Interactions involving within subjects factors - a 2x2 (WW) example
We illustrate two approaches for assessing interactions in an example where each of three subjects tried to recall the same items using two types of memory (working memory and long-term memory) each under two conditions (non-interference and interference). Each subject had four recorded responses corresponding to each memory/condition combination representing the number of times an item was correctly recalled with one memory but not using the other (a 'mismatch'). Ntrials represents the total number of mismatches (ie the sum of the two frequencies corresponding to items recalled correctly using one test but not the other). This can be seen in the table below with each pair of of frequencies having the same value of Ntrials formed by their summation.
Freq_he_eh |
Ntrials |
Pair |
Cond |
Duplicate |
Memory |
||||||
2 |
8 |
1 |
1 |
1 |
1 |
||||||
6 |
8 |
1 |
1 |
1 |
2 |
||||||
10 |
12 |
1 |
2 |
2 |
1 |
||||||
2 |
12 |
1 |
2 |
2 |
2 |
||||||
5 |
8 |
2 |
1 |
1 |
1 |
||||||
3 |
8 |
2 |
1 |
1 |
2 |
||||||
2 |
8 |
2 |
2 |
2 |
1 |
||||||
6 |
8 |
2 |
2 |
2 |
2 |
||||||
7 |
9 |
3 |
1 |
1 |
1 |
||||||
2 |
9 |
3 |
1 |
1 |
2 |
||||||
11 |
14 |
3 |
2 |
2 |
1 |
||||||
3 |
14 |
3 |
2 |
2 |
2 |
||||||
The memory by condition interaction can now be tested using the Cox Model syntax below (usable when one of the factors has only two levels). There is no evidence of a memory by condition interaction (chi-square(1)=0.329, p=0.56) using this model.
WEIGHT BY FREQ_HE_EH. COXREG cond /STATUS=duplic(1) /STRATA=pair /CONTRAST (mem)=Indicator /METHOD=ENTER mem /PRINT=CI(95) /CRITERIA=PIN(.05) POUT(.10) ITERATE(20) .
The interaction can also be tested using generalised estimating equations (GEEs) giving the near identical result of chi-square(1)=0.334, p=0.56 again showing no evidence of a memory by condition interaction. GEEs can be used for testing differences in repeated counts across factors with any number of levels.
weight off.
GENLIN freq_he_eh OF ntrials BY mem cond (ORDER=ASCENDING)
/MODEL mem cond mem*cond INTERCEPT=YES
DISTRIBUTION=BINOMIAL LINK=LOGIT
/CRITERIA METHOD=FISHER(1) SCALE=1 MAXITERATIONS=100 MAXSTEPHALVING=5 PCONVERGE=1E-006(ABSOLUTE)
SINGULAR=1E-012 ANALYSISTYPE=3(WALD) CILEVEL=95 LIKELIHOOD=FULL
/REPEATED SUBJECT=pair WITHINSUBJECT=mem*cond SORT=YES CORRTYPE=INDEPENDENT ADJUSTCORR=YES
COVB=ROBUST MAXITERATIONS=100 PCONVERGE=1e-006(ABSOLUTE) UPDATECORR=1
/MISSING CLASSMISSING=EXCLUDE
/PRINT CPS DESCRIPTIVES MODELINFO FIT SUMMARY SOLUTION.