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From |
David Muller <[email protected]> |

To |
[email protected] |

Subject |
Re: st: Relative Importance of predictors in regression |

Date |
Wed, 6 Nov 2013 15:38:22 +0100 |

I may be misunderstanding or mischaracterising David Hoaglin's problems with the term "holding constant" for describing adjustment for covariates in multiple regression, so forgive me for interjecting if I am off the mark. I think the main issue is that the data used to fit the model won't necessarily support a difference/change in one variable with all other variables held constant. This is trivially the case when, for instance, both x and x^2 are used as predictors. When data are sparse or continuous it is also unlikely that there will be observations that differ on one variable but are _identical_ on all others. Personally, I don't think this is a big deal. If one sees regression coefficients as differences in conditional expectations, then the "held constant" interpretation is just a model-based interpolation or extrapolation. It's up to the person fitting and interpreting the model to justify any such extrapolation. All the best, David Muller On 6 November 2013 01:19, Lucas <[email protected]> wrote: > Dear David, > > I am confused. You first write the following (emphasis capitalization added): > > "I would add a note of caution, however. Nathans et al. (and many > others) interpret a beta weight (or a regression coefficient more > generally) in a way that involves holding all the other predictor > variables constant. The "held constant" part of that interpretation > is not correct. STRAIGHTFORWARD MATHEMATICS shows that it does not > reflect the way that multiple regression actually works." > > In response I wrote: > > "What would be the mathematical expression for "held constant"? And > what is the mathematical expression to which you are comparing it that > leads you to reject "held constant"? Thanks a bunch!" > > It seemed to me both pieces of information would be necessary for > someone to rule that one is appropriate and the other wrong (or, at > least, it should be demonstrable that the wrong one has no formal > expression). To this David replied: > > "I'm not sure what you mean by "the mathematical expression for 'held > constant,'" other than setting each of the other predictors to some > particular value." > > This latter reply suggests David and I agree that a mathematical > expression will be an equation--not a derivation. I responded, > writing: > > "I presumed you had a mathematical representation of the two > interpretations and could then show that the former is wrong because > the actual regression model is accurately represented by the latter. > However, instead of a formula, you provided more text, which is > necessarily somewhat imprecise." > > In that message I introduced a critique of David's use of change when > difference is generally correct--the aim of doing so was to suggest > that maybe we all can cut each other some slack. I had expected David > to just say, "Sure, yeah, that's right, my bad" but David resists this > obvious fact. Okay, fine--it's a general discussion, but he prefers to > use the specific language. Anyway, David does address the request for > a mathematical expression by responding that: > > "I do have all the necessary mathematical expressions for the proper > general interpretation. A plain-text message, however, is not > suitable for displaying them. I am not aware of a mathematical > representation of the "held constant" interpretation in the > n-dimensional geometry in which ordinary least squares operates. It > is easy to represent the "held constant" interpretation in the > p-dimensional geometry, but that is not the relevant geometry. The > absence of a representation for the "held constant" interpretation in > the n-dimensional geometry is evidence for its lack of validity. If > you have a suitable representation in mind, I would be interested in > seeing it." > > I have not offered a representation because I have not maintained one > is right and the other wrong, so it seems I would not be required to > distinguish two things I am not sure can be distinguished. In an > effort to understand David's point, every response I have written > since has been asking for one simple thing: Where can I find this > point made in n-dimensional geometry? > > Other matters are not directly relevant--David won't accept that if > you have 2 terms, one general, and one specific, the general applying > everywhere, the specific applying in a smaller subset, one should use > the general language. Pedagogically and scientifically this seems > obvious. Okay. This just means this is not the ideal speech > community one might have hoped. Still, I ask--which of the two > textbooks David mentioned have the n-dimensional expression David > intimated existed? Do either of them have it? Both? Neither? If > neither, is there another citation to which I (we?) could turn? Just > answering this question with the relevant citation(s) would be > immensely helpful. Of course, it is not your job to be helpful. But > you've made this point several times on statalist, which led me to > think you might want people to get the point. I'm asking for help in > getting the point. Rather than more analogies and your plain text > derivations (which you indicate are intrinsically sub-optimal), a > citation I (and perhaps others) can peruse would be incredibly > helpful. > > Again, thanks a bunch! > > Sam > > On Tue, Nov 5, 2013 at 9:26 AM, David Hoaglin <[email protected]> wrote: >> Dear Sam, >> >> It would help communication if you explained, as specifically as >> possible, what sort of "mathematical expression" you are looking for. >> >> The material in my previous message that you reject as a "mathematical >> manipulation" needs only one further step, involving straightforward >> algebra: In the result of regressing the Y-residuals on the >> X2-residuals, multiply out the right-hand side, rearrange the equation >> to leave only Y on the left-hand side, and compare the result term by >> term against the original model. Since the adjustments for the >> contributions of the other predictors are shown explicitly, the >> interpretation of b2 is clear. Please explain how you would interpret >> the demonstration differently. >> >> The fact that regression coefficients are a type of slope does not >> provide any basis for the "held constant" interpretation. I do not >> see the connection between a regression model and your analogy of the >> position of two people on a hill. Please explain further. >> >> When you said that I "retain one mis-interpretation of the regression >> model that is extremely elementary and easily corrected," I assume you >> are referring to the distinction that you make between "change" and >> "difference." I explained earlier that I would use words appropriate >> to the particular context and application, so I am not making any >> mis-interpretation. >> >> I remind you that you have not offered any mathematical expression for >> the "held constant" interpretation. >> >> Regards, >> >> David Hoaglin >> >> On Tue, Nov 5, 2013 at 9:37 AM, Lucas <[email protected]> wrote: >>> Hi David, >>> >>> I am looking for the mathematical expression you indicated would make >>> it clear which interpretation is correct. The mathematical >>> manipulation isn't very helpful, because someone who interprets the >>> issue differently than you do before can interpret this demonstration >>> differently than you do. So, do either of those books have the >>> mathematical expression you mentioned? If so, I'll check it out. >>> >>> On change vs. difference, discrete things change or do not, and >>> non-discrete things change or do not. The distinction between "change >>> and difference" is orthogonal to the distinction between "discrete and >>> non-discrete." >>> >>> Indeed, the analogy you deploy to support the change interpretation, >>> using slopes and hills, is one reason people say "held constant." The >>> difference (slope) between my height on the hill and Joe's height on >>> the hill is distinct from (and independently estimable given) our >>> horizontal placement on the hill. Horizontal placement, thus, is "held >>> constant." If this is incorrect, it shows why analogies are less >>> helpful than mathematical expressions. Thus, my request for the >>> mathematical expression you indicated was available. >>> >>> I do not understand why you retain one mis-interpretation of the >>> regression model that is extremely elementary and easily corrected, >>> but are adamant that everyone else is wrong if they use (what you >>> call) another mis-interpretation of the model, a mis-interpretation >>> that 1)can be shown with straightforward mathematical expressions but >>> then 2)seems so complex that it cannot be written in plain text. >>> >>> Anyway, please let me know which of those textbooks have the >>> mathematical expression you referenced earlier. I'll pull it from the >>> library and take a look >>> >>> Thanks! >>> >>> Sam >> * >> * For searches and help try: >> * http://www.stata.com/help.cgi?search >> * http://www.stata.com/support/faqs/resources/statalist-faq/ >> * http://www.ats.ucla.edu/stat/stata/ > * > * For searches and help try: > * http://www.stata.com/help.cgi?search > * http://www.stata.com/support/faqs/resources/statalist-faq/ > * http://www.ats.ucla.edu/stat/stata/ * * For searches and help try: * http://www.stata.com/help.cgi?search * http://www.stata.com/support/faqs/resources/statalist-faq/ * http://www.ats.ucla.edu/stat/stata/

**Follow-Ups**:

**References**:**st: Relative Importance of predictors in regression***From:*Nikos Kakouros <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*David Hoaglin <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*Lucas <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*David Hoaglin <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*Lucas <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*David Hoaglin <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*Lucas <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*David Hoaglin <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*Lucas <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*David Hoaglin <[email protected]>

**Re: st: Relative Importance of predictors in regression***From:*Lucas <[email protected]>

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