Do we have some crossed threads here? I can't tell whether this is about normal quantile plots for univariate data or the plotting of residuals vs. predicted y for multivariate data. But, who's counting? Both are interesting. More comments below. I've edited the original messages somewhat.
----- Forwarded message from KINGB@WCSUB.CTSTATEU.EDU -----
Almost three weeks ago, in a discussion on normal probability plots, I commented as follows:
> So why use Y-hat instead of X, if they provide the same information in > the residual plot? Because in multiple regression there is no e_i vs X > plot; you have only the e_i vs Y-hat plot available. So it's (at least > partly) a matter of foreshadowing later developments. >
Chris Olsen subsequently asked me:
> do you think there is > any reason to use this plot with kids given that __multiple__ > regression is not in the AP Stat syllabus?
I am not anxious to continue that thread, but I don't want to ignore a direct question (any longer than I already have!) either. It's easy for me to say "Yes, I think we should teach normal probability plots to AP Statistics students." But, of course, that's because I don't have the privilege of teaching the course, and am not well-attuned to the problems that arise, or to the tradeoffs that must be considered.
All I can say is that I see many places in elementary statistics where I want to say to my students: "Look: sensible analyses take advantage of BOTH visual evidence and numerical evidence." And there are numerical ways of assessing normality (say, a goodness of fit test). So I want my students to encounter at the same time what I understand to be the best available visual device for assessing normality.
And, finally (I promise!): I think we should not forget Bob Hayden's comments during that discussion. Bob asserted that assessing whether or not a curve is "bell shaped" is not important; that what's important is "symmetry, lack of outliers, and weight of the tails".
----- End of forwarded message from KINGB@WCSUB.CTSTATEU.EDU -----
I agree with Bob !-), which is why I don't think goodness-of-fit tests are very useful here. We routinely use normal approximations with 0-1 data, which I expect would flunk any goodness-of-fit test, but we can still get away with it because the tails are so slim and trim. I think we MUST teach students SOME way to assess normality if we teach them any technique that ASSUMES normality. If I were writing exam questions, I would provide some dangerously non-normal data and say "Jones wants to estimate a typical population value for the variable x, and assess the precision of her estimate. Chose an appropriate typical value and find a 95% CI for it." This could even be multiple choice.
BTW, in multiple regression, you CAN plot residuals vs. each independent variable and the graphs have roughly the same meaning as residuals vs. X OR resids. vs. Y-hat in simple linear regression. This is one of those areas where two ideas coincide in lower dimensions but become separate in higher dimensions. Because of this, I would not teach students BOTH in a first course where there is no good way to explain why there are two different graphs telling you the same thing. It would be like talking about partial derivatives in one-variable calculus.
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