-way analyss of varance Balanced versus unbalanced layouts Before we proceed to defne the one-way layout we just need to dstngush between balanced and unbalanced layouts. A balanced one-way ANOVA refer to the specal case of one-way ANOVA n whch there are equal numbers of observatons n each group, say J 1 = J 2 = K = J I. An expermental layout nvolvng dfferent numbers of observatons n each group s referred to as unbalanced. Below we wll specfy the one-way layout n ts most general form, allowng for dfferent numbers of observatons n each group. 1

2 The one-way ANOVA model and assumptons The one-way layout refers to the smplest case n whch analyss of varance s appled and nvolves the comparson of the means of several (unvarate) populatons. One-way analyss of varance gets ts name from the fact that the data are classfed n only one way, namely, by 2 treatment. We shall assume that the I populatons are normal wth equal varance σ, and that we have ndependent random samples of szes J, J 2,, from the respectve populatons 1 K or treatment groups wth J. Furthermore, let µ denote the mean of the th populaton. If Y s the random varable denotng the jth measurement from the th populaton, we can specfy the one-way analyss of varance model as Y J I n = I = 1 2 ( 0, ) = µ + ε = 1, K, I; j = 1, K, J, ε.. d. N σ, (1) Note that ths model has the same man assumpton as the standard lnear model n that the unobservable random errors are ndependent and follow a normal dstrbuton wth mean ε zero and unknown constant varance. If ths assumpton s not satsfed, then the valdty of the results of an ANOVA s n queston. The hypothess that s assocated wth ths model s Alternatvely, model (1) s often wrtten as Y H 0 : µ 1 = µ 2 = K = µ I H : at least two of the µ 's are dfferent A 2 ( 0, ) = µ + α + ε = 1, K, I; j = 1, K, J, ε.. d. N σ, (2) where µ = µ + α. The parameter µ s vewed as a grand mean, whle α s an effect for the th treatment group. The hypothess assocated wth ths model s then specfed as H 0 : α1 = α 2 = K = α I = 0 H : at least oneα 0 A (3) It s mportant to note that the parameters µ and { α } are not unquely defned n model (2). We say that the parameters µ and { } are not completely specfed by the model. However, α we can assume wthout loss of generalty that α. = α = 0, snce we can wrte η I = 1 ( ) = + α = ( µ + α ) + ( α α. E µ. ), = y and take as new µ and { α } the quanttes ~ µ = µ + α. and ~ α α = α., then ~α = 0. 2

3 It follows from the general theory that there s a unque soluton satsfyng every parametrc functon of the new parameters ~ µ and { ~ α } s estmable. ˆ α = 0, and that One-way ANOVA table The results from fttng model (2) are typcally summarsed n an ANOVA table, whch can be obtaned from most statstcal software packages such as SPSS. Below we gve a typcal template of an ANOVA table for the one-way ANOVA classfcaton. Source of Varaton Df Sum of Squares Mean Square F Intercept 1 SSA MSA Treatments I-1 SST MST MST/MSE Error N-I SSE MSE Total N TSS In the table above SSA, SST, SSE, TSS, MST and MSE are calculated from the observed data. Furthermore, I s the number treatments, and N s the total number of observatons. The last column contans the value of the test statstc for the hypothess n (3). We wll dscuss these quanttes below wthout gong nto too much mathematcal detal. In order to gve a short explanaton of the above ANOVA table, we start off by notng that a measure of the overall varaton could have been obtaned by gnorng the separaton nto treatments and calculatng the sample varance for the aggregate of N observatons. Ths would be done by calculatng the total sum of squares of devatons about the overall mean y SSD = I J = 1 j= 1 ( ) 2 y y and by dvdng by the approprate degrees of freedom algebrac dentty ν D = N 1. Based on the SSD = I J = 1 j= 1 I I J 2 2 ( y y) = J ( y y) + ( y y ) = 1 = 1 j= 1 j 2 3

4 or SSD = SST + the total sum of squares of devatons from the overall mean can be dvded nto sum of the between treatment sum of squares (SST) and the wthn treatment or resdual sum of squares (SSE). The SSD can also be wrtten as the total sum of squares (TSS) mnus the sum of squares due to the average or correcton factor (SSA) SSE I J 2 = 1 j= 1 SSD = y Ny 2 = TSS SSA Therefore TSS = SSA + so that, we can splt up the sum of squares of the orgnal N observatons nto three addtve parts: SSD TSS = SSA + SST + SSE. In words ths means that the total sum of squares (TSS) can be wrtten as the sum of squares due to the average, the between treatment sum of squares and the resdual sum of squares. The assocated degrees of freedom are N = 1 + k 1 + N I. Inference and nterpretaton Once we have checked the assumptons underlyng the model, as we wll dscuss n the next subsecton, we can use the analyss of varance table to draw nference about the treatment means. The table provdes the bass for a formal test of the hypothess that the treatment effects are all equal to zero, that s, α1 = α 2 = K = α I = 0. If the null hypothess of zero treatment effects s true and the errors are dentcally ndependently 2 dstrbuted N 0, σ, then the test statstc s gven by the rato ( ) SST SSE F C = = I 1 N 1 MST MSE and follows an F-dstrbuton on I-1 and N-I degrees of freedom. Intutvely, we would expect the test statstc,, to be approxmately 1 f there s no dfference between the F C treatments, and consderably greater than 1 f there s a dfference. If we wsh to test H at the 100( 1 α )% level, then the crteron for the test s to reject H f

5 F C F > 1 α, I 1, N I, where Fα, I 1, N I s the100 ( 1 α ) percentle of a F-dstrbuton wth I-1 and N-I degrees of freedom. Its value can be obtaned from standard tables or from a software package. Software packages usually provde an exact p-value for the test. Dagnostc checkng of the model It s mportant that we check the assumptons underlyng our model, namely, errors that are ndependent and dentcally normally dstrbuted wth constant varance. In order to nvestgate the valdty of these assumptons there are a few standard plots of the resduals that can be used to evaluate ther dstrbuton and to check for systematc patterns n the resduals. When the assumptons concernng the adequacy of the model are true, we expect the resduals to vary randomly. Recall from Lecture 2 that the resduals are smply the dfferences between the observed and ftted values, y ˆ. y Below we consder a few useful plots to nvestgate the above propertes of the resduals. These partcular dscrepances should be looked for as a matter of routne, but the expermenter should also be on the alert for other abnormaltes. Refer to Lecture Notes 3 for a more detaled dscusson. Normal probablty plot of the resduals The frst plot that we consder for use n checkng whether we have satsfed the assumptons s somethng called a normal probablty plot or a quantle-normal plot of the resduals. Ths plot was ntroduced n Lecture 3. Ths plot s used to check the normalty assumpton snce t allows us to nvestgate whether the resduals are normally dstrbuted: snce the resduals can be thought of as estmates of the errors, evdence that the resduals are non-normal mght lead us to suspect that the errors are not normal. The nterpretaton of ths plot was dscussed n Lecture 3. Scatterplot of resduals versus predcted values The varablty of the resduals should be unrelated to the levels of the response, as we assumed constant varance for the resduals. Ths can be nvestgated by plottng the resduals, y ˆ, aganst the ftted values, ŷ. Sometmes the varance ncreases as y the value of the response ncreases, whch s ndcatve of non-constant varance. Plot of the resduals n tme sequence Sometmes an expermental factor may drft or the skll of the expermenter may mprove as the experment proceeds. Tendences of ths knd may be uncovered by plottng the resduals aganst tme order where t s approprate. Example 5

6 Consder an experment that s set-up to compare melon varetes. Each varety s grown 6 tmes under dentcal condtons wth dentcal feedng patterns 1. The yelds n kg are dsplayed n Table 1. Table1: Yelds of four dfferent varetes of melons. Varety A Varety B Varety C Varety D We analysed ths data set and obtaned the followng ANOVA table as output: ANOVA YIELD Between Groups Wthn Groups Total Sum of Squares df Mean Square F Sg Test of Homogenety of Varances YIELD Levene Statstc df1 df2 Sg We now verfy the assumptons underlyng the above model by consderng resdual plots dscussed above and a homogenety of varance test. From the quantle-quantle plot of the standardsed resduals n fgure 1 t seems as though the assumpton of normalty for the resduals s a reasonable one. Fgure 2 shows that varablty of the resduals may be a lttle larger for the frst varety or perhaps a lttle smaller for the thrd varety, but n general the varablty of the resduals seems to be reasonably smlar across the dfferent varetes. For a more formal analyss we can consder the Levne test for equal varances gven above. From the above SPSS output we see that the test does not reject the null hypothess of equal varances at a 5% sgnfcance level, 1 Data taken from Borja, M. C. Introducton to Statstcal Modellng for Research: Course notes at the Department of Statstcs at Oxford Unversty. 6

7 but t does reject t at a 10% level. Thus, t seems that the varablty of the resduals s somewhat dfferent for the dfferent varetes, but that the dfference s not sgnfcant at a 5% level. In ths example, the result of the hypothess test s hghly sgnfcant, as ndcated by the very small p-value. Thus, the null hypothess that the yelds of all varetes are zero wll be rejected n favour of the alternatve that at least one varety has a non-zero effect. Standardsed resduals Quantles of Standard Normal Fgure 1: QQ-plot for the standardsed resduals of the melon yeld example. 7

8 Resduals Ftted : Varety Fgure 2: Resduals plotted aganst the ftted values of the expected response. Treatment contrasts and multple comparsons The analyss of varance test above nvolves only one hypothess, namely, that of equal treatment means (or treatment effects α ). If the hypothess s rejected n an actual µ applcaton of the F-test for the equalty of means n the one-way layout, the resultng concluson that the means µ 1, µ 2, K, µ I are not all equal would by tself usually be nsuffcent to satsfy the expermenter. Methods of makng further nferences about the mean are then desrable. In more complcated stuatons than the one-way ANOVA, the analyss of varance table becomes a very useful tool for dentfyng aspects of a complcated problem that deserve more attenton. It also ntroduces the SST as a measure of treatment dfferences. The SST can be broken nto components correspondng to the sums of squares for ndvdual orthogonal contrasts. These components of the SST can then be used to explan the dfferences n the means. I A contrast among the parameters µ 1, µ 2, K, µ I s a lnear functon of the µ, = 1 λµ, I wth known contrast coeffcents, λ, subject to the condton =1 λ = 0. Furthermore, two contrasts are defned as orthogonal f I λ1λ N = 1 2 = 0. Contrasts are only of nterest when they defne nterestng functons of the s. There are many ways to choose contrasts and these depend on the queston the researcher s nterested n answerng. Treatment contrasts are commonly used contrast to compare all of the treatments to a control. Other commonly used contrasts are polynomal µ 8

9 contrasts and Helmert contrasts. The choce of contrasts can be a rather techncal subject and so we wll not go nto too much detal. As an example, consder a one-way ANOVA set-up where there are fve dfferent dets A, B, C, D and E. Suppose that A s an exstng standard det that serves as the control and that B, C, D and E are new dets. An example of an nterestng contrast may be to compare the control det, A, wth the four new dets. Ths means that we are nterested n comparng the control to the average of the other four dets. Ths contrast would then be µ B µ A µ C µ 4 D µ E and by multplyng the contrast by four gves the equvalent contrast 4 µ. A µ B µ C µ D µ E Software packages such as SPSS accommodate commonly used contrasts such as polynomal contrasts and also allow the user to specfy any a pror contrast that may be of nterest. Ultmately the choce of whch comparson to make s up to the researcher. One very mportant fnal remark we wll make nvolves the ssue of multple comparsons. The specfcaton several dfferent contrasts leads to multple hypothess tests that are performed usng the same data set. Ths means that we have to remember to adjust the sgnfcance level of any multple hypothess tests that we conduct to ensure that the overall level of sgnfcance for carryng out all of the tests s equal to the desred level of sgnfcance. Recall that the reason for ths s that the probablty of makng at least one type I error s greater than the orgnal sgnfcance level when we conduct multple tests usng the same data. There are a number of dfferent sgnfcance correcton methods descrbed n the lterature. Fve of the most common ones nclude adjustments suggested by Bonferron, Scheffe, Dunnett, Tukey and Sdak. The advantages and dsadvantages of usng dfferent methods are qute complex and t s common practce to use all of the avalable methods and then report the most conservatve one. Bonferron s method s perhaps the smplest and most wdely used method. We have dscussed ths method n the Descrptve Statstcs Course. Non-parametrc one-way ANOVA As was the case n the dscusson for lnear models, transformatons of the response varable can be useful tools for dealng wth the problems of heteroscedastcty and non-normalty. However, sometmes a transformaton cannot solve the problem, n cases lke these we can proceed by usng a non-parametrc test. In the case of two samples, the Mann-Whtney U statstc s the non-parametrc equvalent of the twosample t-test, and n the case of more than two samples, the Kruskal-Walls test s the non-parametrc equvalent of the one-way ANOVA. Ths approach can be appled when the assumpton of normalty or equal varance does not hold. 9

10 The ntutve dea for the Kruskal-Walls test s that f you rank all of the data and sum the ranks n each group, then f the groups have no real dfferences, the sum of the ranks should be the same n each group. If there were dfferences between the groups, then you would expect the sum of the ranks wthn each group to be dfferent. As an example, consder the followng experment nvestgatng LDL cholesterol n quals 2. Thrty-nne quals were randomly assgned to four dets, each det contanng a dfferent drug compound, whch would hopefully reduce LDL cholesterol. The drug compounds are labelled I, II, III and IV. At the end of the expermental tme the LDL cholesterol of each qual was measured. Thus, there are two values relatng to each qual, one recordng the det and one ndcatng the LDL cholesterol level. The data are dsplayed n Table 2. Table 2: LDL cholesterol levels n quals exposed to four dfferent dets. Drug I Drug II Drug III Drug IV The man nterest s to see f whether or not det has any effect on the mean value of LDL cholesterol level. If we consder the boxplots n fgure 3 of the LDL level for each det, there appear to be sgnfcant dfferences between the dets. We start of by fttng the usual two-way ANOVA model to the data. It s clear from the normal Q-Q plot of the standardsed resduals n fgure 4 that the resduals do not satsfy the assumpton of normalty. Any nferences drawn from the results of ths ANOVA model wll not be vald. Consequently, we perform the Kruskal-Walls test. The results from ths procedure are also presented below. 2 Data taken from Hettmansperger, T. P. and McKean, J. W. (1998). Robust Nonparametrc Statstcal Methods: Kendall s Lbrary of Statstcs 5. London: Arnold. 10

11 Det I Det II Det III Det IV Fgure 3: Boxplots for the qual data. Standardsed resduals Quantles of Standard Normal Fgure 4: QQ-plot of the standardsed resduals of the one-way ANOVA. Ch-Square df Asymp. Sg. Test Statstcs a,b LDL a. Kruskal Walls Test b. Groupng Varable: DIET 11

12 The p-value of for Kruskal-Walls test, although not sgnfcant at a 5% level, ndcates some dfference between the dets as suggested by the dets. A standard oneway ANOVA gves a p-value of 0.35 for the F-test of equal treatment effects. Ths does not agree wth the boxplots n fgure 3. The reason for ths s that the long rght tal of the errors shown n fgure 4 adversely affects the test statstc. Unfortunately, there s no easy extenson to the problem of multple comparsons that we wll ntroduce n more detals n a later secton. Complete randomsed blocks Introducton In ths secton we extend the deas of the prevous secton by comparng more than two treatments, usng randomsed desgns wth larger block szes. In blocked desgns there are two knds of effects of nterest. The frst s the treatment effects, whch are of prmary nterest to the expermenter. The second s the blocks, whose contrbuton must be accounted for. In practce, blocks mght be, for example, dfferent ltters of anmals, blends of chemcal materal, strps of land, or contguous perods of tme. In the next secton we wll consder a replcated factoral desgn n whch the man effects of two factors and the nteracton are all of equal nterest. Randomsed complete block desgns: (1) dentfy blocks of homogeneous expermental materal (unts) and (2) randomly assgn each treatment to an expermental unt wthn each block. The blocks are complete n the sense that each block contans all of the treatments. Random assgnment of treatments to expermental unts allows us to nfer causaton from a desgned experment. If treatments are randomly assgned to expermental unts, then the only systematc dfferences between the unts are the treatments. In observatonal studes where the treatments are not randomly assgned t s much more dffcult to nfer causaton. So the advantage of ths procedure s that treatment comparsons are subject only to the varablty wthn blocks. Block to block varaton s elmnated n the analyss. In a completely randomsed desgn appled to the same expermental materal, the treatment comparsons would be subject to both wthn block and between block varablty. Let us consder an example. We consder a randomsed block experment n whch a process for the manufacture of penclln was beng nvestgated, and yeld was the response of prmary nterest 3. There were 4 treatments to be studed, denoted by A, B, C and D. It was known that an mportant raw materal, corn steep lquor, was qute varable. Fortunately suffcent blends could be made to allow researchers to run all 4 treatments wthn each of 5 blocks (blends of corn steep lquor). Furthermore, the experment was protected aganst extraneous unknown sources of bas by runnng the 3 Example taken from Box, Hunter and Hunter Statstcs for Expermenters: An Introducton to Desgn, Data Analyss and Model Buldng. John Wley, New York. 12

13 treatments n random order wthn each block. The randomsed block desgn s gven n Table 3. Table 3: Randomsed block desgn on penclln manufacture 4 Block Treatment A B C D Blend 1 89 (1) 88 (3) 97 (2) 94 (4) Blend 2 84 (4) 77 (2) 92 (3) 79 (1) Blend 3 81 (2) 87 (1) 87 (4) 85 (3) Blend 4 87 (1) 92 (3) 89 (2) 84 (4) Blend 5 79 (3) 81 (4) 80 (1) 88 (2) The model and assumptons The model for a randomsed complete block desgn s gven by y 2 ( 0, ) = µ + α + β + ε, = 1, K, I; j = 1, K, J, ε..d. N σ. j There are J blocks wth I treatments observed wthn each block. As before the parameter s vewed as a grand mean, α s an unknown fxed effect for the th treatment, and µ β j s an unknown fxed effect for the jth block. The theoretcal bass for the analyss of ths model s precsely as n the balanced one-way ANOVA. As before the computatons can be summarsed n an ANOVA table, as we wll show n the followng secton. ANOVA table As was the case for the one-way layout, the results of fttng model the model for a complete randomsed block desgn are typcally represented n an ANOVA table, whch s a summary of the modellng procedure and can be calculated usng most statstcal software packages. As we are nterested n the nterpretaton rather than theory, we only consder an example of what an ANOVA table looks lke for a complete block desgn, thereby avodng unnecessary mathematcs. Table 4: ANOVA table for a complete randomsed block desgn 4 The superscrpts n parentheses assocated wth the observatons ndcate the random order n whch the experments were run wthn each blend. 13

14 Source of Varaton Df Sum of Squares Mean Square F Intercept 1 SSA MSA Treatments I-1 SST MST MST/MSE Blocks J-1 SSB MSB MSB/MSE Resduals (I-1)(J-1) SSE MSE Total IJ TSS In the table above SSA, SST, SSE, TSS, MST and MSE are smply summares calculated from the observed data, smlar to what we saw for the one-way ANOVA table. Furthermore, I s the number of treatments and J s the number of blocks. The two test statstcs are produced n a smlar way to the one-way case. We wll consder the two hypotheses nvolved n a lttle more detal n the followng secton. Inference and nterpretaton Table 4 provdes the bass for a formal test of the hypothess that the treatment effects are all equal to zero, as well as a formal test for the hypothess that the block effects are all equal to zero. The F-statstc, F T = MST MSE, s used to test whether there are sgnfcant treatment effects,.e., t s used to test H H 0 A : α1 = α 2 = K = α I = 0. : at least oneα 0 The statstc follows an F-dstrbuton on I-1 and (I-1)(J-1) degrees of freedom. If we wsh to test H at the 100( 1 α )% level, then the crteron for the test s to reject H f 0 0 FT > F1 α, I 1, ( I 1)( J 1). F α, I 1, ( I 1)( J 1) s the100 ( 1 α ) percentle of a F-dstrbuton wth I-1 and (I-1)(J-1) degrees of freedom and ts value can be obtaned from standard tables or from a software package. Smlarly, the F-statstc, F B = MSB block effects,.e., t s used to test MSE, s used to test whether there are sgnfcant 14

15 H H 0 A : β1 = β 2 = K = β J = 0. : at least one β 0 The F-statstc provdes a test of whether we can solate comparatve dfferences n the block effects. Thus, a sgnfcant test ndcates that blockng was a worthwhle exercse. If we wsh to test H at the 100( 1 α )% level, then the crteron for the test s to reject H 0 0 f FT > F1 α, J 1, ( I 1)( J 1). F α, J 1, ( I 1)( J 1) s the100 ( 1 α ) percentle of a F-dstrbuton wth J-1 and (I-1)(J-1) degrees of freedom. j Dagnostc checkng of the model The assumptons underlyng the randomsed complete block desgn model are smlar to those of the one-way ANOVA model. These assumptons need to be verfed n order for the model to be vald. We can use the same methods as for the one-way ANOVA model to acheve ths. Example Let us return to the penclln example that we ntroduced earler n ths secton. We wll ft the randomsed block desgn model on ths data set and valdate the underlyng assumptons. Below we nclude the resultng ANOVA table and some dagnostc plots. Dependent Varable: YIELD Source Intercept BLOCKS TREATMEN Error Total Tests of Between-Subjects Effects Sum of Squares df Mean Square F Sg The resdual plots for the penclln example n fgures 4 and 5 do not reveal anythng of specal nterest. The assumptons of normalty and constant varance for the resduals seem to be reasonable. Sometmes the plot of the resduals versus the predcted values shows a curvlnear relatonshp. For example the resduals may tend to be postve for low values of ŷ, become negatve for ntermedate values, and be postve agan for hgh values. Ths often suggests nonaddtvty between the block and treatment effects and mght be elmnated by a sutable transformaton of the response. However, t s not the case n ths example. 15

16 Resduals Ftted : 1 + TREATMEN + BLOCKS Fgure 5: Resduals versus ftted values for penclln example Quantles of Standard Normal Fgure 6: Q-Q plot of the standardsed resduals n the penclln example. The p-value of for the hypothess of zero treatment effects, suggests that the four dfferent treatments have not resulted n dfferent yelds. The varablty among the treatment averages can be reasonably attrbuted to expermental error. We reject the null hypothess of no blend-to-blend varaton, as suggested by the small p-value (0.041), thus there are sgnfcant block effects. 16

17 The two-way factoral desgn Introducton In ths secton we are nterested n applyng two dfferent treatments (each treatment havng a number of levels). We are tryng to dscover f there are any dfferences wthn each treatment and whether the treatments nteract. The man effects of the two treatments and ther nteracton are all of equal nterest. An easy way to understand ths topc s by means of an example. Consder an agrcultural experment where the nvestgator s nterested n the corn yeld when three dfferent fertlsers are avalable, and corn s planted n four dfferent sol types. The researcher would lke to establsh f the fertlser has an effect on crop yeld, f the sol type has an effect on crop yeld and whether the two treatments nteract. The presence of an nteracton n ths example means that there may be no dfference between fertlser 1 and fertlser 2 n sol type 1, but that fertlser 1 may produce a greater crop yeld than fertlser 2 n sol type 2. We wll only consder balanced desgns here, although the theory extends to non-balanced desgns. The model and assumptons Assume that we have two expermental factors, named A (wth a levels) and B (wth b levels). The model for the balanced two-way factoral desgn wth nteracton s gven by y k = µ + α + β + j ( αβ ) + ε = 1,2, K, a; j = 1,2, K, b; k = 1,2, K, n, k, ε..d. N 2 ( 0, σ ) In the above model y k s the kth response value subject to the th level of factor A, and the jth level of factor B. It s assumed that these 2 (, ) µ y k are ndependent and y ~ N σ. It represents a balanced desgn because we have the same number, n, k observatons for each treatment combnaton. The global mean or ntercept term s agan represented by. In the above model α s the treatment effect of the th level of µ factor A, whle β s the treatment effect of the jth level of factor B. The α are often j referred to as the row effects and the as the column effects. Ths stems from the fact that the th row of the data table often represents the observatons made for the th level of factor A, and the jth column of the data table often represents the observatons made for the jth level of factor B. They are called the treatment man effects. The nteracton effects are represented by the αβ, where ( αβ ) represents the nteracton effect of the ( ) β j th level of factor A and the jth level of factor B. We wll dscuss nteracton n some more detal n a later subsecton. 17

18 ANOVA table Agan we consder the ANOVA table for the above model by avodng unnecessary mathematcal detal. The general form of the ANOVA table s presented n table 5. Table 5: ANOVA table for a balanced two-way factoral desgn wth nteracton Source of Varaton Df Sum of Squares Mean Square F Intercept 1 SSInt MSInt Factor A a-1 SSA MSA MSA/MSE Factor B b-1 SSB MSB MSB/MSE Interacton (a-1)(b-1) SS(AB) MS(AB) MS(AB)/MSE Error ab(n-1) SSE MSE Total TSS As before the quanttes n the table above are smply summares calculated from the observed data, smlar to what we saw for the one-way ANOVA table and the block desgn. The three test statstcs are produced n a smlar way than before, and s based on the same ntutve approach. That s, we are consderng estmatng how much of the overall varaton each factor and the nteracton explan, compared to the resdual (error) varaton. The next subsecton wll look at the relevant hypotheses n some more detal. Inference and nterpretaton Agan the ANOVA table provdes the bass for formal tests of all the relevant hypotheses that we may be nterested n. There are three man hypothess tests of nterest here, namely, the test for sgnfcant treatment effects for factor A, the test for sgnfcant treatment effects for factor B and the test for sgnfcant nteracton effects. As wth the prevous sectons, to test each of the hypotheses, we compare the test statstc gven n the last column wth the F-dstrbuton wth approprate degrees of freedom. The approprate degrees of freedom are the degrees of freedom assocated wth the source of varaton and the degrees of freedom assocated wth the error. 18

19 The F-statstc, F A = MSA effects for factor A,.e. MSE, s used to test whether there are sgnfcant treatment H H 0 A : α1 = α 2 = K = α a = 0. : at least oneα 0 Smlar to the one-way ANOVA the statstc follows an F-dstrbuton on a-1 and (a- 1)(b-1) degrees of freedom. If we wsh to test H at the 100( 1 α )% level, then the crteron for the test s to reject H 0 f FA > F1 α, a 1, ( a 1)( b 1) where F1 α, a 1, ( a 1)( b 1) s the100 ( 1 α ) percentle of a F-dstrbuton wth a-1 and (a-1)(b- 1) degrees of freedom. 0 Smlarly, the F-statstc, F B = MSB treatment effects for factor B,.e. MSE, s used to test whether there are sgnfcant H H 0 A : β1 = β 2 = K = β b = 0. : at least one β 0 If we wsh to test H at the 100 ( 1 α )% level, then the crteron for the test s to reject H 0 0 f FB > F1 α, b 1, ( a 1)( b 1) where F1 α, b 1, ( a 1)( b 1) s the100 ( 1 α ) percentle of a F-dstrbuton wth b-1 and (a-1)(b- 1) degrees of freedom. The thrd hypothess that we want to test s that of no sgnfcant nteracton effects. The F-statstc, F AB = MS( AB) MSE, provdes us wth the bass of dong that. The correspondng hypothess can be formulated as H H 0 A : αβ = 0, = 1,2, K, a; j = 1,2, K, b, : at least oneαβ When we test the null hypothess of no nteracton at the 100 ( 1 α )% level, the crteron for the test s to reject f H 0 0 FAB > F1 α, ab( n 1)(, a 1)( b 1) j 19

20 where F1 α, ab( n 1)(, a 1)( b 1) s the100 ( 1 α ) percentle of a F-dstrbuton wth ab(n-1) and (a-1)(b-1) degrees of freedom. Normally all these values are provded by the software package and thus there s no need to calculate them yourself. Example Our example nvolves an experment n whch haemoglobn levels n the blood of brown trout were measured after treatment wth four rates of sulfamerazne. Two methods of admnsterng the sulfamerazne were used. Ten fsh were measured for each rate and each method. The data are gven n Table 6 5. Table 6: Data for the haemoglobn example. Method Rate A B If we now ft a two-way ANOVA to the data we obtan the followng results: 5 Ths data s taken from Rencher, A. C. (2000). Lnear Models n Statstcs. Wley: New York. 20

21 Dependent Varable: HEMOGLOB Source Intercept RATE METHOD RATE * METHOD Error Total ANOVA output Sum of Squares df Mean Square F Sg The approach s generally to frst test the hypothess that there s an nteracton, snce the sgnfcance of the man effects cannot be tested n the presence of an nteracton. If there s a sgnfcant nteracton we can use somethng called an nteracton plot to allow us to examne the nteracton and explan what s happenng. If there s no nteracton we then consder testng for the effects of the two treatments. In the above example the nteracton term s not sgnfcant. If we just consder the effect of method on ts own, t appears to be strongly nsgnfcant, whereas the effect of rate on ts own s very sgnfcant. The underlyng normalty and constant varance assumpton have to be verfed n order for the ANOVA results to be vald. Ths has been done for ths example, but the results are not reported here. Interactons Although the nteracton term s not sgnfcant n the above example, we present two nteracton plots from the example to llustrate how nteracton affects can be graphcally represented. An nteracton plot bascally plots the mean of each level of one treatment varable, at each level of the other treatment varable. If all the means follow the same general pattern, then there wll be no nteracton. If some levels follow dfferent patterns, there wll be an nteracton. In the haemoglobn example we get the two plots n Fgure 7. 21

22 Margnal mean of haemoglobn A RATE B Margnal mean of haemoglobn METHOD A B 4.00 METHOD RATE Fgure 7: Interacton plots for the haemoglobn example. The means n the above plots all follow the same general pattern, as we would expect from the non-sgnfcant nteracton term. The use of nteracton plots can be very useful to explore the practcal mportance of nteractons between two varables. As an example, let us consder the followng hypothetcal graphs n Fgure 8. Non-sgnfcant A, Sgnfcant B Sgnfcant A, Non-sgnfcant B Average Response Level 1 of B Level 2 of B Level 3 of B Average Response Level 1 of B Level 2 of B Level 3 of B Factor A Factor A Both A,B Sgnfcant, No nteracton Both A,B Sgnfcant and an nteracton Average Response Level 1 of B Level 2 of B Level 3 of B Average Response Level 1 of B Level 2 of B Level 3 of B Factor A Factor A Fgure 8: Some hypothetcal nteracton plots. It s qute clear from these graphs, the effect that a sgnfcant man effect of a varable and an nteracton would have on the nteracton plot. You should always pay partcular attenton to nteracton terms and ther nterpretaton. In partcular, you should never consder a model that has an nteracton 22

23 between two varables wthout havng both of the varables ncluded; not ncludng the man effects of an nteracton does not make ntutve sense and s dangerous, although a lot of software packages wll allow you to do ths. You should also be careful when nterpretng the results of any partcular modellng procedure, f you have a sgnfcant nteracton between two varables, you can not say anythng about the effect of one varable on the response, however, you can say that your varable of nterest and ts nteracton wth another varable has an effect on the response. In many observatonal studes, nteractons can often be one of the most nterestng fndngs, snce they mply that the two varables do not act n solaton on the response, they n fact act together and ths jont nteracton can gve a researcher great nsght nto the effect of the varables on the response. Treatment contrasts and multple comparsons As an example consder a two-factor study where the response s the score on a 20 tem multple choce test over a taped lecture, and the two factors are cogntve style (FI=Feld Independent, FD=Feld Dependent) and study technque (NN=No notes, SN=Student Notes, PO=Partal Outlne Suppled, CO=Complete Outlne) 6. We ntally consder an nteracton plot for ths data to see whether there s any evdence that there may be an nteracton. mean of score studytec SN CO PO NN FD FI cogstyle It appears from the above plot that there s an nteracton between cogntve style and study technque, so we consder the ANOVA wth the nteracton term ncluded. The ANOVA table s as follows: 6 The data were collected by Frank (1984) Effects of feld ndependence-dependence and study technque on learnng from a lecture Amer.Educ.Res.J., 21,

24 ANOVA output Dependent Varable: SCORE Source COGSTYLE STUDYTEC COGSTYLE * STUDYTEC Error Sum of Squares df Mean Square F Sg Clearly from ths table, the nteracton term s sgnfcant. It s now of nterest to perform an analyss of effects. In ths example we are nterested n testng for dfferences n each of the four study technques. However, as these technques occur n each of the two cogntve style groups we wll have to carry out the multple comparsons n each group. The results from ths analyss are gven below. The nterestng result from ths analyss s that t appears that the dfferences n study technque are the same n each of the cogntve method groups; thus suggestng that most of the dfferences n the response varables come from dfferences n study technque rather than cogntve style. 95 % smultaneous confdence ntervals for specfed lnear combnatons, by the Sdak method crtcal pont: response varable: cogstyle ntervals excludng 0 are flagged by '****' Estmate Std.Error Lower Bound Upper Bound CO.adj1-NN.adj **** CO.adj1-PO.adj CO.adj1-SN.adj **** NN.adj1-PO.adj **** NN.adj1-SN.adj **** PO.adj1-SN.adj CO.adj2-NN.adj **** CO.adj2-PO.adj CO.adj2-SN.adj NN.adj2-PO.adj **** NN.adj2-SN.adj **** PO.adj2-SN.adj The confdence ntervals above are represented graphcally n fgure 8 below. 24

25 CO.adj1-NN.adj1 CO.adj1-PO.adj1 CO.adj1-SN.adj1 NN.adj1-PO.adj1 NN.adj1-SN.adj1 PO.adj1-SN.adj1 CO.adj2-NN.adj2 CO.adj2-PO.adj2 CO.adj2-SN.adj2 NN.adj2-PO.adj2 NN.adj2-SN.adj2 PO.adj2-SN.adj2 ( ( ( ( ( ( ) ) ( ) ( ( ) ( ) ) ( ( ) ) ) ) ) ) smultaneous 95 % confdence lmts, Sdak method response varable: cogstyle Fgure 8: Graphcal representaton of confdence ntervals above. Non-parametrc two-way ANOVA Just as n the one-way case, we can equvalently consder a non-parametrc two-way ANOVA. However, t now becomes consderably more complex and the common test statstc n ths settng s restrcted to the followng specal case. Consder the case where you have two factors, one representng the treatment of nterest and the other representng a blockng varable (see the secton on complete block desgns). The nterest s then n assessng whether there are any dfferences between the treatments and to do ths we can use the Fredman rank sum test. The generalsaton to more complex settngs s possble but requres a lot of work and s not generally mplemented n the computng packages. The nterpretaton s dentcal to that of the Kruskal-Walls test for the one-way ANOVA, you wll get a test statstc and also a p- value, and as usual you can conclude that there s a sgnfcant dfference between your treatments f the p-value s less than More than two factors and covarates Although all of the technques descrbed above only cover the case of ether one or twofactor models the same deas can be extended to three- or more factor models or to models that nclude contnuous covarates. The man ssue to consder s whch model you are gong to ft.e. do you ft the full factoral model whch ncludes all the man effects and all of the nteractons (two-way and hgher) or do you only ft a subset of ths model, perhaps one whch only contans the man effects and two-factor nteractons. These decsons wll be guded by the am of your analyss and what hypothess you wsh to test and s called model selecton. You have the same choces f you nclude covarates n the model. In the next secton we wll consder the case where model ncludes contnuous covarates. Ths type of analyss s called analyss of covarance. 25

26 Analyss of covarance Introducton Analyss of covarance ncorporates one or more regresson varables nto an analyss of varance. The regresson varables are typcally contnuous and are referred to as covarates, hence the name analyss of covarance. In ths course we wll only examne the use of a sngle covarate. We dscuss the analyss of covarance at the hand of an example that nvolves one-way analyss of varance and a covarate. Example We consder data on the body weghts (n klograms) and heart weghts (n grams) for domestc cats of both sexes that were gven dgtals. Part of the orgnal data s presented n Table 7 7, 8. The prmary nterest s to determne whether females heart weghts dffer from males when both have receved dgtals. A frst step n the analyss mght be to ft a one-way ANOVA model by gnorng the body weght varable. Such a model would be gven by Y = µ + α + ε = 1,2; j = 1, K,24, where the y are the heart weghts. The ANOVA output for ths model s gven below. Table 7: Body weghts (kg) and heart weghts (g) of domestc cats wth dgtals. 7 The data was orgnally publshed by Fsher, R. A. (1947). The analyss of covarance method for the relaton between a part and the whole. Bometrcs, 3, The data s taken from Chrstensen, R. (1996). Analyss of Varance, Desgn and Regresson: Appled statstcal methods. New York: Chapman & Hall. Here a subset of the orgnal data s used to llustrate the applcaton of the analyss of covarance. 26

27 Females Males Body Heart Body Heart Body Heart Body Heart one-way ANOVA output BRAIN Between Groups Wthn Groups Total Sum of Squares df Mean Square F Sg We see that the effect due to sexes s hghly sgnfcant n the one-way ANOVA. Untl now we have not made use of the extra value of body weght that we have n addton to the bran weght varable. It s natural to consder whether effectve use can be made of ths value. In order to make use of the body weght observatons, we add a regresson term to the one-way ANOVA model above and ft the tradtonal analyss of covarance model, y 2 ( 0, ) = µ + α + γz + ε, = 1,2; j = 1, K,24, ε..d. N σ In the above model the z s are the body weghts and γ s a slope parameter assocated wth body weghts. Note that ths model s an extenson of the smple lnear regresson model between the y s and z s n whch we allow a dfferent ntercept µ for each sex. The ANOVA table for ths model s gven below. 27

28 Dependent Varable: BRAIN Source Intercept SEX BODY Error Total (adjusted) ANOVA output for the covarance model Sum of Squares df Mean Square F Sg The nterpretaton of the above ANOVA table s dfferent from the earler ANOVA tables. We note that the sum of squares for body weghts, sex and error do not add up to the total sum of squares, for example. The sums of squares n the above ANOVA table are referred to as adjusted sums of squares because the body weght sum of squares s adjusted for sexes and the sex sum of squares s adjusted for body weghts. We do not dscuss the computaton here. Interested readers can refer to Chrstansen (1996) for the computatonal detals. The error lne n the table above s smply the error from fttng the covarance model. The only dfference between the one-way model and the covarance model s that the one-way model does not nvolve the regresson of on body weghts, so by testng the models we are testng whether there s a sgnfcant effect due to the regresson on body weghts. The standard way of comparng a full and a reduced model s by comparng ther error terms. We see from the ANOVA table above that there s a major effect due to the regresson on body weghts. Fgures 9, 10 and 11 contan resdual plots for the covarance model. The plot of the resduals versus the predcted values (fgure 9) looks good, whle fgure 10 shows slghtly less varablty for females than males. The dfference s not very large though and we need perhaps not worry about t too much. The normal plot of the resduals (fgure 11) also seems reasonably acceptable. resduals ftted values 28

29 Fgure 9: Resduals versus predcted values of the covarance model resduals FEMALE MALE sex Fgure 10: Box plots of resduals by sex for the resduals of the covarance model Quantles of Standard Normal Fgure 11: Normal Q-Q plot of the standardsed resduals of the covarance model Analyss of covarance n desgned experments 29

30 In the prevous example we are dealng wth an observatonal study as opposed to a desgned experment. In a desgned experment the role of covarates s solely to reduce the error of treatment comparsons. For a covarate to be of help n an analyss t must be related to the dependent varable. Unfortunately, mproper use of covarates can nvaldate, or alter, comparsons among treatments. In the observatonal study above the very nature of what we were comparng changed when we adjusted for body weghts. Orgnally we nvestgated whether heart weghts were dfferent for females and males. The analyss of covarance examned whether there were dfferences between female and male heart weghts beyond what could be accounted for by the regresson on body weghts. These are qute dfferent nterpretatons. In a desgned experment, we want to nvestgate the effects of the treatments and not the treatments adjusted for some covarates. Cox (1958) refers to a supplementary observaton that may be used to ncrease precson as a concomtant observaton. It s stated that an mportant condton for the use of a concomtant observaton s that after ts use, estmated effects for the desred man observaton shall stll be obtaned. Ths condton means that the concomtant observatons should be unaffected by the treatments. In practce ths means that ether the concomtant observatons are taken before the assgnment of the treatments, or the concomtant observatons are made after the assgnment of the treatments, but before the effect of treatments has had tme to develop. These requrements on the nature of covarates n a desgned experment are mposed so that the treatment effects do not depend on the presence or absence of the covarates n the analyss. The treatment effects are logcally ndependent regardless of whether covarates are measured or ncorporated n the analyss. Multvarate analyss of varance (MANOVA) The multvarate approach to analysng data that contan repeated measurements on each subject nvolves usng the repeated measures as separate dependent varables n a collecton of standard analyses of varance. The method of analyss, known as multvarate analyss of varance (MANOVA), then combnes results from the several ANOVAs. A detaled dscusson of MANOVA s beyond the scope of ths course. Readers who are nterested to learn more about the subject are referred to Johnson and Wchern (1992). A. Roddam(2000), K. Javaras and W. Vos (2002) References Box, G., J. S., Hunter, W. G., Hunter, J. S. (1978). Statstcs for Expermenters: an Introducton to desgn, Data Analyss, and Model Buldng. New York: John Wley. Chrstensen, R. (1996). Analyss of Varance, Desgn and Regresson: Appled statstcal methods. New York: Chapman & Hall. 30

31 Cox, D. R. (1958). Plannng of Experments. New York: John Wley. Fsher, R. A. (1947). The analyss of covarance method for the relaton between a part and the whole. Bometrcs, 3, Hettmansperger, T. P. and McKean, J. W. (1998). Robust Nonparametrc Statstcal Methods: Kendall s Lbrary of Statstcs 5. London: Arnold. Johnson, R. A. and Wchern, D. W. (1992). Appled Multvarate Statstcal Analyss. New York: Prentce Hall. Rencher, A. C. (2000). Lnear Models n Statstcs. Wley: New York. Scheffé, H. (1959). The Analyss of Varance. New York: John Wley. 31

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