[R-sig-ME] Standard errors of variances; negative variance

[Ben Bolker]

 From Molenberghs and Verbeke 2011:

In the first view, where the focus is entirely on the resulting marginal 
model (1.2), negative values for λ2 are perfectly acceptable (Nelder, 
1954; Verbeke and Molenberghs, 2000, Section 5.6.2), because this merely 
corresponds to the occurrence of negative within-cluster correlation ρ = 
λ2/(λ2 + ν2). In such a case, the only requirement is that λ2 + ν2 > 
0andVi = λ2 Jni + ν2 Ini is a positive definite, marginal covariance 
matrix. Further discussions on negative variance components, and their 
implications for inferences, can be found in Thompson (1962), Searle et 
al. (1996), Verbeke and Molenberghs (2003) and Molenberghs and Verbeke 
(2007). In the second view, when the link between marginal model (1.2) 
and its generating hierarchical model (1.1) is preserved, thereby 
including the concept of random effects bi and perhaps even requiring 
inferences about them, it has been considered imperative to restrict λ2 
to non-negative values.

   Molenberghs, Geert, and Geert Verbeke. 2011. “A Note on a 
Hierarchical Interpretation for Negative Variance Components.” 
Statistical Modelling 11 (5): 389–408. 
https://doi.org/10.1177/1471082X1001100501.

    Since lme4 deals with the hierarchical model, I don't think that 
negative variances are going to work.

On 2023-06-20 9:21 a.m., Sorkin, John wrote:
> 
> Thierry,
> I have the same question that you posed, how can a variance, a measure 
> that is squared be negative? Will, please enlighten us!
> John
>> John David Sorkin M.D., Ph.D.
>> Professor of Medicine
>> Chief, Biostatistics and Informatics
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>> On Jun 20, 2023, at 3:45 AM, Thierry Onkelinx via R-sig-mixed-models 
>> <r-sig-mixed-models using r-project.org> wrote:
>>
>> Dear all,
>>
>> A negative variance seems odd to me. Isn't a variance positive by
>> definition? How would you interpret a random intercept with a negative
>> variance?
>>
>> Best regards,
>>
>> ir. Thierry Onkelinx
>> Statisticus / Statistician
>>
>> Vlaamse Overheid / Government of Flanders
>> INSTITUUT VOOR NATUUR- EN BOSONDERZOEK / RESEARCH INSTITUTE FOR NATURE AND
>> FOREST
>> Team Biometrie & Kwaliteitszorg / Team Biometrics & Quality Assurance
>> thierry.onkelinx using inbo.be
>> Havenlaan 88 bus 73, 1000 Brussel
>> https://nam11.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.inbo.be%2F&data=05%7C01%7Cjsorkin%40som.umaryland.edu%7C2c0d1047636b43c2bfcc08db71625f82%7C717009a620de461a88940312a395cac9%7C0%7C0%7C638228439560019013%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=sbP2Xemwg0iHjvxcc6zR8vtPbZf5ZoFCv1BIFIeDdBE%3D&reserved=0
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>> what the experiment died of. ~ Sir Ronald Aylmer Fisher
>> The plural of anecdote is not data. ~ Roger Brinner
>> The combination of some data and an aching desire for an answer does not
>> ensure that a reasonable answer can be extracted from a given body of 
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>> <https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.inbo.be%2F&data=05%7C01%7Cjsorkin%40som.umaryland.edu%7C2c0d1047636b43c2bfcc08db71625f82%7C717009a620de461a88940312a395cac9%7C0%7C0%7C638228439560019013%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=I5HJ05o0Bd4ly0GL0Smc1iYm3cuZMc2IGCcxy8OWn%2B8%3D&reserved=0>
>>
>>
>> Op di 20 jun 2023 om 01:36 schreef Ben Bolker <bbolker using gmail.com>:
>>
>>>   Hi, Will,
>>>
>>> Googling
>>>
>>>  site:https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat.ethz.ch%2Fpipermail%2Fr-sig-mixed-models%2F&data=05%7C01%7Cjsorkin%40som.umaryland.edu%7C2c0d1047636b43c2bfcc08db71625f82%7C717009a620de461a88940312a395cac9%7C0%7C0%7C638228439560019013%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=vw%2FnMxnvA7L9A0qYuJfHa9suA2YBTnZ%2BqxPltyR3G%2FU%3D&reserved=0 "negative
>>> variance"
>>>
>>> claims to get you 67 results (although only 11 that Google considers
>>> worth displaying by default),
>>>
>>>   You can filter by date - there are only two hits since 2020:
>>>
>>>
>>> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.google.com%2Fsearch%3Fq%3Dsite%253Ahttps%253A%252F%252Fstat.ethz.ch%252Fpipermail%252Fr-sig-mixed-models%252F%2B%2522negative%2Bvariance%2522%26client%3Dfirefox-b-d%26tbs%3Dcdr%253A1%252Ccd_min%253A01-01-2020%252Ccd_max%253A%26tbm%3D&data=05%7C01%7Cjsorkin%40som.umaryland.edu%7C2c0d1047636b43c2bfcc08db71625f82%7C717009a620de461a88940312a395cac9%7C0%7C0%7C638228439560019013%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=xbWrjRwClEMYFKVkncDFgZga6SPNcmsPvbNs4kpUQjY%3D&reserved=0
>>>
>>>
>>>   and these both look like false positives (they include "negative" and
>>> "variance" but not "negative variance" ...)
>>>
>>>   The short answer is that I am not aware of any mixed effect package
>>> in R that will allow you to return negative values. You can see my
>>> answer here:
>>> https://nam11.safelinks.protection.outlook.com/?url=https%3A%2F%2Fstat.ethz.ch%2Fpipermail%2Fr-sig-mixed-models%2F2018q1%2F026437.html&data=05%7C01%7Cjsorkin%40som.umaryland.edu%7C2c0d1047636b43c2bfcc08db71625f82%7C717009a620de461a88940312a395cac9%7C0%7C0%7C638228439560019013%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=2E5S51uIXiPEsuH11f693MEOLYFb2dh5e39t2GSLor8%3D&reserved=0 ...
>>>
>>>
>>>   As for uncertainties in variances - the merDeriv package (dating back
>>> to 2017) will give you the standard errors of variances and covariances
>>> (although again, note that there's a theoretical issue here - the
>>> standard errors are often extremely poor summaries of the uncertainty or
>>> RE variances, the original authors of lme4 would certainly prefer that
>>> you use profile confidence intervals to summarize the uncertainty ...)
>>>
>>>   That's what I know -- perhaps someone else knows about a package that
>>> allows for negative variances ... ??
>>>
>>> On 2023-06-19 6:13 p.m., Will Hopkins wrote:
>>>> I've just joined this list to get answers to a few questions. I would
>>> have
>>>> searched the archive before posting, but there seems to be no way of
>>>> searching except via quarterly summaries. I searched the last four
>>> quarters
>>>> without success.
>>>>
>>>> I'm a SAS user, but a few years ago I tried the mixed model in R, with
>>> the
>>>> help of an R user (Alice Sweeting). At that time, the lme package 
>>>> did not
>>>> provide standard errors for the variances, nor did it allow negative
>>>> variance. Have these limitations been addressed?  As I recall, Alice
>>> found
>>>> some code that provided standard errors, but it gave values different
>>> from
>>>> those of the mixed model in SAS (Proc Mixed). I therefore opted to stay
>>> with
>>>> SAS, because allowing for negative variance for random effects other 
>>>> than
>>>> residuals and estimating uncertainties in variances are both fundamental
>>> to
>>>> mixed modeling, in my view. I also became fluent with SAS coding 
>>>> over the
>>>> years and did not want to make the effort with R coding. Interestingly,
>>> SAS
>>>> introduced a free cloud version called SAS Studio, evidently modeled 
>>>> on R
>>>> Studio, to try to win back customers!
>>>>
>>>> Will
>>>>
>>>> _______________________________________________
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-- 
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
(Acting) Graduate chair, Mathematics & Statistics
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