Model averaging rB from models which include rBA ≠ rBa

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Model averaging rB from models which include rBA ≠ rBa

Postby pete » Fri Apr 13, 2018 8:50 am

1. When model averaging rB across models where rBA ≠ rBa, Richmond et al. (2010), last paragraph pg. 2041

Does rB = rBA + rBa as seems likely from MacKenzie (2006) p. 237 or is there a more complicated extrapolation process?

2. If so, does it seem plausible that rB calculated as rBA+rBa would become much higher in models where rBA and rBa are freed up to be estimated conditionally than in models where they are forced to be estimated unconditionally?

3. Finally, when calculating standard error for an estimate of rB calculated from rBA + rBa, is the approach recommended elsewhere for combing standard errors of within model averages (SE = sqrt(sum(VAR[i]))/N, where VAR[i] is the variance for each site-specific estimate of psi (ie SE^2) suitable or should some alternative approach be considered?

Any help with some or all of the questions are kindly appreciated. Self-taught beginner without access to expert advice or training, so please keep answer simple and provide textual explanations for equation shorthand if possible.

• MacKenzie, D.I., 2006. Occupancy estimation and modeling: inferring patterns and dynamics of species occurrence. Academic Press, London UK.
• Richmond, O.M.W., Hines, J.E., Beissinger, S.R., 2010. Two-species occupancy models: a new parameterization applied to co-occurrence of secretive rails. Ecol Appl 20, 2036-2046.
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Re: Model averaging rB from models which include rBA ≠ rBa

Postby darryl » Fri Apr 13, 2018 4:05 pm

Hi Pete
1. The Richmond and MacKenzie parameterisations are a bit different. Under the Richmond parameterisation then rB = rBA*rA + rBa*(1-rA) because rBA and rBa are conditional upon whether species A was also detected in the survey or not.

2. NA because of above answer

3. So are you trying to get an average, model averaged estimate? If so, then you also need to account for the covariance between the different model averaged estimates.

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Re: Model averaging rB from models which include rBA ≠ rBa

Postby pete » Mon Apr 16, 2018 10:48 am

Thanks ever so much for the prompt and helpful response.

1. Excellent, this makes a lot more sense. Presumably, where my models are constrained to pA = rA I can just use a given models value for pA in this calculation?
(Species B was present at all sites species A was present so rA = pA was constrained for all models).

3. Apologies for being unclear. I would like to obtain a model average value of rB for each of the 3 levels of a factor that appears in all models (3 values of rB per model but these are not averaged within each model just across models). So 3 model average values of rB (one for each factor level) which can be compared to the corresponding model averages for pB. I would intend to calculate unconditional variance for each average to then be converted into unconditional SE. re: Burnham and Anderson (2002) pg. 150 & 162.

For each conditional model where rB can now be calculated as rB = rBA*rA + rBa*(1-rA), I would have two SE estimates, one for rBA and one for rBa but once combined I am unsure if the covariance between these two standard errors can be identified and how to account for this when achieving one standard error value for a given models rB estimate.

Your comment suggests to me that perhaps there is some additional need to account for covariance between estimates from different models as well but only if I was averaging the 3 rB values within the models? Am I following correctly?

The help is much appreciated
Thanks kindly


• Burnham, K.P., Anderson, D.R., 2002. Model selection and multi-model inference: a practical information-theoretic approach, Second edition edn. Springer-Verlag, New York, New York, USA.
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