Searcher efficiency is modeled as a function of the number of
times a carcass has been missed in previous searches and any number of
covariates. Format and usage parallel that of common R functions
lm, glm, and gam. However, the input data
(data) is structured differently to accommodate the
multiple-search searcher efficiency trials (see Details), and model
formulas may be entered for both p (akin to an intercept) and
k (akin to a slope).
pkm(formula_p, formula_k = NULL, data, obsCol = NULL, kFixed = NULL,
allCombos = FALSE, sizeCol = NULL, CL = 0.9, kInit = 0.7,
quiet = FALSE, ...)pkm0(formula_p, formula_k = NULL, data, obsCol = NULL, kFixed = NULL,
kInit = 0.7, CL = 0.9, quiet = FALSE)
pkmSet(formula_p, formula_k = NULL, data, obsCol = NULL,
kFixed = NULL, kInit = 0.7, CL = 0.9, quiet = FALSE)
pkmSize(formula_p, formula_k = NULL, data, kFixed = NULL,
obsCol = NULL, sizeCol = NULL, allCombos = FALSE, kInit = 0.7,
CL = 0.9, quiet = FALSE)
Formula for p; an object of class formula
(or one that can be coerced to that class): a symbolic description of
the model to be fitted. Details of model specification are given under
"Details".
Formula for k; an object of class formula
(or one that can be coerced to that class): a symbolic description of the
model to be fitted. Details of model specification are given under
"Details".
Data frame with results from searcher efficiency trials and any
covariates included in formula_p or formula_k (required).
Vector of names of columns in data where results
for each search occasion are stored (optional). If obsCol is not
provided, pkm uses as obsCol all columns with names that
begin with an "s" or "S" and end with a number, e.g., "s1",
"s2", "s3", etc. This option is included as a convenience for the user,
but care must be taken that other data are not stored in columns with
names matching that pattern. Alternatively, obsCol may be
entered as a vector of names, like c("s1", "s2", "s3"),
paste0("s", 1:3), or c("initialSearch", "anotherSearch",
"lastSearch"). The columns must be in chronological order, that is, it is
assumed that the first column is for the first search after carcass arrival,
the second column is for the second search, etc.
Parameter for user-specified k value (optional). If a
value is provided, formula_k is ignored and the model is fit under
the assumption that the k parameter is fixed and known to be
kFixed \(\in [0, 1]\). If a sizeCol is provided, kFixed
may either be NULL, a single number in [0, 1], or a vector with
kFixed values for two or more of the carcass size classes. For
example, if there are three sizes (S, M, and L),
kFixed could be c(S = 0.3, M = 0.8, L = 1.0) to assign fixed
k values to each size. To fit k for size S and to assign
values of 0.8 and 1.0 to sizes M and L, resp., use
kFixed = c(S = 0.3, M = 0.8, L = 1.0). If there are more than one size
classes and kFixed is a scalar, then all size classes are assigned the
same kFixed value (unless kFixed is named, e.g.,
kFixed = c(S = 0.5), in which case only the named size is assigned the
kFixed).
logical. If allCombos = FALSE, then the single model
expressed by formula_p and formula_k is fit using a call to
pkm0. If allCombos = TRUE, a full set of pkm
submodels derived from combinations of the given covariates for p
and k is fit. For example, submodels of formula_p = p ~ A * B
would be p ~ A * B, p ~ A + B, p ~ A, p ~ B,
and p ~ 1. Models for each pairing of a p submodel with a
k submodel are fit via pkmSet, which fits each model
combination using successive calls to pkm0, which fits a
single model.
character string. The name of the column in data that
gives the carcass class of the carcasses in the field trials. If
sizeCol = NULL, then models are not segregated by size. If a
sizeCol is provided, then separate models are fit for the data
subsetted by sizeCol.
numeric value in (0, 1). confidence level
numeric value in (0, 1). Initial value used for numerical
optimization of k. Default is kInit = 0.7. It is rarely
(if ever) necessary to use an alternative initial value.
Logical indicator of whether or not to print messsages
additional arguments passed to subfunctions
an object of an object of class pkm, pkmSet,
pkmSize, or pkmSetSize.
pkm0()returns a pkm object, which is a description
of a single, fitted pk model. Due to the large number and complexity of
components of apkm model, only a subset of them is printed
automatically; the rest can be viewed/accessed via the $ operator
if desired. These are described in detail in the 'pkm Components'
section.
pkmSet()returns a list of pkm objects, one for each
of the submodels, as described with parameter allCombos = TRUE.
pkmSize()returns a list of pkmSet objects (one for
each 'size') if allCombos = T, or a list of pkm objects (one
for each 'size') if allCombos = T
pkmreturns a pkm, pkmSet, pkmSize, or
pkmSetSize object:
pkm object if allCombos = FALSE, sizeCol = NULL
pkmSet object if allCombos = TRUE, sizeCol = NULL
pkmSize object if allCombos = FALSE, sizeCol != NULL
pkmSetSize object if allCombos = TRUE, sizeCol != NULL
The following components of a pkm object are displayed automatically:
callthe function call to fit the model
formula_pthe model formula for the p parameter
formula_kthe model formula for the k parameter
predictorslist of covariates of p and/or k
AICcthe AIC value as corrected for small sample size
convergenceconvergence status of the numerical optimization
to find the maximum likelihood estimates of p and k. A
value of 0 indicates that the model was fit successfully. For
help in deciphering other values, see optim.
cell_pksummary statistics for estimated cellwise estimates
of p and k, including the number of carcasses in each cell,
medians and upper & lower bounds on CIs for each parameter, indexed by
cell (or combination of covariate levels).
The following components are not printed automatically but can be accessed
via the $ operator:
datathe data used to fit the model
data0$data with NA rows removed
betahat_p, betahat_kparameter estimates for the terms in the
regression model for for p and k (logit scale). If k
is fixed or not provided, then betahat_k is not calculated.
varbetathe variance-covariance matrix of the estimators
for c(betahat_p, betahat_k).
cellMM_p, cellMM_kcellwise model (design) matrices for
covariate structures of p_formula and k_formula
levels_p, levels_kall levels of each covariate of p
and k
nbeta_p, nbeta_knumber of parameters to fit the p
and k models
cellscell structure of the pk-model, i.e., combinations of
all levels for each covariate of p and k. For example, if
covar1 has levels "a", "b", and "c", and
covar2 has levels "X" and "Y", then the cells
would consist of a.X, a.Y, b.X, b.Y,
c.X, and c.Y.
ncelltotal number of cells
predictors_k, predictors_pcovariates of p and k
observationsobservations used to fit the model
kFixedthe input kFixed
AICthe AIC value for the fitted model
carcCellsthe cell to which each carcass belongs
CLthe input CL
loglikthe log-liklihood for the maximum likelihood estimate
pOnlya logical value telling whether k is included in
the model. pOnly = TRUE if and only if length(obsCol) == 1)
and kFixed = NULL
data_adjdata0 as adjusted for the 2n fix to accommodate
scenarios in which all trial carcasses are either found or all are not
found on the first search occasion (uncommon)
fixBadCellsvector giving the names of cells adjusted for the 2n fix
pkmSize may also be used to fit a single model for each carcass class if
allCombos = FALSE. To do so, formula_p and formula_k
must be a named list of formulas with names matching the sizes listed in
unique(data[, sizeCol]). The return value is then a list of
pkm objects, one for each size.
The probability of finding a carcass that is present at the time of
search is p on the first search after carcass arrival and is
assumed to decrease by a factor of k each time the carcass is
missed in searches. Both p and k may depend on covariates
such as ground cover, season, species, etc., and a separate model format
(formula_p and formula_k) may be entered for each. The
models are entered as they would be in the familiar lm or
glm functions in R. For example, p might vary with
A and B, while k varies
only with A. A user might then enter p ~ A + B
for formula_p and k ~ A for
formula_k. Other R conventions for defining formulas may also be
used, with A:B for the interaction between covariates
A and B and A * B as short-hand for A + B + A:B.
Search trial data must be entered in a data frame with data in
each row giving the fate of a single carcass in the field trials. There
must be a column for each search occassion, with 0, 1, or NA depending on
whether the carcass was missed, found, or not available (typically
because it was found and removed on a previous search, had been earlier
removed by scavengers, or was not searched for) on the given search
occasion. Additional columns with values for categorical covariates
(e.g., visibility = E, M, or D) may also be included.
When all trial carcasses are either found on the first search or
are missed on the first search after carcass placement, pkm effects a
necessary adjustment to the for accuracy; otherwise, the model would not be
able to determine the uncertainty and would substantially over-estimate the
variance of the parameter estimates, giving \(\hat{p}\) essentially equal
to 0 or 1 with approximately equal probability. The adjustment is to fit the
model on an adjusted data set with duplicated copies of the original data
(2n observations) but with one carcass having the opposite fate of the
others. For example, in field trials with very high searcher efficiency and
n = 10 carcasses, all of which are found in the first search after
carcass placement, the original data set would have a carcass observation
column consisting of 1s (rep(1, 10)). The adjusted data set would
have an observation column consisting of 2n - 1 1s and one 0. In this
case, the point estimate of p is 1/(2n) with distribution that
closely resembling the Bayesian posterior distributions of p with a
uniform or a Jeffreys prior. The adjustment is applied on a cellwise basis
in full cell models (e.g., 1, A, B, A * B). In the additive model with two
predictors (A + B), the adjustment is made only when a full level of
covariate A or B is all 0s or 1s.
# NOT RUN {
head(data(wind_RP))
mod1 <- pkm(formula_p = p ~ Season, formula_k = k ~ 1, data = wind_RP$SE)
class(mod1)
mod2 <- pkm(formula_p = p ~ Season, formula_k = k ~ 1, data = wind_RP$SE,
allCombos = TRUE)
class(mod2)
names(mod2)
class(mod2[[1]])
mod3 <- pkm(formula_p = p ~ Season, formula_k = k ~ 1, data = wind_RP$SE,
allCombos = TRUE, sizeCol = "Size")
class(mod3)
names(mod3)
class(mod3[[1]])
class(mod3[[1]][[1]])
# }
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