In this vignette, we will use the `cypdata`

dataset to
illustrate the estimation of **function-based MPMs**. To
reduce file size, we have prevented some statements from running if they
produce long stretches of output. Examples include most
`summary()`

calls - we include hashtagged versions of these
calls, and our text assumes that the user runs these statements without
hashtags. We have also written this vignette using the default Ehrlén
format for all historical MPMs. All examples can be run in deVries
format as well. This vignette is only a sample analysis. Detailed
information and instructions on using `lefko3`

are available
through a free online e-book called *lefko3: a gentle
introduction*, available on
the projects page
of the Shefferson lab website. This website also includes links to
long-format vignettes and other demonstrations of this package.

The white lady’s slipper, *Cypripedium candidum*, is a North
American perennial herb in the family Orchidaceae. It is long-lived and
of conservation concern. The population from which the dataset
originates is located within a state nature preserve located in
northeastern Illinois, USA. The population was monitored annually from
2004 to 2009, with two monitoring sessions per year. Monitoring sessions
took roughly 2 weeks each, and included complete censuses of the
population divided into sections referred to as patches. Each monitoring
session consisted of searches for previously recorded individuals, which
were located according to coordinates relative to fixed stakes at the
site, followed by a search for new individuals. Data recorded per
individual included: the location, the number of non-flowering sprouts,
the number of flowering sprouts, the number of flowers per flowering
sprout, and the number of fruit pods per sprout (only in the second
monitoring session per year, once fruiting had occurred). Location was
used to infer individual identity. More information about this
population and its characteristics is given in Shefferson et al. (2001) and Shefferson, Mizuta, and Hutchings (2017).

Population matrix projection modeling requires an appropriate life
history model showing how all stages and transitions are related. The
figure below shows a very general life history model detailing these
relationships in *Cypripedium candidum*. The first stage of life
is the dormant seed, although a seed may germinate in the year following
seed production, in which case the first stage of life is the protocorm.
The protocorm is the first germinated stage - an underground,
mycoheterotrophic stage unique to the families Orchidaceae and
Pyrolaceae. There are three years of protocorm stages, followed by a
seedling stage, and finally a set of stages that comprise the
size-classified adult portion of life. The figure shows 49 such stages,
each for a different number of stems (including zero for vegetative
dormancy) and one of two reproductive statuses. These stages may be
reorganized for different circumstances (more on this later).

Figure 5.1. Life history model of *Cypripedium candidum*.
Survival transitions are indicated with solid arrows, while fecundity
transitions are indicated with dashed arrows.

We can see a variety of transitions within this figure. The juvenile
stages are limited to fairly simple, progressive transitions. New
recruits may enter the population directly from germination of a seed
produced the previous year, in which case they start in the protocorm 1
stage, or they may begin as dormant seed. Dormant seed may remain
dormant, die, or germinate into the protocorm 1 stage. Protocorms exist
for up to 3 years, yielding the protocorm 1, 2, and 3 stages, without
any possibility of staying within each of these stages for more than a
single year. Protocorm 3 leads to a seedling stage, in which the plant
may persist for many years before becoming mature. Here, maturity does
not really refer to reproduction *per se*, but rather to a
morphology indistinguishable from a reproductive plant except for the
lack of a flower. The first mature stage is usually either vegetative
dormancy (`Dorm`

), during which time the plant does not
sprout, or a small, non-flowering adult (`1V`

). Once in this
portion of the life history, the plant may transition among 49 mature
stages, including vegetative dormancy, 1-24 shoots without flowers, or
1-24 shoots with at least one flower (we have never observed a plant
bigger than this in the population).

The horizontal dataset `cypdata`

, and the ahistorical
vertical dataset `cypvert`

which is the same as
`cypdata`

but is structured differently, both include only
data for the adult stages, and so later we will need to set juvenile
transitions to constants from the literature. Let’s begin by clearing
the memory, loading the package and the dataset, and looking over a
summary of the latter.

```
rm(list=ls(all=TRUE))
library(lefko3)
data(cypdata)
#summary(cypdata)
```

We will first describe the life history characterizing the population,
matching it to our dataset with a *stageframe* for our
*Cypripedium candidum* dataset. Since this analysis will be
function-based and all sizes have reasonable representation, we will
include all observed size classes here. Since size is count-based in the
*Cypripedium candidum* case, we will use all numbers of stems
that occurred, representing the life history diagram shown previously.

```
<- c(0, 0, 0, 0, 0, seq(from = 0, t = 24), seq(from = 1, to = 24))
sizevector <- c("SD", "P1", "P2", "P3", "SL", "D", "V1", "V2", "V3", "V4", "V5",
stagevector "V6", "V7", "V8", "V9", "V10", "V11", "V12", "V13", "V14", "V15", "V16", "V17",
"V18", "V19", "V20", "V21", "V22", "V23", "V24", "F1", "F2", "F3", "F4", "F5",
"F6", "F7", "F8", "F9", "F10", "F11", "F12", "F13", "F14", "F15", "F16", "F17",
"F18", "F19", "F20", "F21", "F22", "F23", "F24")
<- c(0, 0, 0, 0, 0, rep(0, 25), rep(1, 24))
repvector <- c(0, 0, 0, 0, 0, 0, rep(1, 48))
obsvector <- c(0, 0, 0, 0, 0, rep(1, 49))
matvector <- c(0, 1, 1, 1, 1, rep(0, 49))
immvector <- c(1, rep(0, 53))
propvector <- c(0, 0, 0, 0, 0, rep(1, 49))
indataset <- c("Dormant seed", "Yr1 protocorm", "Yr2 protocorm", "Yr3 protocorm",
comments "Seedling", "Veg dorm", "Veg adult 1 stem", "Veg adult 2 stems",
"Veg adult 3 stems", "Veg adult 4 stems", "Veg adult 5 stems",
"Veg adult 6 stems", "Veg adult 7 stems", "Veg adult 8 stems",
"Veg adult 9 stems", "Veg adult 10 stems", "Veg adult 11 stems",
"Veg adult 12 stems", "Veg adult 13 stems", "Veg adult 14 stems",
"Veg adult 15 stems", "Veg adult 16 stems", "Veg adult 17 stems",
"Veg adult 18 stems", "Veg adult 19 stems", "Veg adult 20 stems",
"Veg adult 21 stems", "Veg adult 22 stems", "Veg adult 23 stems",
"Veg adult 24 stems", "Flo adult 1 stem", "Flo adult 2 stems",
"Flo adult 3 stems", "Flo adult 4 stems", "Flo adult 5 stems",
"Flo adult 6 stems", "Flo adult 7 stems", "Flo adult 8 stems",
"Flo adult 9 stems", "Flo adult 10 stems", "Flo adult 11 stems",
"Flo adult 12 stems", "Flo adult 13 stems", "Flo adult 14 stems",
"Flo adult 15 stems", "Flo adult 16 stems", "Flo adult 17 stems",
"Flo adult 18 stems", "Flo adult 19 stems", "Flo adult 20 stems",
"Flo adult 21 stems", "Flo adult 22 stems", "Flo adult 23 stems",
"Flo adult 24 stems")
<- sf_create(sizes = sizevector, stagenames = stagevector,
cypframe repstatus = repvector, obsstatus = obsvector, matstatus = matvector,
propstatus = propvector, immstatus = immvector, indataset = indataset,
comments = comments)
#cypframe
```

A close look at the resulting object, `cypframe`

, shows a
data frame that includes the following information in order for each
stage: the stage’s name; the associated primary, secondary, and tertiary
sizes; the minimum and maximum ages associated with the stage; its
reproductive status; its status as an observable stage; its status as a
propagule stage; its status as an immature stage; its status as a mature
stage; whether it occurs in the dataset; the half-width, minimum,
maximum, centroid, and full width of its primary, secondary, and
tertiary size class bins; stage group number; and a comments field
describing the stage. We only used a primary size variable, so values
for the secondary and tertiary sizes are `NA`

s. Stage names
and combinations of characteristics occurring within the dataset must be
unique to prevent estimation errors, and the comments field may be
edited to include any information deemed pertinent.

Now we will standardize our vertical dataset into a
historically-formatted vertical (*hfv*) data frame using the
`verticalize3()`

function. The resulting dataset will have
each individual’s observed life history broken up into states
corresponding to three consecutive monitoring occasions (years) per row,
with plant identity marked in each row.

```
<- verticalize3(data = cypdata, noyears = 6, firstyear = 2004,
vertdata patchidcol = "patch", individcol = "plantid", blocksize = 4,
sizeacol = "Inf2.04", sizebcol = "Inf.04", sizeccol = "Veg.04",
repstracol = "Inf.04", repstrbcol = "Inf2.04", fecacol = "Pod.04",
stageassign = cypframe, stagesize = "sizeadded", NAas0 = TRUE)
#summary(vertdata)
```

In the above code, we described the input dataset in a way that allows R
to reorganize it appropriately. Note that in the input to the function,
we utilized a repeating pattern of variable names arranged in the same
order for each monitoring occasion. This arrangement allows us to enter
only the first variable in each set, as long as `noyears`

and
`blocksize`

are set properly and no gaps or shuffles appear
among columns in the dataset. `lefko3`

’s data management
functions do not require such repeating patterns, but such patterns do
make the required function input much more succinct. Without these
patterns, column names should be input as vectors in the correct
temporal order for each input option, and the `blocksize`

option should be omitted.

The output dataset includes a number of summary variables, but the data
is essentially broken down into groups of three consecutive monitoring
occasions each (occasions *t*+1, *t*, and *t*-1,
corresponding to `year3`

, `year2`

, and
`year1`

in the output, respectively), with individuals spread
across multiple rows. The output dataset is further limited to those
entries in which the individual is alive in occasion *t*
(`year2`

), meaning that all rows in which an individual is
dead or not yet recruited in occasion *t* are dropped. We end up
with 320 rows of data and 54 variables.

This reorganized dataset includes a set of interesting terms that were
automatically calculated: the `sizeadded`

group of three
variables. These are sums of the size variables for each occasion, such
that `size1added`

is calculated as
`sizea1 + sizeb1 + sizec1`

. This term may or may not make
sense depending on the dataset. In this dataset, the full size of the
individual in each occasion is this sum, because size is determined as
the number of stems per plant, and the three original size columns give
the numbers of three different kinds of stems: `Veg`

,
`Inf`

, and `Inf2`

are the numbers of
non-reproductive stems, single-flowered inflorescences, and
double-flowered inflorescences, respectively, per plant per occasion (an
inflorescence is a single flowering stem, and no inflorescence has more
than two flowers). Since size is given as the total number of stems in
this example, we will use the `sizeadded`

group of variables
to code individual size.

Size and fecundity are both count variables here, and we need to
determine the appropriate distribution to use in both cases. Let’s look
at histograms of size and fecundity in occasion *t*.

```
par(mfrow=c(1,2))
hist(vertdata$size2added, main = "Size in time t", xlab = "No. of stems in time t")
hist(subset(vertdata, repstatus2 == 1)$feca2, main = "Fecundity in time t",
xlab = "No. of fruit pods in time t")
```

Stem number has a very low mean, and the curve declines fairly smoothly.
The Poisson or negative binomial distribution might work for size.
Fecundity appears to have many zeros, suggesting that we might need a
zero-inflated count distribution. Let’s conduct a formal set of tests to
determine the appropriate distributions for size and fecundity. The size
of zero is actually absorbed in an unobservable stage, vegetative
dormancy (stage `D`

), so we will not test for zero-inflation
in the size variable. However, we will test for zero-inflation in
fecundity, since fecundity is measured as the number of fruit pods
produced, while all flowering individuals are treated as reproductive
and so some individuals must have fecundity of zero.

```
sf_distrib(vertdata, sizea = c("size3added", "size2added"), obs3 = "obsstatus3",
fec = c("feca3", "feca2"), repst = c("repstatus3", "repstatus2"),
zisizea = FALSE)
#> Mean sizea is 3.653
#>
#> The variance in sizea is 13.41
#>
#> The probability of this dispersion level by chance assuming that
#> the true mean sizea = variance in sizea,
#> and an alternative hypothesis of overdispersion, is 3.721e-138
#>
#> Primary size is significantly overdispersed.
#>
#> Mean fec is 0.8036
#>
#> The variance in fec is 1.601
#>
#> The probability of this dispersion level by chance assuming that
#> the true mean fec = variance in fec,
#> and an alternative hypothesis of overdispersion, is 0.07088
#>
#> Dispersion level in fecundity matches expectation.
#>
#>
#> Mean lambda in fec is 0.4477
#> The actual number of 0s in fec is 65
#> The expected number of 0s in fec under the null hypothesis is 50.15
#> The probability of this deviation in 0s from expectation by chance is 1.742e-06
#>
#> Fecundity is significantly zero-inflated.
```

The results of these tests show us that size is overdispersed. Since we have no zeros in size, we will use the zero-truncated negative binomial distribution for this vital rate. Our tests of fecundity show no significant overdispersion, but the number of zeros exceeds expectation, suggesting that we use a zero-inflated Poisson distribution.

We may also wish to see how to proceed if our original dataset is
already in vertical, ahistorical format, or a format in which each row
gives the state of an individual in a single monitoring occasion. This
package also includes dataset `cypvert`

, which is the same
dataset as `cypdata`

but set in ahistorical vertical format.
Here, we use the `historicalize3()`

function to deal with
this dataset. First, let’s load the ahistorical vertical raw data file,
look at its dimensions relative to the original, horizontal dataset, and
finally view a summary.

```
data(cypvert)
dim(cypdata)
#> [1] 77 29
dim(cypvert)
#> [1] 322 14
#summary(cypvert)
```

This dataset has more rows and fewer columns, because we constructed the
dataset by splitting the data for each individual across multiple rows.
After three columns of identifying information (`plantid`

,
`patch`

, and `censor`

), a single column designates
the time of occasion *t*, given as `year2`

. This
dataset then includes columns showing individual state in pairs of
consecutive years corresponding to occasions *t* and
*t*+1. State in occasion *t*-1 is not presented because
this is an ahistorical dataset. This dataset includes the
`plantid`

variable, which is an individual identity term that
allows us to associate rows with their individuals and so allows
conversion. The `historicalize3()`

function uses individual
identity to reorganize datasets into historical vertical format.

```
<- historicalize3(data = cypvert, patchidcol = "patch",
vertdata2 individcol = "plantid", year2col = "year2", sizea2col = "Inf2.2",
sizea3col = "Inf2.3", sizeb2col = "Inf.2", sizeb3col = "Inf.3",
sizec2col = "Veg.2", sizec3col = "Veg.3", repstra2col = "Inf2.2",
repstra3col = "Inf2.3", repstrb2col = "Inf.2", repstrb3col = "Inf.3",
feca2col = "Pod.2", feca3col = "Pod.3", repstrrel = 2, stageassign = cypframe,
stagesize = "sizeadded", censorcol = "censor", censorkeep = 1, censor = FALSE,
NAas0 = TRUE, reduce = TRUE)
#summary(vertdata2)
```

We can compare the dimensions of the new vertical datasets. The lengths of the datasets are the same in terms of rows and columns, and the variables and data are the same although the order of the columns and rows might not match (see the summaries for comparison).

```
dim(vertdata)
#> [1] 320 54
dim(vertdata2)
#> [1] 320 54
```

One important consideration is the use of censor variables. Censoring a
demographic dataset refers to the elimination of suspect data points
from analysis. This is typically accomplished by including a binary
variable in the dataset denoting whether an individual datum is to be
kept or excluded. The objects `vertdata`

and
`vertdata2`

were both created without using censoring
variables. However, because the datasets included censor variables (all
data were set to be included, with no suspect data), we wished to
incorporate those variables in the final datasets. Hence, although
`censor = FALSE`

in both the call to
`verticalize3()`

and the call to
`historicalize3()`

, we also noted
`censorcol = "censor"`

and `censorkeep = 1`

in the
call to `historicalize3()`

(the latter option specifies the
value of the censor variable coding for a “good” datum). Failing to add
these options to the call to `historicalize3()`

will produce
approximately the same dataset, but with some zeroes entering variables
`censor1`

, `censor2`

, and `censor3`

in
the historicalized dataset that do not exist in the first, verticalized
dataset.

The next steps involve adding some external data to parameterize the
matrices properly. We will use the `supplemental()`

function
to do so. The `supplemental()`

function provides a means of
inputting four kinds of data into MPM construction:

- fixed transition values derived from other studies and added as constants to matrices;
- proxy transition values when data for particular transitions do not exist and other, estimable transitions will be used as proxies;
- reproductive multipliers to indicate which stages lead to the production of which stages, and at what level relative to estimated fecundity; and
- survival multipliers, in cases in which proxy survival transitions are used but must be set at elevated or reduced levels relative to the original transition.

We will create two supplement tables, the first covering the historical
analysis, and the second covering the ahistorical analysis. Each row
refers to a specific transition, and in the historical case, there are
codes for 23 given transitions (12 for the ahistorical case). The first
nine of the historical transitions are set to specific probabilities
(six for the ahistorical), and the next 12 are transitions that will be
set to other, estimated transitions acting as proxies (four for the
ahistorical; these are the non-NA transitions in `eststage`

set below). The final two terms are fecundity multipliers. Based on the
literature, the proxies for entry into the adult classes are transitions
from dormancy, as below. Where necessary, we also use `rep`

and `mat`

as shorthand to code for all reproductive stages
and all mature stages, respectively.

```
<- supplemental(stage3 = c("SD", "SD", "P1", "P1", "P2", "P3", "SL",
cypsupp3 "SL", "SL", "D", "V1", "V2", "V3", "D", "V1", "V2", "V3", "mat", "mat",
"mat", "mat", "SD", "P1"),
stage2 = c("SD", "SD", "SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL",
"SL", "SL", "SL", "SL", "SL", "SL", "D", "V1", "V2", "V3", "rep", "rep"),
stage1 = c("SD", "rep", "SD", "rep", "SD", "P1", "P2", "P3", "SL", "P3", "P3",
"P3", "P3", "SL", "SL", "SL", "SL", "SL", "SL", "SL", "SL", "mat", "mat"),
eststage3 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", "D",
"V1", "V2", "V3", "mat", "mat", "mat", "mat", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", "D",
"D", "D", "D", "D", "V1", "V2", "V3", NA, NA),
eststage1 = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", "D",
"D", "D", "D", "V1", "V1", "V1", "V1", NA, NA),
givenrate = c(0.08, 0.08, 0.1, 0.1, 0.1, 0.1, 0.125, 0.2, 0.2, NA, NA, NA, NA,
NA, NA, NA, NA, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
NA, 1, 1, 1, 1, 0.5, 0.5),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"S", "S", "S", "S", "S", "S", "S", "R", "R"),
type_t12 = c("S", "F", "S", "F", "S", "S", "S", "S", "S", "S", "S", "S", "S",
"S", "S", "S", "S", "S", "S", "S", "S", "S", "S"), stageframe = cypframe)
<- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
cypsupp2 "V1", "V2", "V3", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL", "SL", "rep",
"rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", NA, NA),
givenrate = c(0.08, 0.1, 0.1, 0.1, 0.125, 0.2, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 0.5, 0.5),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "R", "R"),
stageframe = cypframe, historical = FALSE)
#cypsupp3
#cypsupp2
```

These supplement tables provide the means to add external data to our
MPMs by isolating specific transitions to be updated, and allow the use
of shorthand to identify large groups of transitions for update
(i.e. using `mat`

, `immat`

, `rep`

,
`nrep`

, `prop`

, `npr`

,
`obs`

, `nobs`

, `groupX`

, or
`all`

to signify all mature stages, immature stages,
reproductive stages, non-reproductive stages, propagule stages,
non-propagule stages, observable stages, unobservable stages, stages
within group `X`

, or simply all stages, respectively). They
also allow more powerful changes to MPMs, such as block reset of groups
of transitions to zero or other constants.

Matrix estimation can proceed either via raw matrix methods, as initially outlined in Ehrlén (2000), or via function-based matrix development, essentially equivalent to the creation of complex integral projection models per Ellner and Rees (2006) and as further described in the non-Gaussian case in Shefferson, Warren II, and Pulliam (2014). In the raw MPM case, no vital rate models are estimated, whereas in function-based MPMs, vital rates are first analyzed via linear or non-linear models of the raw demographic dataset, and then functions are created that estimate these vital rates according the inputs given. Matrices are then estimated using these functions, as opposed to the raw data.

Prior to vital rate estimation, a number of key decisions need to be
made regarding the assumptions underlying the vital rates, and their
relationships with the factors under investigation. These decisions
include the **general modeling strategy**, and the
**size and fecundity distributions**.

Most function-based matrices, whether integral projection models or
otherwise, are developed using either a *generalized linear modeling
(GLM)* strategy, or a *generalized linear mixed modeling (GLMM,
or mixed)* strategy. The former is more common and is simpler, but
the latter is more theoretically sound because it provides a means of
correcting the lack of independence inherent in datasets incorporating
repeated sampling of the same individuals. The difference between the
two methods with regards to vital rate modeling is strongly related to
assumptions regarding individual and spatiotemporal variation in vital
rates.

In both GLM and GLMM-based MPMs, the underlying dataset utilized is a
vertical dataset. Each row of data gives the state of the individual in
either two consecutive occasions (the ahistorical case), or three
consecutive occasions (the historical case). Under the GLM framework,
time in occasion *t* is a fixed categorical variable, and
individual identity is ignored. Treating time as fixed implies that the
actual monitoring occasions are the only times for which inference is
sought. Thus, it would be inappropriate to infer future population
dynamics after 2009 for a dataset collected between 2004 and 2009.
Ignoring individual identity treats all transitions as independent, even
though data originating from the same sampled individual is clearly not
independent of that individual’s previous transitions. This may be
interpreted as a form of pseudoreplication because strongly related data
is used to estimate matrix elements and matrices that are assumed to be
statistically independent of each other. This might impact demographic
modeling by inflating Type 1 error in the linear modeling, yielding more
significant terms in the chosen best-fit model and causing the retention
of more terms than is warranted.

Under the GLMM (generalized linear mixed model, or mixed) framework, both time and individual identity can be treated as random categorical terms. This has two major implications. First, both time and individuals can be assumed to be random samples from a broader population of times and individuals for which inference is sought. Thus, sampled monitoring occasions represent a greater universe of years for which inference can be made. Second, treating individual as a random categorical term eliminates the pseudoreplication that is inherent in the GLM approach when individuals are monitored potentially many times, because each individual is assumed to be randomly drawn and associated with its own response distribution. Subpopulations may also be considered random, in which case they are assumed to have been sampled from all possible subpopulations whether or not other subpopulations actually exist. We encourage researchers to use the GLMM approach in their demographic work, but we have also included easy-to-use GLM functionality, since many will find the GLM approach particularly useful in cases where mixed modeling breaks down or takes too long.

Once a general approach is decided upon, the next step is to choose the underlying distributions. The probabilities of survival, observation, and reproductive status are automatically set to the binomial distribution, and this cannot be altered. However, size transition probability and fecundity rate can be set to the Gaussian, Gamma, Poisson, or negative binomial distributions, with zero-inflated and zero-truncated versions of the Poisson and negative binomial also available. If size or fecundity rate is a continuous variable, then it should be set to the Gaussian distribution if not too right-skewed; otherwise, the Gamma distribution may be better (although the Gamma distribution requires only positive data). If size or fecundity is a count variable, then it should be set to the Poisson distribution if the mean equals the variance. The negative binomial distribution is provided in cases where the assumption that the mean equals the variance is clearly broken.

The Poisson and the negative binomial distributions both predict specific numbers of zeros in the response variable. If excess zeros occur within the dataset, then a zero-inflated Poisson or negative binomial distribution may be used. These modeling approaches work by parameterizing a binomial model, typically with a logit link, to predict zero responses. The Poisson or negative binomial is then used to predict non-zero responses. This model is really two separate models in which zeros are assumed to be predicted under potentially different processes than the remaining counts. Because an extra model is built to cover the occurrence of zeros, zero-inflated models are more complex and can include many more parameters than their non-inflated counterparts. The principle of parsimony suggests that they should only be used when there are significantly more zeros than expected.

Cases may arise in which zeros do not exist in either size or fecundity.
For these situations, we provide zero-truncated distributions. This may
occur in size if all cases of `size = 0`

are absorbed by
observation status, leaving only positive integers for the size of
observed individuals. For example, if an unobservable stage such as
vegetative dormancy occurs and absorbs all cases of
`size = 0`

, then a zero-truncated Poisson or negative
binomial distribution will be more appropriate than the equivalent
distribution without zero-truncation. It can also occur if all cases of
`fecundity = 0`

are absorbed by reproductive status.

In *lefko3*, function `modelsearch()`

is the workhorse
that conducts vital rate model estimation. Here, we will create a full
suite of vital rate models for the *Cypripedium candidum*
dataset. Before proceeding, we need to decide on the linear model
building strategy, the correct vital rates to model, the proper
statistical distributions for estimated vital rates, the proper
parameterization of each vital rate, and the strategy for determination
of the best-fit model for each vital rate.

First, we must determine the model building strategy. In most cases, the
best procedure will be through mixed linear models in which monitoring
occasion and individual identity are random terms. We will set
monitoring occasion as random because we wish to make inferences for the
population as a whole and do not wish to restrict ourselves to inference
only for the years monitored. We will set individual identity as random
because many or most of the individuals that we have sampled to produce
our dataset yield multiple observation data points across time. Thus, we
will set `approach = "mixed"`

. To make sure that time and
individual identity are treated as random, we will set the proper
variable names for `indiv`

and `year`

,
corresponding to individual identity (`individ`

by default),
and to occasion *t* (`year2`

by default). The
`year.as.random`

option is set to random by default, and
leaving it this way also means that R will randomly draw coefficient
values for years with inestimable coefficients. Setting
`year.as.random`

to `FALSE`

would make time a
fixed categorical variable.

Next, we must determine which vital rates to model. Function
`modelsearch()`

estimates up to fourteen vital rate models:

survival probability from occasion

*t*to occasion*t*+1,observation probability in occasion

*t*+1 assuming survival until that time,primary size in occasion

*t*+1 assuming survival and observation in that time,secondary size in occasion

*t*+1 assuming survival and observation in that time,tertiary size in occasion

*t*+1 assuming survival and observation in that time,reproduction status in occasion

*t*+1 assuming survival and observation until that time,fecundity rate assuming survival until and observation and reproduction in the occasion of production of offspring (occasion

*t*or*t*+1; mature only),juvenile survival probability from occasion

*t*to occasion*t*+1,juvenile observation probability in occasion

*t*+1 assuming survival until that time,juvenile primary size in occasion

*t*+1 assuming survival and observation in that time,juvenile secondary size in occasion

*t*+1 assuming survival and observation in that time,juvenile tertiary size in occasion

*t*+1 assuming survival and observation in that time,reproduction status in occasion

*t*+1 assuming survival and observation until that time of a juvenile in occasion*t*that is becoming mature in occasion*t*+1, andmaturity status in occasion

*t*+1 assuming survival until that time of a juvenile in occasion*t*.

The default settings for `modelsearch`

estimate 1) survival
probability, 3) primary size distribution, and 7) fecundity, which are
the minimum three vital rates required for a full MPM. Observation
probability (option `obs`

in `vitalrates`

) should
only be included when a life history stage or size exists that cannot be
observed. For example, in the case of a plant with vegetative dormancy,
the observation probability can be thought of as the sprouting
probability, which is a biologically meaningful vital rate (Shefferson et al. 2001). Reproduction status
(option `repst`

in `vitalrates`

) should only be
modeled if size classification needs to be stratified by the ability to
reproduce, as when equivalently sized stages exist differing in the
ability to reproduce. Secondary and tertiary size should only be modeled
when two or three size variables are used in stage classification. Since
*Cypripedium candidum* is capable of long bouts of vegetative
dormancy, since we wish to stratify the population into reproductive and
non-reproductive adults of the same size classes, and since we have no
data derived from juvenile individuals, we will set
`vitalrates = c("surv", "obs", "size", "repst", "fec")`

and
leave the juvenile-related options at their defaults.

Third, we need to set the proper statistical distribution for each parameter. Survival probability, observation probability, and reproductive status (and maturity status) are all modeled as binomial variables, and this cannot be changed. Size was measured as the number of stems and so is a count variable. Likewise, fecundity is actually estimated as the number of fruits produced per plant, and so is also a count variable. We have already performed tests for overdispersion and zero-inflation, and we are also aware that size in observed stages cannot equal 0, requiring zero truncation in that parameter. So we will set size to the zero-truncated negative binomial distribution, and fecundity to the zero-inflated Poisson distribution.

Fourth, we need the proper model parameterizations for each vital rate,
using the `suite`

option. The default,
`suite = "main"`

, under the mixed model setting
(`approach = "mixed"`

) starts with the estimation of global
models that include size and reproductive status in occasions *t*
and *t*-1 as fixed factors, with individual identity and time in
occasion *t* (year *t*) set as random categorical terms.
Other terms can be specified, including individual covariates, spatial
density, and age. Setting `suite = "full"`

will yield global
models that also include all two-way interactions. The default under the
GLM setting (`approach = "glm"`

) makes time in occasion
*t* a fixed term and drops individual identity from
consideration. The global model under `suite = "full"`

then
includes all fixed factors noted before, plus all two-way interactions
between fixed factors (`"full"`

is the only setting with
interaction terms). If the population is not stratified by reproductive
status, then `suite = "size"`

will eliminate reproductive
status terms and use all others in the global model. If size is not
important, then `suite = "rep"`

will eliminate size but keep
reproductive status and all other terms. Finally,
`suite = "cons"`

will result in a global model in which
neither reproductive status nor size are considered, although optional
terms such as age or spatial density may be still be incorporated. We
will set `suite = "full"`

.

Finally, we need to determine the proper strategy for the determination
of the best-fit model. Model building proceeds through the
`dredge`

function in package `MuMIn`

(Bartoń 2014), and each model has an associated
AICc value. The default setting in `lefko3`

(`bestfit = "AICc&k"`

) will compare all models within 2.0
AICc units of the model with \(\Delta AICc =
0\), and choose the one with the lowest degrees of freedom. This
approach is generally better than the alternative, which simply uses the
model with \(\Delta AICc = 0\)
(`bestfit = "AICc"`

), as all models within 2.0 AICc units of
that model are equally parsimonious and so fewer degrees of freedom
result from fewer parameters estimated (Burnham
and Anderson 2002).

```
<- modelsearch(vertdata, historical = TRUE, approach = "mixed",
cypmodels3 vitalrates = c("surv", "obs", "size", "repst", "fec"), sizedist = "negbin",
size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE, suite = "full",
size = c("size3added", "size2added", "size1added"), quiet = TRUE)
#summary(cypmodels3)
```

As `modelsearch`

works, it produces text describing its
current processes and any issues encountered. It is quite likely that
many warning messages will appear, and these may be of use in
understanding the dataset and how well it conforms to analytical
assumptions. We have silenced most of this with the
`quiet = TRUE`

option, although some red text still indicates
a few warnings that have slipped through. These are not to be terribly
concerned about. However, we encourage users to allow the function to
run unsilenced in case of severe modeling problems. Please read the
documentation for functions `lm()`

(`stats`

package), `glm()`

(`stats`

package),
`glm.nb()`

(`MASS`

package), `lmer()`

(`lme4`

package), `glmer()`

(`lme4`

package), `glmmTMB()`

(`glmmTMB`

package),
`zeroinfl()`

(`pscl`

package), `vglm()`

(`VGAM`

package), and `dredge()`

(`MuMIn`

package) for further information on the sources of
problems in such models.

In the output, we see historical terms determining primary size,
suggesting that we cannot ignore history. As such, the models created
above are actually only usable for the construction of historical
models. We can also see that although the accuracy of our survival and
observation status models is very high (94.7% and 95% respectively), the
accuracy of our reproductive status model is only fair (71.5%), and the
pseudo R^{2} for fecundity is only 32%. For comparison, we may
wish to estimate ahistorical models. In that case, we also need linear
models in which the global models tested do not include state at
occasion *t*-1, as below.

```
<- modelsearch(vertdata, historical = FALSE, approach = "mixed",
cypmodels2 vitalrates = c("surv", "obs", "size", "repst", "fec"), sizedist = "negbin",
size.trunc = TRUE, fecdist = "poisson", fec.zero = TRUE, suite = "full",
size = c("size3added", "size2added"), quiet = TRUE)
#summary(cypmodels2)
```

Fewer models were estimated per dredge, since fewer parameters were
tested in the global models (size and reproductive status in occasion
*t*-1 were not included). So, the best-fit models should look a
little bit different. However, a more thorough comparison will show that
many of the best-fit models are similar between historical and
ahistorical analysis, and that the only model different is the primary
size model, which now shows only a y-intercept and random terms for
monitoring occasion and individual. This is not guaranteed - in this
case, it may be that the relatively small number of years and small
overall sample size leaves too little power to find an impact of
historical status on most vital rates. However, with size impacted by
history, the choice of a historical model is better here.

Now we will create function-based MPMs. Let’s first create a set of ahistorical matrices.

```
<- flefko2(stageframe = cypframe, supplement = cypsupp2,
cypmatrix2 modelsuite = cypmodels2, data = vertdata)
#summary(cypmatrix2)
```

A quick glance at the summary output will highlight that many more elements are estimated for function-based matrices than for raw matrices - here, 2,459 out of 2,916 total elements (84.3%). In raw matrices, elements associated with transitions from specific stages are only estimated when individuals actually exist within those particular stages, and so the number of stages is strongly limited by the size of the dataset. In function-based matrices, in contrast, the linear models estimated allow the estimation of all elements that are theoretically possible (i.e. only structural zeros are not estimated), and so stages may be developed for the full range of sizes experienced by individuals.

The survival probability summaries at the bottom of the output provide a useful quality control check. These are calculated as the column sums from the associated survival-transition matrices. The use of externally-supplied rates and probabilities, as supplied via the supplement and overwrite tables, can result in some stages or stage-pairs having survival probabilities greater than 1.0, and if that occurs, then the supplement/overwrite tables need to be corrected to reduce survival to something reasonable. Our matrices here look fine.

The output of the `flefko2()`

call includes several key
elements. One of the most important is a data frame showing the order of
matrices produced. Let’s take a look at this object.

```
$labels
cypmatrix2#> pop patch year2
#> 1 1 1 2004
#> 2 1 1 2005
#> 3 1 1 2006
#> 4 1 1 2007
#> 5 1 1 2008
```

Here we see that our five matrices are annual matrices corresponding to
the same population/patch. Other elements in this object include
summaries of the stages used, in the order that they occur in the
matrix. Since this is an ahistorical MPM, the key element here is
`cypmatrix2$ahstages`

. If this were a historical MPM, then
`cypmatrix2$ahstages`

would give the order of stages, and
`cypmatrix2$hstages`

would include the order of stage-pairs
corresponding to the rows and columns in such MPMs.

Next, we will create a set of historical Lefkovitch matrices.

```
<- flefko3(stageframe = cypframe, supplement = cypsupp3,
cypmatrix3 modelsuite = cypmodels3, data = vertdata)
#summary(cypmatrix3)
```

We see many more total elements per matrix (over 8.5 million, in
contrast to 2,916 in the ahistorical case), and many more rows and
columns (54 rows and columns in the ahistorical case, and 54^{2}
= 2,916 rows and columns in the historical case). However, historical
matrices are composed primarily of structural zeros. Thus, with only
117,908 + 2,352 = 120,260 elements estimated per matrix, only 1.4% of
elements are non-zero (the equivalent percentage for the ahistorical
case is 84.3%).

Now let’s estimate some element-wise mean matrices. Here, we will create the ahistorical and historical mean MPMs and look at their summaries.

```
<- lmean(cypmatrix2)
tmeans2r <- lmean(cypmatrix3)
tmeans3r #summary(tmeans2r)
#summary(tmeans3r)
```

It might help to do a more holistic comparison of the ahistorical
vs. historical MPMs. Let’s create matrix plots of the ahistorical and
historical means. For this purpose, we will use the
`image3()`

function. Let’s first view the ahistorical matrix
image.

`image3(tmeans2r, used = 1)`

```
#> [[1]]
#> NULL
```

This is a fairly large matrix, and rather dense though with obviously large clusters of zero elements particularly at the edges. Now let’s see the historical matrix.

`image3(tmeans3r, used = 1)`

```
#> [[1]]
#> NULL
```

Historical matrices are often huge and always sparse, just as our historical matrix is in the plot above. Although there are roughly 8.5 million elements, very few are non-zero and those are identified as the red streaks in the matrix image above.

For further comparison, let’s decompose the mean historical matrix into
conditional historical matrices. We’ll take a look at the matrix
conditional on vegetative dormancy in occasion *t*-1.

```
<- cond_hmpm(tmeans3r)
condm3r #print(condm3r$Mcond[[1]]$D, digits = 3)
```

This conditional matrix essentially takes the logically possible
elements in which stage in time *t*-1 is `D`

(vegetative dormancy), and creates a matrix with the dimensions and row
and column order of an ahistorical matrix with them. Since vegetative
dormancy is an adult stage, we do not see survival transitions from
juvenile stages - only adult portions of the matrix are shown, including
both survival and fecundity transitions.

Let’s now conduct some projection analyses. First, we will estimate the
deterministic and stochastic population growth rates for these means.
The `set.seed()`

function makes our stochastic results
reproducible.

```
set.seed(42)
<- slambda3(cypmatrix2)
cypann2s set.seed(42)
<- slambda3(cypmatrix3)
cypann3s
lambda3(tmeans2r)
#> pop patch lambda
#> 1 1 1 0.9603441
lambda3(tmeans3r)
#> pop patch lambda
#> 1 1 1 0.9609416
cypann2s#> pop patch a var sd se
#> 1 1 1 -0.04043409 0.002352962 0.04850734 0.0004850734
cypann3s#> pop patch a var sd se
#> 1 1 1 -0.0397941 0.002286954 0.04782211 0.0004782211
```

These \(\lambda\) values are similar to one another, as are estimates of \(a = \text{log} \lambda _{S}\). Let’s take a look at a plot of annual deterministic growth rates (stochastic growth rate is calculated across time so is not included).

```
<- lambda3(cypmatrix3)
cypann3 <- lambda3(cypmatrix2)
cypann2
plot(lambda ~ year2, data = cypann2, xlab = "Year", ylab = "Lambda",
ylim = c(0.93, 0.98), type = "l", lty= 1, lwd = 2, bty = "n")
lines(lambda ~ year2, data = cypann3, lty = 2, lwd= 2, col = "red")
legend("bottomright", c("ahistorical", "historical"), lty = c(1, 2),
col = c("black", "red"), lwd = 2, bty = "n")
```

We find that annual \(\lambda\) estimates differ only slightly between ahistorical and historical analyses.

Now let’s take a look at the stable stage distributions. We will look at deterministic and stochastic versions of the ahistorical and historical MPMs.

```
<- stablestage3(tmeans2r)
tm2ss <- stablestage3(tmeans3r)
tm3ss <- stablestage3(cypmatrix2, stochastic = TRUE, seed = 42)
tm2ss_s <- stablestage3(cypmatrix3, stochastic = TRUE, seed = 42)
tm3ss_s
<- cbind.data.frame(tm2ss$ss_prop, tm3ss$ahist$ss_prop,
ss_put_together $ss_prop, tm3ss_s$ahist$ss_prop)
tm2ss_snames(ss_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(ss_put_together) <- tm2ss$stage_id
barplot(t(ss_put_together), beside=T, ylab = "Proportion", xlab = "Stage",
ylim = c(0, 0.15), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Overall, these are very similar patterns. Whether ahistorical or historical, deterministic or stochastic, our analyses suggest that small non-reproductive adults take up the greatest share of the stable stage structure, with small reproductive adults coming next, and then dormant seed and 1st-year protocorms.

Next let’s look at the reproductive values associated with both ahistorical and historical approaches. We will standardize against the maximum value in each case to make these comparable in a single plot (normally we would not standardize across analyses like this, using the first or minimum reproductive value in each case for standardization instead).

```
<- repvalue3(tmeans2r)
tm2rv <- repvalue3(tmeans3r)
tm3rv <- repvalue3(cypmatrix2, stochastic = TRUE, seed = 42)
tm2rv_s <- repvalue3(cypmatrix3, stochastic = TRUE, seed = 42)
tm3rv_s
<- cbind.data.frame((tm2rv$rep_value / max(tm2rv$rep_value)),
rv_put_together $ahist$rep_value / max(tm3rv$ahist$rep_value)),
(tm3rv$rep_value / max(tm2rv_s$rep_value)),
(tm2rv_s$ahist$rep_value / max(tm3rv_s$ahist$rep_value)))
(tm3rv_snames(rv_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(rv_put_together) <- tm2rv$stage_id
barplot(t(rv_put_together), beside=T, ylab = "Relative rep value", xlab = "Stage",
ylim = c(0, 1.3), col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topleft", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

We see roughly equivalent patterns here. All analyses show the smallest reproductive value in the juvenile stages, while adults have the greatest value, particularly medium- and large-sized adults of both flowering and non-flowering kinds. Interesting results in need of further study!

Let’s now conduct a sensitivity analysis, again using both deterministic and stochastic approaches. First we will look at the ahistorical MPM.

```
<- sensitivity3(tmeans2r)
tm2sens set.seed(42)
<- sensitivity3(cypmatrix2, stochastic = TRUE)
tm2sens_s
writeLines(paste0("\nThe highest deterministic sensitivity value: ",
max(tm2sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])))
#>
#> The highest deterministic sensitivity value: 0.200249793902156
writeLines(paste0("This value is associated with element: ",
which(tm2sens$ah_sensmats[[1]] ==
max(tm2sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))))
#> This value is associated with element: 378
writeLines(paste0("\nThe highest stochastic sensitivity value: ",
max(tm2sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])))
#>
#> The highest stochastic sensitivity value: 0.207109655095675
writeLines(paste0("This value is associated with element: ",
which(tm2sens_s$ah_sensmats[[1]] ==
max(tm2sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))))
#> This value is associated with element: 378
```

The highest sensitivity value in both cases appears to be associated with the transition from 1-sprouted non-flowering adult to the largest flowering adult (element 378, in the 7th column and 54th row). Inspecting each sensitivity matrix also shows that transitions near that element in both matrices are also associated with rather high sensitivities.

Now let’s compare with the historical case. We will compare both the full historical sensitivity matrices, and also look at the historically-corrected deterministic sensitivities.

```
<- sensitivity3(tmeans3r)
tm3sens set.seed(42)
<- sensitivity3(cypmatrix3, stochastic = TRUE)
tm3sens_s
writeLines(paste0("The highest deterministic sensitivity value: ",
max(tm3sens$h_sensmats[[1]][which(tmeans3r$A[[1]] > 0)])))
#> The highest deterministic sensitivity value: 0.0507110864421899
writeLines(paste0("This value is associated with element: ",
which(tm3sens$h_sensmats[[1]] ==
max(tm3sens$h_sensmats[[1]][which(tmeans3r$A[[1]] > 0)]))))
#> This value is associated with element: 962658
writeLines(paste0("The highest stochastic sensitivity value: ",
max(tm3sens_s$h_sensmats[[1]][which(tmeans3r$A[[1]] > 0)])))
#> The highest stochastic sensitivity value: 0.0521953997830354
writeLines(paste0("This value is associated with element: ",
which(tm3sens_s$h_sensmats[[1]] ==
max(tm3sens_s$h_sensmats[[1]][which(tmeans3r$A[[1]] > 0)]))))
#> This value is associated with element: 962658
writeLines(paste0("The highest deterministic sensitivity value (historically-corrected): ",
max(tm3sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])))
#> The highest deterministic sensitivity value (historically-corrected): 0.198337403966719
writeLines(paste0("This value is associated with element: ",
which(tm3sens$ah_sensmats[[1]] ==
max(tm3sens$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))))
#> This value is associated with element: 378
writeLines(paste0("The highest stochastic sensitivity value (historically-corrected): ",
max(tm3sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)])))
#> The highest stochastic sensitivity value (historically-corrected): 0.205686867202455
writeLines(paste0("This value is associated with element: ",
which(tm3sens_s$ah_sensmats[[1]] ==
max(tm3sens_s$ah_sensmats[[1]][which(tmeans2r$A[[1]] > 0)]))))
#> This value is associated with element: 378
```

We see both deterministic and stochastic sensitivity analyses supporting
the transition from 1-sprouted non-flowering adult in occasions
*t*-1 and *t* to the largest flowering adult in occasion
*t*+1 (we can see this by dividing 962,658 by the number of stage
pairs, which is 2916 - rounding up the answer gives us the matrix column
[331], and the remainder from this division problem gives us the row
[378]; the row and column numbers can then be checked against the
`hstages`

portion of the historical `lefkoMat`

object used as input). The highest historically-corrected deterministic
and stochastic sensitivity value is associated with the transition from
1-sprouted non-flowering adult to the largest flowering adult stage
(element 378). Thus, our results reinforce the results of the
ahistorical case.

Let’s now assess the elasticity of \(\lambda\) to matrix elements, comparing the ahistorical to the historically-corrected case in both deterministic and stochastic analyses.

```
<- elasticity3(tmeans2r)
tm2elas <- elasticity3(tmeans3r)
tm3elas set.seed(42)
<- elasticity3(cypmatrix2, stochastic = TRUE)
tm2elas_s set.seed(42)
<- elasticity3(cypmatrix3, stochastic = TRUE)
tm3elas_s
writeLines(paste0("The largest ahistorical deterministic elasticity is associated with element: ",
which(tm2elas$ah_elasmats[[1]] == max(tm2elas$ah_elasmats[[1]]))))
#> The largest ahistorical deterministic elasticity is associated with element: 331
writeLines(paste0("The largest historically-corrected deterministic elasticity is associated with element: ",
which(tm3elas$ah_elasmats[[1]] == max(tm3elas$ah_elasmats[[1]]))))
#> The largest historically-corrected deterministic elasticity is associated with element: 331
writeLines(paste0("The largest historical deterministic elasticity is associated with element: ",
which(tm3elas$h_elasmats[[1]] == max(tm3elas$h_elasmats[[1]]))))
#> The largest historical deterministic elasticity is associated with element: 962611
writeLines(paste0("The largest ahistorical stochastic elasticity is associated with element: ",
which(tm2elas_s$ah_elasmats[[1]] == max(tm2elas_s$ah_elasmats[[1]]))))
#> The largest ahistorical stochastic elasticity is associated with element: 331
writeLines(paste0("The largest historically-corrected stochastic elasticity is associated with element: ",
which(tm3elas_s$ah_elasmats[[1]] == max(tm3elas_s$ah_elasmats[[1]]))))
#> The largest historically-corrected stochastic elasticity is associated with element: 331
writeLines(paste0("The largest historical stochastic elasticity is associated with element: ",
which(tm3elas_s$h_elasmats[[1]] == max(tm3elas_s$h_elasmats[[1]]))))
#> The largest historical stochastic elasticity is associated with element: 962611
```

Ahistorical and historical, deterministic and stochastic analyses
suggest that population growth rate is most elastic in response to
stasis as 1-sprouted non-flowering adult (element 331). Indeed, element
962,611 in the historical case is actually stasis in this stage across
times *t*-1, *t*, and *t*+1 (column:
`round(962611/2916) = 331`

; row:
`962611 %% 2916 = 331`

). This is a remarkably consistent
result, reinforcing the importance of this stage as seen in sensitivity
analyses.

Now let’s compare the elasticity of population growth rate in relation to the core life history stages, via a barplot comparison.

```
<- cbind.data.frame(colSums(tm2elas$ah_elasmats[[1]]),
elas_put_together colSums(tm3elas$ah_elasmats[[1]]), colSums(tm2elas_s$ah_elasmats[[1]]),
colSums(tm3elas_s$ah_elasmats[[1]]))
names(elas_put_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_put_together) <- tm2elas$ah_stages$stage_id
barplot(t(elas_put_together), beside=T, ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Elasticity analyses in these plots look very similar. Population growth rates are most elastic in response to changes in stage 7, which is the one-sprouted non-flowering adult stage. Small adult stages have higher elasticity than larger adults and juveniles. Most of the differences between analyses here appear to lie in elasticity of small non-flowering vs. flowering adults in the ahistorical vs. historical comparison.

Finally, let’s take a look at how the importance of different kinds of transitions changes, by looking at elasticity sums. The plot will show that population growth rates are most elastic in response to changes in growth and shrinkage, with virtually no influence of fecundity.

```
<- summary(tm2elas)
tm2elas_sums <- summary(tm3elas)
tm3elas_sums <- summary(tm2elas_s)
tm2elas_s_sums <- summary(tm3elas_s)
tm3elas_s_sums
<- cbind.data.frame(tm2elas_sums$ahist[,2],
elas_sums_together $ahist[,2], tm2elas_s_sums$ahist[,2], tm3elas_s_sums$ahist[,2])
tm3elas_sumsnames(elas_sums_together) <- c("det ahist", "det hist", "sto ahist", "sto hist")
rownames(elas_sums_together) <- tm2elas_sums$ahist$category
barplot(t(elas_sums_together), beside=T, ylab = "Elasticity", xlab = "Transition",
col = c("black", "orangered", "grey", "darkred"), bty = "n")
legend("topright", c("det ahist", "det hist", "sto ahist", "sto hist"),
col = c("black", "orangered", "grey", "darkred"), pch = 15, bty = "n")
```

Some users of `lefko3`

may wish to use conduct custom
projections, perhaps involving density dependence or projected
individual covariate inputs. Beginning with version 5.0,
`lefko3`

includes function `f_projection3()`

,
which allows users to conduct projection simulations of function-based
MPMs. This function takes the vital rate models that constitute the
`lefkoMod`

object input into matrix generating functions
`flefko3()`

, `flefko2()`

, `aflefko2()`

,
and `afleslie()`

, and builds matrices projecting the
population forward according to whatever inputs the user has in mind.
Function `f_projection3()`

is different from function
`projection3()`

in that the latter requires a
`lefkoMat`

object as input, and uses the matrices in whatever
projection the user has in mind. The former actually creates these
matrices from scratch at each time step and so does not take matrices as
inputs.

To illustrate how we might use this function, let’s run a
conservation-focused analysis using the ahistorical vital rate models
created above. We will ask what happens if we artificially pollinate
these plants enough to double their fecundity. Let’s first create a
`lefkoDens`

object, which holds the density dependence
information. Let’s also create an alternative supplement table with
doubled fecundity.

```
<- density_input(cypmatrix2, stage3 = c("SD", "P1", "SD", "P1"),
c2d stage2 = c("SD", "SD", "rep", "rep"), style = 1,
time_delay = 1, alpha = 1, beta = 0, type = c(1, 1, 2, 2))
#c2d
<- supplemental(stage3 = c("SD", "P1", "P2", "P3", "SL", "SL", "D",
cypsupp2alt "V1", "V2", "V3", "SD", "P1"),
stage2 = c("SD", "SD", "P1", "P2", "P3", "SL", "SL", "SL", "SL", "SL", "rep",
"rep"),
eststage3 = c(NA, NA, NA, NA, NA, NA, "D", "V1", "V2", "V3", NA, NA),
eststage2 = c(NA, NA, NA, NA, NA, NA, "D", "D", "D", "D", NA, NA),
givenrate = c(0.08, 0.1, 0.1, 0.1, 0.125, 0.2, NA, NA, NA, NA, NA, NA),
multiplier = c(NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, 0.5*2, 0.5*2),
type = c("S", "S", "S", "S", "S", "S", "S", "S", "S", "S", "R", "R"),
stageframe = cypframe, historical = FALSE)
#cypsupp2alt
```

Now let’s run two projections, the first using the original ahistorical information, and the second using the altered supplement table. We will run both as stochastic and density dependent, with ten replicates of each and 100 time steps.

```
set.seed(42)
<- f_projection3(format = 3, data = vertdata, supplement = cypsupp2,
basetrial stageframe = cypframe, modelsuite = cypmodels2, nreps = 10, times = 100,
stochastic = TRUE, growthonly = TRUE, substoch = 2, density = c2d, err_check = TRUE)
#> Warning: Option patch not set, so will set to first patch/population.
<- f_projection3(format = 3, data = vertdata, supplement = cypsupp2alt,
alttrial stageframe = cypframe, modelsuite = cypmodels2, nreps = 10, times = 100,
stochastic = TRUE, growthonly = TRUE, substoch = 2, density = c2d)
#> Warning: Option patch not set, so will set to first patch/population.
summary(basetrial)
#>
#> The input lefkoProj object covers 1 population-patches.
#> It includes 100 projected steps per replicate and 10 replicates.
#> The number of replicates with population size above the threshold size of 1 is as in
#> the following matrix, with pop-patches given by column and milepost times given by row:
#> 1 1
#> 1 10
#> 26 10
#> 51 10
#> 76 10
#> 101 10
summary(alttrial)
#>
#> The input lefkoProj object covers 1 population-patches.
#> It includes 100 projected steps per replicate and 10 replicates.
#> The number of replicates with population size above the threshold size of 1 is as in
#> the following matrix, with pop-patches given by column and milepost times given by row:
#> 1 1
#> 1 10
#> 26 10
#> 51 10
#> 76 10
#> 101 10
```

Let’s compare these two projections with plots.

```
par(mfrow = c(1,2))
plot(basetrial, ylim = c(0, 100))
plot(alttrial, ylim = c(0, 100))
```

Each plot shows 10 lines, one for each replicate. Both projections show decline, but the population appears to reach a higher high size early on under the boosted fecundity scenario. So, while boosting fecundity does not prevent extinction, it does seem to boost the population and probably lowers the overall chances of extinction within a short time window.

We have also created other literature to help users familiarize
themselves with population analyses in general and using the package.
Users should see our free e-manual called ** lefko3: a gentle
introduction**, as well as other vignettes including
long-format and video vignettes, on
the projects page
of the Shefferson lab website.

We are grateful to Joyce and George Proper, Ken Klick, and the many
other volunteers who have helped amass the *Cypripedium dataset*,
as well as to two anonymous reviewers whose scrutiny improved the
quality of this vignette. The project resulting in this package and this
tutorial was funded by Grant-In-Aid 19H03298 from the Japan Society for
the Promotion of Science.

Bartoń, Kamil A. 2014. “MuMIn: Multi-Model
Inference.” https://CRAN.R-project.org/package=MuMIn.

Burnham, Kenneth P., and David R. Anderson. 2002. *Model Selection
and Multimodel Inference: A Practical Information-Theoretic
Approach*. New York, New York, USA: Springer-Verlag New York, Inc.

Ehrlén, Johan. 2000. “The Dynamics of Plant Populations: Does the
History of Individuals Matter?” *Ecology* 81 (6): 1675–84.
https://doi.org/10.1890/0012-9658(2000)081[1675:TDOPPD]2.0.CO;2.

Ellner, Stephen P., and Mark Rees. 2006. “Integral Projection
Models for Species with Complex Demography.” *American
Naturalist* 167 (3): 410–28. https://doi.org/10.1086/499438.

Shefferson, Richard P., Ryo Mizuta, and Michael J. Hutchings. 2017.
“Predicting Evolution in Response to Climate Change: The Example
of Sprouting Probability in Three Dormancy-Prone Orchid Species.”
*Royal Society Open Science* 4 (1): 160647. https://doi.org/10.1098/rsos.160647.

Shefferson, Richard P., Brett K. Sandercock, Joyce Proper, and Steven R.
Beissinger. 2001. “Estimating Dormancy and Survival of a Rare
Herbaceous Perennial Using Mark-Recapture Models.”
*Ecology* 82 (1): 145–56. https://doi.org/10.1890/0012-9658(2001)082[0145:EDASOA]2.0.CO;2.

Shefferson, Richard P., Robert J. Warren II, and H. Ronald Pulliam.
2014. “Life History Costs Make Perfect Sprouting Maladaptive in
Two Herbaceous Perennials.” *Journal of Ecology* 102 (5):
1318–28. https://doi.org/10.1111/1365-2745.12281.