In this vignette, we use the `lathyrus`

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

calls. Remove the hashtags to run these calls.
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.

*Lathyrus vernus* (family Fabaceae) is a long-lived forest herb,
native to Europe and large parts of northern Asia. Please see our
description of the plant, study site, and methods in our vignette on raw ahistorical MPM creation and
analysis.

The dataset that we have provided is organized in horizontal format,
meaning that each row holds all of the data for a single, unique
individual, and columns correspond to individual condition in particular
monitoring occasions (which we refer to as *years* here, since
there was one main census in each year). The original spreadsheet file
used to keep the dataset has a repeating pattern to these columns, with
each year having a similarly arranged group of variables. Package
`lefko3`

includes functions to handle data in horizontal
format, as well as vertically formatted data (i.e. data for individuals
is broken up across rows, where each row is a unique combination of
individual and year in occasion *t*). Let’s load the dataset.

```
data(lathyrus)
dim(lathyrus)
#> [1] 1119 38
#summary(lathyrus)
```

This dataset includes information on 1,119 individuals, so there are
1,119 rows with data (not counting the header). There are 38 columns.
The first two columns are variables identifying each individual
(`SUBPLOT`

refers to the patch, and `GENET`

refers
to individual identity), with each individual’s data entirely restricted
to one row. This is followed by four sets of nine columns, each named
`VolumeXX`

, `lnVolXX`

, `FCODEXX`

,
`FlowXX`

, `IntactseedXX`

, `Dead19XX`

,
`DormantXX`

, `Missing19XX`

, and
`SeedlingXX`

, where `XX`

corresponds to the year
of observation and with years organized consecutively. Thus, columns
3-11 refer to year 1988, columns 12-20 refer to year 1989, etc. This
strictly repeated pattern allows us to manipulate the original dataset
quickly and efficiently via `lefko3`

. There are four years of
data - 1988 to 1991. Our data columns are also arranged in the same
order for each year, with years in consecutive order with no extra
columns between them. Note that this order is not required, but it makes
life easier because following a strictly repeating pattern allows us to
skip inputting the names or numbers of each column of data directly
later during demographic data formatting.

To begin, we will create a **stageframe** for this dataset.
A stageframe is a data frame that describes all stages in the life
history of the organism, in a way usable by the functions in this
package and using stage names and classifications that completely match
those used in the dataset. It links the dataset and our analyses to our
life history model. In this case, the life history model is a life cycle
graph (Figure 8.1). This model is based on the life history model
provided in Ehrlén (2000), but we utilize
a different size classification based on the log leaf volume to make the
size distribution more closely match a symmetric and somewhat normal
distribution.

Figure 8.1. Life history model of *Lathyrus vernus* using the
log leaf volume as the size classification metric.

Our stageframe will include complete descriptions of all stages that occur in the dataset and in the life history model, with each stage defined uniquely. Since this object can be used for automated classification of individuals, all sizes, reproductive states, and other characteristics defining each stage in the dataset need to be accounted for explicitly. The final description of each stage occurring in the dataset must not completely overlap with any other stage also found in the dataset, although partial overlap is allowed and expected.

Before creating the stageframe, let’s explore the possible size variables. We will particularly look at summaries of the distribution of original and log sizes. Size here is given as leaf volume, and because this metric may have allometric relationships to vital rates, we will also look at log size.

```
summary(c(lathyrus$Volume88, lathyrus$Volume89, lathyrus$Volume90, lathyrus$Volume91))
#> Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
#> 1.8 14.7 123.0 484.2 732.5 7032.0 1248
summary(c(lathyrus$lnVol88, lathyrus$lnVol89, lathyrus$lnVol90, lathyrus$lnVol91))
#> Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
#> 0.600 2.700 4.800 4.777 6.600 8.900 1248
```

The upper summary shows the original size, while the lower line shows the size given in logarithmic terms. We should note the size minima and maxima, because we have been using 0 as the size of vegetatively dormant individuals. The lowest uncorrected size is 1.8, with a maximum of 7032. The minimum corrected size is 0.6, and the maximum corrected size is 8.9. Since the minimum corrected size is positive, and since the number of NAs has not increased (increased NAs would suggest that some unusable log sizes occur in the dataset), we are still able to use the log size value 0 as an indicator of vegetative dormancy. Note, however, that vegetative dormancy is currently included in the many NAs that occur in size variables in this dataset.

It can also help to take a look at plots of these distributions. We will plot raw and log volume.

```
par(mfrow=c(1,2))
plot(density(c(lathyrus$Volume88, lathyrus$Volume89, lathyrus$Volume90,
$Volume91), na.rm = TRUE), main = "", xlab = "Volume", bty = "n")
lathyrusplot(density(c(lathyrus$lnVol88, lathyrus$lnVol89, lathyrus$lnVol90,
$lnVol91), na.rm = TRUE), main = "", xlab = "Log volume", bty = "n") lathyrus
```

The raw volume distribution is highly skewed. This might cause difficulty if we used raw size untransformed and with a Gaussian distribution. A Gamma distribution might be justified. However, we will use the log volume here, which looks ‘better’ than the raw volume distribution in the sense that it is closer to some semblance of a Gaussian distribution, mostly through an increased level of symmetry. We will assume that log volume is Gaussian-distributed and that the mean bears no relationship to the variance.

We will now develop a stageframe that incorporates the log volume of
size. We will build this with vectors of the values describing each
stage, always in the same order. Note that we are using the midpoint
approach to determining the size bins here, using the default bin
halfwidths of 0.5 (since all size bins are using this default, we do not
include a `binhalfwidth`

vector option in the
`sf_create()`

input). However, we could have used the
`sizemin`

and `sizemax`

options to more
deliberately set the size bin minima and maxima instead.

```
<- c(0, 4.6, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9)
sizevector <- c("Sd", "Sdl", "Dorm", "Sz1nr", "Sz2nr", "Sz3nr", "Sz4nr",
stagevector "Sz5nr","Sz6nr", "Sz7nr", "Sz8nr", "Sz9nr", "Sz1r", "Sz2r", "Sz3r", "Sz4r",
"Sz5r", "Sz6r", "Sz7r", "Sz8r", "Sz9r")
<- c(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1)
repvector <- c(0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
obsvector <- c(0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
matvector <- c(1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
immvector <- c(1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)
propvector <- c(0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
indataset <- c(0, 4.6, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
binvec 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5)
<- c(0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)
minima <- c(NA, 1, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA,
maxima NA, NA, NA, NA, NA)
<- c("Dormant seed", "Seedling", "Dormant", "Size 1 Veg", "Size 2 Veg",
comments "Size 3 Veg", "Size 4 Veg", "Size 5 Veg", "Size 6 Veg", "Size 7 Veg",
"Size 8 Veg", "Size 9 Veg", "Size 1 Flo", "Size 2 Flo", "Size 3 Flo",
"Size 4 Flo", "Size 5 Flo", "Size 6 Flo", "Size 7 Flo", "Size 8 Flo",
"Size 9 Flo")
<- sf_create(sizes = sizevector, stagenames = stagevector,
lathframeln repstatus = repvector, obsstatus = obsvector, propstatus = propvector,
immstatus = immvector, matstatus = matvector, indataset = indataset,
binhalfwidth = binvec, minage = minima, maxage = maxima, comments = comments)
#lathframeln
```

Now we will standardize the dataset into historically-formatted vertical
(*hfv*) format. Here, we will use the `verticalize3()`

function, which creates historically-formatted vertical datasets from
horizontally structured raw data, as below. We will also replace NAs in
size and fecundity variables with zeros for `modelsearch`

to
work properly when we build models of vital rates, so we will now set
`NAas0 = TRUE`

. Some care needs to be taken with this last
step, since some authors give missing values extra meaning not present
in a value of zero. In our case, a missing value indicates that a plant
was dead (both size and fecundity are missing), was alive but not
sprouting (size was missing), or was alive but did not produce seed
(fecundity was missing). In all cases, these NA values may be replaced
by 0, because other variables indicate those conditions.

We also have two choices for use as our reproductive status and
fecundity variables. The first choice, `FCODE88`

indicates
whether a plant flowered. The second choice, `Intactseed88`

,
indicates the number of seed produced. Flowering plants typically have
different demographic characteristics than non-flowering plants, either
reflecting reproductive costs, or, conversely, because flowering plants
might have more resources and hence higher survival than non-flowering
plants. We should separate transitions among these two groups. So, let’s
use `FCODE88`

to indicate reproductive status, and
`Intactseed88`

to indicate fecundity.

Finally, note that in the input to the following function, we utilize a
strictly 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
in the dataset. The data management functions that we have created for
`lefko3`

do not require such repeating patterns, but they do
make the required input in the function much shorter and more succinct.
To see a detailed summary of the resulting hfv dataset, remove
`full = FALSE`

from the `summary_hfv()`

call. Note
that prior to standardizing the dataset, we will create a new variable
to code individual identity, since different plants in different
subpopulations use the same identifiers.

```
$indiv_id <- paste(lathyrus$SUBPLOT, lathyrus$GENET)
lathyrus
<- verticalize3(lathyrus, noyears = 4, firstyear = 1988,
lathvertln patchidcol = "SUBPLOT", individcol = "indiv_id", blocksize = 9,
juvcol = "Seedling1988", sizeacol = "lnVol88", repstracol = "FCODE88",
fecacol = "Intactseed88", deadacol = "Dead1988",
nonobsacol = "Dormant1988", stageassign = lathframeln,
stagesize = "sizea", censorcol = "Missing1988", censorkeep = NA,
NAas0 = TRUE, censorRepeat = TRUE, censor = TRUE)
summary_hfv(lathvertln, full = FALSE)
#>
#> This hfv dataset contains 2527 rows, 54 variables, 1 population,
#> 6 patches, 1053 individuals, and 3 time steps.
```

Before we move on to the next key steps in analysis, let’s take a closer look at fecundity. In this dataset, fecundity is mostly a count of intact seeds, and only differs in six cases where the seed output was estimated based on other models. To see this, try the following code.

```
# Length of fecundity vaiable in t:
length(lathvertln$feca2)
#> [1] 2527
# Number of non-integer entries:
length(which(lathvertln$feca2 != round(lathvertln$feca2)))
#> [1] 6
```

We see that we have quite a bit of fecundity data, and that it is overwhelmingly but not exclusively composed of integer values. The six non-integer values force us to make a decision - should we treat fecundity as a continuous variable, or round the values and treat it as a count variable? Here, we will round fecundity so that we can treat fecundity as a count variable in the analysis.

```
$feca3 <- round(lathvertln$feca3)
lathvertln$feca2 <- round(lathvertln$feca2)
lathvertln$feca1 <- round(lathvertln$feca1) lathvertln
```

Fecundity is now an integer variable, allowing us to use a count-based
distribution. `lefko3`

currently allows six choices of count
distributions: Poisson, negative binomial, zero-inflated Poisson,
zero-inflated negative binomial, zero-truncated Poisson, and
zero-truncated negative binomial. To assess which to use, we should
first assess whether the mean and variance of the count are equal using
a dispersion test. This test allows us to test whether the variance is
greater than that expected under our mean value for fecundity using a
chi-squared test. If it is not significantly different, then we may use
some variant of the Poisson distribution. If the data are overdispersed,
then we should use the negative binomial distribution. We should also
test whether the number of zeroes is more than expected under these
distributions, and make the distribution zero-inflated if so. Note that,
because we have not excluded zeros from fecundity using reproductive
status, we should not use a zero-truncated distribution.

We will use the `hfv_qc()`

function to assess the quality of
the data and decide on the correct distributions. We will use most of
the input from the `modelsearch()`

call that we will conduct
later.

```
hfv_qc(data = lathvertln, vitalrates = c("surv", "obs", "size", "repst", "fec"),
juvestimate = "Sdl", indiv = "individ", year = "year2", age = "obsage")
#> Survival:
#>
#> Data subset has 54 variables and 2246 transitions.
#>
#> Variable alive3 has 0 missing values.
#> Variable alive3 is a binomial variable.
#>
#>
#> Observation status:
#>
#> Data subset has 54 variables and 2121 transitions.
#>
#> Variable obsstatus3 has 0 missing values.
#> Variable obsstatus3 is a binomial variable.
#>
#>
#> Primary size:
#>
#> Data subset has 54 variables and 1916 transitions.
#>
#> Variable sizea3 has 0 missing values.
#> Variable sizea3 appears to be a floating point variable.
#> 1753 elements are not integers.
#> The minimum value of sizea3 is 1.2 and the maximum is 8.8.
#> The mean value of sizea3 is 5.099 and the variance is 3.093.
#> The value of the Shapiro-Wilk test of normality is 0.9551 with P = 7.719e-24.
#> Variable sizea3 differs significantly from a Gaussian distribution.
#>
#> Variable sizea3 is fully positive, lacking even 0s.
#>
#>
#> Reproductive status:
#>
#> Data subset has 54 variables and 1916 transitions.
#>
#> Variable repstatus3 has 0 missing values.
#> Variable repstatus3 is a binomial variable.
#>
#>
#> Fecundity:
#>
#> Data subset has 54 variables and 599 transitions.
#>
#> Variable feca2 has 0 missing values.
#> Variable feca2 appears to be an integer variable.
#>
#> Variable feca2 is fully non-negative.
#>
#> Overdispersion test:
#> Mean feca2 is 4.791
#> The variance in feca2 is 70.14
#> The probability of this dispersion level by chance assuming that
#> the true mean feca2 = variance in feca2,
#> and an alternative hypothesis of overdispersion, is 0
#> Variable feca2 is significantly overdispersed.
#>
#> Zero-inflation and truncation tests:
#> Mean lambda in feca2 is 0.008302
#> The actual number of 0s in feca2 is 334
#> The expected number of 0s in feca2 under the null hypothesis is 4.973
#> The probability of this deviation in 0s from expectation by chance is 0
#> Variable feca2 is significantly zero-inflated.
#>
#>
#> Juvenile survival:
#>
#> Data subset has 54 variables and 281 transitions.
#>
#> Variable alive3 has 0 missing values.
#> Variable alive3 is a binomial variable.
#>
#>
#> Juvenile observation status:
#>
#> Data subset has 54 variables and 210 transitions.
#>
#> Variable obsstatus3 has 0 missing values.
#> Variable obsstatus3 is a binomial variable.
#>
#>
#> Juvenile primary size:
#>
#> Data subset has 54 variables and 193 transitions.
#>
#> Variable sizea3 has 0 missing values.
#> Variable sizea3 appears to be a floating point variable.
#> 183 elements are not integers.
#> The minimum value of sizea3 is 0.7 and the maximum is 4.1.
#> The mean value of sizea3 is 2.307 and the variance is 0.2051.
#> The value of the Shapiro-Wilk test of normality is 0.9273 with P = 3.278e-08.
#> Variable sizea3 differs significantly from a Gaussian distribution.
#>
#> Variable sizea3 is fully positive, lacking even 0s.
#>
#>
#> Juvenile reproductive status:
#>
#> Data subset has 54 variables and 193 transitions.
#>
#> Variable repstatus3 has 0 missing values.
#> Variable repstatus3 is a binomial variable.
#>
#>
#> Juvenile maturity status:
#>
#> Data subset has 54 variables and 210 transitions.
#>
#> Variable matstatus3 has 0 missing values.
#> Variable matstatus3 is a binomial variable.
```

All of the probability variables look like binomials, so we have no problem there. Size and juvenile size appear to be continuous variables that differ significantly from the assumptions of the Gaussian distribution. However, we will still use a Gaussian distribution in this case, for instructional purposes. Fecundity is a count variable with significant overdispersion and significantly more zeros than expected, so we should use a zero-inflated negative binomial distribution.

Matrix estimation functions in package `lefko3`

are made to
work with **supplement tables**, which provide extra data
for matrix estimation that is not included in the main demographic
dataset. The `supplemental()`

function provides a means of
incorporating four additional 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 need to be used as proxies,
- reproductive multipliers indicating which stages lead to the production of which stages, and at what level relative to estimated fecundity, as well as multipliers for proxy transitions, 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.

Here, we will create an ahistorical supplemental table organizing some
of these sorts of data. Each row refers to a specific transition. The
first two of these transitions are set to specific probabilities, which
are the probabilities of germination and seed dormancy, estimated from a
separate study. The final two terms are fecundity multipliers, which
mark which transitions correspond to fecundity and provide information
on what multiple of fecundity estimated via linear modeling applies to
each case. Note that `type = 3`

multipliers cannot be
age-specific, while fecundity multipliers incorporated as
`type = 2`

may be set to specific ages.

```
<- supplemental(stage3 = c("Sd", "Sdl", "Sd", "Sdl"),
lathsupp2 stage2 = c("Sd", "Sd", "rep", "rep"), givenrate = c(0.345, 0.054, NA, NA),
multiplier = c(NA, NA, 0.345, 0.054), type = c(1, 1, 3, 3),
stageframe = lathframeln, historical = FALSE, agebased = TRUE)
#lathsupp2
```

Next we will run the `modelsearch`

function with the new
vertical dataset. This function will develop our best-fit vital rate
models for us. We provide descriptions of this function in the
documentation, in some of the other vignettes, and in *lefko3: a
gentle introduction*. Please see those materials for further
information.

Here, we will create two ahistorical model sets. The first will be a
model set for the entire population, without separating patches. The
second will include patch as a random factor, and will thus allow us to
explore patch dynamics as well as population dynamics. We will not
create a historical set this time because we are producing an age x
stage MPM only - `lefko3`

does not currently estimate or
support historical age x stage MPMs. Note the use of
`age = "obsage"`

to identify the correct variable name, and
`test.age = TRUE`

to tell R to include age in the global
models of vital rates.

```
<- modelsearch(lathvertln, historical = FALSE,
lathmodelsln2 approach = "mixed", suite = "main",
vitalrates = c("surv", "obs", "size", "repst", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "negbin",
indiv = "individ", year = "year2", age = "obsage", test.age = TRUE,
year.as.random = TRUE, patch.as.random = TRUE, show.model.tables = TRUE,
fec.zero = TRUE, quiet = "partial")
#>
#> Developing global model of survival probability...
#>
#> Global model of survival probability developed. Proceeding with model dredge...
#>
#> Developing global model of observation probability...
#>
#> Global model of observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of primary size...
#>
#> Global model of primary size developed. Proceeding with model dredge...
#>
#> Developing global model of reproduction probability...
#>
#> Global model of reproduction probability developed. Proceeding with model dredge...
#>
#> Developing global model of fecundity...
#>
#> Global model of fecundity developed. Proceeding with model dredge...
#>
#> Developing global model of juvenile survival probability...
#>
#> Global model of juvenile survival probability developed. Proceeding with model dredge...
#> Warning: Juvenile maturity status in time t+1 appears to be constant, and so will be set to constant.
#>
#> Developing global model of juvenile observation probability...
#>
#> Global model of juvenile observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of juvenile primary size...
#>
#> Could not properly estimate a global model for juvenile size.
#> Warning: Juvenile reproductive status in time t+1 appears to be constant, and so will be set to constant.
#>
#> Finished selecting best-fit models.
<- modelsearch(lathvertln, historical = FALSE,
lathmodelsln2p approach = "mixed", suite = "main",
vitalrates = c("surv", "obs", "size", "repst", "fec"), juvestimate = "Sdl",
bestfit = "AICc&k", sizedist = "gaussian", fecdist = "negbin",
indiv = "individ", patch = "patchid", year = "year2", age = "obsage",
test.age = TRUE, year.as.random = TRUE, patch.as.random = TRUE,
show.model.tables = TRUE, fec.zero = TRUE, quiet = "partial")
#>
#> Developing global model of survival probability...
#>
#> Global model of survival probability developed. Proceeding with model dredge...
#>
#> Developing global model of observation probability...
#>
#> Global model of observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of primary size...
#>
#> Global model of primary size developed. Proceeding with model dredge...
#>
#> Developing global model of reproduction probability...
#>
#> Global model of reproduction probability developed. Proceeding with model dredge...
#>
#> Developing global model of fecundity...
#>
#> Global model of fecundity developed. Proceeding with model dredge...
#>
#> Developing global model of juvenile survival probability...
#>
#> Global model of juvenile survival probability developed. Proceeding with model dredge...
#> Warning: Juvenile maturity status in time t+1 appears to be constant, and so will be set to constant.
#>
#> Developing global model of juvenile observation probability...
#>
#> Global model of juvenile observation probability developed. Proceeding with model dredge...
#>
#> Developing global model of juvenile primary size...
#>
#> Could not properly estimate a global model for juvenile size.
#> Warning: Juvenile reproductive status in time t+1 appears to be constant, and so will be set to constant.
#>
#> Finished selecting best-fit models.
#summary(lathmodelsln2)
#summary(lathmodelsln2p)
```

The output can be rather verbose, and so we have limited it with
`quiet = "partial"`

. The function was developed to provide
text marker posts of what the function is doing and what it has
accomplished, as well as to show all warnings from all workhorse
functions used. Because we used the `mixed`

approach here,
this includes warnings originating from estimating mixed linear models
with package `lme4`

(Bates et al.
2015) and, in the case of fecundity, the package
`glmmTMB`

(Brooks et al. 2017).
It also shows warnings originating from the `dredge()`

function of package `MuMIn`

(Bartoń
2014), which is the core function used in model building and AICc
estimation. We encourage users to get familiar with interpreting these
warnings and assessing the degree to which they impact their own
analyses.

A look at the summaries shows that the best-fit models vary in complexity. Age is important in size, reproductive status, and fecundity in both the population-only model set as well as the patch model set. For example, the conditional model for fecundity is influenced by size and age in the current year, as well as by patch, individual identity, and year, while observation status is not influenced by age. We can see these models explicitly, as well as the model tables developed, by calling them directly from the lefkoMod object. The accuracy of many of our models is high, but some are mid-range and juvenile size is abysmally low, meaning that it is likely that we have not included some important factors accounting for variability in at least some of our vital rates.

Next, we will estimate the ahistorical sets of matrices. We will match
the ahistorical age-by-stage matrix estimation function,
`aflefko2()`

, with the appropriate ahistorical input,
including the ahistorical lefkoMod objects `lathmodelsln2`

and `lathmodelsln2p`

. Model sets that include historical
terms should not be used to create ahistorical matrices, since the
coefficients in the best-fit models are estimated assuming a specific
model structure that either relies on historical terms or does not.
Historical vital rate models may yield biased results if used to
construct ahistorical matrices. Also note that `lefko3`

does
not currently allow the construction of historical age-by-stage MPMs. We
will assume a prebreeding model, and set the maximum age to 3, per our
dataset, but note that individuals may move past this age with
`continue = TRUE`

.

```
<- aflefko2(year = "all", stageframe = lathframeln,
lathmat2age modelsuite = lathmodelsln2, data = lathvertln, supplement = lathsupp2,
final_age = 3, continue = TRUE, reduce = FALSE)
#> Warning: Supplement table contains multiple entries for the same transition(s).
<- aflefko2(year = "all", patch = "all", stageframe = lathframeln,
lathmat2agep modelsuite = lathmodelsln2p, data = lathvertln, supplement = lathsupp2,
final_age = 3, continue = TRUE, reduce = FALSE)
#> Warning: Supplement table contains multiple entries for the same transition(s).
#summary(lathmat2age)
#summary(lathmat2agep)
```

The first model set led to the development of three matrices, reflecting
the four years of data. The second model set led to the development of
18 matrices, reflecting four years and six patches. The quality control
section gives us a sense of the amount of data used to model each vital
rate, and also shows us that the survival-transition (`U`

)
matrices are composed entirely of proper probabilities yielding stage
survival probability falling between 0.0 and 1.0. Matrix estimation
protocols can sometimes create spurious values, such as stage survival
greater than 1.0. Such values can occur for a variety of reasons, but
the most common is the inclusion through a supplement table of
externally-determined survival probabilities that are too high. Make
sure to check your matrix column sums each time you estimate MPMs to
prevent this problem. Survival greater than 1.0 can lead to strange
effects on metrics of population dynamics.

We can get a sense of what these matrices look like by visualizing them.
Let’s use the `image3()`

function to look at the first image.

```
image3(lathmat2age, used = 1)
#> [[1]]
```

The clear squares refer to zero elements, and the red elements refer to non-zero values corresponding to survival transitions and fecundity. The vast number of zeros may be surprising, but this matrix is a supermatrix organized by age first, with stage organizing within-age blocks. The first age is age 0, which cannot be adult, and age 1 corresponds to seedlings, leading to most non-zero elements in the adult portion. The adult block occurs from age 2, and this block can perpetuate indefinitely. The number of elements estimated is greater now than in the typical ahistorical MPM, because now we have added age as a major factor for analysis. This matrix is overwhelmingly composed of elements that must be 0, and so it is a rather sparse matrix ((1070.67 + 54) / 3969 = 28.3% of elements).

We can view the order of ages and stages using the
`agestages`

element of the lefkoMat object we produced, as
below. Note that our matrix is 63 rows by 63 columns, and this object
gives us the exact order used.

`#lathmat2age$agestages`

Now let’s estimate the element-wise arithmetic mean matrices. The first
`lefkoMat`

object created will include a single mean matrix,
while the second will include six patch-level mean matrices followed by
a grand mean matrix, yielding seven matrices (remember to look at the
`labels`

element of the output to see the exact order of
matrices).

```
<- lmean(lathmat2age)
lathmat2mean <- lmean(lathmat2agep)
lathmat2pmean #summary(lathmat2mean)
#summary(lathmat2pmean)
```

Now let’s estimate the deterministic population growth rate \(\lambda\), and the stochastic population growth rate, \(a = \text{log} \lambda _{S}\), in our age x stage MPM. We’ll plot them for comparison, making sure to take the natural exponent of the stochastic growth rate to make it comparable to \(\lambda\). We will show the patch means and the grand population mean.

```
<- lambda3(lathmat2age)
lambda2 <- lambda3(lathmat2mean)
lambda2m <- lambda3(lathmat2pmean)
lambda2mp set.seed(42)
<- slambda3(lathmat2age) #Stochastic growth rate
sl2 $expa <- exp(sl2$a)
sl2
par(mfrow = c(1,2))
plot(lambda ~ year2, data = lambda2, ylim = c(0.90, 1.02),xlab = "Year",
ylab = expression(lambda), type = "l", lty= 1, lwd = 2, bty = "n")
abline(a = lambda2m$lambda[1], b = 0, lty = 2, lwd= 2, col = "orangered")
abline(a = sl2$expa[1], b = 0, lty = 2, lwd= 3, col = "darkred")
legend("bottomleft", c("det annual", "det mean", "stochastic"), lty = c(1, 2, 2),
col = c("black", "orangered", "darkred"), lwd = c(2, 2, 3), bty = "n")
plot(lambda ~ patch, data = lambda2mp[1:6,], ylim = c(0.90, 1.02), xlab = "Patch",
ylab = expression(lambda), type = "l", lty= 1, lwd = 2, bty = "n")
abline(a = lambda2m$lambda[1], b = 0, lty = 2, lwd= 2, col = "orangered")
abline(a = sl2$expa[1], b = 0, lty = 2, lwd= 2, col = "darkred")
legend("bottomleft", c("patch det mean", "pop det mean", "pop sto"), lty = c(1, 2, 2),
col = c("black", "orangered", "darkred"), lwd = 2, bty = "n")
```

Deterministic and stochastic analyses suggest that the population and
all patches are declining, with all years and patches associated with
\(\lambda < 1\) and \(a = \text{log} \lambda _{S} < 0\).
Qualitatively, they agree with the \(\lambda\) and \(a
= \text{log} \lambda _{S}\) estimates from our other
*Lathyrus* vignettes.

Now let’s look at the stable stage distribution, using the population
matrices rather than the patch matrices. The output for the ahistorical
MPM is a data frame with matrix of origin, stage name, and stable stage
proportion in each row. Because the default output for the
`stablestage3()`

function will give us the stable
distribution of age-stages, we will collapse the output across age to
produce a true stable stage distribution.

```
<- stablestage3(lathmat2mean)
ehrlen2ss <- stablestage3(lathmat2age, stochastic = TRUE, seed = 42)
ehrlen2ss_s
<- cbind.data.frame(ehrlen2ss$ss_prop, ehrlen2ss_s$ss_prop)
ss_props names(ss_props) <- c("det", "sto")
rownames(ss_props) <- paste(ehrlen2ss$age, ehrlen2ss$stage)
<- apply(as.matrix(c(1:21)), 1, function(X) {
det_dist <- ss_props$det[X] + ss_props$det[21 + X] + ss_props$det[42 + X]
ss_sum return(ss_sum)
})<- apply(as.matrix(c(1:21)), 1, function(X) {
sto_dist <- ss_props$sto[X] + ss_props$sto[21 + X] + ss_props$sto[42 + X]
ss_sum return(ss_sum)
})
barplot(t(cbind.data.frame(det_dist, sto_dist)), beside = T, ylab = "Proportion",
xaxt = "n", ylim = c(0, 0.2), col = c("black", "red"), bty = "n")
text(cex=1, x=seq(from = 0.5, to = 3 * length(lathframeln$stage), by = 3),
y=-0.055, lathframeln$stage, xpd=TRUE, srt=45)
legend("topright", c("deterministic", "stochastic"), col = c("black", "red"),
pch = 15, bty = "n")
```

The stable stage distribution shows high levels of dormant seeds and mid-sized adults. Small-sized adults comprise the smallest part of the stable age-stage structure. Deterministic and stochastic analyses show more or less the same results.

Now let’s take look at the reproductive values. We’ll go straight to the plots, as with the stable stage distribution.

```
<- repvalue3(lathmat2mean)
ehrlen2rv <- repvalue3(lathmat2age, stochastic = TRUE, seed = 42)
ehrlen2rv_s
barplot(t(cbind.data.frame(ehrlen2rv$rep_value, ehrlen2rv_s$rep_value)),
beside = T, ylab = "Relative rep value", xlab = "Age-Stage", ylim = c(0, 1.5),
col = c("black", "red"), bty = "n")
text(cex=0.5, y = -0.06, x = seq(from = 0, to = 2.98*length(rownames(ss_props)),
by = 3), rownames(ss_props), xpd=TRUE, srt=45)
legend("topleft", c("deterministic", "stochastic"), col = c("black", "red"),
pch = 15, bty = "n")
```

We see reproductive values above 0 starting with dormant adults, and generally staying roughly equivalent with minor changes in ages 2 and 3. These patterns hold in both deterministic and stochastic analyses.

We will next assess to which matrix elements \(\lambda\) and \(\lambda_S\) are most sensitive to changes in via deterministic and stochastic sensitivity analysis.

```
<- sensitivity3(lathmat2mean)
lathmat2msens set.seed(42)
<- sensitivity3(lathmat2age, stochastic = TRUE)
lathmat2msens_s
# The highest deterministic sensitivity value:
max(lathmat2msens$ah_sensmats[[1]][which(lathmat2mean$A[[1]] > 0)])
#> [1] 0.1699623
# This value is associated with element: "
which(lathmat2msens$ah_sensmats[[1]] ==
max(lathmat2msens$ah_sensmats[[1]][which(lathmat2mean$A[[1]] > 0)]))
#> [1] 3762
# The highest stochastic sensitivity value:
max(lathmat2msens_s$ah_sensmats[[1]][which(lathmat2mean$A[[1]] > 0)])
#> [1] 0.4561818
# This value is associated with element:
which(as.matrix(lathmat2msens_s$ah_sensmats[[1]]) ==
max(as.matrix(lathmat2msens_s$ah_sensmats[[1]])[which(lathmat2mean$A[[1]] > 0)]))
#> [1] 3888
```

The highest deterministic value appears to be associated with the
transition from flowering size 6 adults in age 3 or higher to dormant
adult (element 3762, in column 45, row 60). The stochastic sensitivity
analysis suggests element 3888, which is associated with the transition
from three year old flowering size 8 adults to dormant adult (column 62,
row 45). Inspecting the sensitivity matrix (type
`lathmat2msens$ah_sensmats[[1]]`

to view the full
deterministic sensitivity matrix, or
`lathmat2msens_s$ah_sensmats[[1]]`

to view the full
stochastic sensitivity matrix) also shows that transitions near these
elements are associated with rather high sensitivities.

Let’s now assess the elasticity of \(\lambda\) or \(\lambda_S\) to matrix elements.

```
<- elasticity3(lathmat2mean)
lathmat2melas set.seed(42)
<- elasticity3(lathmat2age, stochastic = TRUE)
lathmat2melas_s
# The highest deterministic elasticity value:
max(lathmat2melas$ah_elasmats[[1]][which(lathmat2mean$A[[1]] > 0)])
#> [1] 0.03229501
# The largest determnistic elasticity is associated with element:
which(lathmat2melas$ah_elasmats[[1]] == max(lathmat2melas$ah_elasmats[[1]]))
#> [1] 3201
# The highest stochastic elasticity value:
max(lathmat2melas_s$ah_elasmats[[1]][which(lathmat2mean$A[[1]] > 0)])
#> [1] 0.2167045
# The largest stochastic elasticity is associated with element:
which(as.matrix(lathmat2melas_s$ah_elasmats[[1]]) == max(lathmat2melas_s$ah_elasmats[[1]]))
#> [1] 3905
```

The deterministic analysis shows the highest elasticity associated with element 3201 (col 51, row 51, stasis as a size 6 non-reproductive 3yr old or older adult), while the stochastic analysis show that population growth rate is most elastic to element 3905, which is in column 62 and row 62 (transition from three year old or older size 8 reproductive adult to the same stage).

Elasticity values can be treated as additive and sum to 1.0 within single matrices. This allows us to make interesting comparisons, such as to compare the elasticity of population growth rate to changes in transitions associated with specific stages. Let’s conduct such an analysis, and plot the results.

```
<- cbind.data.frame(colSums(lathmat2melas$ah_elasmats[[1]]),
elas_put_together ::colSums(lathmat2melas_s$ah_elasmats[[1]]))
Matrixnames(elas_put_together) <- c("det", "sto")
rownames(elas_put_together) <- apply(as.matrix(c(1:dim(lathmat2melas$agestages)[1])),
1, function(X) {
paste(lathmat2melas$agestages$stage[X], lathmat2melas$agestages$age[X])
})
barplot(t(elas_put_together), beside=T, names.arg = rep(NA,
length(rownames(elas_put_together))), ylab = "Elasticity", xlab = "Stage",
col = c("black", "orangered"), bty = "n")
text(cex=0.5, y = -0.007, x = seq(from = 0, to = 2.98*length(rownames(elas_put_together)),
by = 3), rownames(elas_put_together), xpd=TRUE, srt=45)
legend("topleft", c("deterministic", "stochastic"), col = c("black", "orangered"),
pch = 15, bty = "n")
```

We see in the analysis above that in the deterministic case, the population growth rate is most elastic to changes in flowering and non-flowering mid-sized older adults. In the stochastic case, the population growth rate is most elastic to changes in the oldest and largest flowering adults. So, the differences are quite notable, and we would infer that environmental stochasticity makes a big impact here.

Our estimated elasticities can also be summarized by transition type.
Let’s take a look at a plot, using the `summary.lefkoElas()`

function to help us assess these transition types.

```
<- summary(lathmat2melas)
lathmat2m_sums <- summary(lathmat2melas_s)
lathmat2m_s_sums
<- cbind.data.frame(lathmat2m_sums$ahist[,2],
elas_sums_together $ahist[,2])
lathmat2m_s_sumsnames(elas_sums_together) <- c("det", "sto")
rownames(elas_sums_together) <- lathmat2m_sums$ahist$category
barplot(t(elas_sums_together), beside=T, ylab = "Elasticity", xlab = "Transition",
col = c("black", "orangered"), bty = "n")
legend("topright", c("deterministic", "stochastic"), col = c("black", "orangered"),
pch = 15, bty = "n")
```

We see in the plot above that deterministic and stochastic analyses agree strongly about the importance and influence of transition types to population growth rate. Specifically, the population growth rate is most strongly elastic in response to changes in growth transitions, and almost completely inelastic to changes in fecundity. Shrinkage is almost as influential as growth, and stasis is almost as important in stochastic analysis.

In addition to sensitivity and elasticity analyses, we can use package
`lefko3`

to assess the actual impacts of demographic shifts
or differences on the population growth rate. Two tools for this purpose
are the life table response experiment (LTRE), and its stochastic
versions the stochastic life table response experiment (sLTRE) and the
small noise approximation LTRE (SNA-LTRE). All three are available via
the `ltre3()`

function. Below, we perform a standard
deterministic LTRE to assess the impact of space on the population
growth rate. This is done via a comparison of patch-level demography to
the grand mean matrix, which is the default comparison if no reference
matrices are provided. Remove the hashtag from the second line to see
the full structure of the resulting object. Particularly, you will
notice that the first element, `ltre_det`

, is a list of six
matrices. These matrices show the actual impact of the difference in
elements between the respective patch-level matrix and the reference
matrix (here, the grand population mean matrix) on the population growth
rate \(\lambda\). After this, we see
similar structure to the input `lefkoMat`

object, including
the stageframe and order of matrices.

```
<- ltre3(lathmat2pmean)
lathmat2m_ltre #> Warning: Matrices input as mats will also be used in reference matrix calculation.
#> Using all refmats matrices in reference matrix calculation.
#lathmat2m_ltre
```

The sLTRE produces output that is a bit different. Here, we see two
lists of matrices prior to the MPM metadata. The first,
`ltre_mean`

, is a list of matrices showing the impact of
differences in mean elements between the patch-level temporal mean
matrices and the reference temporal mean matrix. The second,
`ltre_sd`

, is a list of matrices showing the impact of
differences in the temporal standard deviation of each element between
the patch-level and reference matrix sets. In other words, while a
standard LTRE shows the impact of changes in matrix elements on \(\lambda\), the sLTRE shows the impacts of
changes in the mean and the variability of matrix elements on \(\text{log} \lambda\).

```
<- ltre3(lathmat2agep, stochastic = TRUE, rseed = 42)
lathmat2m_sltre #> Warning: Matrices input as mats will also be used in reference matrix calculation.
#> Using all refmats matrices in reference matrix calculation.
#lathmat2m_sltre
```

Let’s identify which age-stages exert the strongest impact on the population growth rate.

```
<- lathmat2m_ltre$cont_mean[[1]]
ltre_pos <- lathmat2m_ltre$cont_mean[[1]]
ltre_neg < 0)] <- 0
ltre_pos[(ltre_pos > 0)] <- 0
ltre_neg[(ltre_neg
<- lathmat2m_sltre$cont_mean[[1]]
sltre_meanpos <- lathmat2m_sltre$cont_mean[[1]]
sltre_meanneg < 0)] <- 0
sltre_meanpos[(sltre_meanpos > 0)] <- 0
sltre_meanneg[(sltre_meanneg
<- lathmat2m_sltre$cont_sd[[1]]
sltre_sdpos <- lathmat2m_sltre$cont_sd[[1]]
sltre_sdneg < 0)] <- 0
sltre_sdpos[(sltre_sdpos > 0)] <- 0
sltre_sdneg[(sltre_sdneg
<- cbind(Matrix::colSums(ltre_pos), Matrix::colSums(sltre_meanpos),
ltresums_pos ::colSums(sltre_sdpos))
Matrix<- cbind(Matrix::colSums(ltre_neg), Matrix::colSums(sltre_meanneg),
ltresums_neg ::colSums(sltre_sdneg))
Matrix
<- apply(as.matrix(c(1:length(lathmat2agep$agestages$stage))), 1,
ltre_as_names function(X) {
paste(lathmat2agep$agestages$stage[X], lathmat2agep$agestages$age[X])
})barplot(t(ltresums_pos), beside = T, col = c("black", "grey", "red"),
ylim = c(-0.010, 0.010))
barplot(t(ltresums_neg), beside = T, col = c("black", "grey", "red"), add = TRUE)
text(cex=0.5, y = -0.0095, x = seq(from = 0, to = 3.98*length(ltre_as_names),
by = 4), ltre_as_names, xpd=TRUE, srt=45)
legend("topleft", c("deterministic", "stochastic mean", "stochastic SD"),
col = c("black", "grey", "red"), pch = 15, bty = "n")
```

The output above shows that mid-sized non-flowering and flowering adults and seedlings exert strong influences on both \(\lambda\) and \(\text{log} \lambda\). This appears to be primarily through changes to the mean transition values, with only a small contribution due to changes in the variability of transition values in the stochastic case.

Finally, we will assess what transition types exert the greatest impact on population growth rate.

```
<- summary(lathmat2m_ltre)
lathmat2m_ltre_summary <- summary(lathmat2m_sltre)
lathmat2m_sltre_summary
<- cbind(lathmat2m_ltre_summary$ahist_mean$matrix1_pos,
ltresums_tpos $ahist_mean$matrix1_pos,
lathmat2m_sltre_summary$ahist_sd$matrix1_pos)
lathmat2m_sltre_summary<- cbind(lathmat2m_ltre_summary$ahist_mean$matrix1_neg,
ltresums_tneg $ahist_mean$matrix1_neg,
lathmat2m_sltre_summary$ahist_sd$matrix1_neg)
lathmat2m_sltre_summary
barplot(t(ltresums_tpos), beside = T, col = c("black", "grey", "red"),
ylim = c(-0.04, 0.04))
barplot(t(ltresums_tneg), beside = T, col = c("black", "grey", "red"),
add = TRUE)
abline(0, 0, lty = 3)
text(cex=0.85, y = -0.050, x = seq(from = 2,
to = 3.98*length(lathmat2m_ltre_summary$ahist_mean$category), by = 4),
$ahist_mean$category, xpd=TRUE, srt=45)
lathmat2m_ltre_summarylegend("topleft", c("deterministic", "stochastic mean", "stochastic SD"),
col = c("black", "grey", "red"), pch = 15, bty = "n")
```

The overall greatest impact on the population growth rate appears to come from growth and shrinkage transitions, with growth transitions having strongly negative influences in general.

LTREs and sLTREs are powerful tools, and more complex versions of both
analyses exist. Particularly view our other online resources for the
SNA-LTRE, and please consult Caswell
(2001) and Davison et al. (2013)
for more information. Users wishing to conduct these analyses should see
our free e-manual called ** lefko3: a gentle
introduction** and other vignettes on
the projects
page of the Shefferson lab website.

We are grateful 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.

Bates, Douglas, Martin Maechler, Ben Bolker, and Steve Walker. 2015.
“Fitting Linear Mixed-Effects Models Using *Lme4*.”
*Journal of Statistical Software* 67 (1): 1–48. https://doi.org/10.18637/jss.v067.i01.

Brooks, Mollie E., Kasper Kristensen, Koen J. van Benthem, Arni
Magnusson, Casper W. Berg, Anders Nielsen, Hans J. Skaug, Martin
Machler, and Benjamin M. Bolker. 2017. “glmmTMB Balances Speed and Flexibility Among
Packages for Zero-Inflated Generalized Linear Mixed Modeling.”
*The R Journal* 9 (2): 378–400. https://doi.org/10.32614/RJ-2017-066.

Caswell, Hal. 2001. *Matrix Population Models: Construction,
Analysis, and Interpretation*. Second edition. Sunderland,
Massachusetts, USA: Sinauer Associates, Inc.

Davison, Raziel, Florence Nicole, Hans Jacquemyn, and Shripad
Tuljapurkar. 2013. “Contributions of Covariance: Decomposing the
Components of Stochastic Population Growth in
*Cypripedium Calceolus*.” *The American
Naturalist* 181 (3): 410–20. https://doi.org/10.1086/669155.

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.