How to Perform a Two-Way (Within-Between) ANOVA in SPSS

8/9/2019 How to Perform a Two-Way (Within-Between) ANOVA in SPSS

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How to Perform a Two-Way (Within-Between) ANOVA in SPSS

When to use a two-way, within-between, ANOVA

A two-way ANOVA, also called a two factor ANOVA, can be used when there are two

independent variables (factors) influencing one dependent variable. In the ea!ple below, onefactor is method of training , and the second factor is time. "raining !ethod is the #between$

factor, because we are loo%ing at differences between groups using different training. "i!e is the

#within$ factor, because we are !easuring each group twice (pre-test and post-test). "herefore,we are also interested in the difference within each group over ti!e.

Why ANOVA and not T-test

&. 'o!paring three groups using t-tests would reuire that t-tests be conducted. *roup &

vs. *roup +, *roup & vs. *roup , and *roup + vs. *roup . "his increases the chances

of !a%ing a type I error. Only a single ANOVA is reuired to deter!ine if there aredifferences between !ultiple groups.

+. "he t-test does not !a%e use of all of the available infor!ation fro! which the sa!pleswere drawn. or ea!ple, in a co!parison of *roup & vs. *roup +, the infor!ation fro!

*roup is neglected. An ANOVA !a%es use of the entire data set.

. It is !uch easier to perfor! a single ANOVA then it is to perfor! !ultiple t-tests. "his isespecially true when a co!puter and statistical software progra! are used.

The Theory in Brief

i%e the t-test, the ANOVA calculates the ratio of the actual difference to the difference epecteddue to chance alone. "his ratio is called the F ratio and it can be co!pared to an F distribution,in the sa!e !anner as a t ratio is co!pared to a t distribution. or an F ratio, the actual

difference is the variance between groups, and the epected difference is the variance within

groups. lease read the ANOVA handout for !ore infor!ation.

!et"s #o$$

/et-up your data so that one colu!n categori0es your data into groups and a separate colu!n is

assigned for each !easure!ent ti!e. or this ea!ple, three groups (N1+2) of trac% athletes

participated in a study to ascertain the effects of traditional strength training (/uats) and power training ('leans). "he /uats group underwent &+ wee%s of strength training, the 'leans group

underwent &+ wee%s of power training, and a 'ontrol group did nothing. Vertical 3u!p was

!easured for each group before (re "est) and at the conclusion (ost "est) of the training.4ased on this infor!ation, there is one colu!n (*roup) that indicates which of the groups the

data belongs to. "here are also + colu!ns of actual data, one for the re "est and one for the

ost "est. "his is shown in igure &. At this point, you should assign #values$ to the nu!bers in

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the *roup colu!n. or ea!ple, #& 1 'ontrol$, #+ 1 /uat$, # 1 'leans$.


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After you clic% #5efine$, the window shown in igure B should appear. At this point, the

within-sub3ect factors, and the between-sub3ects factors should be assigned to specific colu!ns

of data. /ince the sa!e sub3ects undergo a re "est and a ost "est, this is the within factor. Itshows changes within a sub3ect. 5ifferent sub3ects were assigned to the three groups ('ontrol,

/uat, and 'leans), therefore *roup is the between factor. It shows changes that occur between

different sub3ects due to training. /ee igure B for clarification on actor assign!ent.

igure B= "he indow

After associating the factors with the appropriate colu!ns of data, select the #'ontrasts$ button

and the window shown in igure ; will appear. "his window provides features for contrastingthe levels of a certain factor. "his was done with the #4eer 'onsu!ption 8 *olf erfor!ance$

data. 9owever, for this ea!ple and with !ost >ithin-4etween designs, contrasting levels of a

certain factor is not useful. "his is because all levels of the other factors are collapsed together.or ea!ple, the contrast window would allow us to co!pare the difference between the re "est

and ost "est by selecting #test(


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ro! the #indow

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ro! the #ithin-4etween ANOVA. "hefollowing should help you analy0e your data.

"he first i!portant, and non-obvious, chart in the /// output is the test for sphericity. Arepeated !easures ANOVA !ust !eet the assu!ption of sphericity. /phericity reuires that the

repeated !easures de!onstrate ho!ogeneity of variance (each group of data has si!ilar

variance) and ho!ogeneity of covariance (the correlation of each repeated !easure with thedependent variable is si!ilar). In this case, there is only a re and ost test, so sphericity is not

so!ething that we have to worry about (we would if there was also a 6id test).

igure := 6auchlys "est of /pericity

Mauchly's Test of Sphericity

Measure: vjump

Within SubjectsEffect

Mauchly'sW

Approx.Chi-Suare !f Si". Epsilon#a$

%reenhouse-

%eisser &uynh-el!t(o)er-boun!

test *.+++ .+++ + . *.+++ *.+++ *.+++

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igure &2 displays the !ain results of the >ithin-4etween ANOVA. "he analysis of the !ain

effect of #test$ is significant. "his !eans that there is a significant difference between the re

and ost tests when all groups are averaged together. "his is not very helpful, because we areinterested in %nowing how the /uats, 'leans and 'ontrol groups i!proved relative to each

otherJ The test%&rou' ana$ysis is the most im'ortant 'art of the entire Within-Between

ANOVA out'ut As shown below, the interaction between test and &rou' was significant.4asically, this !eans that the groups had significantly different changes fro! re to ost. It

could be that there was no change for the 'ontrol and /uat group, but there was an increase for

the 'leans group. Another possibility could be that the /uat group actually decreased in 3u!pheight, while the 'leans and 'ontrol re!ained unchanged. Analy0ing the plot in igure & and

using co!!on sense will give us the specific answer.

igure &2= "ests of >ithin-/ub3ects Hffects

Tests of Within-Subjects Effects

Measure: vjump

,.+ * ,.+ /.,0 .+12 .+,

,.+ *.+++ ,.+ /.,0 .+12 .+,

,.+ *.+++ ,.+ /.,0 .+12 .+,

,.+ *.+++ ,.+ /.,0 .+12 .+,

*+*.*0 1 /+.3+3 2.** .+* .**

*+*.*0 1.+++ /+.3+3 2.** .+* .**

*+*.*0 1.+++ /+.3+3 2.** .+* .**

*+*.*0 1.+++ /+.3+3 2.** .+* .**

,0.+, /0 **.+3

,0.+, /0.+++ **.+3

,0.+, /0.+++ **.+3

,0.+, /0.+++ **.+3

Sphericity Assume!

%reenhouse-%eisser

&uynh-el!t

(o)er-boun!

Sphericity Assume!

%reenhouse-%eisser

&uynh-el!t

(o)er-boun!

Sphericity Assume!

%reenhouse-%eisser

&uynh-el!t

(o)er-boun!

Source

test

test 4 %roup

Error#test$

5ype 666 Sum

of Suares !f Mean Suare Si".

7artial Eta

Suare!

igure && co!pares each trial with the ad3acent trials. >ith a re "estCost "est design, as in this

ea!ple, the table provides redundant infor!ation. /ince there are only + !easure!ent ti!es to

co!pare the results are identical to the >ithin-/ub3ects Hffects shown in igure &2.

igure &&= "ests of >ithin-/ub3ects 'ontrasts

Tests of Within-Subjects Contrasts

Measure: vjump

,.+ * ,.+ /.,0 .+12 .+,

*+*.*0 1 /+.3+3 2.** .+* .**

,0.+, /0 **.+3

test

(inear

(inear

(inear

Source

test

test 4 %roup

Error#test$

5ype 666 Sum


7artial Eta

Suare!

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igure &+ shows the test of 4etween-/ub3ects Hffects. In our case, this tests if there are

significant differences between the groups. 9owever, this test co!bines the re "est and ost

"est data for all groups. "herefore, for our ea!ple, this is not a useful test. or ea!ple, if the'ontrol group averaged F2 c! on both the re and ost test, while the 'leans group scored D;

and F c! on the re and ost tests respectively, then the i!prove!ents of the 'leans group

would not show up in the test. >e can see in igure &+, that when the re and ost tests areco!bined the groups are not significantly different (p 1 .;2&).

igure &+= "ests of >ithin-/ub3ects Hffects

Tests of Between-Subjects Effects

Measure: vjump

5ransforme! 8ariable: Avera"e

0+2/.1,0 * 0+2/.1,0 */3*.02 .+++ .33,

/1.+3 1 1,.+2, ./* .,+* .+*

13*.,31 /0 /+.0*

Source

6ntercept

%roup

Error

5ype 666 Sum


7artial Eta

Suare!

igure & shows the plot of the !ean scores for each co!bination of factor level. ut another way, it shows how each group perfor!ed on the re "est and ost "est with each line

representing a group. ro! this graph it is clear that one should reco!!end perfor!ing 'leans

instead of /uats to i!prove vertical 3u!p perfor!ance. 9owever, it is not clear where thestatistical significant differences lie with respect to the rate of i!prove!ent. >e %now that so!e

groups i!proved !ore than other due to the significant interaction, but which onesJ

igure &= lot of 6eans

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5id the /uat group show a significant i!prove!ent relative to 'ontrolJ 5id the 'leans group

i!prove significantly !ore than the /uats groupJ "hese were uestions we wanted answered

right fro! the beginning of the study, before any data was collected or analy0ed. As such, the!ethods used to answer these uestions are referred to as planned comparisons.


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ro! this point, you can follow the instructions in the handout #9ow to erfor! a One->ay

ANOVA in ///$. 'hange will be the 5ependent Variable. igure &; shows all the i!portant

output to deter!ine which groups are significantly different with respect to the interaction. Asyou can see, the -value and p-value for this ANOVA are the sa!e as those found for the

interaction ter! in igure && above. "he /cheffe test shows that the only significant difference is

between the 'leans and the 'ontrol group. /o statistically spea%ing, the i!prove!ent the 'leansgroup showed was not significant in co!parison to the i!prove!ent the /uat group showed.

igure &;= Output fro! ANOVA used as a ost-9oc for the InteractionANOA

Chan"e

1+.,2 1 *+*.*0 2.** .+*

*2,.*0* /0 1.,*0

*/23.+, /3

9et)een %roups

Within %roups

5otal

Sum of

Suares !f Mean Suare Si".

Multiple Co!parisons

epen!ent 8ariable: Chan"e

Scheffe

-*.++/+ *./,0 .033 -2.31 1.11

-2.1+++4 *./,0 .+1/ -.*10 -.2/0

*.++/+ *./,0 .033 -1.11 2.31

-.13/+ *./,0 .**+ -0.*/11 ./01

2.1+++4 *./,0 .+1/ .2/0 .*10

.13/+ *./,0 .**+ -./01 0.*/11

#;$ %roup

Suat

Cleans

Control

Cleans

Control

Suat

#6$ %roup

Control

Suat

Cleans

Mean

ifference

#6-;$ St!. Error Si". (o)er 9oun!


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igure &F= "he #If condition$

$aire% Sa!ples Test

-.0++/+ 2.*2,2 .310*/ -1.,2*+/ *.12++/ -.0/, *3 .2/37re>5est - 7ost>5est7air *

Mean St!. eviation

St!. Error

Mean (o)er

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How to Perform a Two-Way (Within-Between) ANOVA in SPSS