Anova

Study Notes for Applied Research

TWO FACTOR ANALYSIS OF VARIANCE
Jose Luis Ibave G., Ph.D.
H E R E A R E T W O I M P O R T A N T considerations when calculating the two factor ANOVA: first, it is necessary to lay out the data correctly and second, the correct error terms must be chosen for the variance ratios. In this study notes three different types of two factor ANOVA are dealt with: thetwo factor independent measures ANOVA where both the factors, A and B, are independent measures; the two factor mixed design ANOVA where Factor A is independent measures and Factor B is repeated measures, and the two factor repeated measures ANOVA where both Factor A and Factor B are repeated measures.

T

The two factor independent measures ANOVA The simplest two factor ANOVA to calculate iswhere both factors are independent measures. Here the between conditions variance has to be separated into that arising from Factor A, Factor B and the interaction A × B, as in all two factor ANOVAs. As there are individual differences in all sums of squares calculations we can use the within conditions variance as the error term for all three variance ratios. This makes the calculations relativelyeasy. We, therefore, complete the following ANOVA summary table.
THE ANOVA SUMMARY TABLE

Source of variation Factor A Factor B Interaction A × B Error (Within conditions) Total

Degrees of freedom

Sums of squares

Mean square

Variance ratio (F )

Probability

dfA dfB dfA ×B dferror dftotal

SSA SSB SSA ×B SSerror SStotal

MSA MSB MSA ×B

FA FB FA ×B

pA pB pA ×B

1CALCULATING THE TWO FACTOR ANOVA

The results table Organising the results table is important for all ANOVAs but which factor we choose as the rows and which as the columns is not as crucial for the two factor independent measures ANOVA as for the other types of two factor ANOVA, but it is important to get the various totals of the different conditions and combination of conditions correct. Thefollowing data layout is a good example to use for clarity and organisation.11
THE RESULTS TABLE

Factor B Factor A Condition A1 Condition B1 Condition B2 ... ... ... ... ... ... ... ... ... ... Condition Bb

X1 X2 Xn TA1B1

X... X... X... TA1B2 X... X... X... TA2B2

X... X... X... TA1Bb X... X... X... TA2Bb TA2 TA1

Condition A2

X... X... X... TA2B1

Condition Aa

X... X...X... TAa B1 TB1

X... X... X... TAa B2 TB2 ...

X... X... Xabn TAaBb TBb TAa
∑X

2

The formulae for calculation Degrees of freedom: dfA = a − 1 dfB = b − 1 dfA×B = (a − 1)(b − 1) dferror = ab(n − 1) dftotal = N − 1 where n is the number of scores in an AB condition. where N is the total number of scores in the data. where a is the number of condition of Factor A. where b is the number ofconditions of Factor B.

Sums of squares:

SStotal =

∑ X2 − ∑T
nb
2 A

(∑ X )2 N (∑ X )2 N (∑ X )2 N (∑ X )2 N
2 where ∑ TA is TA21 + TA22 + . . . + TA2a

SSA =

−

SSB =

∑T
na

2 B

−

2 where ∑ TB is TB21 + TB22 + . . . + TB2b

SSA×B

∑T =
n

2 AB

−

− SSA − SSB

2 where ∑ T AB is 2 2 2 T A1B1 + T A1B2 + . . . + T Aa Bb

SSerror = SStotal − SSA − SSB −SSA×B (There is an alternative formula for SSerror:
SSerror = SSwith.conds =

∑X

2

∑T −
n

2 AB

Both formulae should give the same answer.)

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CALCULATING THE TWO FACTOR ANOVA

Mean square:
MSA = SSA dfA

MSB =

SSB dfB SSA ×B dfA ×B
SSerror dferror

MSA ×B =

MSerror =

Variance ratio:
FA (dfA ,dferror ) = MSA MSerror MSB MSerror

FB (dfB ,dferror ) =

FA×B(dfA ×B,dferror ) =

MSA ×B MSerror

The F values are then compared to the table values (Table A.1 in the Appendix) at the chosen level of significance. (The above calculations are based on equal numbers of scores, n, in each of the AB conditions. It is possible to perform this analysis with unequal numbers of scores in each condition, as with the single factor independent measures ANOVA....