{"id":275,"date":"2017-04-18T03:40:31","date_gmt":"2017-04-18T03:40:31","guid":{"rendered":"http:\/\/sites.warnercnr.colostate.edu\/gwhite\/?page_id=275"},"modified":"2017-04-18T03:40:31","modified_gmt":"2017-04-18T03:40:31","slug":"occupancy-estimation-robust-design","status":"publish","type":"page","link":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/occupancy-estimation-robust-design\/","title":{"rendered":"Occupancy Estimation Robust Design"},"content":{"rendered":"

Contents<\/a> – Index<\/a><\/span><\/p>\n

\n

Occupancy Estimation Robust Design<\/b><\/span><\/p>\n

The robust design occupancy estimation model (McKenzie et al. 2003<\/a>) provides a method to estimate the rate (epsilon) at which occupied sites (plots) go extinct, and the rate (gamma) at which unoccupied sites are occupied.\u00a0 The parameters of the models implemented in MARK are psi (proportion of sites occupied), epsilon (probability of an occupied site becoming unoccupied), gamma (probability of an unoccupied site becoming occupied), and p (detection probability on a visit to the site).\u00a0 Three implementations of the model are present in MARK.\u00a0 The default, which is the parameterization used when the “Robust Design Occupancy Estimation” button is clicked in initiating a new analysis, is {psi(1), epsilon(t), gamma(t), p(session, t)}.\u00a0 The time intervals specified when the data are first read into MARK<\/a> determine the number of intervals where epsilon applies, and the number of primary sessions that psi applies.\u00a0 The time intervals are specified the same as for the robust design<\/a> data type.<\/span><\/p>\n

The default parameterization {psi(1), epsilon(t), gamma(t), p(session, t)} generates estimates of the time-specific psi(t), lambda(t) and lambda'(t) parameters as derived parameters (see page 200 of the McKenzie et al. occupancy book).\u00a0 The lambda(t) estimates are ratios of consecutive occupancy rates, and are equivalent to the rate of change in occupancy.\u00a0 The lambda'(t) estimates are the log-odds ratio of consecutive occupancy rates (and are not technically defined for psi = 0 or psi = 1.\u00a0 The main advantage of this parameterization is that gamma and epsilon can be constrained to the [0, 1] interval with the link function, and not affect convergence properties of the optimization.\u00a0 The {psi(t} epsilon{t} p(session, t)} produces derived parameters of gamma(t) and lambda(t), the ratios of consecutive occupancy rates.\u00a0 However, gamma is constrained to the [0, 1] interval through the penalty function<\/a> approach, and for some problems, numerical convergence may be problematic.\u00a0 The {psi(t} gamma{t} p(session, t)} produces derived parameters of epsilon(t) and lambda(t), the ratios of consecutive occupancy rates.\u00a0 Here, epsilon is constrained to the [0, 1] interval through the penalty function<\/a> approach, and for some problems, numerical convergence may be problematic.<\/span><\/p>\n

To build models for the {psi(t} epsilon{t} p(session, t)} and {psi(t} gamma{t} p(session, t)} parameterizations, you change the data type<\/a> from the PIM main menu.\u00a0 All three of these models have the same likelihood, so AICc<\/a> values are comparable between them.\u00a0 In addition, the Pledger mixture models<\/a> have been added for all 3 of the robust design occupancy models, and are also available with the change data type menu choice from the PIM main menu.\u00a0 All 6 of these parameterizations produce estimates of lambda(t) and lambda'(t).<\/span><\/p>\n

The usual occupancy model<\/a> can also be used with robust design data by treating the primary sessions as different attribute groups, and psi estimated for each group.\u00a0 However, this approach to the analysis would not provide estimates of the extinction (epsilon) and recolonization (gamma) rates.<\/span><\/p>\n

One model often of interest is the random extinction and colonization model, where the probability of a site being occupied at time t+1 is the same regardless of whether or not the site was occupied at time t.\u00a0 You can obtain estimates for this model in MARK by cleaver coding of the <\/span>design matrix<\/a><\/span>.\u00a0 Suppose the epsilon of interest is parameter 2 in the PIM, and the gamma of interest is parameter 3 in the PIM.\u00a0 Use a common beta parameter for both epsilon and gamma, i.e., there will be a single column that is modeled by both rows 2 and 3 of the design matrix.\u00a0 If you specify a <\/span>logit link<\/a><\/span> for both epsilon and gamma, and code epsilon as -1 in the design matrix and gamma as +1, the result is the model with 1 – epsilon = gamma.\u00a0 The design matrix looks like:<\/span>
\nColumns: B1 B2 B3<\/span>
\nRow 1: 1 0 0<\/span>
\nRow 2: 0 -1 0 \/* This is epsilon *\/<\/span>
\nRow 3: 0 1 0 \/* This is gamma *\/<\/span><\/p>\n

You also get the same results if you use a <\/span>sin link<\/a><\/span> instead of a logit link, because both link functions are symmetric about 0.5.<\/span>
\n\u00a0<\/span>
\nFor those of you that want to be more complicated, you can also code epsilon with -1 and use a cloglog link, and gamma with 1 and use a loglog link, and get the same result as you would with the logit link.\u00a0 Likewise, you can reverse the 2 link functions, or reverse the -1 and 1, and still get the same model, but with different interpretations of the estimate of beta.<\/span><\/p>\n

The -1 logit link approach also works with covariates, either temporal, group, or individual covariates.\u00a0 Assume the following PIMs:<\/span><\/p>\n

\u00a0 Epsilon PIM<\/span>
\n\u00a0 2 3 4<\/span>
\n\u00a0\u00a0 3 4<\/span>
\n\u00a0\u00a0\u00a0 4<\/span><\/p>\n

\u00a0 Gamma PIM<\/span>
\n\u00a0 5 6 7<\/span>
\n\u00a0\u00a0 6 7<\/span>
\n\u00a0\u00a0\u00a0 7<\/span><\/p>\n

Further assume that you have the time-varying covariates with values 11, 12, and 13 for the 3 time intervals.\u00a0 The design matrix then looks like:<\/span><\/p>\n

Row 1: 1 0 0<\/span>
\nRow 2: 0 -1 -11<\/span>
\nRow 3: 0 -1 -12<\/span>
\nRow 4 0 -1 -13<\/span>
\nRow 5 0 1 11<\/span>
\nRow 6: 0 1 12<\/span>
\nRow 7: 0 1 13<\/span><\/p>\n

Similarly, if you have an individual covariate named “covariate”, you have to use the <\/span>design matrix function<\/a><\/span> product to specify a negative value for the covariate:<\/span><\/p>\n

Row 1: 1 0 0<\/span>
\nRow 2: 0 -1 product(-1,covariate)<\/span>
\nRow 3: 0 -1 product(-1,covariate)<\/span>
\nRow 4 0 -1 product(-1,covariate)<\/span>
\nRow 5 0 1 covariate<\/span>
\nRow 6: 0 1 covariate<\/span>
\nRow 7: 0 1 covariate<\/span><\/p>\n

In all of these cases, the estimate of gamma = 1 – epsilon.<\/span><\/p>\n

An extension of the occupancy robust design model is the data type with multiple states, i.e., more than just simple occupied\/not occupied.\u00a0 See <\/span>Occupancy Estimation Multiple States Robust Design<\/a><\/span> for the corresponding model.<\/span><\/p>\n

Individual Heterogeneity for p<\/b><\/span><\/p>\n

Two additional sets of models have been incorporated in MARK to handle individual heterogeneity of p.\u00a0 Pledger mixture models (<\/span>Pledger 2000<\/a><\/span>), and normally distributed random errors on the logit scale (<\/span>McClintock and White 2009, Gimenez and Choquet 2010<\/a><\/span>).\u00a0 Both of these approaches would typically require more than 2 secondary occasions in a primary occasion to be able to detect individual heterogeneity.\u00a0 However, one advantage of the robust design is that you might look at individual heterogeneity across multiple primary occasions.<\/span><\/p>\n

Derived Parameters<\/b><\/span><\/p>\n

Three <\/span>derived parameters<\/a><\/span> are provided.\u00a0 First is occupancy (psi) through time.\u00a0 Second is lambda, the ratio of consecutive psi estimates.\u00a0 Finally is lambda’ (lambda prime), that is i.e., [psi(t)\/(1 – psi(t))] \/ [psi(t – 1)\/(1 – psi(t – 1))] = [psi(t)*(1-psi(t-1)] \/ [psi(t – 1)*psi(1 – psi(t))] (<\/span>MacKenzie et al 2006<\/a><\/span>, page 200).<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"

Contents – Index Occupancy Estimation Robust Design The robust design occupancy estimation model (McKenzie et al. 2003) provides a method to estimate the rate (epsilon) at which occupied sites (plots) go extinct, and the rate (gamma) at which unoccupied sites …<\/p>\n

Occupancy Estimation Robust Design<\/span> Read More »<\/a><\/p>\n","protected":false},"author":117,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-275","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/275","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/users\/117"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/comments?post=275"}],"version-history":[{"count":1,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/275\/revisions"}],"predecessor-version":[{"id":276,"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/pages\/275\/revisions\/276"}],"wp:attachment":[{"href":"https:\/\/sites.warnercnr.colostate.edu\/gwhite\/wp-json\/wp\/v2\/media?parent=275"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}