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I am working in ramdom effects model. when I compute the within-study variance/between-study variance, I find the negative value. Can? for this model. If we find in simulation how should we do?

Thanks.

Harvey Motulsky
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alice
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3 Answers3

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It is an artifact of the methodology that you are using. You could avoid this by using a Bayesian model with a prior probability of non-positive variance of zero percent. Technically, an impossible answer is impossible using a Bayesian methodology. It is possible to get impossible answers using a Frequentist methodology. The defense of this is that you are protected against false positives $1-\alpha$ percent of the time, but the price is that you can get strange or impossible answers from time to time. The literature is full of weird effects you can create. Technically, a negative variance would imply the data is drawn from the complex numbers, but the complex numbers are not ordered so you couldn't create an ordinary probability distribution over them. In practice it is due to small samples, bad models or weird outliers. I would go down the bad model path. SAS provides a brief explanation at https://v8doc.sas.com/sashtml/stat/chap69/sect12.htm

You can dig through their bibliography to get original source material. Still, if I were you I would presume you had a bad model. There are many problems out there in real world models that people often miss and you see them as weird results. It could be a weird sample or too small a sample, but I am prejudiced toward presupposing bad models. It is so simple for there to be something hidden in the real world that has an impact on a calculation.

Frequentist models can be fragile or robust. The same is true for Bayesian models. This should be a warning of a fragility. Bayesian models cannot give impossible answers if they are properly formed, but they can have other sources of fragility. If I were you, I would assume that something in your model made it fragile. Think of a new way to ask a similar question.

Dave Harris
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The answer is yes. This question has come up many times on this site. Of course no random variable can have a variance < 0. Yet there are many instances where estimates of variance come out negative. If you search this site using the key words negative variance there are probably hundreds of questions where this has been discovered in a host of applications. When I just did a search for "negative variance" among questions and answers I got 1105 hits.

Michael R. Chernick
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  • Thank you very much. However it hard to interpret if it is negative. – alice Dec 27 '16 at 03:43
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    If a question has come up many times on the site (and has been answered), don't answer a new version of the same question. Instead, stackexchange policy is to vote to close as a duplicate. That way instead of having the site littered with brief 5-line answers to dozens of copies of teh same question, we can get them all pointing to a single good version of the question with good, substantive (preferably canonical) answers. If there are numerous old nearly-identical questions, you should also be trying to consolidate those by voting to close the least-canonical ones. – Glen_b Dec 27 '16 at 05:09
  • Generally I think the reason is either that there is a very poor model being used or the actual variance though positive is very small. I think if you check out some of the major questions iyou will get more comfortable with the idea. – Michael R. Chernick Dec 27 '16 at 05:09
  • @Glen_b Your comment is typical for moderators. I know that typically a moderator will find a question that is in his or her judgement an exact duplicate and the question will be quickly closed. Sometimes the OP will argue about it and I suppose in some cases it will be reopened. I think this is not so satisfactory to the OP. I think the moderators should encourage the questioner to check the site for answers before submitting the question. In fact the system is automated to make such suggestions. But we still get these near duplicates. – Michael R. Chernick Dec 27 '16 at 05:17
  • The rules should be followed for good reasons but human judgment is always involved, – Michael R. Chernick Dec 27 '16 at 05:18
  • I got 2 favorables on this but just now got a negative too. – Michael R. Chernick Dec 27 '16 at 05:19
  • I don't consider fresh ideas clutter and often someone viewing the question will offer something worthwhile to the discussion. I know it is the job of the moderators to spend time cleaning up the site but it is time consuming and you shouldn't have to do so much of it. But achieving the ideal is really not possible. Sometimes the quick decision to put close/put on hold a question is a turnoff to the new members. This is probably better to be made a discussion on meta. – Michael R. Chernick Dec 27 '16 at 05:28
  • In response to a comment here, https://stats.stackexchange.com/a/14185/3601, I looked for questions about this topic, and didn't find one that was posed clearly or that had a clear canonical answer. Do you know of one? – Aaron left Stack Overflow Oct 17 '17 at 13:10
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Think about the distribution of any unbiased estimate when the parameter is 0. The mean estimate has to be 0 so some estimates must be negative.

David Lane
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  • I am not sure about your answer. The estimator that led to a negative estimate may not have been unbiased. But I do agree that when the true variance is small seemingly logical estimates will not be restricted to be positive. An example would be an estimate of residual variance that is obtained by subtracting it from another variance estimate. Look at examples where the estimate of Rsquare can be greater than 1 or less than 0. – Michael R. Chernick Dec 27 '16 at 05:57
  • That's kind of my point. Unbiased estimates of Rsquare often lead to negative estimates. – David Lane Dec 27 '16 at 14:57