[R] Compare two normal to one normal

Mark Leeds markleeds2 at gmail.com
Wed Sep 23 06:38:27 CEST 2015

```John: After I sent what I wrote, I read Rolf's intelligent response. I
didn't realize that
there are boundary issues so yes, he's correct and  my approach is EL
WRONGO. I feel very not good that I just sent that email being that it's
totally wrong. My apologies for noise
and thanks Rolf for the correct response.

Oh,  thing that does still hold in my response is  the AIC approach unless
Rolf
tells us that it's not valid also. I don't see why it wouldn't be though
because you're
not doing a hypothesis test when you go the AIC route.

On Wed, Sep 23, 2015 at 12:33 AM, Mark Leeds <markleeds2 at gmail.com> wrote:

> Hi John:  For the log likelihood in the single case, you can just
> calculate it directly
> using the normal density, so the sum from i = 1 to n of f(x_i, uhat,
> sigmahat)
> where f(x_i, uhat, sigma hat)  is the density of the normal with that mean
> and variance.
> so you can use dnorm with log = TRUE.  Of course you need to estimate the
> parameters uhat and sigma hat first but for the single normal case, they
> are of course just the sample mean and sample variance
>
> Note though: If you going to calculate a log likelihood ratio, make sure
> you compare
> apples and apples and not apples and oranges in the sense that the
> loglikelihood
> that comes out of the mixture case may include constants such
> So you need to know EXACTLY how the mixture algorithm is calculating it's
> log likelihood.
>
> In fact, it may be better and safer to just calculate the loglikelihood
> for the mixture yourself also so sum  from i = 1 to n of [ lambda*f(x_i,
> mu1hat, sigma1hat) + (1-lambda)*f(x_i, mu2hat, sigma2hat) By calculating it
> yourself and being consistent, you then know that you will be calculating
> apples and applies.
>
> As I said earlier, another way is by comparing AICs. in that case, you
> calculate it
> in both cases and see which AIC is lower. Lower wins and it penalizes for
> number of parameters. There are asymptotics required in both the LRT
> approach and the AIC
> approach so you can pick your poison !!! :).
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> On Tue, Sep 22, 2015 at 6:01 PM, John Sorkin <JSorkin at grecc.umaryland.edu>
> wrote:
>
>> Bert
>> I am surprised by your response. Statistics serves two purposes:
>> estimation and hypothesis testing. Sometimes we are fortunate and theory,
>> physiology, physics, or something else tell us what is the correct, or
>> perhaps I should same most adequate model. Sometimes theory fails us and we
>> wish to choose between two competing models. This is my case.  The cell
>> sizes may come from one normal distribution (theory 1) or two (theory 2).
>> Choosing between the models will help us postulate about physiology. I want
>> to use statistics to help me decide between the two competing models, and
>> thus inform my understanding of physiology. It is true that statistics
>> can't tell me which model is the "correct" or "true" model, but it should
>> be able to help me select the more "adequate" or "appropriate" or "closer
>> to he truth" model.
>>
>> In any event, I still don't know how to fit a single normal distribution
>> and get a measure of fit e.g. log likelihood.
>>
>> John
>>
>>
>> John David Sorkin M.D., Ph.D.
>> Professor of Medicine
>> Chief, Biostatistics and Informatics
>> University of Maryland School of Medicine Division of Gerontology and
>> Geriatric Medicine
>> Baltimore VA Medical Center
>> 10 North Greene Street
>> GRECC (BT/18/GR)
>> Baltimore, MD 21201-1524
>> (Phone) 410-605-7119
>> (Fax) 410-605-7913 (Please call phone number above prior to faxing)
>>
>> >>> Bert Gunter <bgunter.4567 at gmail.com> 09/22/15 4:48 PM >>>
>> I'll be brief in my reply to you both, as this is off topic.
>>
>> So what? All this statistical stuff is irrelevant baloney(and of
>> questionable accuracy, since based on asymptotics and strong
>> assumptions, anyway) . The question of interest is whether a mixture
>> fit better suits the context, which only the OP knows and which none
>>
>> I know that many will disagree with this -- maybe a few might agree --
>> but please send all replies, insults, praise, and learned discourse to
>> me privately, as I have already occupied more space on the list than
>> I should.
>>
>> Cheers,
>> Bert
>>
>>
>> Bert Gunter
>>
>> "Data is not information. Information is not knowledge. And knowledge
>> is certainly not wisdom."
>> -- Clifford Stoll
>>
>>
>> On Tue, Sep 22, 2015 at 1:35 PM, Mark Leeds <markleeds2 at gmail.com> wrote:
>> > That's true but if he uses some AIC or BIC criterion that penalizes the
>> > number of parameters,
>> > then he might see something else ? This ( comparing mixtures to not
>> mixtures
>> > ) is not something I deal with so I'm just throwing it out there.
>> >
>> >
>> >
>> >
>> > On Tue, Sep 22, 2015 at 4:30 PM, Bert Gunter <bgunter.4567 at gmail.com>
>> wrote:
>> >>
>> >> Two normals will **always** be a better fit than one, as the latter
>> >> must be a subset of the former (with identical parameters for both
>> >> normals).
>> >>
>> >> Cheers,
>> >> Bert
>> >>
>> >>
>> >> Bert Gunter
>> >>
>> >> "Data is not information. Information is not knowledge. And knowledge
>> >> is certainly not wisdom."
>> >> -- Clifford Stoll
>> >>
>> >>
>> >> On Tue, Sep 22, 2015 at 1:21 PM, John Sorkin
>> >> <JSorkin at grecc.umaryland.edu> wrote:
>> >> > I have data that may be the mixture of two normal distributions (one
>> >> > contained within the other) vs. a single normal.
>> >> > I used normalmixEM to get estimates of parameters assuming two
>> normals:
>> >> >
>> >> >
>> >> > GLUT <- scale(na.omit(data[,"FCW_glut"]))
>> >> > GLUT
>> >> > mixmdl = normalmixEM(GLUT,k=1,arbmean=TRUE)
>> >> > summary(mixmdl)
>> >> > plot(mixmdl,which=2)
>> >> > lines(density(data[,"GLUT"]), lty=2, lwd=2)
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> > summary of normalmixEM object:
>> >> > comp 1 comp 2
>> >> > lambda 0.7035179 0.296482
>> >> > mu -0.0592302 0.140545
>> >> > sigma 1.1271620 0.536076
>> >> > loglik at estimate: -110.8037
>> >> >
>> >> >
>> >> >
>> >> > I would like to see if the two normal distributions are a better fit
>> >> > that one normal. I have two problems
>> >> > (1) normalmixEM does not seem to what to fit a single normal (even
>> if I
>> >> > address the error message produced):
>> >> >
>> >> >
>> >> >> mixmdl = normalmixEM(GLUT,k=1)
>> >> > Error in normalmix.init(x = x, lambda = lambda, mu = mu, s = sigma,
>> k =
>> >> > k, :
>> >> > arbmean and arbvar cannot both be FALSE
>> >> >> mixmdl = normalmixEM(GLUT,k=1,arbmean=TRUE)
>> >> > Error in normalmix.init(x = x, lambda = lambda, mu = mu, s = sigma,
>> k =
>> >> > k, :
>> >> > arbmean and arbvar cannot both be FALSE
>> >> >
>> >> >
>> >> >
>> >> > (2) Even if I had the loglik from a single normal, I am not sure how
>> >> > many DFs to use when computing the -2LL ratio test.
>> >> >
>> >> >
>> >> > Any suggestions for comparing the two-normal vs. one normal
>> distribution
>> >> > would be appreciated.
>> >> >
>> >> >
>> >> > Thanks
>> >> > John
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> >
>> >> > John David Sorkin M.D., Ph.D.
>> >> > Professor of Medicine
>> >> > Chief, Biostatistics and Informatics
>> >> > University of Maryland School of Medicine Division of Gerontology and
>> >> > Geriatric Medicine
>> >> > Baltimore VA Medical Center
>> >> > 10 North Greene Street
>> >> > GRECC (BT/18/GR)
>> >> > Baltimore, MD 21201-1524
>> >> > (Phone) 410-605-7119410-605-7119
>> >> > (Fax) 410-605-7913 (Please call phone number above prior to faxing)
>> >> >
>> >> >
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