Factorial Analysis of Factorial Analysis of Variance (ANOVA) on SPSSVariance (ANOVA) on SPSSPractice reproducing the analyses Practice reproducing the analyses yourself:yourself:2 Factor Between (2 leels ! 2 leels)"sa 2 Factor Between (2 leels ! 2 leels)"sa2 Factor Between (2 leels !# leels)"sa 2 Factor Between (2 leels !# leels)"sa# Factor Between (2 leels ! 2 leels ! 2 leels)"sa # Factor Between (2 leels ! 2 leels ! 2 leels)"sa2 Factor $ithin (2 leels ! 2 leels)"sa 2 Factor $ithin (2 leels ! 2 leels)"saAll on Portal All on Portal %eading%eadinghttp:&&www"socialresearch'ethods"net&()&e!pfact"ht' * a si'ple su''ary of factorial designshttp:&&daid'lane"co'&hyperstat&inde!"ht'l* see sections ++ , +2 for )etween su)-ects designs and section +# for within su)-ects (repeated 'easures) designs" .his is reco''ended /its concise0 clear and to the point" 1t also contains a ery good glossary fro' which you can 2uic(ly refresh your 'e'ory for de3nitions of such things as the Standard 4rror etc"5hapters +60++0+2 of 7raetter, For8ano coer )etween0 within0 and factorial design issues"5hapters +#0+90+: of 7raetter , $allnau coer the stats / ANOVA etc" ;oweer don.hat the degrees of freedo' for a test of an interaction )etween two or 'ore factors ? the nu')er of leels in one factor ! the nu')er of leels in the other !=etc" .hus the AF for a # way interaction )etween factors haing 202 and 9 factor leels is + ! + ! #?#">.hat ANOVA uses F tests and that the F statistic for any e@ect is the Bean S2uare for the 4@ect diided )y the 4rror 'ean S2uare: BScondition&BSerror>.hat when you hae an alpha leel of "6: this 'eans that the pro)a)ility of not'a(ing a .ype + error is C:D ("C:) for each test you do>.hus if you hae 26 F tests in your ANOVA ta)le the pro)a)ility of none of the' )eing spurious is "C: ! "C: ! "C: ! "C:==or "C:20 or (1-)20>.hisactually ? "#E or #ED which is why (in co'ple! designs especially) you should stic( to e!a'ining a few predictions" .hings you needn.he precise way that Su's of S2uares are calculated(But it will help your understanding of ANOVA if you at least understand the gist of how aria)ility is partitioned)";ow FeeneFogic of the analysis is the sa'e )ut we now hae:> # possi)le main efects : >Auration>Bodality>Noise># possi)le 2-way interactions >!uration " modality>!uration " noise>Modality " noise>+ possi)le #-way interaction>!uration " modality " noise Both 'ain e@ects of duration and noise signi3cant" #*way interaction also signi3cant" 1nterpreting #*$ay interactions">Buch easier if you hae so'e predictions a)out the e!pected pattern>For instance in this e!a'ple we 'ight predict that as well as generally decreasing perfor'ance high leels of noise 'ight o)scure any di@erences )etween the picture and word conditions: #*way interaction is a di@erence in the pattern of a 2*way interaction at leels of the third factor .here was a signi3cant # way interaction )etween duration0 'odality and noise0 F (+0#2)?9":0 p ? "69+" 1n the low noise condition pictures and words produced opposite e@ects on perfor'ance at the two durations" At sti'ulus presentations of 266'sec words gae rise to perfor'ance so'e 26 points lower than pictures whereas the reerse pattern was true for the H66 'sec duration" $ith high noise0 howeer0 there was ery little eidence of any interaction" 1f you want to proide a )it 'ore Iweight< to your conclusions concerning the interpretation of the #*way interaction you could perfor' a simple interaction efects analysis.>.his is actually ery easy> Lou -ust run two separate ANOVAs / one at each leel of (in this e!a'ple) the noise factor">4ach of these analyses has the factors duration and 'odality )ut one uses the data fro' the high noise condition and the other fro' the low noise condition">Lou then interpret the 2*way interactions )etween duration and 'odality at each leel of noise One ANOVA on this data One ANOVA on this data5an then say whether it is true that the interaction on the left (low noise) is signi3cant whilst the one on the right (high noise) is not".here is one catch / the Fratio for the 2*way interactions in each separate analysis needs to )e co'puted using the BSerror fro' the original analysis" Original # Factor ANOVABSerror fro' the original analysis ? 966"H on #2 AF Lou now need to run the two separate 2 way ANOVAS on the data fro' the high and low noise conditions"On SPSS the easiest way to do this is to 3rst split the data using the split data co''and" Any su)se2uent co''ands )e they .a)les0 Plots or0 as in this case ANOVAs0 will now )e done separately for each le$el of the %roupin% $aria&le 'noise( ;aing split the data )le &y the noise $aria&le you now si'ply perfor' a 2 way ANOVA0 with factors duration and 'odality as )efore:(nalyse ) *eneral Linear Model ) +nivariate .his factor is left out as it is the one used to split the 3leSPSS will now co'pute the two 2 way ANOVAs .his ta)le is si'ply 2 ANOVA ta)les put together / one for the low noise data and one for the high noise data" ;oweer the F ratios are wrong as they need to )e co'puted using the BSerror fro' the original # way ANOVA Original # Factor ANOVABSerror fro' the original analysis ? 966"H on #2 AF % ratios are si'ply the result of diiding the Bean S2uare for the e@ect )y the error Bean S2uare (BSerror)4"g" the duration F ratio is si'ply BSduration & BSerrorFor the si'ple interaction e@ects follow up we need to co'pute our own F ratios for the 'odality )y duration interactions at each noise leel )y su)stituting the BSerror fro' the original analysis" For the low noise interactionthe correct Fratio is2#C#"CJH & 966"H ?:"CJFor the high noise interaction the correct Fratio is +2C"CJ &966"H ? "#2BSerror fro' the original analysis ? 966"H on #2 AF For the low noise interactionthe correct Fratio is2#C#"CJH & 966"H ?*.+,For the high noise interaction the correct Fratio is +2C"CJ &966"H ? "32*.+,.32 .o wor( out the p alue you need either to loo( it up in F ta)les"Or to calculate the e!act pro)a)ility (ery easlily) using a pac(age such as 4!cel: 4"g" .o calculate the p alue associated with the low noise 'odality ! duration interaction:.he alue we got was :"CJ.his is )ased on + df for the e@ect and #2 df for error5lic( in any cell in 4!cel and type:?FA1S.(:"CJ0+0#2) and press returnNB" Aon1nterpretation of e@ects fro' the ANOVA ta)le is the sa'e>Bain di@erence is in the data entry>Aesigns can )e all repeated 'easures or a 'i!ture>4"g" A two factor repeated 'easures design could hae:>Both factors as repeated 'easures (or within su)-ects) Or >One repeated 'easure and one )etween su)-ects 'easure 4ach su)-ect is tested under eery co')ination.he order of the co')inations would nor'ally )e rando'ised for each su)-ectOrPseudo*rando'ised so that e2ual nu')ers of su)-ects receie each order (this is the 'ost co''on 'ethod)Both factors as repeated 'easures: Bodality )y Sti'ulus duration dataAssu'ing this e!peri'ent was carried out with )oth factors as repeated 'easures:.his is how the data is entered into SPSS"4ach row represents scores fro' a single su)-ect" 4ach su)-ect has 9 data points".hese could )e single scores or the a$era%e of many trials under that condition. .he latter is co''on with 'easures such as %. which are Iinherently noisy< (i"e" you need to ta(e the aerage of 'any raw data points to get a good esti'ate for that su)-ect under those conditions)" 7ie the colu'ns 'eaningful na'es / the 3rst colu'n contains data fro' the Auration leel + (266'sec) and Bodality leel + (picture)" Lou can use short hand for the actual colu'n na'es and put the longer0 'ore 'eaningful0 description as the $aria&le la&el .o aoid confusion later the colu'ns should always )e ordered in a hierarchy * ta(e a # Factor e!a'ple (all with 2 leels and where F+(+) ? Factor + Feel +):%3(1) %3(2)%2(1)%3(1) %3(2)%2(2)%3(1) %3(2)%2(1)%3(1) %3(2)%2(2)%1(1)%1(2) .o run the analysis: First Factor is Auration and this has two leelsNB / the 3rst factor is the one at the top of the hierarchy: %3(1) %3(2)%2(1)%3(1) %3(2)%2(2)%3(1) %3(2)%2(1)%3(1) %3(2)%2(2)%1(1)%1(2)123Order in which you de3ne the factors in SPSS Second factor is 'odality / with twoleels.his sets up all the factors / now clic( Ae3ne to tell SPSS where the colu'ns are that correspond to each factor leel co')ination The -rst .uestion mar/ is as/ing 0'here is the column containing the data !rom le"el 1 o! !actor 1 and le"el 1 o! !actor 212 This is our column 1 (d1m1)Note at the top where it says $ithin*Su)-ects Varia)les you get a re'inder of which is the 3rst and second factors" .he order we de3ned the factors in was duration then 'odality hence at the top we hae (duration0 'odality)" .he nu')ers in the )rac(ets refer to the leels of the corresponding factors" .he process continues until all the within su)-ect aria)les hae )een set up" NB: only when you set up the factors in the data sheet according to the hierarchy and de3ne the factors starting fro' the top of the hierarchy will they )e in the correct order already" Once set up you can use the plots and options (display 'eans) in e!actly the sa'e way as with )etween su)-ects designs" SPSS Output.his is not 2uite the sa'e as for )etween su)-ects designs".he 3rst )o! -ust su''arises the within*su)-ects factors and allows you to chec( that they hae )een entered in the right order: Lou can ignore the 'ultiariate tests output unless you hae special reason to 2uestion certain assu'ptions"Bauchly Lou want to test for a particular trend (e"g" that perfor'ance increases in a straight line (linear) fashion as drug dosage increases" Plots and ta)les of 'eans can )e interpreted in e!actly the sa'e way as )etween su)-ects designs"