Interpretation of a regression with only one independent variable: a 2x2 interaction.

#1
Hello all,

This is my first post. I am a newly-begun career-researcher and while I have come here now for help, I intend to continue to use this forum and offer my assistance to others where I can in the future.

Here is the issue I'm having right now. I am analyzing a study on Down Syndrome (DS) children, looking at counting scores. The study has a 2x2 design; group (DS v TD (typically developing) and age (young v old). Two independent-samples t-tests, one conducted for each group, show that within the DS group older Ps perform significantly better than younger (p<.001), while in the TD group there is no difference between ages (p=.222).

What I now want is the cleanest way to demonstrate that there is a significant difference of-the-effect-of-age-on-counting between groups. Now, you might tell me, just add group, age, and group*age into a regression with counting as independent variable, and read off the significance of the interactive variable.

My issue with this is that this will actually tell me whether the interactive variable is a significant unique predictor over and above age and group by themselves. This is not what I want to know. For theoretical reasons I just want to know whether there is an interactive effect all by itself.

My initial answer to this was that I would run a regression with just one independent variable, the interaction itself. This lumps DS young with TD old, and DS old with TD young. However, I have been told by an old lecturer who has enough time to tell me I'm wrong but not enough to tell me why that this is incorrect. He says that you must include all simple variables that went into producing an interaction in a regression in order to be able to justifiably interpret the interaction variable itself.

Is he correct? If so, why am I wrong? And if I am wrong, is there some other, legitimate way of checking the interaction without it being over and above the effects of the simple variables?

Thank you very much for reading and/or replying,

Stephen
 

Karabiner

TS Contributor
#2
Re: Interpretation of a regression with only one independent variable: a 2x2 interact

My issue with this is that this will actually tell me whether the interactive variable is a significant unique predictor over and above age and group by themselves.
It's not simply additive. The parameters for age and group
will change after adding the interaction term. The "main"
effects are then conditional on the interaction effect.
This is not what I want to know. For theoretical reasons I just want to know whether there is an interactive effect all by itself.
This makes not enough sense. How can there be an
interaction if you don't include the respective variables?
If you only use the group(0/1)*age(0/1) interaction as
predictor, then you will actually perform a t-test, comparing
(e.g.) older (code=1) DS (code=1)-subjects with all
other subjects.

With kind regards

K.
 
#3
Re: Interpretation of a regression with only one independent variable: a 2x2 interact

Hi Karabiner,

Thank you for your reply. In regressional analysis you can make an interaction variable by , in my case, coding (TD old and DS young) as one group, and (TD young and DS old) as another group. This does not compare older DS to 'all other groups', it compares the groups as described above. The point in the variable is that it mixes the combination of age and condition (DS or TD), such that both groups have TD, DS, old and young participants - only the combination of these factors have changed. Therefore if you get a significant effect you know that there is an interaction between age and condition (i.e. the combination of these two variables is important).

What I don't understand is why it would be incorrect to only include this variable in a regression all by itself, without age or condition.

edit: Also the groups are not coded (0/1), I use (-1/1). This allows you to make an interaction variable by simply multiplying.

Thank you,

Stephen
 
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#4
Re: Interpretation of a regression with only one independent variable: a 2x2 interact

Excluding the constituitive terms is a big no-no. I would say that your interpretation of interaction terms in regression models is flawed. Hopefully the discussion below helps you understand why you must always include x1 and x2 if you use the product term (x1*x2).

Your model should look like...

y = b0 + b1*x1 + b2*x2 + b3*x1*x2

If you want to know the effect of x1 in the above specification you take the partial derivative of y with respect to x1 and you get...

(partial y/partial x1) = b1 + b3*x2

In the above specification when x2 = 0, then the effect of x1 on y equals b1 (something you estimate in your model).

What you are proposing is the following...

y = b0 + b1*x1*x2

In this case, if you want to know the effect of x1 on y then you take the partial derivative of y with respect to x1 and find....

(partial y/partial x1) = b1*x2

Now when x2 equals zero the effect of x1 on y is zero. This is not something you want to assume, because the probability that it is actually the case is quite literally zero.

Long story short, you must include x1 and x2 if you include x1*x2!