Delta method for log SE

[dwc]

I am using the Pradel survival and lambda model in MARK and having trouble getting SEs to match between the Delta method for var(log) and those of real function estimates of lambda generated by MARK.

For the Delta method of var(log) I am using var(y)=[β*(expβ)]2*(varβ) and then taking the square-root of var(y) for the SE.

To validate this solution, I have applied the above solution to log-link function beta SE estimates from a candidate model. When I compare the lambda SEs’ of the Delta method and the real function SEs’ generated by MARK for the same model, the SE estimates are very different. I am not sure how to resolve this discrepancy.

[cooch]

I am using the Pradel survival and lambda model in MARK and having trouble getting SEs to match between the Delta method for var(log) and those of real function estimates of lambda generated by MARK.

For the Delta method of var(log) I am using var(y)=[β*(expβ)]2*(varβ) and then taking the square-root of var(y) for the SE.

To validate this solution, I have applied the above solution to log-link function beta SE estimates from a candidate model. When I compare the lambda SEs’ of the Delta method and the real function SEs’ generated by MARK for the same model, the SE estimates are very different. I am not sure how to resolve this discrepancy.

For a log transform, the back transform of the estimated beta to the paramter of interest (call it theta) is

theta=exp(beta) (i.e., e^beta).

The first order Delta approximation is then the partial derivative of theta wrt beta, squared, multiplied by the estimated variance for beta. Of course, the nice property of the log transform is that the derivative of the transform is the same as the transform. So

d(theta)/d(beta) = exp(beta).

Thus, estimated var (theta) = (exp(beta))^2*var(beta)

In your equation, you seem to have an extra beta in the product term, which isn't necessary - you write

var(theta) = (beta*exp(beta))^2*var(beta)

This is incorrect - the first beta in the squared term shouldn't be there.

Try it with the correct expression, and see how it works.

[dwc]

Thanks for the corrections. I have tried using the Delta method solution you provided and the SE is closer than what I was getting before but still not the same (within rounding error). For example, the Delta method SE estimate is 0.047671 while the real function estimate from MARK output is 0.0695561. I used the output option for parameter estimates in an excel spreadsheet for use with the Delta method as recommended by the MARK text. Any suggestions about resolving the difference?

[cooch]

Thanks for the corrections. I have tried using the Delta method solution you provided and the SE is closer than what I was getting before but still not the same (within rounding error). For example, the Delta method SE estimate is 0.047671 while the real function estimate from MARK output is 0.0695561. I used the output option for parameter estimates in an excel spreadsheet for use with the Delta method as recommended by the MARK text. Any suggestions about resolving the difference?

Not without seeing all the numbers. Make sure you're multiplying the derivative by variance of the Beta estimate (and not the variance of the real parameter).

[cooch]

Thanks for the corrections. I have tried using the Delta method solution you provided and the SE is closer than what I was getting before but still not the same (within rounding error). For example, the Delta method SE estimate is 0.047671 while the real function estimate from MARK output is 0.0695561. I used the output option for parameter estimates in an excel spreadsheet for use with the Delta method as recommended by the MARK text. Any suggestions about resolving the difference?

Not without seeing all the numbers. Make sure you're multiplying the derivative by variance of the Beta estimate (and not the variance of the real parameter).

The other question would be - does your transformation involve a single parameter, or multiple parameters? If the latter (i.e., if the parameter is back-transformed from an equation with multiple beta terms), then you need to use the approach described in example (4) (near the end of the Delta method appendix).