Factorial Within-Subject Design. Full Model and F tests. R example Y ijk = µ + j + k + i +( ) jk +( ) ji +( ) ki +( ) jki + ijk PSYCH 710

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1 Facrial Within-Subject Design PSYCH 70 Higher-Order Within-Subjects NOV Week 3 Pr. Patrick Bennett x3 within-subjects facrial design & B are crossed, fixed facrs subjects a rom facr typically, observation per cell not possible measure withincell cannot dtinguh contributions error subject x treatment interaction -cell variation data from each subject are correlated B B B3 B B B3 subject n n n n n n subject n n n n n n subject 3 n n n n n n subject n n n n n n Full Model F tests Y ijk µ + j + k + i +( ) jk +( ) ji +( ) ki +( ) jki + ijk Stattical significance all parameters evaluated by comparing SSresiduals obtained with nested models error terms F tests slightly more complicated: - effect evaluated with x Subjects term - effect B evaluated with B x Subjects term - x B interaction evaluated with x B x Subjects term Effects noe (dtracrs) stimulus orientation on letter dcrimination (noe) x 3 (orientation) within-subject facrial design Note order columns in wide data mat: > rtdata subj absent.a0 absent.a absent.a8 present.a0 present.a present.a8 s s s s s s s s s s

2 Extract dependent s & sre in matrix dd Gaussian noe make NOV realtic Create data frame that describes within-subjects design > rt.mat <- as.matrix(rtdata[:0,:7]) > set.seed(509); > rt.nz <- matrix(dataround(rnorm(n0*6,sd00)),nrow0,ncol6) > rt.mat <- rt.mat + rt.nz Create multivariate linear model object with -subjects s: > rt.mlm <- lm(rt.mat~) N.B. No -subjects s so only have intercept in mula Note how order levels corresponds order columns in my matrix dependent s: > rt.mat absent.a0 absent.a absent.a8 present.a0 present.a present.a Use nova in car library create within-subjects mula Summary assuming sphericity: > summary(rt.aov,multivariatef) > library(car) > rt.aov <- nova(rt.mlm,idatart.idata,idesign~noe*,type"iii") > summary(rt.aov,multivariatef) data frame that contains within subjects facrs mula that tells R that facrs were crossed same as noe++noe: Univariate Type III Repeated-Measures NOV ssuming Sphericity SS num Df Error SS den Df (Intercept) e-09 *** noe ** ** noe: ** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' '

3 output. Greenhouse-Geser (G-G) epsilon, ˆ, 0.73 in table. However, I still prefer estimate useestimate corrected p ˆ0.76 lted final 0.73 output. Greenhouse-Geser (G-G) epsilon,,values 0.76 in part 85 NOV output. Greenhouse-Geser (G-G) estimate epsilon, ˆ, epsilon,, 0.89 noe: noe: epsilon,, 0.89 output. Greenhouse-Geser (G-G) estimate epsilon,(h-f) ˆ, (H-F) estimate 0.76estimate 0.73 noe: (H-F) estimate epsilon,, noe: Occasionally, > :>In situations itstard mula noe: Occasionally, In0.89 such situations stard (H-F) estimate epsilon, :, such itit In nova comm, notice how I specify within-subjects design withnoe: one-sided practice 0.8 noe: Occasionally, > : In such situations stard set set. Eir G-GG-G or H-FH-F adjusted values aresuch acceptable, but but I prefer use noe:. Eir or adjusted p:values are acceptable, I stard prefer use noe : 0.8practice set pp > In situations practice.because Eir G-G orless Occasionally, H-F adjustedso, values acceptable, butiti prefer use H-F adjustment it itslightly conservative. noe: interaction significant, H-F adjustment because slightly less conservative. So, are noe: interaction significant, > library(car) practice set adjustment. Eir because G-G H-F adjusted p values are acceptable, but I prefer use H-F or less conservative. So,,, noe: interaction p significant, (, 8)7.5, 7.5, 0.8, slightly 0.007, as are s (, 8)7.79, 7.79, 0.89, 0.005, 0.8, p itp0.007, as are s F (,F8) 0.89, p0.005, > rt.aov <- nova(rt.mlm,idatart.idata,idesign~noe*,type"iii") F (,F8) H-F adjustment it0.8, slightly less conservative. So, noe: significant, F (, because 7.5, F p0.007, as are s, F (, 8) interaction 7.79, 0.89, p 0.005, noe, 9)5.69, p5.69, ) noe, F (, 9)(, p > summary(rt.aov,multivariatef) F (, 8) 7.5, 0.8, 0.007, s, F (, 8) 7.79, 0.89, p 0.005, noe, F (,p 9) 5.69, pasare Univariate Type III Repeated-Measures NOV ssuming Sphericity noe, F (, 9) 5.69, p association.. association SS num Df Error SS den (Intercept) noe noe: Signif. codes: 0 '***' 0.00 '**' 0.0 '*' PSY70 Df..Your association textbook equation which an relative Your textbook givesgives equation which an relative e-09 ***.. association sum, nor common measure ** sumtextbook errorerror, due due subjects. nor measure Your gives,, equation which subjects. ancommon relative association dependent levels a within-subject facr ** association dependent levels a within-subject facr sum, error, due subjects. nor common measure key di erence measure your textbook gives equation which an defined relativein textbook Your **, which relative sum error one defined here ho, which, relative sum error association dependent a within-subject facr due subjects levels hled. Variation due measure subjects included in denominar sum, error due subjects. nor common (Keppel Wickens, 00; Kirk, 995). For example, (Keppel Wickens, 00; Kirk, 995). example, denominar 0.05 '.' 0. ' ', which For sum included error defined here. your textbook, but it in association dependent relative levels a not within-subject facr Sphericity tests p adjustments: (Keppel Wickens, 00; Kirk, 995). For example, association indexed by Equation will be than value, which relative sum errorlarger!y (B, () B) () can be used calculate Cohen Mauchly Tests Sphericity Equation 6 in chapter your textbook. Partial (B, B) in + (Keppel Wickens, 00; Kirk, 995).!YFor example, e + e - f ()!Y (B, 995). following equation shows how Cohen s calculate f : B) + Test stattic p-value can beeffects calculated from an NOV table with mula: Sphericity applies all within-subjects e can be calculated from an NOV table with mula: v! () that have more than degree--freedom u (a )(F ) Y (B, noe: B) can be calculated from an NOV with mula: u!y (B,!Ytable (3) B) )(F ) e (a (B, B) (a )(F ) + nab ˆ t!y (B, (3) f B) (a (a)(f ) +)nab! )(F with mula: Y (B, B) can be calculated an NOV Greenhouse-Geser Corrections (3) Partialfrom table B! calculated by: Strength ssociation & Effect Size Y (B, B) Partial B calculated by: Departure from Sphericity Chapter PSY ** noe: * Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Partial ** noe: ** --Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Partial Y B (, B)!Y B (, Chapter RT !Y (B, B) 0 e + can be calculated from an NOV table with mula:!y (B, B) (a (a )(F ) )(F ) + nab (b (b )(FB ) )(FB ) + nab 6 8 () Figure : Interaction plot data in rt.mat. (3) Partial B calculated by:!y B (, B) Partial B calculated by: () 6 ()! Partial 0.96 () (5) 0.85 (5) 0.78 Cohen s f (5) (a )(b )(FB )Table : Strength association sizes rt.mat data. (5) (a )(b )(FB ) + nab..3 simple s :noe interaction was significant, so we should examine simple s. First plot data get an idea what interaction might mean. following comms w create Figure. PSY70 Chapter > rt.mat Source noe noe: Your equation which an relative textbook do notgives recalculate F with sum MS, error, due subjects. nor common measure error dferror from association dependent levels a within-subject facr analys, which relative sum error (Keppel Wickens, 00; Kirk, each simple effect a 995). - For example, B) B B. Par (b simple effects<- colmeans(rt.mat[:0,:3]) ) > ( absent.means way within-subject NOV ) + nab )(FB ) + nab )(FB ) Partial B calculated by: (b )(FB ) + nab (a )(b) )(FB )!Y B (, B)(b )(FB!Y B (, Partial B B) calculated by: (a (a )(b )(b )(F )(FB ) + nab B )!Y B (, B)(b )(FB ) + nab (a )(b )(FB ) + nab (a )(b )(FB )!Y B (, Partial B calculated by:b) (a )(b )(F ) + nab B simple effects terms each simple )(F Partial calculated by:(b B calculated by:!yb B (, B) first part summary lts NOV table. noe: interaction significant, F (, 8) 7.5, p 0.003, as are s, F (, 8) 7.79, p , noe, F (, 9) 5.69, p Note that denominar degrees freedom are di erent two s because di erent error terms are used evaluate significance noe. sphericity assumption was introduced in Chapter. It applies here, o. results in PSY70 NOV table assume that sphericity valid, but course we need evaluate it bee accepting p values lted in table. second part nova output shows results Mauchly test sphericity. Notice that sphericity tests are done only noe: terms, not noe. reason deviations from sphericity are not examined noe that that facr has only two levels ree one degree freedom: sphericity necessarily valid F tests that have one degree freedom in numerar, ree sphericity need not be evaluated that. Mauchly tests sphericity are not significant (p > 0. in both cases), so we could use p values lted in NOV table. However, I still prefer use corrected p values lted in final part output. Greenhouse-Geser (G-G) estimate epsilon, ˆ, noe: (H-F) estimate epsilon,, 0.89 use simple Occasionally, > : In such situations it stard 0.8 effects noe: decompose practice set an. interaction Eir G-G or H-F adjusted p values are acceptable, but I prefer use absent H-F adjustment because it slightly less conservative. So, noe: interaction significant, -subjects designs, present Funlike (, 8) 7.5, 0.8, p 0.007, as are s, F (, 8) 7.79, 0.89, p 0.005, it better use separate noe, F (, 9) 5.69, p error.. effect association (a )(Fbe ) (bobvious It(ashould )(FB how ) change Equation 6 calculate!yb (B, (3) () s sizes (b )(FB ) B)!Y B (, calculated by: B) (a )(F ) + nab f data analyzed in section.. are lted in Table. (b )(F ) + nab B! () absent.a0 absent.a absent.a8 absent.a0 absent.a absent.a8 present.a0 present.a present.a ) > ( present.means 3<- colmeans(rt.mat[:0,:6]) present.a0 present.a present.a > x. <- c(0,,8); analyze simple effect at each level noe first, separate data from noe present & noe absent conditions > rt.absent <- rt.mat[,:3] > plot(xc(0,,8),absent.means,"b",ylimc(50,800),ylab"rt",xlab"") > rt.present <- rt.mat[,:6] > points(xc(0,,8),present.means,"b",pch9) dependent s are sred in matrix rt.mat. syntax rt.mat[, :3] a way specifying > legend(x0,y750,legendc("absent","present"),pchc(,9)) all rows in columns through 3, equivalent rt[:0,:3]. If we wanted access data from data frame rtdata, we could use following comms: It appears that di erence noe absent present conditions increased with > rt.absent <- as.matrix(rtdata[,:]) stimulus. lternatively, can say that appears be larger in no > rt.present we <- as.matrix(rtdata[,5:7]) condition than in However, noe absent I willin evaluate idea tests we s we added condition. noe numbers rtdata, th so I will use by conducting numbers in rt.mat. Next, one-way NOV evaluate when noe present. First, we use lm at eachconduct levela noe. create a multivariate lm object: In a two-way, -subjects facrial design, simple s were evaluated by doin > ang.present.mlm <- lm(rt.present~) one-way NOVs, but using M Sresiduals from overall analys as error term. However, fo we create a datause frame separate that contains one three-level facr names analys. ree, subjects designs itnext, better error terms each <- facr(xc(,,3),labelc("a0","a","a8")) simple s > essentially identical conducting a set one-way, within-subject NO

4 simple effect (noe present) > ang.present.mlm <- lm(rt.present~) > <- facr(xc(,,3),labelc("a0","a","a8")) >.idata <- data.frame() compute anova: multivariate linear model create data frame that describes level within-subject facr > ang.present.aov <- nova(ang.present.mlm,idata.idata,idesign~,type"iii") > summary(ang.present.aov,multivariatef) Univariate Type III Repeated-Measures NOV ssuming Sphericity SS num Df Error SS den Df (Intercept) e-08 *** e-0 *** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' print summary simple effect (noe present) Mauchly Tests Sphericity Test stattic p-value Greenhouse-Geser Corrections Departure from Sphericity *** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' '.7 e-0 *** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' N.B. We do not recalculate F using MSerror dferror from original NOV simple effect (noe absent) > ang.absent.mlm <- lm(rt.absent~) > ang.absent.aov <- nova(ang.absent.mlm,idata.idata,idesign~,type"iii") > summary(ang.absent.aov,multivariatef) Univariate Type III Repeated-Measures NOV ssuming Sphericity SS num Df Error SS den Df (Intercept) e-09 *** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' linear contrasts similar procedures used with -way within-subjects NOV use contrast weights create composite scores - converts multivariate analys univariate analys use t test evaluate null hypos Mauchly Tests Sphericity Test stattic p-value Greenhouse-Geser Corrections Departure from Sphericity

5 evaluate linear trend RT across separately at each level noe > lin.c <- c(-,0,) contrast weights > rt.pres.lin <- rt.present %*% lin.c > t.test(rt.pres.lin) -tailed t test data: rt.pres.lin t.80, df 9, p-value alternative hypos: true mean not equal mean x 86.8 composite scores (noe present) evaluate linear trend RT across separately at each level noe > rt.absent.lin <- rt.absent %*% lin.c > t.test(rt.absent.lin) -tailed t test data: rt.absent.lin t 0.37, df 9, p-value alternative hypos: true mean not equal mean x.7 composite scores (noe absent) evaluate linear trend RT across on entire data set > rt.mat absent.a0 absent.a absent.a8 present.a0 present.a present.a > lin.c <- c(-,0,,-,0,) > rt.lin <- rt.mat %*% lin.c data matrix contrast weights (linear trend ignoring noe) composite scores evaluate linear trend RT across ignoring noe > lin.c <- c(-,0,,-,0,) > rt.lin <- rt.mat %*% lin.c > t.test(rt.lin) contrast weights (linear trend ignoring noe) composite scores -tailed t test data: rt.lin t 3.8, df 9, p-value linear trend ignoring noe significant alternative hypos: true mean not equal mean x 99.5

6 does linear trend RT across differ across noe levels? weights (-,0,) - (-,0,) (-,0,,,0,-) > myc <- c(-,0,,,0,-) contrast weights (linear trend x noe interaction) > rt.lin.x.noe <- rt.mat %*% myc > t.test(rt.lin.x.noe) composite scores -tailed t test data: rt.lin.x.noe t -5.05, df 9, p-value noe x linear trend interaction significant alternative hypos: true mean not equal using NOV evaluate linear trend x noe interaction > lin.c <- c(-,0,) > rt.pres.lin <- rt.present %*% lin.c > rt.absent.lin <- rt.absent %*% lin.c > lin.scores <- cbind(rt.pres.lin,rt.absent.lin) > dimnames(lin.scores)[[]] <- c("nz.pres","nz.absnt") > lin.scores nz.pres nz.absnt convert 6-column data matrix a -column data matrix composite scores using NOV evaluate linear trend x noe interaction split-plot designs evaluate effect noe with -way within-subjects NOV > lin.scores.mlm <- lm(lin.scores~) > nz <- as.facr(c("present","absent")) > lin.scores.aov <- nova(lin.scores.mlm,idatadata.frame(nz),idesign~nz,type"iii") > summary(lin.scores.aov,multivariatef) Univariate Type III Repeated-Measures NOV ssuming Sphericity SS num Df Error SS den Df (Intercept) ** nz *** Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' what do se effects mean? split-plot designs have -subject & within-subject facrs analyzed same way as within-subjects design except we include -subjects facrs in multivariate linear model

7 a within-subjects. I will use same sry, but I ve created my own fake data: case test inmative: it tells us that average linear trend score di ers significantly from zero. > mydata Note again that F value intercept t value obtained in our earlier test th same hypos (t 3.8.). If we get same values as our t tests, why would we ever want use an a a a3 F test? F test useful in situations where second has more than two levels. In current young situation, example, it young not clear how3a t test could be used evaluate a trend x noe interaction if 57 noe had more than two 3levels. In fact, young 63it0 not possible test such an interaction using a single t test, but it possible perm F 6 test667see if trend (or or linear contrast) varies across n levels young or within-subject 5. young Univariate Type III Repeated-Measures SS num Df Error SS den (Intercept) : Signif. codes: 0 '***' 0.00 '**' young PSY70 Chapter 7 old Split plot designs 8 old Some experiments use mixtures within9 old 5 5 -subjects facrs. Such designs ten are called split Signif. codes: 0 '***'00.00 old'**' '*' 0.05 '.' 0. ' ' plot designs. oldanalys Your textbook illustrates data from split-plot a experiment using data presented in Tables old hypotical experiment measured RT in a young subjects (Table.7) * senior subjects (Table.5). Each subject participated in three in which vual stimuli were Next, I use lm Note that data frame contains facrconditions, which a -subjects. --presented at di erent s. th design, Note that -subject ( age) stimulus ainmultivariate-lm object.'.' Signif. codes: 0 '***'create 0.00 '**' 0.0 '*' ' ' -subjects mula now includes -subjects a within-subjects. I will use same sry, but I ve created my own fake data: : > summary(dat.old.aov,multivariatef) > mydata.mlm <- lm(cbind(a,a,a3)~+,datamydata) > mydata split-plot designs split-plot designs reing partsnov ssuming analys aresphericity same as bee: Univariate III Repeated-Measures atype a a3 Error SS den Df young 50 7 SS 5 num >Df <- as.facr(c("a","a","a3")) (Intercept) e-05 *** > mydata.idata <- 5 data.frame() young mydata.aov 77 <- nova(mydata.mlm,idatamydata.idata,idesign~,type"iii") > Bennett, 3 young 63 0 PJ PSY70 --> summary(mydata.aov,multivariatef) Signif. youngcodes: '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' 5 young ssuming Sphericity Univariate Type III Repeated-Measures NOV 6 young SS num Df Error SS den Df 7Mauchly oldtests Sphericity (Intercept) e- *** 8 old old 5 Test5stattic p-value old : * old old codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Greenhouse-Geser Signif. Corrections Note data frame contains facr, which a -subjects. Next, I use lm that Departure from Sphericity create a multivariate-lm object. Note that -subjects mula now includes -subjects eps Pr(>F[GG])Mauchly Tests Sphericity GG: Test stattic p-value > mydata.mlm <- lm(cbind(a,a,a3)~+,datamydata) reing parts : analys are same as bee: > <- as.facr(c("a","a","a3")) that sum squares error terms in two analyses are 37 77, that sum >Notice mydata.idata <-data.frame() Corrections se values, <3, equalsgreenhouse-geser sum squares s error term in original split-plot analys. > mydata.aov nova(mydata.mlm,idatamydata.idata,idesign~,type"iii") Departure lso note that average two mean from squaresphericity values, ( )/ 57.5, same as > summary(mydata.aov,multivariatef) PSY '*' 0.05 '.' 0. ' ' Test stattic p-value Greenhouse-Geser Corrections Departure from Sphericity : Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Chapter PSY70 0 NOV ssuming Sphericity Df e- *** * Mauchly Tests Sphericity : sphericity assumption applies all effects that Chapter a within-subjects include facr (i.e., even x within interactions) : * Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Notice that F p values are same ( within rounding error) as values lted in split-plot NOV table. In fact, two analyses areshows equivalent: analys p NOV table uncorrected values s, -subjects facr in a split-plot design same as a -subjects NOV on average score : Examination table will show that s angl each subject. sums squares means squares residuals lted in two NOV evaluated with di erent error terms. nova comm assumes that - within-sub tables di er because di erent numbers data points are analyzed in two cases, but ratio sare uses appropriate error term generate unbiased F tests (see Table.7, mean squares are same. One more thing: if evaluation fixed, -subjects equivalent 596not in your textbook Chapter expected meanlevel squares Bennett, PJ n in each a one-way, -subjects NOV, PSY70 n it should matter if we have di erent th design). second part output con matter: results having unequal Mauchly tests sphericity: Note that a test done interaction as well a -subject. nd, indeed, it does not n on -subjects. In general, sphericity assumption applies within-subjects facrs an doesdfnot cause significant problems with within-subjects analys. Sum Sq Mean Sq F value Pr(>F) Next, let s consider error term that ** used interactions evaluate within-subjectsfacrs. facr. third part output shows G-G that contain within-subjects In case where both s were within-subject adjusted facrs, each error term so I will use H-F adjusted p values. re a signifi p values. was evaluated Mauchlywith test an significant, was interaction that with subjects: was evaluated with S, B was evaluated that : interaction, F (, 0)., 0.65, p 0.07, but s angl Signif. codes: '***' '*' 0.05 '.' 0. ' 'S. In current case, if B within-subjects Chapter with B S, 0 B was'**' evaluated with B not significant., n it evaluated with B S/, which mean square interaction B subjects nested > summary(aov(a~+,datamydata)) Let s think about what di erent components NOV table actually mean. First conside within. Th error term equivalent weighted average values MSB S at each level -subjects. To illustrate what th represents, I am going Df Sum Sq, Mean Sq F.value -subjects Pr(>F) following code calculates NOV within-subject, one-way -subjects analys on averaged within-subject scores: series -way within subjects NOVs, separately each : mean square error term in original analys. Hence, error term that used evaluate GG eps within-subjects in a split-plot analys can Pr(>F[GG]) be thought Bennett, as an average error terms in a series PJ PSY within-subjects NOVs. : simple s Signif. codes: '**' 0.0 '*' 0.05 '.' ' series -subjects NOVs0 '***' ' ** Our previous analys found a significant : interaction, so we should examine simple s >> summary(aov(a3~+,datamydata)) dat.young <- subset(mydata,"young") > y.mat <- mydata[,:] We start by examining simple Pr(>F[HF]) -subject facr, at each level withinhf eps > y.avg <- rowmeans(y.mat) --> dat.old <- subset(mydata,"old") subject facr,. Each analys uses a0.67 separate error term calculated on particular subset data 0.3 Mean Sq F value Pr(>F) > dat.young.mlm <-lm(cbind(a,a,a3)~,datadat.young) > summary(aov(y.avg~,datamydata)) Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' Df Sum Sq begin examined, ree equivalent0.67 a one-way : * subjects NOV. > dat.old.mlm <-lm(cbind(a,a,a3)~,datadat.old) > dat.young.aov <- nova(dat.young.mlm,idatamydata.idata,idesign~,type"iii") > summary(aov(a~+,datamydata)) > names(mydata) analyses indicate simple age significant only in first level, >dat.old.aov <-that nova(dat.old.mlm,idatamydata.idata,idesign~,type"iii") Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' F (, 0).3, p [] "" "a" "a" "a3" Here tables or parts output): Weare can alsonov look at simple(i ve deleted within-subject at each level NOV table shows uncorrected p values 08 08s ,. Notice that each analys simply a one-way within-subjects NOV, that we use PSY70 Examination Chapter s >subject Bennett, PJ PSY70 : 0 table 8 will show that are 8 summary(dat.young.aov,multivariatef) a separate error term each analys. We did se analyses in previous section, so I will just > summary(aov(a~+,datamydata)) reprint summaries here: evaluated with di erent error terms. nova comm assumes that - within-subjects > summary(aov(a3~+,datamydata)) s are fixed, uses appropriate error term generate unbiased F tests (see Table Univariate page Type III Repeated-Measures NOV ssuming Sphericity >.7, summary(dat.young.aov,multivariatef) > summary(dat.old.aov,multivariatef) SS num Df Error SS den Df 596 in your textbook expected mean squares th design). second part output contains Univariate Type III Repeated-Measures NOV ssuming Sphericity Univariate Type0 III Repeated-Measures NOV ssuming Sphericity ** (Intercept) e-08 *** Df Sum Sq Mean Sq F value Pr(>F) results Mauchly 5 tests sphericity: Note that a test done interaction as well as SS num Df Error SS den Df SS num Df Error SS den Df ** assumption within-subjects. In general, sphericity applies within-subjects facrs all0 (Intercept) e-08 *** (Intercept) e-05 *** ** interactions that contain within-subjects facrs. third part output shows G-G H-F Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' -- Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' --adjusted p values. Mauchly test significant, so I will use H-F adjusted p values. re asignif. significant codes: 0only '***' in 0.00 '*' 0.05 '.' 0. ' ' Signif. codes: 0 '***' 0.00 '**' 0.0 '*' 0.05 '.' 0. ' ' analyses indicate that simple age significant '**' first0.0 level, > summary(aov(a~+,datamydata)) : interaction, F (, 0)., 0.65, p 0.07, but s are F (, 0).3, p Mauchly Tests Sphericity not significant. We can also look at simple within-subject at each level Mauchly Tests Sphericity Df Sum Sq Mean Sq F value Pr(>F)what di erent Mauchly Tests Sphericity Let s think about components NOV table actually mean. First consider subject. Notice that each analys simply a one-way within-subjects NOV, that we use Test p-value -subjects. To illustrate what th represents, I am going Test do stattic a stattic p-value Test stattic p-value a separate error term each analys. We did se analyses in previous section, 0.8 so I will just 0.9 one-way -subjects analys on averaged within-subject scores: reprint summaries here: > summary(aov(a3~+,datamydata)) > y.mat <- mydata[,:] > summary(dat.young.aov,multivariatef) Greenhouse-Geser Corrections Greenhouse-Geser Corrections Greenhouse-Geser Corrections > y.avg <- rowmeans(y.mat) Df Sum Sq Mean Sq F value Pr(>F) Departure from Sphericity Departure from Sphericity Departure from Sphericity > summary(aov(y.avg~,datamydata)) Univariate Type III Repeated-Measures NOV ssuming Sphericity simple effect at each Chapt simple effect in each SS num Df Error SS den Df * (Intercept) e-08 *** * analyses indicate that simple age significant only in first level, Signif. ** codes: 0 '***' 0.00 '**' F (, 0).3, p '*' 0.05 '.' 0. ' ' Chapter

8 linear contrasts on -subject : - calculate mean score each subject - analyze mean scores as -way -subject design on within-subject : - use contrast weights convert measures composite scores - use t-test or anova determine if scores differ across s (i.e., contrast x interaction) > y.mat<-as.matrix( mydata[,:] ) > lin.c <- c(-,0,) > mydata$lin.scores <- y.mat %*% lin.c > mydata contrast (linear trend) does not differ significantly s a a a3 lin.scores young young young young young young old old old old old old > t.test(lin.scores~,datamydata) Welch Two Sample t-test data: lin.scores by t.779, df 9.99, p-value alternative hypos: true difference in means not equal mean in young mean in old test overall contrast ignoring differences > y.mat<-as.matrix( mydata[,:] ) > lin.c <- c(-,0,) > mydata$lin.scores <- y.mat %*% lin.c > mydata a a a3 lin.scores young young young young young young old old old old old old mean contrast (linear trend) does not differ significantly from zero > t.test(mydata$lin.scores) data: mydata$lin.scores t -.30, df, p-value 0.99 alternative hypos: true mean not equal mean x - test overall contrast while controlling difference > lin.scores.aov <- aov(lin.scores~,datamydata) > summary(lin.scores.aov,interceptt) (Intercept) Intercept gr mean when using sum--zero definition effects test mean contrast test x contrast interaction