# How to interpret 3-way interaction with one continuous variable in mixed model

I have problems interpreting the direction of the effects in my model. Can you help me with this?

I conducted an experiment which includes

• between subject factor: group=2 (coded as 0-1)

• within subject factor:

• stimulus type=3 (coded as -1 0-1)
• empathy questionnaire which I centred (subtracted the mean). Participants could answer 1-4 on each item and I calculated their mean answer.
• another questionnaire which I centred (subtracted the mean)

The depenendent variable is categorical: yes/no (0-1)

The SPSS syntax I used is shown below:

MIXED in_team_yes_no BY stimulustype group WITH empathy PT_centred
/CRITERIA=CIN(95) MXITER(100) MXSTEP(10) SCORING(1) SINGULAR(0.000000000001) HCONVERGE(0, ABSOLUTE) LCONVERGE(0, ABSOLUTE) PCONVERGE(0.000001, ABSOLUTE)
/FIXED=stimulustype group empathy stimulustype*group stimulustype*empathy group*empathy stimulustype*group*empathy PT_centred stimulustype*PT_centred group*PT_centred stimulustype*group*PT_centred | NOINT SSTYPE(3)
/METHOD=ML
/PRINT=COVB DESCRIPTIVES  SOLUTION TESTCOV
/RANDOM=INTERCEPT | SUBJECT(id) COVTYPE(VC)
/EMMEANS=TABLES(OVERALL)
/EMMEANS=TABLES(stimulustype*group) .


In the output, there is an interaction between group*stimulus and type * empathy. But what is significantly larger versus what? I hope you can clarify this for me.

This is part of the output:

**Type III Tests of Fixed Effectsa**

stimulustypeNumerator df=2;Denominator df=175.999999999999;F=25.1625067300411;Sig.=2.44708998141032E-10;=;

groupNumerator df=1;Denominator df=88.0000000000013;F=0.182104461000037;Sig.=0.670613087293204;=;

empathyNumerator df=1;Denominator df=88.0000000000006;F=2.64490681880741;Sig.=0.107458165008125;=;

stimulustype * groupNumerator df=2;Denominator df=176.000000000002;F=0.393575958791748;Sig.=0.675232333852088;=;

stimulustype * empathyNumerator df=2;Denominator df=175.999999999999;F=0.436679732184867;Sig.=0.646876527043373;=;

group * empathyNumerator df=1;Denominator df=88.0000000000007;F=0.0205898398771826;Sig.=0.886230015228424;=;

stimulustype * group * empathyNumerator df=2;Denominator df=176.000000000002;F=3.85067166016453;**Sig.=0.023**0796672084849;=;

PT_centredNumerator df=1;Denominator df=88.0000000000006;F=0.780733693066422;Sig.=0.379324440222066;=;

stimulustype * PT_centredNumerator df=2;Denominator df=176.000000000002;F=0.0268007378853535;Sig.=0.973559187279457;=;

group * PT_centredNumerator df=1;Denominator df=88.0000000000007;F=0.265321415628692;Sig.=0.60777909887368;=;

stimulustype * group * PT_centredNumerator df=2;Denominator df=176.000000000002;F=0.357575654993585;Sig.=0.699876675215139;=;


Table: Estimates of Fixed Effects (this is output from the same model)

[stimulustype=-1];Estimate=3.29244254001684;Std Error=0.225268732947286;df=216.95359688355;t=14.6156215154248;Sig.=3.9122956206142E-34;

[stimulustype=0];Estimate=3.11592342177773;Std Error=0.225268732947286;df=216.95359688355;t=13.8320280005609;Sig.=1.2853156112195E-31;

[stimulustype=1];Estimate=2.18642210821693;Std Error=0.225268732947286;df=216.95359688355;t=9.70583924191808;Sig.=9.98676090898553E-19;

[group=.00];Estimate=-0.0871216158027804;Std Error=0.330560977978059;df=216.953596883549;t=-0.263556867285657;Sig.=0.79237155076331;

[group=1.00];Estimate=0b;Std Error=0;df=;t=;Sig.=;

empathy;Estimate=1.17744019177937;Std
Error=0.563748889564515;df=216.953596883548;t=2.08858981999755;Sig.=0.0379111000598545;

[stimulustype=-1] * [group=.00];Estimate=0.322580523902738;Std Error=0.382857667976268;df=176;t=0.842559914257049;Sig.=0.400618137515222;

[stimulustype=-1] * [group=1.00];Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=0] * [group=.00];Estimate=0.253441749589722;Std Error=0.382857667976268;df=176;t=0.661973811127722;Sig.=0.508853817603762;

[stimulustype=0] * [group=1.00];Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=.00];Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=1.00];Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=-1] * empathy;Estimate=-1.34747175897676;Std
Error=0.65293727802683;df=176;t=-2.06370780827342;Sig.=0.0405131556011519;

[stimulustype=0] * empathy;Estimate=-0.940798167490195;Std Error=0.65293727802683;df=176;t=-1.44087066116561;Sig.=0.151397919015566;

[stimulustype=1] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[group=.00] * empathy;Estimate=-1.31554938904041;Std Error=0.75223224868924;df=216.953596883547;t=-1.74886066282421;Sig.=0.0817293498796308;

[group=1.00] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=-1] * [group=.00] * empathy;Estimate=2.10908428142564;Std Error=0.87123981260977;df=176;t=2.42078501337989;**Sig.=0.01**65032903786237;

[stimulustype=-1] * [group=1.00] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=0] * [group=.00] * empathy;Estimate=2.07833508789358;Std Error=0.87123981260977;df=176.000000000001;t=2.38549140869492;**Sig.=0.01**81184114665423;

[stimulustype=0] * [group=1.00] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=.00] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=1.00] * empathy;Estimate=0b;Std Error=0;df=;t=;Sig.=;

PT_centred;Estimate=0.155115894939361;Std Error=0.589136705351013;df=216.953596883549;t=0.263293550597806;Sig.=0.792574217512385;

[stimulustype=-1] * PT_centred;Estimate=0.314817967290445;Std Error=0.682341595519122;df=176.000000000002;t=0.461378830424273;Sig.=0.645096534364194;

[stimulustype=0] * PT_centred;Estimate=0.489619040733416;Std Error=0.682341595519122;df=176.000000000002;t=0.717557076907961;Sig.=0.473981736575375;

[stimulustype=1] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[group=.00] * PT_centred;Estimate=0.134214472802144;Std Error=0.813980628541158;df=216.953596883548;t=0.164886568667718;Sig.=0.869186792494091;

[group=1.00] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=-1] * [group=.00] * PT_centred;Estimate=-0.570241358210639;Std Error=0.942757149156938;df=176.000000000001;t=-0.604865588895908;Sig.=0.546047353847579;

[stimulustype=-1] * [group=1.00] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=0] * [group=.00] * PT_centred;Estimate=-0.767649075872404;Std Error=0.942757149156938;df=176.000000000001;t=-0.814259617716902;Sig.=0.416596578011685;

[stimulustype=0] * [group=1.00] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=.00] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

[stimulustype=1] * [group=1.00] * PT_centred;Estimate=0b;Std Error=0;df=;t=;Sig.=;

a. Dependent Variable: in_team_yes_no.
b. This parameter is set to zero because it is redundant.


What do these t-tests mean, for example the significant three-way interaction which I made bold? What t-tests are conducted exactly? What comparisons are made?

Apart from the interpretation, how should I report these results?

• You can put a link to your image, we will convert this as an inline image for you. – chl Feb 14 '12 at 21:28

It's not exactly clear where these effect estimates in your output are coming from. I'm unfamiliar with SPSS, so bear with me.

I will assume that SPSS has treated all the 0 levels of your factors as "referent" levels. So the intercept represents the mean effect among people having group=0, stimulus=0, and empathy=0. I'm also assuming that empathy (or empathy score) is treated as "continuous".

1. I think the first model is testing for the global interaction between empathy and stimulus. Which is why it's a 2-df numerator F-test. The full model has the additional parameters for levels where [stimulus==-1]*empathy and [stimulus==1]*empathy, hence a difference of parameters of tow.

2. I think the second model is a pairwise test for the individual parameters I mentioned above. In the full model, we have three possible effects for empathy stratified by stimulus whereas in the reduced model there is only one. We examine the differences of empathy across each strata empirically, but there are only two t-tests available to us to describe the pairwise disparity in such effects relative to the null model.

3. I do not understand your output. It's unclear to me whether there were insufficient samples to estimate effects in these groups, or whether SPSS has determined some tests as redundant as before (but is unable to report them).

At any rate, you want to focus on global tests, as was the case in model 1, because the scientific question when testing for interaction is stated as follows, "Is there any stratum in the sample for which the estimated effect of our predictor of interest is significantly different than the stratified effect estimate?" This leads us to the proper interpretation for the p-value of the F-test reported above. You can follow this up by reporting the most significantly different estimated difference in effects based on the pairwise comparisons, but this is unreliable and can be confusing.

I should mention again that, if you're interested in the test of 3-way interaction, then the F-test should have 1 numerator degree of freedom. This is because you should have all 2-way interactions included in the null model. Or perhaps you're interested in whether there are any 2-way interactions instead (giving a 5 numerator degrees of freedom F test). It begs the question whether you're asking for the test you really want here.

• Dear Adam, thank you for your reply! I made some changes in my post. It is a shame I can't upload an image. Do you think for stimulus -1 will be used as the reference you are talking about? Or you think both 'one' and 'minus one' are tested against zero? Maybe my model is incorrect in that case. -1 refers to low threatening faces, 0 to neutral, 1 to highly threatening face stimuli. I have 88 participants equally divided over the two groups. – user9203 Feb 14 '12 at 20:16