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, groud texture), 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:
head(data_SE)
#> pkID Season s1 s2 s3 s4 s5
#> 1 pk1 spring 0 NA NA NA NA
#> 2 pk2 spring 0 0 NA NA NA
#> 3 pk3 spring 1 NA NA NA NA
#> 4 pk4 spring 1 NA NA NA NA
#> 5 pk5 spring 0 NA NA NA NA
#> 6 pk6 spring 1 NA NA NA NA
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:
head(data_CP)
#> cpID Season LastPresent FirstAbsent
#> 1 cp1 spring 0.00 0.93
#> 2 cp2 spring 0.98 1.97
#> 3 cp3 spring 0.00 1.01
#> 4 cp4 spring 13.99 21.13
#> 5 cp5 spring 0.00 1.17
#> 6 cp6 spring 20.95 27.97
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 our fictitious site. 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).
head(data_SS[, 1:10])
#> SearchDate Season t1 t2 t3 t4 t5 t6 t7 t8
#> 1 1955-04-15 spring 0 0 0 1 0 0 0 1
#> 2 1955-04-18 spring 1 0 0 0 1 0 0 0
#> 3 1955-04-21 spring 0 1 0 0 0 1 0 0
#> 4 1955-04-24 spring 0 0 1 0 0 0 1 0
#> 5 1955-04-27 spring 0 0 0 1 0 0 0 1
#> 6 1955-04-30 spring 1 0 0 0 1 0 0 0
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 or 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.
head(data_DWP)
#> Turbine bat
#> 1 t1 0.183
#> 2 t2 0.192
#> 3 t3 0.205
#> 4 t4 0.194
#> 5 t5 0.179
#> 6 t6 0.171
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.
head(data_CO)
#> carcID Turbine TurbineType DateFound Species SpeciesGroup Distance
#> 1 x79 t86 Z 1955-05-03 BA bat1 7.1
#> 2 x37 t19 X 1955-05-06 BB bat1 5.3
#> 3 x175 t37 Y 1955-05-12 BB bat1 9.8
#> 4 x181 t91 Z 1955-05-12 BA bat1 2.7
#> 5 x203 t38 Y 1955-05-15 BA bat1 3.8
#> 6 x291 t7 X 1955-05-18 BB bat1 112.3
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.
model_SE <- pkm(p ~ Season, k ~ 1, data = data_SE)
class(model_SE)
#> [1] "pkm" "list"
model_SE
#> $call
#> pkm0(formula_p = formula_p, formula_k = formula_k, data = data,
#> obsCol = obsCol, kFixed = kFixed, kInit = kInit, CL = CL,
#> quiet = quiet)
#>
#> $formula_p
#> p ~ Season
#>
#> $formula_k
#> k ~ 1
#>
#> $predictors
#> [1] "Season"
#>
#> $AICc
#> [1] 104.42
#>
#> $convergence
#> [1] 0
#>
#> $cell_pk
#> cell n p_median p_lwr p_upr k_median k_lwr k_upr
#> 1 fall 20 0.618034 0.430798 0.775741 0.920744 0.176705 0.998412
#> 2 spring 20 0.439879 0.280244 0.613000 0.920744 0.176705 0.998412
#> 3 summer 20 0.619048 0.436933 0.772879 0.920744 0.176705 0.998412
#>
#> $CL
#> [1] 0.9
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.
model_CP <- cpm(l ~ Season, s ~ Season, data = data_CP, dist = "weibull",
left = "LastPresent", right = "FirstAbsent")
class(model_CP)
#> [1] "cpm" "list"
model_CP
#> $call
#> cpm0(formula_l = formula_l, formula_s = formula_s, data = data,
#> left = left, right = right, dist = dist, CL = CL, quiet = quiet)
#>
#> $formula_l
#> l ~ Season
#>
#> $formula_s
#> s ~ Season
#>
#> $distribution
#> [1] "weibull"
#>
#> $predictors
#> [1] "Season"
#>
#> $AICc
#> [1] 235.51
#>
#> $convergence
#> [1] 0
#>
#> $cell_ls
#> cell n l_median l_lwr l_upr s_median s_lwr s_upr
#> 1 fall 20 1.163 0.612 1.715 1.360 0.932 1.986
#> 2 spring 20 0.857 0.169 1.545 1.679 1.198 2.354
#> 3 summer 20 1.246 0.561 1.930 1.688 1.155 2.467
#>
#> $cell_ab
#> cell n pda_median pda_lwr pda_upr pdb_median pdb_lwr pdb_upr
#> 1 fall 20 0.735 0.504 1.073 3.200 1.844 5.557
#> 2 spring 20 0.596 0.425 0.835 2.356 1.184 4.688
#> 3 summer 20 0.592 0.405 0.866 3.476 1.752 6.890
#>
#> $CL
#> [1] 0.9
#>
#> $cell_desc
#> cell medianCP r1 r3 r7 r14 r28
#> 1 fall 1.943506 0.7877998 0.5968500 0.4035271 0.2504831 0.1362684
#> 2 spring 1.273808 0.6946923 0.5067622 0.3406993 0.2163474 0.1221627
#> 3 summer 1.871560 0.7459730 0.5779176 0.4168571 0.2838980 0.1714068
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.
Mhat <- estM(nsim = 1000, data_CO = data_CO, data_SS = data_SS,
data_DWP = data_DWP, model_SE = model_SE, model_CP = model_CP,
unitCol = "Turbine", COdate = "DateFound")
summary(Mhat)
#> median 5% 95%
#> 1460.08 1030.07 2239.22
plot(Mhat)
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):
M_species <- calcSplits(M = Mhat, split_CO = "Species", data_CO = data_CO)
summary(M_species)
#> X 5% 25% 50% 75% 95%
#> BA 42 561.99570 697.24682 820.06018 988.90168 1284.71595
#> BB 14 189.75151 271.14240 347.72692 447.05644 652.84809
#> BC 9 69.63661 114.33633 151.78800 202.72047 296.85181
#> BD 5 27.74551 58.50035 85.24975 121.66288 193.71199
#> BE 1 1.00000 1.00000 14.30777 25.96535 46.89764
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "Species"
#> attr(,"type")
#> [1] "CO"
#> attr(,"class")
#> [1] "splitSummary"
plot(M_species)
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):
M_season <- calcSplits(M = Mhat, split_SS = "Season", data_SS = SSdat,
split_CO = NULL, data_CO = data_CO)
summary(M_season)
#> X 5% 25% 50% 75% 95%
#> spring 18.10538 307.28436 445.7743 587.4128 778.5566 1255.5490
#> summer 10.16162 92.90686 157.0354 205.4157 267.7957 391.2335
#> fall 42.73300 422.65705 536.4184 618.5544 726.5111 921.8572
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "Season"
#> attr(,"type")
#> [1] "SS"
#> attr(,"times")
#> [1] 0 60 130 200
#> attr(,"class")
#> [1] "splitSummary"
plot(M_season)
Mortality by month (a temporal split that spans the monitoring period):
M_month <- calcSplits(M = Mhat, split_time = seq(0, max(SSdat$days), by = 28),
data_SS = SSdat, data_CO = data_CO)
summary(M_month)
#> X 5% 25% 50% 75% 95%
#> 28 6.257583 87.491617 152.39761 222.60817 322.5496 571.3611
#> 56 11.713295 168.756967 254.53087 349.03225 478.3012 785.1703
#> 84 2.121836 2.121836 34.18265 66.03484 106.5027 199.4422
#> 112 4.570143 23.393471 49.85770 74.03236 101.8694 163.8731
#> 140 10.042429 73.339240 115.78569 143.12471 177.1878 238.5340
#> 168 27.842905 267.336851 351.60254 416.80881 496.2453 628.2497
#> 196 8.451810 51.167158 85.18927 113.84717 149.6160 205.3721
#> 200 0.000000 0.000000 0.00000 0.00000 0.0000 0.0000
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "time"
#> attr(,"type")
#> [1] "time"
#> attr(,"times")
#> [1] 0 28 56 84 112 140 168 196 200
#> attr(,"class")
#> [1] "splitSummary"
plot(M_month)
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).
M_various_times <- calcSplits(M = Mhat,
split_time = c(seq(0, 90, by = 15), 120, 150, seq(155, 200, by = 5)),
data_SS = SSdat, data_CO = data_CO)
plot(M_various_times)
plot(M_various_times, 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).
M_species_by_season <- calcSplits(M = Mhat,
split_CO = "Species", data_CO = data_CO,
split_SS = "Season", data_SS = SSdat)
plot(M_species_by_season)
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:
head(data_SE)
#> pkID Size Season Visibility s1 s2 s3 s4 s5
#> 1 pk1 bat spring RP 0 NA NA NA NA
#> 2 pk2 bat spring RP 0 0 NA NA NA
#> 3 pk3 bat spring RP 1 NA NA NA NA
#> 4 pk4 bat spring RP 1 NA NA NA NA
#> 5 pk5 bat spring RP 0 NA NA NA NA
#> 6 pk6 bat spring RP 1 NA NA NA NA
head(data_CP)
#> cpID Size Season Visibility LastPresent FirstAbsent
#> 1 cp1 bat spring RP 0.00 0.93
#> 2 cp2 bat spring RP 0.98 1.97
#> 3 cp3 bat spring RP 0.00 1.01
#> 4 cp4 bat spring RP 13.99 21.13
#> 5 cp5 bat spring RP 0.00 1.17
#> 6 cp6 bat spring RP 20.95 27.97
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.
pkModels <- pkm(p ~ Visibility * Season, k ~ Visibility * Season, data = data_SE,
allCombos = TRUE, sizeCol = "Size")
class(pkModels)
#> [1] "pkmSetSize" "list"
names(pkModels)
#> [1] "bat" "lrg" "med" "sml"
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.
names(pkModels[["sml"]])
#> [1] "p ~ Visibility * Season; k ~ Visibility * Season" "p ~ Visibility + Season; k ~ Visibility * Season"
#> [3] "p ~ Season; k ~ Visibility * Season" "p ~ Visibility; k ~ Visibility * Season"
#> [5] "p ~ 1; k ~ Visibility * Season" "p ~ Visibility * Season; k ~ Visibility + Season"
#> [7] "p ~ Visibility + Season; k ~ Visibility + Season" "p ~ Season; k ~ Visibility + Season"
#> [9] "p ~ Visibility; k ~ Visibility + Season" "p ~ 1; k ~ Visibility + Season"
#> [11] "p ~ Visibility * Season; k ~ Season" "p ~ Visibility + Season; k ~ Season"
#> [13] "p ~ Season; k ~ Season" "p ~ Visibility; k ~ Season"
#> [15] "p ~ 1; k ~ Season" "p ~ Visibility * Season; k ~ Visibility"
#> [17] "p ~ Visibility + Season; k ~ Visibility" "p ~ Season; k ~ Visibility"
#> [19] "p ~ Visibility; k ~ Visibility" "p ~ 1; k ~ Visibility"
#> [21] "p ~ Visibility * Season; k ~ 1" "p ~ Visibility + Season; k ~ 1"
#> [23] "p ~ Season; k ~ 1" "p ~ Visibility; k ~ 1"
#> [25] "p ~ 1; k ~ 1"
class(pkModels[["sml"]])
#> [1] "pkmSet" "list"
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.
aicc(pkModels[["sml"]])
#> p Formula k Formula AICc ΔAICc
#> 24 p ~ Visibility k ~ 1 318.55 0.00
#> 19 p ~ Visibility k ~ Visibility 319.69 1.14
#> 14 p ~ Visibility k ~ Season 321.42 2.87
#> 21 p ~ Visibility * Season k ~ 1 321.53 2.98
#> 22 p ~ Visibility + Season k ~ 1 322.17 3.62
#> 16 p ~ Visibility * Season k ~ Visibility 322.98 4.43
#> 17 p ~ Visibility + Season k ~ Visibility 323.14 4.59
#> 9 p ~ Visibility k ~ Visibility + Season 323.44 4.89
#> 11 p ~ Visibility * Season k ~ Season 323.97 5.42
#> 12 p ~ Visibility + Season k ~ Season 324.33 5.78
#> 7 p ~ Visibility + Season k ~ Visibility + Season 326.33 7.78
#> 6 p ~ Visibility * Season k ~ Visibility + Season 326.51 7.96
#> 4 p ~ Visibility k ~ Visibility * Season 327.98 9.43
#> 2 p ~ Visibility + Season k ~ Visibility * Season 331.07 12.52
#> 1 p ~ Visibility * Season k ~ Visibility * Season 333.47 14.92
#> 20 p ~ 1 k ~ Visibility 339.51 20.96
#> 10 p ~ 1 k ~ Visibility + Season 342.41 23.86
#> 18 p ~ Season k ~ Visibility 342.74 24.19
#> 8 p ~ Season k ~ Visibility + Season 344.10 25.55
#> 25 p ~ 1 k ~ 1 346.45 27.90
#> 5 p ~ 1 k ~ Visibility * Season 347.47 28.92
#> 15 p ~ 1 k ~ Season 347.66 29.11
#> 23 p ~ Season k ~ 1 350.35 31.80
#> 13 p ~ Season k ~ Season 350.44 31.89
#> 3 p ~ Season k ~ Visibility * Season 350.49 31.94
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:
plot(pkModels[["sml"]], specificModel = "p ~ Visibility; k ~ 1")
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 (blue) 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!
plot(pkModels[["sml"]], specificModel = "p ~ Season; k ~ Visibility * Season")
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.
pkMods <- list(
sml = pkModels[["sml"]][["p ~ Visibility; k ~ 1"]],
med = pkModels[["med"]][["p ~ Visibility; k ~ 1"]],
lrg = pkModels[["lrg"]][["p ~ Visibility; k ~ 1"]],
bat = pkModels[["bat"]][["p ~ Visibility; k ~ 1"]]
)
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.
cpModels <- cpm(
l ~ Visibility * Season, s ~ Visibility * Season,
data = data_CP, left = "LastPresent", right = "FirstAbsent",
dist = c( "weibull", "lognormal", "loglogistic", "exponential"),
allCombos = TRUE, sizeCol = "Size"
)
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 ΔAICc
#> 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"]][["dist: lognormal; 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:
cp_smlCandidates <-
names(cpModels[["sml"]])[c(25, 24, 15, 19, 20, 10, 14, 22, 74, 21)]
cp_smlCandidates
#> [1] "dist: weibull; l ~ 1; s ~ 1" "dist: weibull; l ~ Visibility; s ~ 1"
#> [3] "dist: weibull; l ~ 1; s ~ Season" "dist: weibull; l ~ Visibility; s ~ Visibility"
#> [5] "dist: weibull; l ~ 1; s ~ Visibility" "dist: weibull; l ~ 1; s ~ Visibility + Season"
#> [7] "dist: weibull; l ~ Visibility; s ~ Season" "dist: weibull; l ~ Visibility + Season; s ~ 1"
#> [9] "dist: loglogistic; l ~ Visibility; s ~ 1" "dist: weibull; l ~ Visibility * Season; s ~ 1"
These can be compared in graphs as follows:
plot(cpModels[["sml"]], specificModel = cp_smlCandidates)
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:
cp_sml <- cpModels[["sml"]][[cp_smlCandidates[1]]]
Following a similar model selection process for the other size classes, we select the following:
cp_med <- cpModels[["med"]][["dist: weibull; l ~ Visibility; s ~ Season"]]
cp_lrg <- cpModels[["lrg"]][["dist: exponential; l ~ Visibility + Season; NULL"]]
cp_bat <- cpModels[["bat"]][["dist: weibull; l ~ Visibility + Season; s ~ 1"]]
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.
cpMods <- list(
sml = cp_sml,
med = cp_med,
lrg = cp_lrg,
bat = cp_bat
)
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.
Mhat <- estM(nsim = 1000, data_CO = data_CO, data_SS = data_SS, frac = 0.23,
data_DWP = data_DWP, model_SE = pkMods, model_CP = cpMods,
sizeCol = "Size", unitCol = "Turbine", COdate = "DateFound")
summary(Mhat)
#> median 5% 95%
#> 1916.46 1483.34 2483.46
plot(Mhat)
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:
M_speciesGroup <- calcSplits(M = Mhat,
split_CO = "SpeciesGroup", data_CO = data_CO)
summary(M_speciesGroup)
#> X 5% 25% 50% 75% 95%
#> bat0 4 16.24120 41.68399 62.86273 85.55249 129.0495
#> bat1 49 804.49620 981.44825 1110.85088 1246.64839 1527.2471
#> brd1 19 302.19653 424.59709 519.07928 637.02494 864.5506
#> brd2 10 67.02341 104.28074 138.98593 172.36072 233.9164
#> brd3 5 20.20642 37.43876 53.14112 70.31839 100.5767
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "SpeciesGroup"
#> attr(,"type")
#> [1] "CO"
#> attr(,"class")
#> [1] "splitSummary"
plot(M_speciesGroup)
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 76.29369 143.88691 196.3865 263.4324 362.3624
#> summer 3.349 15.51143 49.49931 78.5921 112.6984 173.9983
#> fall 22.651 278.15381 350.50173 419.0914 497.6210 627.5099
#>
#> $BB
#> X 5% 25% 50% 75% 95%
#> spring 3.000 27.29299 74.94659 118.61772 169.30998 274.51683
#> summer 1.162 1.16200 1.16200 25.84782 48.50449 90.50564
#> fall 6.838 46.77286 90.94127 126.24433 163.85150 231.00670
#>
#> $BC
#> X 5% 25% 50% 75% 95%
#> spring 1.024 1.024000 1.02400 36.26280 67.05830 116.85445
#> summer 0.979 0.979000 0.97900 22.30766 37.13601 64.88834
#> fall 2.997 8.709672 27.21028 48.02210 73.48793 115.26898
#>
#> $BD
#> X 5% 25% 50% 75% 95%
#> spring 0.000 0.000 0.00000 0.00000 0.000 0.00000
#> summer 0.002 0.002 0.00200 0.00200 0.002 0.00200
#> fall 1.998 1.998 11.74564 27.16038 45.259 77.75083
#>
#> $BE
#> X 5% 25% 50% 75% 95%
#> spring 0 0 0.0000 0.00000 0.00000 0.00000
#> summer 0 0 0.0000 0.00000 0.00000 0.00000
#> fall 2 2 15.9172 31.80083 49.39509 81.22952
#>
#> $LA
#> X 5% 25% 50% 75% 95%
#> spring 2 2 9.396080 18.50725 27.81603 39.86108
#> summer 0 0 0.000000 0.00000 0.00000 0.00000
#> fall 2 2 8.888873 19.90504 30.01280 48.62278
#>
#> $LB
#> X 5% 25% 50% 75% 95%
#> spring 1 1 1 10.12099 12.23168 30.38922
#> summer 0 0 0 0.00000 0.00000 0.00000
#> fall 0 0 0 0.00000 0.00000 0.00000
#>
#> $LC
#> X 5% 25% 50% 75% 95%
#> spring 1.0163 1.0163 1.0163 9.776032 17.07357 28.92895
#> summer 0.9837 0.9837 0.9837 9.228934 11.76402 27.35423
#> fall 0.0000 0.0000 0.0000 0.000000 0.00000 0.00000
#>
#> $LE
#> X 5% 25% 50% 75% 95%
#> spring 0.086 0.086 0.086 0.086000 0.08600 9.065552
#> summer 0.914 0.914 0.914 9.110979 10.08301 27.969946
#> fall 0.000 0.000 0.000 0.000000 0.00000 0.000000
#>
#> $MA
#> X 5% 25% 50% 75% 95%
#> spring 1.001 1.001 1.001 17.71843 30.69199 53.26644
#> summer 1.000 1.000 1.000 12.56920 17.25909 37.40257
#> fall 0.999 0.999 0.999 12.33023 22.44299 37.85175
#>
#> $MB
#> X 5% 25% 50% 75% 95%
#> spring 0.000 0.000 0.000 0.00000 0.00000 0.00000
#> summer 0.017 0.017 0.017 0.01700 0.01700 0.01700
#> fall 0.983 0.983 0.983 11.52945 14.06774 34.50067
#>
#> $MC
#> X 5% 25% 50% 75% 95%
#> spring 1.262 1.262 1.262 17.80856 34.44527 64.21060
#> summer 0.738 0.738 0.738 0.73800 23.08790 50.74883
#> fall 0.000 0.000 0.000 0.00000 0.00000 0.00000
#>
#> $MD
#> X 5% 25% 50% 75% 95%
#> spring 1 1 1 13.81511 24.1853 41.26872
#> summer 0 0 0 0.00000 0.0000 0.00000
#> fall 0 0 0 0.00000 0.0000 0.00000
#>
#> $ME
#> X 5% 25% 50% 75% 95%
#> spring 0.000 0.000 0.000 0.00000 0.00000 0.00000
#> summer 0.004 0.004 0.004 0.00400 0.00400 0.00400
#> fall 0.996 0.996 0.996 11.60267 15.75852 34.64525
#>
#> $MF
#> X 5% 25% 50% 75% 95%
#> spring 0 0.000000 0.00000 0.00000 0.00000 0.000
#> summer 0 0.000000 0.00000 0.00000 0.00000 0.000
#> fall 3 9.070984 25.45591 44.32282 64.55343 100.523
#>
#> $MH
#> X 5% 25% 50% 75% 95%
#> spring 1 1 1 13.72109 25.43013 40.26018
#> summer 1 1 1 11.76750 21.36353 36.27610
#> fall 0 0 0 0.00000 0.00000 0.00000
#>
#> $SA
#> X 5% 25% 50% 75% 95%
#> spring 1 1 1 37.12315 63.23947 130.5581
#> summer 0 0 0 0.00000 0.00000 0.0000
#> fall 0 0 0 0.00000 0.00000 0.0000
#>
#> $SB
#> X 5% 25% 50% 75% 95%
#> spring 0.437 0.437 0.437 0.437 30.52122 95.47931
#> summer 0.563 0.563 0.563 0.563 31.16079 80.60159
#> fall 0.000 0.000 0.000 0.000 0.00000 0.00000
#>
#> $SC
#> X 5% 25% 50% 75% 95%
#> spring 3 3 58.55331 99.03001 147.3858 251.4919
#> summer 0 0 0.00000 0.00000 0.0000 0.0000
#> fall 0 0 0.00000 0.00000 0.0000 0.0000
#>
#> $SD
#> X 5% 25% 50% 75% 95%
#> spring 1.1489231 1.1489231 1.1489231 25.61468 48.84667 96.62752
#> summer 0.8510769 0.8510769 0.8510769 25.89714 46.81687 87.08501
#> fall 0.0000000 0.0000000 0.0000000 0.00000 0.00000 0.00000
#>
#> $SE
#> X 5% 25% 50% 75% 95%
#> spring 0.003 0.003 0.003 0.00300 0.00300 0.0030
#> summer 0.997 0.997 0.997 30.01611 51.17306 104.7512
#> fall 0.000 0.000 0.000 0.00000 0.00000 0.0000
#>
#> $SG
#> X 5% 25% 50% 75% 95%
#> spring 3.04875 20.61098 62.67863 105.80862 156.01118 260.1824
#> summer 1.95125 1.95125 31.89793 58.47863 96.00748 162.1811
#> fall 0.00000 0.00000 0.00000 0.00000 0.00000 0.0000
#>
#> attr(,"CL")
#> [1] 0.9
#> attr(,"vars")
#> [1] "Season" "Species"
#> attr(,"type")
#> [1] "SS" "CO"
#> attr(,"times")
#> [1] 0 60 130 200
#> attr(,"class")
#> [1] "splitSummary"
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:
plot(transposeSplits(M_speciesseason))
