Robust Design Multi-State with State Uncertainty
Robust Design Multi-State with State Uncertainty
The basic robust-design multi-state data types, as well as the open robust-design multi-state data type, assume that all encounters are correctly classified to their state.
Four additional robust design multi-state with state uncertainty data types (2 closed, 2 open) are also available where encounters cannot be always classified into one of the available states. The 2nd open data type includes parameters for when an attribute for determining the state becomes available. This open model is discussed in more detail below. What follows pertains to the 2 closed data types and the open data type that includes the phi parameter. These 3 data types are labeled as:
119. ORDMSState: Open Robust Design Multi-state with State Probabilities
121. RDMSMisClass: Closed Robust Design Multi-state with State Uncertainty
122. RDMS2MisClass: Closed Robust Design Multi-state with 2 States Uncertain
An additional parameter, delta, is included in these models to estimate the probability of correct classification given that the animal is in a specific state. Two of these data types only allow a single uncertain category. For example, Kendall et al. (2004) were interested in the survival of manatees conditional on their reproductive status: cows with calves versus cows without calves. When a calf was observed with a cow, the state is known unambiguously. However, a cow observed without a calf may have a calf, but the calf just isn’t visible, e.g., because of water clarity. On the other hand, in perfectly clear water, maybe the cow can be classified unambiguously as not having a calf. So, 2 states can be defined: C is with calf, and N is without calf. For these data types with a single uncertain category, a u (lower case u) is used to indicate that a cow was observed, but not sure about whether a calf is present or not. So, the encounter history consists of C, N, u, 0, and dots can be included.
For three of the data types and using the example above to illustrate, the parameters would be SC, SN, psi C to N, psi N to C, pi, omega, pC, pN, deltaC, and deltaN. Most of these parameters are obvious from other robust design multi-state data types, but some are new because of the uncertain states. The parameter pi is the proportion of the population released in a specific state, with 1 less pi parameters than their are states. So, pi(1) is the proportion of the population released as state C, pi(2) is obtained by subtraction as 1 – pi(1), the proportion of the population released in state N. Because animals can be released during any primary period, there is a value of pi for all primary periods except the last. In addition, in April, 2013, a change was made to the 3 data types. The mixture parameters for the first primary period, pi1 was reparameterized, so that pi1^s = w1^s*p1^*s / sum[w1^s*p1^*s] (see Kendall et al. 2012 Ecology). Therefore pi1 no longer exists as a parameter in the likelihood, and there are now K-2 parameters in the pi PIMs, where K is the number of primary periods. The first parameter listed is for primary period 2, and the last pi is for primary period K-1. There is a pi estimate only for the first S – 1 states, where S is the number of states. The pi for the last state is obtained by subtraction. We made this change because for the common case where a given state is never known with certainty, pi1 and therefore the survival and transition probabilities for primary period 1 for that state were not estimable.
The omega parameter is proportion of the population in each state d at each primary period. So again the omega values have to sum to 1, and the omega for the last state is obtained by subtraction. The parameters deltaC and deltaN are the probabilities of correctly classifying the state of an encountered animal, given the true state of either C or N. So, the probability of encountering a cow with calf and correctly determining that the cow had a calf would be pC deltaC, whereas encountering a cow with a calf but not seeing the calf would be p C(1 – deltaC).
For the data type where there is only one uncertain state in the encounter history, which must be the lower-case letter u, the above parameters are all that are needed to parametrize the model. For the robust-design multi-state open data type with an uncertain state, 2 additional parameters are required for each state: the of entry to the study area (pent) and the probability of remaining on the study area (phi). Both are as defined for the robust-design multi-state open model without uncertain states. Note that the pent estimates within a primary period must sum to <= 1, so if there are more than 2 secondary occasions within a primary period, you should either use the mlogit link function, or else initiate these estimates to a small value that meets the criterion of the sum <= 1. The last pent parameter is obtained by subtraction. So, if you want all the animals to be present on occasion 1, i.e., already in the population when sampling starts, fix pent(1) to 1 and the rest of the pent values to zero.
There are no additional parameters for the data type that assumes closed primary sessions with more than 1 uncertain state. This data type assumes that you correctly classify an encounter into a state and specify the upper-case state identifier, e.g., C and N. For observations where you are uncertain, the lower-case state identifier is used, e.g., c and n (although the example breaks down here).
At this time, there is no open data type equivalent for the >1 uncertain states closed data type.
Open Data Type with Parameter for when the state becomes Detectable.
This data type is labeled in the output as 142. RDMSOpenMCSeas: Robust Design Multi-state Open with State Uncertainty and Seasonal Effects.
The parameters between primary sessions are identical to the 3 data types described above: S for each state, appropriate transition parameters (psi), and pi (described above). A PIM of pent parameters appear for each primary session and each state. Each set must sum to one, so the last pent value for each primary session obtained by subtraction. The mlogit link function should be used for pent. There is also a parameter for remaining on the study area, except that this data type defines this parameter opposite of the open model described above. In this data type, instead of phi, the parameter d is used, with d = 1 – phi. Think of the d parameter as departure (rather than remain as for phi). The estimates of d correspond to 1 – phi, with identical SE. There is another difference between the 2 open data types. The d PIM is not upper-triangular as for phi. That is, the phi parameter can be made age-specific. However, the d parameter is only time-specific. The reason for this difference is because of the number of states that would be possible if both the d and c (defined below) were potentially age-specific.
Likewise, there is a set of p PIMs for each primary period for each state. There are also PIMs for the delta parameters (probability that the state is determined, given that the attribute is present to distinguish the state), again for each primary period and each state.
The 2 additional parameters for this more complex open model are alpha and c. PIMs for each appear for each primary occasion and each state. Alpha is the probability that the attribute to assign the state has appeared. In the example above for manatees, this might be the birth of the calf. So, the adult manatee is present on the study area, but until she gives birth, she cannot be classified as with calf (C), or no calf (N). The alpha parameters in each PIM must sum to <=1, with the last alpha obtained by subtraction. So, if sampling starts after all pregnant females have given birth, alpha(1) is fixed to 1 and the rest fixed to zero.
The parameter c is the probability that the attribute allowing the state to be determined still exists. For example, manatee cows might wean their calves towards the end of the sampling period, so that the attribute of a calf present is no long available. The attribute “ceases”, so that the state can no longer be determined. The c parameter has upper-triangular PIMs for each primary period and each state. These PIMs can be time-specific or age-specific if the attribute disappears as a function of the length of time it has been present.
To make the 2 open models exactly equivalent, fix alpha(1) = 1 for each primary session and each state, and the rest of the alpha parameters fixed to zero. Also fix all the c parameters to zero, because the attribute can never disappear. The phi PIMs must be time-specific (not age-specific) to match the time-specific d parameters of the more complex open model.
Residence and duration times for this model are descibed in Duration Times.