[R-sig-ME] choice of reference category only changes coefficient with uncorrelated random intercept and slope

Peter Dalgaard pdalgd at gmail.com
Wed Sep 13 18:31:49 CEST 2017 

> On 13 Sep 2017, at 17:52 , Emmanuel Curis <emmanuel.curis at parisdescartes.fr> wrote:
> 
> Hi,
> 
> I may be slow or missing something, but I do not clearly understand
> the first point. 
> 
> I mean, even if slope and intercept are correlated, there will be
> dependance between the slope and the value (cov(beta, alpha + beta*x)
> = cov(alpha,beta) + x*V(beta)), and there still will be a special
> point where there is uncorrelation (for x = -cov(alpha,beta)/V(beta)).
> 
> So I don't see why it's less weird than the model without correlation,
> there's just an additional assumption on the location of this special
> point — which of course may be or not sensible depending on the
> interpretation of x.
> 
> Am I missing something crucial?

Not really, and notice that I said "kind of" weird. The weirdness (or not) is exactly the assumption that point of uncorrelatedness falls at x==0. This can make sense - the change due to treatment can be independent of initial value, for instance. In other cases, it makes no sense at all, like when the intercept refers to the value for a 0 kg adult male. 

(In older times, these discussions came up for balanced-data random coefficient regression, where each subject has measurements at x1, ..., xn.  You might get the idea of splitting data into the average and the empirical slope for each person, because they are uncorrelated if slope and intercept are considered fixed effects. You might then move on to parametrizing the model as Yij = Ai + Bi (xj - xbar) and assume that Ai and Bi are also independent. This has neat technical properties, but the model is generally unrealistic, e.g. the minimum variance of Y occurs exactly at xbar, which depends on the chosen design points.)

-pd


> 
> On Wed, Sep 13, 2017 at 03:35:50PM +0200, peter dalgaard wrote:
> « Just stumbled over this thread, but it seems pretty obvious to me: A model where slope and intercept are independent will not have independence between the slope and the value anywhere else on the line (cov(beta,alpha+beta*x) = x*V(beta)). This is why that model is kind of weird... 
> « 
> « In particular, shifting the x-axis, thus changing the definition of the intercept will make the variance model substantially different in the independence case. If intercept and slope are correlated, you just get the same model parametrized differently. 
> « 
> « -pd
> « 
> « > On 12 Sep 2017, at 23:08 , Fox, John <jfox at mcmaster.ca> wrote:
> « > 
> « > Dear David and Ben,
> « > 
> « > I haven't worked out the implications specifically, but even in a linear model fit by least-squares, with no constraints on the inter-coefficient correlations, the correlation between the coefficients is influenced by the choice of reference level for a factor. That suggests to me that constraining the correlation to zero would affect the coefficients.
> « > 
> « > As I said, this is far short of a proof, but the result seems intuitively plausible.
> « > 
> « > Best,
> « > John
> « > 
> « > --------------------------------------
> « > John Fox, Professor Emeritus
> « > McMaster University
> « > Hamilton, Ontario, Canada
> « > Web: socserv.mcmaster.ca/jfox
> « > 
> « > 
> « > 
> « >> -----Original Message-----
> « >> From: R-sig-mixed-models [mailto:r-sig-mixed-models-bounces at r-
> « >> project.org] On Behalf Of David Sidhu
> « >> Sent: Friday, September 8, 2017 8:19 PM
> « >> To: Ben Bolker <bbolker at gmail.com>
> « >> Cc: r-sig-mixed-models at r-project.org
> « >> Subject: Re: [R-sig-ME] choice of reference category only changes
> « >> coefficient with uncorrelated random intercept and slope
> « >> 
> « >> Hi Ben
> « >> 
> « >> Thanks for the reply.
> « >> Just to follow up, I tried running an lmer instead of a glmer and the
> « >> same thing happens: when a random slope and intercept are uncorrelated,
> « >> the choice of the reference category affects the absolutely value of
> « >> that predictor’s coefficient.
> « >> 
> « >> Dave
> « >> 
> « >> ---
> « >> David M. Sidhu, MSc<http://davidmsidhu.com/> PhD Candidate Department of
> « >> Psychology University of Calgary
> « >> 
> « >> 
> « >> 
> « >> 
> « >> 
> « >> 
> « >> On Sep 8, 2017, at 12:04 PM, Ben Bolker
> « >> <bbolker at gmail.com<mailto:bbolker at gmail.com>> wrote:
> « >> 
> « >> Not sure, but ...
> « >> 
> « >> I think this is real. (If I were going to pursue it further I would
> « >> probably try running some simulations.)  I think the asymmetry you're
> « >> seeing is most likely related to the nonlinearity inherent in a GLMM; if
> « >> that's true, then the effect should go away if you were using a LMM
> « >> instead of a GLMM ...
> « >> 
> « >> 
> « >> On Tue, Sep 5, 2017 at 7:45 PM, David Sidhu
> « >> <dsidhu at ucalgary.ca<mailto:dsidhu at ucalgary.ca>> wrote:
> « >> 
> « >> Hi Everyone
> « >> 
> « >> I have noticed something strange...
> « >> 
> « >> I am running a glmer with a single dichotomous predictor (coded 1/0).
> « >> The model also includes a random subject intercept, as well as a random
> « >> item intercept and slope.
> « >> 
> « >> Changing which level of the predictor serves as the reference category
> « >> doesn’t change the absolute value of the coefficient, EXCEPT when the
> « >> random intercept and slope are uncorrelated.
> « >> 
> « >> This happens whether I keep the predictor as a numeric variable, or
> « >> change the predictor into a factor and use the following code:
> « >> 
> « >> t1<-glmer(DV~IV+(1|PPT)+(0+dummy(IV, "1")|Item)+(1|Item), data = data,
> « >> family = "binomial”)
> « >> 
> « >> Is this a genuine result? If so, can anyone explain why the uncorrelated
> « >> random intercept and slope allow it to emerge? If not, how can I run a
> « >> model that has an uncorrelated random intercept and slope that would
> « >> prevent the choice of reference category from affecting the result?
> « >> 
> « >> Thank you very much!
> « >> 
> « >> Dave
> « >> 
> « >> ---
> « >> David M. Sidhu, MSc<http://davidmsidhu.com/> PhD Candidate Department of
> « >> Psychology University of Calgary
> « >> 
> « >> 
> « >> 
> « >> 
> « >> 
> « >> 
> « >> 
> « >>       [[alternative HTML version deleted]]
> « >> 
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> « >> 
> « >> 
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> « 
> « -- 
> « Peter Dalgaard, Professor,
> « Center for Statistics, Copenhagen Business School
> « Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> « Phone: (+45)38153501
> « Office: A 4.23
> « Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com
> « 
> « _______________________________________________
> « R-sig-mixed-models at r-project.org mailing list
> « https://stat.ethz.ch/mailman/listinfo/r-sig-mixed-models
> 
> -- 
>                                Emmanuel CURIS
>                                emmanuel.curis at parisdescartes.fr
> 
> Page WWW: http://emmanuel.curis.online.fr/index.html

-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com

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