topic Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated in Statistical Procedures

satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Mon, 20 Jun 2022 21:14:46 GMT

I'm looking for a way to calculate adjusted degrees of freedom for a linear combination of variance components, with possible correlations. I know that there can be some distributional issues with this, so at best it's an approximate solution. The estimate statement does not work for this in MIXED.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

Rick_SAS — Tue, 21 Jun 2022 09:53:25 GMT

Please post the code that you are using (even though it does not work) so we can see the model you are fitting and the estimates you are trying to make. If you have sample data, that's even better.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Tue, 21 Jun 2022 16:53:49 GMT

Rick I did discuss this with Jill Tao and she mentioned that the calculation i want to make is not available with the ESTIMATE statement. So I was wondering if someone had looked at this. I can send you the code, but it doesn't show the linear combination of fixed and variance components that I want to carry out. If you send me your email, I can send you the expression on a powerpoint slide. I can't copy the equations here. My email is saltan@its,jnj.com.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

Rick_SAS — Tue, 21 Jun 2022 17:08:45 GMT

Sorry, I don't give personal advice. Post the code here if you want someone to help.

Jill is top notch, so if she says that the ESTIMATE statement doesn't cover your use case, you can believe her. Sometimes, this situation is because you are trying to estimate a nonlinear function of the parameter. If you know the expression that you are trying to estimate, you might try PROC NLMIXED. For an example, see https://blogs.sas.com/content/iml/2018/10/17/parameter-estimates-for-different-parameterizations.html

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Tue, 21 Jun 2022 18:20:22 GMT

The problem is that I'm trying to describe an inference space that is not covered by SAS. SAS has broad, intermediate and narrow. But the space I am trying to describe does not fit into this trichotomization. So for example, if I have mixed model, and to make it simple, say just a single variance component, say v_b, and then residual error, v_r. I have say a linear combination of the fixed effects, say L'B0, then I want an interval estimate based on Var(L'B0_hat) + v_b_hat. It's not a confidence interval, it's a population interval of the population of units belonging to random factor b. I don't know how to describe this in words very clearly I know, but I cannot copy notation to this page.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

Rick_SAS — Tue, 21 Jun 2022 23:35:23 GMT

I think your chances of obtaining an answer will be improved if you post some sample data, show the model you are considering, and then ask the question in terms of the variables in the problem.

> I'm trying to describe an inference space that is not covered by SAS.

That remains to be seen. The first step is to define the estimator. Even if there is not a built-in procedure that can estimate the interval, SAS supports programming languages such as SAS/IML that can implement any well-formulated algorithm.

> I don't know how to describe this in words very clearly

If it is easier, perhaps you could link to a journal article or textbook in which your problem is described.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Wed, 22 Jun 2022 21:25:09 GMT

Here's some sample data and proc mixed code that describes the model and a provisional ESTIMATE statement.

/* read in raw data for 9 batches */ 

data a0 ; 
input Month B01 B02 B05 B06 B08 B09 B10 B12 B20 ; 
datalines ; 
0  103.7 103.7 103.7 100.8 101.2 100.1  99.9 102.7 101.5
3   99.4 103.1 100.9 100.8  98.1 101.3 101.6 101.7 104.7
6  102.4 102.4 102.4  93.4  99.9 100.1 	95.1  99.4 100.8
9  101.0  98.9  99.7 101.2  98.2  98.4  99.5  98.7  99.9
12 100.0 101.2  99.3  96.5  92.7  92.3  98.1  98.7  97.8
18  99.2  91.7  98.7  94.3  95.7  96.1  95.9 100.1  95.1
24  95.5  96.2  94.3  93.4  94.4  92.8  96.1  94.8  95.9
; 
run; 

proc print data=a0 ; 
title1 "Raw Data 9 Batches Page 8/22" ;
run ; 

/* Create vector of responses by batch */ 

data a1 ; set a0 ; 
array batx (9) B01 B02 B05 B06 B08 B09 B10 B12 B20 ;
array baty (9) $3 ('B01' 'B02' 'B05' 'B06' 'B08' 'B09' 'B10' 'B12' 'B20') ; 
do i = 1 to 9 ; 
y = batx(i) ; 
batch =baty(i) ; 
output; 
end ; 
keep batch month y; 
run ;   

proc print data=a1 ; 
title1 "Data for analysis" ;
run; 

proc mixed data=a1 nobound covtest ranks mmeq mmeqsol ;
   class  batch ;
   model y = month / solution covb outpm = out1 alphap=0.10 solution  ddfm=kr ; 
   random intercept / subject=batch;
   estimate "A2 - Process Model 12 months Including batch component"
            intercept 1 month 12 | Intercept 1
            / cl alpha=0.10 e  ;
  ods output mmeqsol=invmmeq coef=e;
  title1 "Mixed regression model with random batch intercept only";
  title2 "Batch mean and Product calculations for 0.50 OOS, 95% Coverage" ;
run;

the point estimate is correct in the estimate card. But what I want is

sqrt[Var(InterceptF + 12*Month) + Var(InterceptR)]

where Var=Variance, InterceptF is the fixed intercept estimate, Month is the fixed time covariate and InterceptR is the estimate of the random component on intercept due to Batch.

This is not really a standard error as such, it's the standard deviation of the population of batches at time 12 months. The challenge here is to get the Satterthwaite adjusted degrees of freedom for this sum of variances. Actually this is the simple case. If we add the random component due to batch on the time covariate, Month, and assume an unstructured covariance matrix, then we have the sum of 3 variances where 2 are correlated. In addition, note that Var(InterceptF + 12*Month) is not a chi-square.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

Rick_SAS — Thu, 23 Jun 2022 10:47:41 GMT

@jiltao @SteveDenham @lvm

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

SteveDenham — Thu, 23 Jun 2022 15:19:50 GMT

Not real sure how this is going to work. It appears to me that you could get Var(interceptR) by adding solution as an option for the random statement. From the BLUP, you could then calculate the combined variance.

SteveDenham

(this is a real shot in the dark, though)

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Thu, 23 Jun 2022 15:43:31 GMT

Yes that could be done and in fact is what i would do. The problem is getting a degrees of freedom for that to put an uncertainty limit on it. This is one of the reasons why a frequentist approach to this kind of problem is not good. You have to deal with these kinds of complications.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

lvm — Thu, 23 Jun 2022 16:14:10 GMT

I see you are trying to calculate prediction intervals rather than confidence intervals. These are not straightforward for mixed models in SAS. I usually do this "brute force", but I have not dealt with the complexity of Satterthwaite or Kenward-Roger df adjustments (although I am a huge fan of KR adjustments, in general). These calculations can be quite challenging, but I think there may be some IML programs out there to do this (but maybe not for your problem).

Your question made me think of the following article that may be of help. I have not studied it -- it is sitting on my desk to study at some point. It deals with prediction intervals for mixed models, and the online supplement has extensive SAS code.

https://www.tandfonline.com/doi/full/10.1080/19466315.2020.1776762

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

lvm — Thu, 23 Jun 2022 16:28:26 GMT

And I just remembered this article that deals with prediction intervals for mixed models, also with extensive SAS code in the online supplement. I know that address Satterthwaite.

https://onlinelibrary.wiley.com/doi/epdf/10.1002/sim.8386

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

SteveDenham — Thu, 23 Jun 2022 16:38:51 GMT

This one isn't behind a paywall, and looks like it covers much of the same material as the other Francq paper.

SteveDenham

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Thu, 23 Jun 2022 16:47:43 GMT

Thank you for that reference. I am familiar with the Francq paper, he lays out the algebra in a clear way. Unfortunately he doesn't address the df question.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

SteveDenham — Thu, 23 Jun 2022 16:53:30 GMT

If you have 3 components, you might try doing 3 stepwise Satterthwaite calculations (A with B, and then the result with C; A with C, and then the result with B; and B with C, and then the result with A), and then averaging those to get approximate df. Sounds a bit brute force, though.

SteveDenham

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Thu, 23 Jun 2022 17:13:38 GMT

For 3 variances, with 2 correlated, that might be a heuristic approach that gets to an approximate df. But right now, we're struggling with finding a way to get it for 2. I think we're all agreed now that SAS doesn't provide such a calculation in MIXED or GLIMMIX. At this point, we have to resort to writing our own code to carry this out. I wonder if it's worth it given that it's straightforward to just use the PRIOR option in MIXED and generate it from the posteriors.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

lvm — Thu, 23 Jun 2022 17:23:24 GMT

The text indicated df calculated using the Satterthwaite method. But I have only scanned this.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

lvm — Thu, 23 Jun 2022 17:25:16 GMT

If you have a recent version of SAS/STAT, you could directly use PROC BGLIMM for a full Bayesian analysis.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

CMC-Stan — Thu, 23 Jun 2022 17:48:29 GMT

Better to stay with empirical Bayes estimates from MIXED for a problem like this, since the posteriors will give results based on the Jeffreys priors.

Re: satterthwaitie adjustment for linear combination of variance components, possibly correlated

tianlin_wang — Thu, 23 Jun 2022 18:15:23 GMT

Obviously, PROC MIXED does not provide direct answer. But you can get it outside of PROC MIXED.

Ask PROC MIXED to save/print var(L'B0) and v_b_hat. Once you have these two items, then you can construct your own interval limits.

Hope trhis helps,

Tianlin Wanhg