***How to Think About Continuous x Categorical Predictor Interactions in Behavioral Neuroscience***

1) is it appropriate to use ancova to "correct for" or "control" a cov when it differs across groups or conditions? {cf. Miller & Chapman, JAP, 2001}
	* if the slopes are homogeneous (ie, the Cov x IV interaction is NOT significant) and there is NOT a significant main effect of grps/condition on Cov, an unequivocal "yes"
	
	* if the conditions/grps are matched on the cov, but the slopes are NOT homogeneous, then "yes kinda" in that you can legitmately examine the main effect of grp/condition (residualized on cov) so long as you also interpret the interaction...as with any normal interaction
	
	* but if the groups/conditions differ, "no" -- esp. in cases where pre-existing/non-randomized (patients/controls) groups differ on the cov (iq) 

2) what about Lord's Paradox (1967) -- that sounds intimidating {cf. Miller & Chapman, JAP, 2001}
	* The Question: Does college diet impact weight differentially in boys (who are pre/post heavier) vs. girls (pre/post lighter)
	* The Underlying Truth: Neither group changes from pre to post, on average
	* The Paradox: ANOVA/TTest on pre vs. post-scores show no main effect or interaction, whereas an ANCOVA (using pre as cov) reveals a Sex effect, such that boys weight gain is greater than girls after removing the influence of pre weight
	* The Interpretation: regression to the mean, boys who start the year lighter (more like girls) tend to gain more weight, whereas the opposite is true for heavier girls
	* The Lesson: don't use ancova in the presence of a grp difference (esp w/o randomization) in the cov

	Note: If we repeat the ANCOVA this time using SCALED pre weights- that is we subtract the group mean from every individual- the sex effect disappears! That is, there is no difference in expected weight gain for a man and woman who are at the same weight in september RELATIVE to their group mean (eg, 180 for men, 130 for women). Z-Scores w/in-grp would have the same effect.

	* But
		* one key issue in my mind is that Lord did not include a control condition for the impact of college diet (and so could not distinguish a true effect from that of the passage of time per se)
		* if he had, the pro-ANCOVA statistician would have found similar effects across diets -- indicating ns difference from the passage of time alone
		* a second key issue is that the pro-ANCOVA statistician did not report or interpret the grp x cov interaction...if s/he had, s/he would have reported (for the original design w/ no control condition) that the impact of diet was such that lighter boys tend to gain weight, whereas heavier girls tend to lose weight
		* this would provide a more accurate assessment of what really happened and been statistically legitimate -- it's an open (substantive/scientific rather than statistical) question whether that result is "interesting"

3) so what can i do if the grp's/conditions differ on the cov? {cf. Miller & Chapman, JAP, 2001}
	* there are several approaches, but one is to match across grps/conditions on the cov
	* problem = Lord's Paradox = regression to the (true pre-matching population) mean



4) 

5) what about matching groups/conditions 

6) what about the case of correcting changes in acc'y for changes in RT?

7) what about the case of correcting changes in alpha-band eeg for changes in emg?
	
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