Probability of a false absence - multi-scale models

[Alan Dextrase]

For single season, site occupancy models, the expected probability of not detecting a species at an occupied site (probability of a false absence) can be estimated as (1 - p)^k where p is the probability of detecting the species at an occupied site in a single survey and k is the number of repeat surveys at the site.

I would like to estimate the probability of a false absence at larger scale sample units from multi-scale models (where sub-sites are sampled repeatedly as spatial replicates within a larger sample unit). I would like to use the following formula for this purpose and wanted to check with the forum to see if my logic makes sense:

probability of false absence in larger sample unit = [1 - (Pk x θ)]^j
Where,
• Pk = probability of detecting species at the sub-site level in k surveys of the sub-site given the sub-site is occupied = 1 - (1 - p)^k.

• θ = the probability that an individual sub-site is occupied given the larger sample unit is occupied (estimated from multi-scale models).

• j = the number of sub-sites sampled within the larger sample unit with a sampling effort of k surveys/sub-site.

Thank you,

Alan