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Dear All,

I have a logistic regression model as below

logit(y)=b0+b1*x1+b2*x2+b3*x3+b4*x4+b5*x3*x4


PS: There were other scores in the model whose correlations were higher than the above but their interaction didn't turn out to be significant.

You can have a significant interaction without significant main effects and without collinearity; collinearity is a relationship among the independent variables; effects (whether main or interaction) are about relations between the dependent and independent variables

In your case, one IV is significant and the other is not, and the interaction is. It is easy to create data that matches this. In R

x1 <- rnorm(100)
x2 <- rnorm(100)
int <- x1*x2
ylin <- 10*x1 + 5*x1*x2 + rnorm(100, .1, .02)
ylog <- ylin > 0.8
model <- glm(ylog~x1+x2+x1*x2)

and there is no correlation between x1 and x2


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Peter Flom
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Peter is absolutely correct.

Interaction terms may, however, be collinear with main effects. You can alleviate this by centering the variables around their mean before you create the interaction term. I think you simply have a significant interaction, but I thought I would pass along this information as well.

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JuliAnna Smith
Methodology Consultant
Center for Research on Families
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Original Message:
Sent: 07-04-2012 16:32
From: Peter Flom
Subject: Interaction Significant, Simple Main Effects Not Significant


You can have a significant interaction without significant main effects and without collinearity; collinearity is a relationship among the independent variables; effects (whether main or interaction) are about relations between the dependent and independent variables

In your case, one IV is significant and the other is not, and the interaction is. It is easy to create data that matches this. In R

x1 <- rnorm(100)
x2 <- rnorm(100)
int <- x1*x2
ylin <- 10*x1 + 5*x1*x2 + rnorm(100, .1, .02)
ylog <- ylin > 0.8
model <- glm(ylog~x1+x2+x1*x2)

and there is no correlation between x1 and x2


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Peter Flom
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A significant interaction essentially indicates that the response for one variable depends on the level of another variable. Removing the interaction from the model does not make this relationship go away. Because of this relationship, significance of  the variables (ie main effects) should not be tested because such significance is overall statement; an overall statement can not be made because the interaction exists.
David

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David Bristol
Statistical Consulting Services
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here is my take:

1. In the scope of studies we do, you can almost never prove that no clinically important interaction exists.  There is no statistical test that can prove this, outside of huge studies whose coinfidence regions are tightly wound about zero. So assume interactions exisr.

2. Main effects can be hugely impoortant even when interactions exist.  For example in a 2 by 2 factorial study, main effects represent the average treatment A effect over the two settings of B.  For example, whether or not there is a treatment by gender interaction in a clinical trial, the main effect of treatment is an important parameter to estimate, and likely is the primary outcome variable in the study.

Jon

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Jon Shuster
University of Florida
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Original Message:
Sent: 07-04-2012 16:44
From: David Bristol
Subject: Interaction Significant, Simple Main Effects Not Significant

A significant interaction essentially indicates that the response for one variable depends on the level of another variable. Removing the interaction from the model does not make this relationship go away. Because of this relationship, significance of  the variables (ie main effects) should not be tested because such significance is overall statement; an overall statement can not be made because the interaction exists.
David

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David Bristol
Statistical Consulting Services
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Both answers sound reasonable to me. In an "intention to treat" analysis, report the main effects. In a "per protocol" analysis, emphasize the interactions.

Chuck

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Charles White
Senior Biostatistician
Walter Reed Army Institute of Research
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Original Message:
Sent: 07-05-2012 08:20
From: Jon Shuster
Subject: Interaction Significant, Simple Main Effects Not Significant

here is my take:

1. In the scope of studies we do, you can almost never prove that no clinically important interaction exists.  There is no statistical test that can prove this, outside of huge studies whose coinfidence regions are tightly wound about zero. So assume interactions exisr.

2. Main effects can be hugely impoortant even when interactions exist.  For example in a 2 by 2 factorial study, main effects represent the average treatment A effect over the two settings of B.  For example, whether or not there is a treatment by gender interaction in a clinical trial, the main effect of treatment is an important parameter to estimate, and likely is the primary outcome variable in the study.

Jon

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Jon Shuster
University of Florida
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There may be counterexamples, but when I have encountered that type of pattern, it is typically related to collinearity as you suggest, but usually involving the crossproduct term even more strongly than the correlation of the two linear terms.  To confirm I would look at the correlation matrix (and if possible the eigenvalues) of the three variables x3, x4, and x3*x4.  It is sometimes possible to reduce the collinearity by expressing the crossproduct term as (x3-x3mean)*(x4-x4mean).

But regardless, using the hierarchical principle of including all lower order terms if a higher order term is significant, I would keep all three terms x3, x4, and x3*x4 in the model.

Best regards,

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Michael Morton
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Dear All,
Thank you for your suggestions and comments, they are very helpful.
I really appreciate your time.
Best regards,
Tasneem

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[Tasneem] [Zaihra]
[Post Doctoral Fellow]
[McGill University]
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Dear Tasneem,

We need to remind ourselves that we must always think about the data generating process when addressing questions like this. My favorite example is to imagine that we are sampling from a population of rectangles and we have data on A=area, L=length, and W=width.

Clearly, the best model is A=L*W. No main effects and no intercept. Just an interaction term.

I hope this helps you think through your problem.

Best regards,

-- Tom

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Thomas Sexton
Professor and Associate Dean
Stony Brook University
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