If you are talking about a designed experiment then you should start with the full model (all interactions included). You might look at:

https://www.amazon.com/Design-Analysis-Experiments-Douglas-Montgomery-ebook/dp/B071K7NZ7V/ref=sr_1_3?keywords=montgomery+design+and+analysis+of+experiments&qid=1557957009&s=gateway&sr=8-3

Best, David Booth

Dear Seema,

while I would agree to David in the sense that defining something different than the full design in a test-model is very uncommon, I would still say that there are ways to justify such a reduction, which depends on theoretical/methodological information you did not provide. I try to illustrate this in an example:

Assume your DV is 'response (option A vs. option B)', which is assessed in

X:condition factor (dichotomous)

Y: training trials (continuous variable)

So Y implies that there is some response training, which has some impact on the DV. And a question could be, whether being in one of the conditions impacts this learning (X:Y).

So, here the full model would be

1) DV~ X + Y + X:Y

but this model:

2) DV ~ Y + X:Y

would be theoretically more meaningful if the assumption is valid that in each group (X) of participants there is no difference in the overall response variable in the beginning of training (i.e. they have the same intercept). This would be theoretically more meaningful in the definition of the above research question, because the overall 'response average' in each group emerges as a function of 'basic learning Y' and learning:condition interaction. In the end, thus, there will be overall (or marginal) differences between the levels of X, but this would be fully accounted by X:Y.

If this is analogue to what your design or question looks like you could simply focus on X:Y, indeed. However, statistically, this is kind of tricky, also depending on how the variables are coded. For instance, if Y would be mean centered, then the difference between the factor levels of X would be (marginally) evaluated in the middle of learning, which, thus, would reflect the case of partial redundancy, if different learning curves in each condition have led to some difference thus far. Alternatively, Y could be defined as having an actual 'zero-point', and in this case, the X levels would be evaluated as intercepts at the beginning of the training. In this case, it is more likely to 'get the X:Y interaction significant' (at least in likelihood ratio tests [LRT] between full vs. restricted model), because X does not capture variance of the training effect (in the LRT procedure, explained variance is crucial), but asks whether there is a difference in the onset in each group. And you see, in this case, it is actually still meaningful to have X in there, and to separate X:Y from X, is always a matter of how to define the model.

Hope this helps.

Best, René

Ps:

And if you are concerned, e.g. in cases other than in the example above, that including additional (Y, Z) predictors in a model actually reduces the chance to detect an effect of a very particular predictor (e.g. X:Y:Z), then there are two answers: 1) no, the opposite is true in standard multiple regressions: Including covariates reduces error variance, which means, that the 'to-be-explained' variance by your predictor becomes less. And the test-criterion of 'significant' predictor-terms in standard regression is this reduced error-variance. And with decreasing error distributions, effects are more likely to be considered as non-accidental. and 2) If you do LRT model tests, then this should not matter much (I think..., at least if the predictors are uncorrelated) because it is always and only about changes in explained variance, when adding or removing predictors, which does not depend on how much (uncorrelated) covariates are in the model. Which is a different question.

When you test a model with interactions you DO have to include ALL the constitutive terms. If not, your estimations will be biased.

This is not a matter of opinion, it can be demonstrated. See Brambor, Clark & Golder 2006: https://doi.org/10.1093/pan/mpi014.

I hope this was useful Seema Bihari, René Schlegelmilch, David Eugene Booth

@andres.the result you cite is well-known. However just because an estimator is biased doesn't mean it is of no use. As an example see the early papers on ridge regression. D. Booth

David Eugene Booth you are right of course with your 1st answer (which also points to a full model) and also regarding biased estimators. For instance one might trade bias for efficiency. But assuming a large database, one more coefficient would be almost costless.

On leaving out a main effect Z (constituents of an interaction XZ) the authors in the referenced paper say, that there are two common arguments when this is actually done:

"First, some claim that they do not believe that Z has any effect on Y on average and that, as a result, they do not need to include it as a separate term in the model. Second, others claim that they do not believe that Z has an effect on Y when X is zero and this means they can exclude it as a separate variable from their model. Below, we show that the first of these claims is never justified and that the second claim is rarely defensible."

The example I give above is one of those rare defensible cases... Which is actually not that rare... (every learning process starts at 0)

Best, René

Ps: The abstract reveals that the arguments are tailored for a political science audience. Taking a psychological perspective, for instance, changes a lot in their plausibility arguments.

Pps: Including (nonsensical) main effects (i.e. where Z, by the definition of the research design, can not have any effect on Y if X is 0) leads to nonsensical parameter estimates in an ANOVA regression with interactions.

Seema Bihari

I think, if you do so as you asked, you want to cite the paper posted by Andrés, to show that you are aware of the debate :).

Anyway, you can easily sanity-check whether removing Z is "ok" or not.

Assume you have Y~X+Z+X:Z, where X is continuous, and Z is categorical.

If the X scale has a natural zero-point (like temperature), then you can think about dropping Z. If X has no natural 0 point, keep it.

Reason: Let X have a natural zero-point like 'temperature'. If the temperature is 0 (i.e. 0 Kelvin; leaving quantum physics aside...) then manipulating a gas in two conditions somehow (z1 and z2) will have no effect, because 0 molecular movement, and z1=z2 by definition. But the crucial aspect is: you conduct a logistic regression, in which X=0 (without Z) will always predict 50%, and if this is a viable assumption in your case, you can drop Z, otherwise not. In other words, if you "need to find out at which location on the X-scale, the data is likely to be 50%, then you have to keep an intercept, and if you don't know whether the 50% threshold on X differs depending on Z, then you also need to keep Z, because that's what Z is estimating. But if there is no other possibility that X=0 is your 50% anchor ... :)

Ps:

Note, that in this case of a natural scale (with an actual 0-point, and no negative values), and without Z, your logistic regression in praxis is not an effective sigmoid anymore, but grows from 50% to 100% as a limited growth function, which might be or not be a good way to model data... This also implies, that there is a very strong assumption in the model, that Z affects X in that it more or less strongly decelerates the growth function, meaning that the influence of Z on the outcome is -only- defined by the question whether the growth on X is rather linear or rather decelerating (exponentially) in the Z conditions, which might be another way of looking at: "Is this the model you want?" With Z intercepts, the question adds, as described above, whether the inflection point of the sigmoid differs between the Z-conditions.

Best, René

René Schlegelmilch Andrés Santana David Eugene Booth I am assuming that when all the three independent variable are present only then we will have an effect on dependent variable. Though I do not have ant theoretical backing for it, it appears to be logical.

@Seema, It may seem logical BUT IT IS NOT TRUE.SEE for example Douglas Montgomery, Design and Analysis of Experiments. Best, David Booth

David Eugene Booth René Schlegelmilch Andrés Santana theoretically I am convinced that i should take all the interactions into consideration. However, I was looking at some explanation for looking at only the interaction that I am interested in. Looking at the suggested literature and literature by Jaccard ( 2001, 2003), it seems that taking all interaction would take care of bias.

@Seema Bihari You must consider all terms in your model at the start unless you have a hypothesis that says some interactions may be ignored in which case the hypothesis is that the coefficient of that term in the model =0. To not do so means you are not testing your original research question. David Booth BTW dropping terms that belong in a linear statistical model increases not decreases bias as you seem to have supposed(if I understood your comment as you meant it). D. Booth

Dear David, I assume that dropping terms would increase bias and not otherwise. David Eugene Booth René Schlegelmilch Andrés Santana 'I appreciate your inputs and thank you for the help.

Glad we helped. good luck with your research.