## Estimating Process Variance with 2 occasions

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### Estimating Process Variance with 2 occasions

We have been trapping foxes for the past 3 years. We have a robust design with 3 primary sessions (years: 2007, 2008, 2009) and 4 secondary occasions (days) within each year. I am running a multi-state robust design where ages (pup and non-pup) are the states.

I want to get a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09). I was told that MARK cannot calculate this directly. On p. 84 of Chapter 6, there is a sidebar describing a variance components approach. I tried this and MARK froze - probably because I only have 2 time occasions.

Can anyone help explain how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09)?
brp

Posts: 26
Joined: Tue Sep 29, 2009 12:54 pm
Location: Bozeman, MT

### Re: Estimating Process Variance with 2 occasions

brp wrote:We have been trapping foxes for the past 3 years. We have a robust design with 3 primary sessions (years: 2007, 2008, 2009) and 4 secondary occasions (days) within each year. I am running a multi-state robust design where ages (pup and non-pup) are the states.

I want to get a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09). I was told that MARK cannot calculate this directly. On p. 84 of Chapter 6, there is a sidebar describing a variance components approach. I tried this and MARK froze - probably because I only have 2 time occasions.

Can anyone help explain how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09)?

Trying to estimate process variance with only 2 years is more or less impossible (some of the smart folks have suggested you only get decent estimates if you gave >15 years to work with). If you think about it, the reasons would be obvious - you wouldn't estimate a standard variance with only two points, so why would you imagine you could estimate anything else with only two points?
cooch

Posts: 1613
Joined: Thu May 15, 2003 4:11 pm
Location: Cornell University

I understand that estimating variance from only 2 points doesn't work. I would appreciate advice on how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09).

Is this possible?

Thanks!
brp

Posts: 26
Joined: Tue Sep 29, 2009 12:54 pm
Location: Bozeman, MT

brp wrote:I understand that estimating variance from only 2 points doesn't work. I would appreciate advice on how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09).

Is this possible?

Thanks!

Mean estimated over two points is no more meaningful (unless I misunderstand how many points you have). If you have two parameter estimates, and want the mean of those two, then my point stands.

If you had a reasonable number of means, then you'd be faced with the question of whether or not the means represented sample means calculated from random samples from some common underlying distribution (population). And, whether or not there was covariance amongst the means, and so on.
cooch

Posts: 1613
Joined: Thu May 15, 2003 4:11 pm
Location: Cornell University

### Estimating Process Variance with 2 occasions

brp wrote:I understand that estimating variance from only 2 points doesn't work. I would appreciate advice on how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09).

Is this possible?

Thanks!

I think you got the discussion pointed in the wrong direction by bringing up variance components. What I think you want to do is to model average the estimates for 07-08 ad 08-09. When you do this model averaging, make sure the also check the box to have the variance-covariance of the model averaged estimates dumped to a DBF file. Then, compute the mean of the 2 model-averaged values. The variance of this mean is the sum of the 2X2 variance covariance matrix dvided by 4 (=2^2). SE of the mean is sqrt of the variance. CI could be computed in the usual way, i.e., +/1 2SD. You could also convert the mean and variance to the logit scale and compute CI on the logit scale and then back-transform.

Gary
gwhite

Posts: 325
Joined: Fri May 16, 2003 9:05 am

### Re: Estimating Process Variance with 2 occasions

gwhite wrote:
brp wrote:I understand that estimating variance from only 2 points doesn't work. I would appreciate advice on how to obtain a model-averaged estimate (and 95% CI) for pup survival (averaged across years: 07-08 and 08-09).

Is this possible?

Thanks!

I think you got the discussion pointed in the wrong direction by bringing up variance components. What I think you want to do is to model average the estimates for 07-08 ad 08-09. When you do this model averaging, make sure the also check the box to have the variance-covariance of the model averaged estimates dumped to a DBF file. Then, compute the mean of the 2 model-averaged values. The variance of this mean is the sum of the 2X2 variance covariance matrix dvided by 4 (=2^2). SE of the mean is sqrt of the variance. CI could be computed in the usual way, i.e., +/1 2SD. You could also convert the mean and variance to the logit scale and compute CI on the logit scale and then back-transform.

Gary

If Gary has turned the boat in the right direction, then I see the issue. You can get there from here by applying the Delta method (appendix 2) to a function for the mean (X1+X2)/2. If you take a first-order Delta approximation to this function, you get

[0.5,0.5] * VC * T[0.5,0.5]

which amounts to taking the sum of the VC matrix and dividing by 4 (as per Gary's note).

There is an example of the application of the Delta method to a function of random variables in Chapter 6 (sidebar beginning on p. 68).
cooch

Posts: 1613
Joined: Thu May 15, 2003 4:11 pm
Location: Cornell University