Graphing versus F-test: contradictory outcome Following the advice here regarding statistical significance versus substantive meaning of a model, I did a scatterplot with 'fit line at subgroups'.
My graph clearly shows that there is an interaction between X1 and X2 on Y. However, the F test of the X1*X2 interaction is insignificant. I am using SPSS GLM. X1 is significant but X2 is insignificant and the product term (X1*X2) is insignificant as well.
Question: How do I reconcile these contradictory outcomes? (i.e. see an interaction in the graph but not seeing a significant F test result in GLM)
I understand the importance of substantive meaning. However, sooner or later, someone may ask me why I am graphing the X1 and X2 on Y when the product term is insignificant. 
(In the Parameter Estimates table in the SPSS GLM outcome, the t test for each level of X1 with X2 (e.g. maleX1 and femaleX1) is signficant.) 
Please see graph below:

 A: Well... the slopes on the graph you show are the sames. Well, not exactly the same, but too close to be sanely considered as really different (taking into account the small number of data points).
The following R code generates a plot roughly similar to your plot. The slopes of the two regression lines are estimates of the same "true value" of 0.8. They are slightly different of 0.8, it’s because they are just estimates of an unknown true value.
X1 <- rnorm(100, 2.5)
X2 <- rnorm(100, 2.5)
Y1 <- 0.8*X1 + rnorm(100)
Y2 <- 0.8*X2 + rnorm(100)
plot(X1, Y1, pch=15)
points(X2, Y2, pch=19)
abline(lm(Y1~X1))
abline(lm(Y2~X2),lty=2)


I hope that seeing this phenomenon on simulated data (where you can see the "true value") will help you to understand.
A: Statistical significance is partly a function of sample size.  Large effects are interesting, whether they are significant or not. Even Fisher said that you  should test everything with the same p-value (I read a quote from him somewhere along the lines of "no sane researcher would use the same critical value all the time" but I can't find it right now).
You could read Ziliak and McCloskey The Cult of Statistical Significance or, for a less polemical view, Abelson's Statistics as Principled Argument. I reviewed the latter on my blog; but, briefly, Abelson argues that we should evaluate statistical arguments on the following criteria

*

*Magnitude How big is the effect?

*Articulation How precisely stated is it?

*Generality How widely does it apply?

*Interesting How interesting is it?

*Credibility How believable is it?

Notice that "significance" is not listed.
