I have one continuous variable, Variable, and two categorical
variables, Factor1 and Factor2, each comprising two levels. What does
it mean if 


1) According to a t-test, the difference in Variable between the two levels in
Factor2 are statistically significant only for a subset of Factor1
(only within one of its levels). 


2) The pairwise comparisons according to Tukey's Honest Significant
Difference test and the F-statistic for each of the categorical
variables in a 2-way ANOVA both show that the influence of these
categorical variables are not significant (and interactions also not significant).


?


This seems like an elementary textbook question but my lack of formal statistical training is beginning to show...


Thank you in advance for your responses.



Interpretations among t-test, ANOVA, Tukey HSD results?
It is very often the case that some subset of your data will come out to be statistically significant by random chance. If you are running t-tests among the levels given the level of each other factor, that's four tests. Your chance of one of those four comparisons being significant at $\alpha=0.05$ is $1 - (0.95)^4$, or about 20%.


The ANOVA, F-test, and HSD all account for this kind of multiple comparison. It's what they were designed to handle.


It's actually quite a nice example of why multiple comparisons are important to take into account.


