GenEst - A Tutorial with Wind Examples

Daniel Dalthorp

2018-12-18

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Introduction: Tutorial with Examples

This tutorial provides an introduction to the array of command line tools GenEst provides for estimating carcass arrivals and detection probabilities. The approach is to walk through analyses of realistic but fictitious data sets representing studies of bird and bat mortality at a wind power facility.

The general steps in the analysis are:

  1. Construct a model for Searcher Efficiency
  2. Construct a model for Carcass Persistance
  3. Estimate mortality
  4. Specify the type of summary desired (e.g., mortality by species and season)

Data required for a full analysis include results of searcher efficiency trials, results of carcass persistence trials, search schedules for all units searched, the search coverage within each unit, and results of periodic carcass surveys. More information about the kinds of data required can be found in the User Guide. For convenience, data required in this tutorial are available in R as packaged lists, which can readily be loaded as described below. Alternatively, the data may be downloaded into zipped .csv files from the GUI.

To perform the analyses illustrated in the tutorial, begin by starting R and loading GenEst from the command line: library(GenEst)

Example 1: Estimating Bat Mortality from Searches on Roads and Pads

Searcher efficiency and carcass persistence would be expected to vary with carcass size (sparrow, eagle, bat), ground characteristics (road & pad, cleared field, vegetation type), season, etc. In this first example, we limit the analysis to one carcass size (bat) and one ground visibility class (RP = road and pad). A more complicated scenario is analyzed in example 2.

The required data is stored in wind_RPbat, a list of data frames with results for searcher efficiency (SE) and carcass persistence trials (CP), search schedules for all turbines (SS), the search coverage or density weighted proportion (DWP) of area searched at each turbine (i.e., the fraction of carcasses expected to fall in the search plots), and carcass observation (CO) data.

Load the full data set into R:

data(wind_RPbat)
names(wind_RPbat)
#> [1] "SE"  "CP"  "SS"  "DWP" "CO"

To streamline the notation, extract the data from the wind_RPbat list into its components:

data_SE <- wind_RPbat$SE
data_CP <- wind_RPbat$CP
data_SS <- wind_RPbat$SS
data_DWP <- wind_RPbat$DWP
data_CO <- wind_RPbat$CO

Searcher Efficiency (SE)

Searcher efficiency trials were conducted on roads and pads, with a total of 60 fresh carcasses placed in the field over the course of the entire monitoring period, evenly divided among seasons (spring, summer, fall). Carcasses that were later discovered by search teams during the course of normal carcass surveys were removed from the field. Carcasses were left in the field for up to 5 searches after carcass placement.

Results of the SE field trials are stored in the data_SE data frame:

Columns s1, s2, ..., s5 show the fate of carcass pkID on the 1st, 2nd, … 5th searches after the carcass was placed. A 1 indicates that the carcass was discovered, a 0 indicates that the carcass was present but not discovered, and NA indicates that the carcass was no longer present for discovery or no search was conducted.

Carcass Persistence (CP)

Carcass persistence trials were conducted on roads and pads, with a total of 60 fresh carcasses placed in the field over the course of the entire monitoring period, evenly divided among seasons (spring, summer, fall). Carcasses were checked approximately 1, 2, 3, 4, 7, 10, 14, 21, and 28 days after placement in the field. Exact times were entered as decimal fractions of days after placement.

Results of the SE field trials are stored in the data_CP data frame:

Exact persistence times are not known, but a carcass that was present at one check and absent at the next check is assumed to have been removed at some point in the interval. The left endpoint of the interval was entered as LastPresent and the right endpoint as FirstAbsent. For carcasses that had not been scavenged by the end of the study, LastPresent is the time of the last check and FirstAbsent is Inf. For carcasses whose removal time is known exactly (e.g., scavenging was recorded by camera), LastPresent = FirstAbsent. The Season column gives the season at the time the carcass was placed in the field.

Search Schedules (SS)

Carcass searches were conducted on roads and pads within a 120 m radius from all 100 turbines at a fictitious wind power facility. Monitoring began on 1955-04-15 and continued through 1955-11-01. Searches spanned 3 seasons: spring, summer, fall. Search intervals varied by turbine and by time of year, ranging from daily searches at some turbines in the fall and searches once every 12 days in the spring at some other turbines. Search schedules for all turbines are stored in data_SS, which is a data frame with a column for search dates (including all dates that any turbine was searched); a column of 0s and 1s for each turbine, indication whether it was searched on the given date; and zero or more optional columns giving additional information about the date (e.g., season).

Note that we have only displayed a few of the turbine columns - there are 100 turbine columns altogether (t1, …, t100).

Density Weighted Proportion (DWP)

The density-weighted proportion (DWP) is the expected fraction of carcasses that fell in the searched area. Carcass density is not the same at all distances from a turbine, but typically rises over a short distance then decreases eventually to 0. Searches were conducted on roads and pads within a 120 m radius from all 100 turbines to provide sufficient data with which to model the change in density with distance and from this, accurately calculate the fraction of all carcasses that are expected to land on road and pad surrounding each turbine (density-weighted proportion or DWP). The configuration of the roads and pads differs among turbines, hence the DWP must be calculated for each turbine. DWPs for bats at each turbine are stored in data_DWP, which is a data frame with a column for turbine name (note that turbine names also must be included among column names in data_SS, which gives the search schedule at each turbine) and a column of DWP labeled bat. In other studies where, for example, mortality of birds might be of interest, DWP would be expected vary with carcass size or species (e.g., the spatial distributions of bats and large birds around a turbine would likely differ from one another). In that case, each carcass size class (large, medium, small, bat) would have its own column.

Carcass Observations (CO)

Information about each (non-trial) carcass observed during searches is stored in CO_data which is a data frame with at least 4 columns: carcass ID, the turbine (or unit) at which it was found, the date it was found and its distance from the turbine center. In data_CO we also have turbine type, species, and species group variables by which we will later summarize mortality estimates.

Estimating Searcher Efficiency and Carcass Persistence Parameters

Searcher efficiency and carcass persistence parameters are estimated by fitting models using functions pkm and cpm (carcass persistence model) which are patterned after familar R functions such as lm and glm. The “pk” in pkm refers to GenEst’s model of searcher efficiency which includes two parameters: p, which is the initial searcher efficiency for carcasses on the first search after they have arrived, and k, which is a parameter governing the decrease in searcher efficiency in later searches. In this relatively simple example, our SE and CP field trials were conducted for one carcass size (bat) on one type of terrain (roads and pads) in three seasons (spring, summer, fall). The only potential predictor variable we have is Season, which is entered as a column in both data_SE and data_CP.

Searcher efficiency is the probability of detection of a carcass that is present in the searched area at the time of search. Searcher efficiency typically decreases with carcass age because older carcasses tend to become harder to find as they accumulate dust or debris, fall deeper into vegetation, get blown against objects or into holes, decay, or get partially scavenged. In addition, carcasses missed in one search tend to be more likely to be missed in subsequent searches because the relatively easy-to-find carcasses are preferentially removed in the first searches after carcass arrival, leaving mostly the harder-to-find carcasses available in subsequences searches. GenEst accounts for a non-constant searcher efficiency using two parameters, p (searcher efficiency on the first search after carcass arrivals) and k (proportional change in searcher efficiency with each successive search). The k parameter can be estimated from field trials if carcasses that are not discovered in the first search after arrival are left in the field for possible discovery in later searches.

p and k may both depend on covariates such as season, visibility class, or carcass size, and GenEst allows for them to be modeled as functions of different covariate combinations.

NOTE: The pkm family of functions by default interprets columns with names that begin with and “s” or “S” and end with a number contain search results data (carcass found = 0, not found = 1). A user can override the auto-parsing by explicitly listing the names of the search data columns in a vector of character strings in the obsCol argument.

The probability of a carcass persisting a given length of time without being removed by scavengers (or other factors) is modeled as a Weibull, lognormal, loglogistic, or exponential distribution. Like the p and k parameters for searcher efficiency, the location and scale parameters (Therneau 2015) of the persistence distribution may depend on covariates. GenEst allows for them to be modeled as separate functions of predictor combinations.

The model summary shows descriptive statistics for the cellwise estimates of the l and s parameters. The location and scale parameterization is common in survival analysis, but the pda and pdb parameterization (Dalthorp and Huso, 2014) is also shown. These parameterizations are convenient to work with in statistical calculations but are not as convenient for giving users quick insight into the distributions, so a third set of summary statistics about the fitted distributions is also given. Namely, the median persistence time and the r statistic, which is the probability that a carcass persists until the first search after arrival (assuming uniformly distributed arrival times within the interval). Clearly, r depends on the length of the search interval, and the table shows r for intervals of 1, 3, 7, 14, and 28 days. A rough, back-of-the-envelope calculation for the probability of observing a carcass that arrives at a site during the monitored period would be DWP * r * p * f, which is DWP = fraction of carcasses that arrive in the area searched at a unit, r = fraction of carcasses that persist until a search, p = fraction of carcasses found on the first search after arrival (given that they persisted), and f = fraction of carcasses that arrive at the units searched.

In other scenarios we might consider other predictors, like the visibility of the ground searched or the search team. We might also be interested in carcasses of different sizes (e.g., large, medium, and small birds instead of or in addition to bats). We are not restricted to using the same predictors of both SE and CP. The modeling complexity increases with each additional predictor, but, in theory, any number of predictors can be used. The only rule is that sufficient numbers of trial carcasses must be placed in each cell combination of factor levels among the selected predictors. For example, if we were to place 15 carcasses for each cell for predictors that include season (spring, summer, fall, winter), size (S, M, L, B), visibility (RP, M, D), search team (dogs, humans), and turbine type (small, medium, large), we’d need 15 x 4 x 4 x 3 x 2 x 2 = 2880 carcasses. Typically, the number of predictors is limited to a few key variables.

Mortality Estimation

Each carcass’s contribution to the total mortality in each search interval is estimated using the estM function.

Mortality estimates may be partitioned or split into desired categories, such as species, season, or turbine type. Splits may be performed according to characteristics of the carcasses or where they were found (e.g., species, turbine or other variable found in data_CO) or when they were found (e.g., season or other variable associated with search schedule and found in data_SS, or a vector of specific times).

Mortality by Species (a CO split because it is a column in the CO file):

Mortality estimates may also be split by temporal variables that are represented as columns in data_SS or as numeric vectors spanning the monitoring season (from day 0 to length of monitoring season). If several temporal splits are to be calculated, creating a specially formatted prepSS object for the search schedule can streamline the calculations.

Mortality by Season (an SS split because it is a column in the SS file):

Mortality by month (a temporal split that spans the monitoring period):

Temporal splits that divide the monitoring season into separate time intervals (like season or month) can be plotted as the number per interval (rate = FALSE, which is the default arg in calcSplits) or the number per unit time (rate = TRUE).

Finally, splits can be calculated for combinations of splitting covariates, like species by season or species group by turbine type. No more than two splitting covariates may be used in one call to calcSplits and at most one temporal split may be used (whether it is an SS split or a vector of times).

Example 2: Estimating Bird and Bat Mortality from Searches on Varied Ground

Thorough searches out to a radius of 60 m from each turbine were conducted at 23 out of 100 turbines. The searched area was divided into three visibility classes (RP, M, D) according the difficulty of finding carcasses.

Searcher efficiency and carcass persistence would be expected to vary with carcass size (sparrow, eagle, bat), ground characteristics (road & pad, cleared field, vegetation type), season, etc. In this example, we perform a full analysis of scenario with four classes of carcass (lrg, med, sml, and bat), three visibility classes (difficult = D, moderate = M, and road & pad = RP), and three seasons (spring, summer, and fall).

The required data is stored in wind_cleared, a list of data frames with results for searcher efficiency (SE) and carcass persistence trials (CP), search schedules for all turbines (SS), the search coverage or density weighted proportion (DWP) of area searched at each turbine (i.e., the fraction of carcasses expected to fall in the search plots), and carcass observation (CO) data.

Load the full data set into R:

data(wind_cleared)
names(wind_cleared)
#> [1] "SE"  "CP"  "SS"  "DWP" "CO"

To streamline the notation, extract the data from the wind_cleared list into its components:

data_SE <- wind_cleared$SE
data_CP <- wind_cleared$CP
data_SS <- wind_cleared$SS
data_DWP <- wind_cleared$DWP
data_CO <- wind_cleared$CO

Searcher Efficiency and Carcass Persistence Trials

In searcher efficiency and carcass persistence trials, 15 trial carcasses were placed in each combination of visibility class (D, M, RP), season (spring, summer, fall), and size class (lrg, med, sml, bat). Data formats are like those of example 1:

Searcher Efficiency Modeling

With 36 combinations of covariate levels (3 visibilities x 3 seasons x 4 sizes) and two parameters (p and k), the number of possible models to consider for searcher efficiency is unwieldy using simple calls to pkm, but pkm has powerful model building and model selection capabilities that can be accessed via the arg list: allCombos and sizeCol, which are discussed below.

When allCombos = TRUE, pkm fits the set of submodels of the given covariate combinations, including the full model, the null model, and everything in between. For example, if the parameter models are p ~ Visibility * Season and k ~ Visibility, pkm with allCombos = TRUE would fit all combinations of possible p models (p ~ Visibility * Season, p ~ Visibility + Season, p ~ Visibility, p ~ Season, and p ~ 1) and possible k models (k ~ Visibility and k ~ 1), or 10 models in all.

Carcasses in different size classes would be expected to have different searcher efficiency and carcass persistence parameters and would even be likely to be affected by covariates in different ways. When analyzing data with carcasses in different size classes, it is recommended that separate CP and SE models be fit for size classes separately. This can be accomplished using the sizeCol argument in pkm and cpm, where sizeCol gives the name of the column that gives the carcass size classes in data_SE.

When allCombos = TRUE and sizeCol is defined, pkm returns a list of sets of models for each size class. The sets of models for each size class include the full spectrum of models that can be constructed using simple combinations of the covariates.

To estimate mortality, one model for each size class must be selected from the long list of models fit. GenEst provides several tools for guiding the selection. First, the models can be listed by AICc, which gives a score for the quality of the model for the given data. Complicated models that use many parameters may fit the data more closely than a simpler model but are penalized because of their complexity and relative instability. The scores have meaning only in comparison with other models’. AICc provides a rough but useful guide for model selection, but should in no way be relied upon as definitive. Its utility is in identifying relatively poor models and in narrowing the choice of plausible models to a manageable number.

The aicc function lists the fitted models in order of \({\small \Delta}\)AICc for each size class (if applicable). For this discussion, we will focus on sml only, but for a full analysis, all size classes would need to be similarly analyzed.

Preference should normally be given to models with \({\small \Delta}\)AICc less than 6 or 7. Models with differences of less than 3 or 4 are generally considered indistinguishable by this measure. Choices among such models should be based on other criteria.

Diagnostic plots can be used to identify potential problems with model fits and to help distinguish between models with similar AICc scores. The plot function is defined for pkm (one model) and pkmSet (set of models for a given size class) objects. For example, plot(pkModels[["sml"]][[1]]) would produce the single figure for the first model for the sml size class. plot(pkModels[["sml"]]) would create plots for each model fit for the sml size class. To plot a specific single model from the full set, use the specificModel argument. For example, diagnostic plots for the model with the lowest AICc score are shown below:

The top row shows box plots of estimated p and k parameters for all cells (i.e., combinations of covariate levels, like D.fall for difficult visibility in the fall) for both the selected model (black) and the full model (gray). With the full model, the fits for each cell are based solely on data from that specific cell. The advantage is that each cell’s estimates are untainted by data from other cells. The disadvantage is that the sample size for each estimate is relatively small and the error bars large. In the reduced models, estimates for one cell borrow strength from estimates in related cells. This gives smaller error bars but can lead to errors if the model structure does not properly reflect the dependence of searcher efficiency parameters on cell characteristics.

In the figure, the estimates of p from the selected model are markedly less variable than the estimates from the full model, while the locations of the boxes are very similar for the two models. Thus, this selected model (the one with the lowest AICc) appears to be an improvement over the full cell model for estimating p.

The \({\small \Delta}\)AICc for the full model is 12.52, which indicates a serious deficiency in comparison to the model with the best fit, "p ~ Visibility; k ~ 1". The boxplots for k highlight one particular problem with the fit of the reference model. In some of the cells, the boxes extend from 0 to 1, which suggests that the reference model is unable to estimate k for the given cell. Selecting a simpler model for k often remedies this problem. If all models display this 0-1 phenomenon, a fixed k of 1 is appropriate if a smaller proportion of carcasses was found on the first search occasion than on later searches.

By comparison, the model with the highest \({\small \Delta}\)AICc (= {r max(aicc(pkModels[["sml"]]))} routinely estimates p and k either well above or well below the reference model but has fairly tight error bars–bad estimates but quite confident about them!

A similar model selection exercise gives the same form of model (p ~ Visibility; k ~ 1) for large birds, medium birds, and bats. These can all be collated into a list for later analysis of detection probabilities and mortality rates.

Carcass Persistence Modeling

The work flow for carcass persistence modeling is similar to that for searcher efficiency except that in addition to selecting covariates for two different parameters (location = l and scale = s), there are four model forms to choose from: Weibull, lognormal, loglogistic, and exponential.

The list of models is long:

lapply(aicc(cpModels), nrow)
#> $bat
#> [1] 80
#> 
#> $lrg
#> [1] 80
#> 
#> $med
#> [1] 80
#> 
#> $sml
#> [1] 80
aicc(cpModels[["sml"]])
#>    Distribution Location Formula        Scale Formula             AICc <U+0394>AICc
#> 25 weibull      l ~ 1                   s ~ 1                   780.00  0.00
#> 24 weibull      l ~ Visibility          s ~ 1                   780.16  0.16
#> 15 weibull      l ~ 1                   s ~ Season              780.38  0.38
#> 19 weibull      l ~ Visibility          s ~ Visibility          780.97  0.97
#> 20 weibull      l ~ 1                   s ~ Visibility          781.32  1.32
#> 10 weibull      l ~ 1                   s ~ Visibility + Season 782.00  2.00
#> 14 weibull      l ~ Visibility          s ~ Season              782.04  2.04
#> 22 weibull      l ~ Visibility + Season s ~ 1                   782.55  2.55
#> 74 loglogistic  l ~ Visibility          s ~ 1                   782.57  2.57
#> 21 weibull      l ~ Visibility * Season s ~ 1                   782.77  2.77
#> 49 lognormal    l ~ Visibility          s ~ 1                   782.84  2.84
#> 23 weibull      l ~ Season              s ~ 1                   782.88  2.88
#> 71 loglogistic  l ~ Visibility * Season s ~ 1                   783.06  3.06
#> 13 weibull      l ~ Season              s ~ Season              783.16  3.16
#> 50 lognormal    l ~ 1                   s ~ 1                   783.18  3.18
#> 75 loglogistic  l ~ 1                   s ~ 1                   783.30  3.30
#> 9  weibull      l ~ Visibility          s ~ Visibility + Season 783.33  3.33
#> 16 weibull      l ~ Visibility * Season s ~ Visibility          783.42  3.42
#> 46 lognormal    l ~ Visibility * Season s ~ 1                   783.86  3.86
#> 47 lognormal    l ~ Visibility + Season s ~ 1                   783.86  3.86
#> 40 lognormal    l ~ 1                   s ~ Season              783.88  3.88
#> 11 weibull      l ~ Visibility * Season s ~ Season              783.95  3.95
#> 72 loglogistic  l ~ Visibility + Season s ~ 1                   784.00  4.00
#> 17 weibull      l ~ Visibility + Season s ~ Visibility          784.09  4.09
#> 48 lognormal    l ~ Season              s ~ 1                   784.13  4.13
#> 69 loglogistic  l ~ Visibility          s ~ Visibility          784.28  4.28
#> 65 loglogistic  l ~ 1                   s ~ Season              784.30  4.30
#> 12 weibull      l ~ Visibility + Season s ~ Season              784.48  4.48
#> 73 loglogistic  l ~ Season              s ~ 1                   784.54  4.54
#> 44 lognormal    l ~ Visibility          s ~ Visibility          784.58  4.58
#> 18 weibull      l ~ Season              s ~ Visibility          784.85  4.85
#> 66 loglogistic  l ~ Visibility * Season s ~ Visibility          784.85  4.85
#> 64 loglogistic  l ~ Visibility          s ~ Season              784.95  4.95
#> 39 lognormal    l ~ Visibility          s ~ Season              785.01  5.01
#> 38 lognormal    l ~ Season              s ~ Season              785.09  5.09
#> 45 lognormal    l ~ 1                   s ~ Visibility          785.14  5.14
#> 6  weibull      l ~ Visibility * Season s ~ Visibility + Season 785.29  5.29
#> 70 loglogistic  l ~ 1                   s ~ Visibility          785.31  5.31
#> 8  weibull      l ~ Season              s ~ Visibility + Season 785.43  5.43
#> 61 loglogistic  l ~ Visibility * Season s ~ Season              785.47  5.47
#> 41 lognormal    l ~ Visibility * Season s ~ Visibility          785.68  5.68
#> 63 loglogistic  l ~ Season              s ~ Season              785.71  5.71
#> 36 lognormal    l ~ Visibility * Season s ~ Season              785.89  5.89
#> 35 lognormal    l ~ 1                   s ~ Visibility + Season 786.07  6.07
#> 42 lognormal    l ~ Visibility + Season s ~ Visibility          786.15  6.15
#> 37 lognormal    l ~ Visibility + Season s ~ Season              786.29  6.29
#> 67 loglogistic  l ~ Visibility + Season s ~ Visibility          786.30  6.30
#> 7  weibull      l ~ Visibility + Season s ~ Visibility + Season 786.48  6.48
#> 43 lognormal    l ~ Season              s ~ Visibility          786.48  6.48
#> 60 loglogistic  l ~ 1                   s ~ Visibility + Season 786.56  6.56
#> 62 loglogistic  l ~ Visibility + Season s ~ Season              786.56  6.56
#> 68 loglogistic  l ~ Season              s ~ Visibility          786.89  6.89
#> 59 loglogistic  l ~ Visibility          s ~ Visibility + Season 786.94  6.94
#> 34 lognormal    l ~ Visibility          s ~ Visibility + Season 787.01  7.01
#> 56 loglogistic  l ~ Visibility * Season s ~ Visibility + Season 787.41  7.41
#> 33 lognormal    l ~ Season              s ~ Visibility + Season 787.71  7.71
#> 31 lognormal    l ~ Visibility * Season s ~ Visibility + Season 787.87  7.87
#> 58 loglogistic  l ~ Season              s ~ Visibility + Season 788.35  8.35
#> 32 lognormal    l ~ Visibility + Season s ~ Visibility + Season 788.83  8.83
#> 57 loglogistic  l ~ Visibility + Season s ~ Visibility + Season 789.12  9.12
#> 5  weibull      l ~ 1                   s ~ Visibility * Season 789.98  9.98
#> 4  weibull      l ~ Visibility          s ~ Visibility * Season 791.45 11.45
#> 3  weibull      l ~ Season              s ~ Visibility * Season 793.01 13.01
#> 1  weibull      l ~ Visibility * Season s ~ Visibility * Season 794.00 14.00
#> 30 lognormal    l ~ 1                   s ~ Visibility * Season 794.47 14.47
#> 2  weibull      l ~ Visibility + Season s ~ Visibility * Season 794.50 14.50
#> 55 loglogistic  l ~ 1                   s ~ Visibility * Season 794.95 14.95
#> 29 lognormal    l ~ Visibility          s ~ Visibility * Season 795.13 15.13
#> 54 loglogistic  l ~ Visibility          s ~ Visibility * Season 795.14 15.14
#> 28 lognormal    l ~ Season              s ~ Visibility * Season 795.65 15.65
#> 53 loglogistic  l ~ Season              s ~ Visibility * Season 796.37 16.37
#> 51 loglogistic  l ~ Visibility * Season s ~ Visibility * Season 796.80 16.80
#> 26 lognormal    l ~ Visibility * Season s ~ Visibility * Season 797.20 17.20
#> 27 lognormal    l ~ Visibility + Season s ~ Visibility * Season 797.36 17.36
#> 52 loglogistic  l ~ Visibility + Season s ~ Visibility * Season 797.70 17.70
#> 76 exponential  l ~ Visibility * Season NULL                    835.23 55.23
#> 79 exponential  l ~ Visibility          NULL                    839.78 59.78
#> 77 exponential  l ~ Visibility + Season NULL                    840.96 60.96
#> 80 exponential  l ~ 1                   NULL                    842.24 62.24
#> 78 exponential  l ~ Season              NULL                    844.75 64.75

It is not uncommon to see the fits for the exponential distribution at the bottom of the AIC list. The exponential distribution has only one parameter and does not have nearly as much flexibility as the others, which each have two-parameters. An implicit assumption of the exponential model is that the scavenging rate is constant, regardless of carcass age. When that assumption is not met, the exponential provides an inferior fit.

To compare among a set of cp models with plausible AICs, use, for example, plot(cpModels[["sml"]]) to browse through all the models or plot(cpModels[["sml"]][["p ~ Visibility; k ~ 1"]]) to view a single model specified by name. It can be seen from the AICc table (aicc(cpModels[["sml"]])) that the top 10 models according to AICc are:

These can be compared in graphs as follows:

The figure shows the raw persistence data (fraction of carcasses remaining after the given time) for each cell, as a black stair case with Kaplan-Meier confidence intervals as dashed lines. In addition, the fitted curves for each of the distributions are shown in color, with the specificModel distribution having a thicker than the others. The two-parameter models (Weibull, lognormal, and loglogistic) tend to be very similar and a relatively close fit to the data. The exponential model tends to be somewhat removed from the others. Clicking on the graphing window brings up the next set of figures.

We’re looking for a good fit between the selected model and the data in as many cells as possible. The first several models among the cp_smlCandidates seems to provide a reasonably good fit in all cells, although there seems to be a trade-off between fitting the RP.spring cell well or fitting the RP.summer cell well. Visibility occurs frequently in the top models, while Season appears more frequently in the bottom models. This indicates that Season is probably not a strong predictor of carcass persistence, while Visibility is.

Selecting the top AICc model for use in mortality estimation:

Following a similar model selection process for the other size classes, we select the following:

NOTE: For large carcasses, the exponential distribution was at the top of the AICc list. The exponential requires only one parameter (l), so no scale parameter is provided in the model.

Again, we collate the models into a list for later analysis of detection probabilities and mortality rates.

Mortality Estimation

Each carcass’s contribution to the total mortality in each search interval is estimated using the estM function. The function call is largely similar that used in the simple scenario discussed in example 1. However, there are some important differences. First, model_SE and model_CP are lists of models, one element for each size class. In addition, the name of the size class variable is provided as sizeCol = "Size". Finally, the frac argument represents the sampling fraction or the fraction of carcasses expected to fall at the units that were searched. In this example, 23 out of 100 turbines were searched, and frac is set equal to 0.23 under the assumption that the mortality rates at the unsearched turbines did not differ substantially from the rates at the searched turbines.

This estimate is for the total number of fatalities among all size classes combined, from hummingbirds and bats to eagles and may be too vague to be very useful. Fortunately, mortality estimates may be partitioned or split into desired categories, such as species, size, or season. Splits may be performed according to characteristics of the carcasses or where they were found (e.g., species, turbine or other variable found in data_CO) or when they were found (e.g., season or other variable associated with search schedule and found in data_SS, or a vector of specific times).

Carcasses were categorized not only by size but also by species, species group, the type of turbine they were found at, the visibility class of the ground where they were found, and distance from nearest turbine.

Although species groups may extend across different size classes (e.g., raptors could include kestrels, red-tailed hawks, and golden eagles; passerines could include sparrows and ravens), splits according to species group can easily be accomplished:

Split by species and season:

M_speciesseason <- calcSplits(M = Mhat,
  split_CO = "Species",  data_CO = data_CO, split_SS = "Season", data_SS = data_SS)
summary(M_speciesseason)
#> $BA
#>             X        5%       25%       50%      75%      95%
#> spring  7.000  79.78687 140.74452 193.01414 255.9246 368.4583
#> summer  3.305  17.12462  46.57742  77.37656 111.7534 177.6963
#> fall   22.695 275.29760 362.73586 425.41640 499.9752 620.1810
#> 
#> $BB
#>            X       5%       25%       50%       75%       95%
#> spring 3.000 23.83836 72.446337 118.90613 169.46163 272.23480
#> summer 1.164  1.16400  9.082954  25.21979  45.92947  85.11504
#> fall   6.836 56.50402 96.391312 130.88062 167.55657 235.82484
#> 
#> $BC
#>                X        5%       25%      50%      75%       95%
#> spring 1.0266154 1.0266154  8.945471 35.14838 61.13031 114.69359
#> summer 0.9753846 0.9753846  6.308332 21.93598 40.71403  78.18403
#> fall   2.9980000 6.2359272 28.998829 49.52778 74.61682 119.58061
#> 
#> $BD
#>            X    5%     25%      50%      75%     95%
#> spring 0.000 0.000  0.0000  0.00000  0.00000  0.0000
#> summer 0.004 0.004  0.0040  0.00400  0.00400  0.0040
#> fall   1.996 1.996 15.2163 28.96353 45.55864 79.9047
#> 
#> $BE
#>        X 5%      25%      50%      75%      95%
#> spring 0  0  0.00000  0.00000  0.00000  0.00000
#> summer 0  0  0.00000  0.00000  0.00000  0.00000
#> fall   2  2 16.60294 30.80079 46.81699 72.23181
#> 
#> $LA
#>        X 5%       25%      50%      75%      95%
#> spring 2  2  9.615851 17.13755 25.95716 40.47364
#> summer 0  0  0.000000  0.00000  0.00000  0.00000
#> fall   2  2 11.374518 19.96605 30.41664 49.09592
#> 
#> $LB
#>        X 5%      25%      50%     75%      95%
#> spring 1  1 3.182334 9.450272 16.3018 29.83401
#> summer 0  0 0.000000 0.000000  0.0000  0.00000
#> fall   0  0 0.000000 0.000000  0.0000  0.00000
#> 
#> $LC
#>             X     5%      25%      50%      75%      95%
#> spring 1.0165 1.0165 2.582502 8.540035 16.06504 27.46958
#> summer 0.9835 0.9835 2.658027 7.935527 14.78285 25.81352
#> fall   0.0000 0.0000 0.000000 0.000000  0.00000  0.00000
#> 
#> $LE
#>            X    5%      25%      50%      75%       95%
#> spring 0.068 0.068 0.068000 0.068000  0.06800  3.037928
#> summer 0.932 0.932 1.613348 7.265885 13.52336 25.750592
#> fall   0.000 0.000 0.000000 0.000000  0.00000  0.000000
#> 
#> $MA
#>            X    5%      25%      50%      75%      95%
#> spring 1.000 1.000 4.925198 15.35570 29.61561 52.94076
#> summer 1.001 1.001 4.003494 11.22003 20.76573 35.29052
#> fall   0.999 0.999 3.780151 11.41379 20.57565 36.68053
#> 
#> $MB
#>            X    5%      25%      50%      75%      95%
#> spring 0.000 0.000 0.000000 0.000000  0.00000  0.00000
#> summer 0.028 0.028 0.028000 0.028000  0.02800  0.02800
#> fall   0.972 0.972 2.463254 8.745307 17.01218 33.40032
#> 
#> $MC
#>            X    5%      25%       50%      75%      95%
#> spring 1.268 1.268 7.375381 19.369473 34.35903 64.17469
#> summer 0.732 0.732 0.732000  9.827538 26.81004 54.24315
#> fall   0.000 0.000 0.000000  0.000000  0.00000  0.00000
#> 
#> $MD
#>        X 5%      25%      50%      75%      95%
#> spring 1  1 4.162654 12.48321 21.84037 36.99237
#> summer 0  0 0.000000  0.00000  0.00000  0.00000
#> fall   0  0 0.000000  0.00000  0.00000  0.00000
#> 
#> $ME
#>            X    5%      25%      50%     75%      95%
#> spring 0.000 0.000 0.000000 0.000000  0.0000  0.00000
#> summer 0.006 0.006 0.006000 0.006000  0.0060  0.00600
#> fall   0.994 0.994 3.270641 9.865803 18.2534 33.55189
#> 
#> $MF
#>        X       5%      25%      50%      75%      95%
#> spring 0 0.000000  0.00000  0.00000  0.00000  0.00000
#> summer 0 0.000000  0.00000  0.00000  0.00000  0.00000
#> fall   3 5.897952 24.93285 42.64625 61.32292 98.50439
#> 
#> $MH
#>        X 5%      25%      50%      75%      95%
#> spring 1  1 3.741557 12.79184 22.52991 39.69005
#> summer 1  1 3.947382 11.51164 20.45287 34.84962
#> fall   0  0 0.000000  0.00000  0.00000  0.00000
#> 
#> $SA
#>        X 5%      25%     50%      75%     95%
#> spring 1  1 11.09376 35.4269 69.23817 131.454
#> summer 0  0  0.00000  0.0000  0.00000   0.000
#> fall   0  0  0.00000  0.0000  0.00000   0.000
#> 
#> $SB
#>            X    5%   25%   50%      75%      95%
#> spring 0.425 0.425 0.425 0.425 25.85557 91.96835
#> summer 0.575 0.575 0.575 0.575 36.43546 82.78368
#> fall   0.000 0.000 0.000 0.000  0.00000  0.00000
#> 
#> $SC
#>        X       5%      25%     50%      75%      95%
#> spring 3 17.57968 65.28107 107.363 155.2183 253.1432
#> summer 0  0.00000  0.00000   0.000   0.0000   0.0000
#> fall   0  0.00000  0.00000   0.000   0.0000   0.0000
#> 
#> $SD
#>                X        5%      25%      50%      75%      95%
#> spring 1.1348462 1.1348462 9.906165 24.80756 47.39757 96.11444
#> summer 0.8651538 0.8651538 4.769733 23.36322 47.79312 98.11767
#> fall   0.0000000 0.0000000 0.000000  0.00000  0.00000  0.00000
#> 
#> $SE
#>            X    5%      25%      50%      75%      95%
#> spring 0.002 0.002  0.00200  0.00200  0.00200   0.0020
#> summer 0.998 0.998 10.16706 26.69874 52.89831 100.0759
#> fall   0.000 0.000  0.00000  0.00000  0.00000   0.0000
#> 
#> $SG
#>               X        5%      25%       50%      75%      95%
#> spring 3.054187 19.322687 69.24094 109.13170 161.3351 262.3597
#> summer 1.945812  1.945812 32.85280  59.92559  98.6255 173.5754
#> fall   0.000000  0.000000  0.00000   0.00000   0.0000   0.0000
#> 
#> attr(,"class")
#> [1] "splitSummary"
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "Season"  "Species"
#> attr(,"type")
#> [1] "SS" "CO"
#> attr(,"times")
#> [1]   0  60 130 200
plot(M_speciesseason)

There are so many vertical panels that it is difficult to glean any useful information out of the graph. However, the panels may be transposed and graphed for better interpretability: