Study site
The dropwort population is located in a single Delmarva bay on the Eastern Shore of Maryland near the Delaware state line at a preserve managed by The Nature Conservancy (TNC) (Fig. 1). This preserve was acquired in 1983, although monitoring of the dropwort began in 1982, with 36 individual stems (Boone et al. 1984). The location of this preserve is not shared with the public, thus trampling and other disturbances are kept to a minimum.
Soils at this location consist of Corsica mucky loam, with 0–2% slope (Soil Survey Staff, Natural Resources Conservation Service, United States Department of Agriculture. Web Soil Survey. Available online at http://websoilsurvey.nrcs.usda.gov/ accessed January 19, 2022). This is consistent with other dropwort sites, which tend to be sandy loam or loam soil with a medium to high organic content, high water table, and deep, poorly drained and acidic soils underlain by a clay layer (NatureServe 2022).
Propagation and girdling
In 1989, an attempt was made to artificially increase the dropwort population in Maryland by removing two of four remaining plants and propagating them at the North Carolina Botanical Garden. Sixty cultivated plants were subsequently re-introduced to the site in 1992 in 13 locations in the wetland and monitored annually. Transplants declined every year until no transplants were observed in 1999. Thus, propagation was not found to be a viable tool for dropwort protection.
When the preserve was acquired in 1983, woody vegetation encroachment by red maple (Acer rubrum) and sweet gum (Liquidambar styraciflua) was substantial. The trees were shading out herbaceous vegetation, with portions of the wetland completely under a tree canopy. Additionally, there was concern about increased evapotranspiration as the trees increased in size, lowering the water levels in the wetland. In 2002, the majority of trees within the wetland were removed or girdled. Cut stumps and girdled trees were treated with 50% glyphosate (cutting phase). Re-sprouting seedlings were manually removed in subsequent years.
Intervention with prescribed fire
It was determined from multiple sources (Kirkman 1995; Chafin 2007) that a controlled burn might be the most effective means of maintaining the open character of this wetland. Additionally, the native grass Panicum hemitomon had spread substantially following woody vegetation removal (Fig. 1a). This grass was encroaching on the dropwort area, creating a thick layer of thatch and possibly preventing dropwort seeds from landing on a suitable substrate for germination. Planning for a burn was problematic, however, as the sensitive nature of the wetland made the creation of standard fire breaks undesirable. Thus, a burn prescription was prepared that allowed for fire to burn off the vegetation in the open wetland but not spread into the surrounding shaded woods, utilizing the fuel type transition as a natural firebreak.
Prescribed fires were successfully implemented in October 2015 and 2017 after the dropwort had senesced. The burn prescription required that the relative humidity and fuel moisture be high enough that fire would not carry into the woods, thus only burning the herbaceous vegetation and other fine fuels within the pond where the dropwort was present. On the day of each burn, a test fire was ignited in the woods and observed. If it self-extinguished, it was determined within the prescription and the burn proceeded. Each burn was conducted at midday and lasted approximately 1 h (burn #1: 23C, winds S 8MPH, RH 52%, burn #2: 28C, winds N 6MPH, RH 58%). Immediately post-burn, the dropwort site was inspected; on each occasion, above-ground vegetation was consumed by fire up to the base of the dropwort stems, where soil was still damp (Fig. 1b, c).
Monitoring
Monitoring was conducted in mid-August when the dropwort was in bloom (Fig. 1a), making identification of individual stems easier. This study analyzes the results of plant counts conducted from 1982 to 2021. Initial monitoring consisted of counting each individual dropwort stem, measuring its height, and recording whether it was in bloom, fruit, or vegetative. However, due to the high number of stems after 2016, it was no longer possible to count non-flowering stems without damaging the population; thus, only flowering stems were counted from 2017 to 2021.
Precipitation and temperature data
Data from nearby weather stations was obtained from the National Climate Data Center Archive (NCDC 2022). Monthly summaries of total precipitation were divided into fall, winter, spring, and summer periods and analyzed for correlations with stem count data. The Palmer Drought Severity Index (PDSI) and Standardized Precipitation-Evapotranspiration Index (SPEI) values were obtained from the North American Drought Atlas (Cook et al. 2010; Beguería et al. 2014).
After we determined important factors and confounders for the dropwort data, we constructed a directed acyclic graph (DAG, Fig. 2). A DAG is a visual representation of the study subject based on the structural causal model framework (Pearl 2009) that makes assumptions explicit and reduces bias for causal inference from observational data (Arif and Aaron MacNeil 2022). Two different multivariable model selection processes were used to account for confounders shown by the DAG, test for interactions and predict a counterfactual.
Model selection
An interrupted time series model with first-order autocorrelation (ITSA, (Natesan 2019)) was determined to be best suited for analyzing our dataset. The data are time series, taking place over a 40-year period of time with repeated annual measurements. The data were assumed to be autocorrelated because dropwort is a perennial; the number of plant stems in a given year has an impact on the number of stems the following year. The ITSA model type is appropriate because the interruptions in the time series are the introduction of interventions such as cutting and prescribed fire. The dependent variable was the stem count of dropwort. Bayesian estimation was used to provide more information about the parameters in the form of a posterior distribution. The posterior distribution contains all probable values of the parameter and therefore the credible interval estimates obtained are straightforward to interpret. More details about the methodology specifically for this kind of data can be found in Natesan (2019) and Natesan Batley (in press). The Bayesian rate ratio (BRR) effect size proposed in Natesan Batley et al. (2021) was computed to measure the magnitude of the effect of each intervention on stem count. Therefore, two BRRs were estimated. BRR 1 measured the ratio of the rate of stem count increase between the cutting phase and the baseline phase where no treatment was applied. BRR 2 measured the ratio of the rate of stem count increase between the fire phase and the cutting phase. Larger BRR values would indicate that there is a higher stem count increase in the phase under consideration than the previous phase. The method of calculating this ratio is given in the following equations:
Let θjt be the expected number of stems counted at time t in phase j where for the baseline phase, j = 0. The number of stem counts in j at time t is given as \({Y}_{jt}\sim Pois\left({\hat{\theta}}_{jt}\right)\).
$$\widehat\theta{}_{jt}=\left\{\begin{array}{l}\quad\quad\quad\quad\;\;\theta_{jt},\;if\;t=1\\\theta_{jt}\;(1-\rho)+\rho Y_{j(t-1)},\;otherwise\end{array}\right..$$
(1)
If λj is the stem count rate in j, and ρ the autocorrelation,
$$\theta_{jt}=\left\{\begin{array}{l}\quad\quad\quad\quad\;\;\lambda_je_j,\;if\;t=1\\\lambda_je_{j\;}(1-\rho)+\rho Y_{j(t-1)},\;otherwise\end{array}.\right.$$
(2)
In Eqs. 1 and 2, autocorrelation is applied within a given phase only past the first time point. The number of events at the proceeding time points depends on the number of events at the previous time point. The stem count rate λj is given as
$$\lambda_j=\xi\exp\left(\omega j\right),$$
(3)
where ξ is the stem count rate when j = 0 and exp(ω) is the rate ratio effect size (Kunz et al. 2015). The rate ratio effect size is computed as,
$$\exp\left(\omega\right)=\frac{\lambda_{1}}{\lambda_{0}}\;=\frac{Y_{1}/e_{1}}{Y_{0}/e_{0}}$$
(4)
When using predictor variables, Eq. 1 was expanded as:
$$\widehat\theta{}_{jt}=\left\{\begin{array}{l}\quad\quad\quad\quad\quad\quad\quad\;\;\quad\theta_{jt}+\sum\nolimits_{K=1}^K\;b_k\gamma_k,\;if\;t=1\\(\theta_{jt}+b_0(t-1)+\sum\nolimits_{K=1}^K\;b_k\gamma_k\;(1-\rho)+\rho Y_{j(t-1)},\;otherwise\end{array}\right.$$
(5)
In Eq. 5, γk refers to the kth effect with a total of K one-way and two-way interaction effects.
The priors that were used are specified below:
$${\displaystyle \begin{array}{c}{\omega}_j\sim gamma\left(.1,.1\right)\\ {}{\lambda}_0\sim unif\left(0,20\right)\\ {}{\lambda}_j={\lambda}_{\left(\textrm{j}-1\right)}\mathit{{e}^{\omega_j}},\;where\;j=1,\dots \\ {}\rho \sim unif\left(-.3,1\right)\\ {}{b}_k\sim \mathit{\operatorname{norm}}\left(0,3\right),\;where\;k=0,\dots,\;K+1\end{array}}$$
Four models were fitted to the time series using a Poisson distribution. Poisson distribution was used to model the dependent variable due to the nature of the count data. Model 1 is a simple ITSA model estimated without any slopes, that is, time (year) was not included in the model. The interruption in the ITSA model is change in phase. Model 2 is an ITSA model with time included as a predictor. Each phase was allowed to have its own intercept and slope. In model 3, relevant predictor variables were added to model 2. These variables included October precipitation, October temperature, March temperature, the minimum temperature in March, previous year’s PDSI, the interaction between all these variables and phase indicator (i.e., cutting phase, fire phase), and the interaction between October temperature and October precipitation. For model 4, variables that were not statistically significant were removed from model 3 to form a more parsimonious model. Deviance information criterion (DIC) was used to retain the models. Similar to the Akaike information criterion (AIC), lower values of DIC are used to select models which have better fit and fewer effective parameters (Spiegelhalter et al. 2002; Spiegelhalter et al. 2014).
A plant condition index was calculated using the first year of the counts as the baseline by dividing each subsequent year by the value for the first year (1982 = 1 in the index). The Bayesian rate ratio from the previously described ITSA model was used to pick the most important intervention phase (reintroduction of fire). Using the R statistical software (version 4.2.1, R core team 2022), four additional models were fitted, using the BSTS package (Scott and Varian 2013). Model 5 is a state space that includes auto-regression (AR), whereas model 6 includes AR and a student linear local trend. The state space for model 7 included AR and dynamic regression on covariates (October precipitation, October temperature, SPEI, and minimum March temperature). Model 8 state space included AR and covariates as model terms (October precipitation, October temperature, SPEI, and minimum March temperature).
The BSTS package was used to compare one ahead errors through each time step for all four models. The model with the lowest overall error (model 8) was then used in the CausalImpact package to predict a counterfactual for comparison to the actual data after the intervention (Brodersen et al. 2015).