### Plot and census

The 50 ha plot on Barro Colorado Island was established in 1981 and fully censused in 1982 then every 5 years from 1985 through 2015. All stems ≥1 cm diameter-at-breast-height (dbh) were mapped, measured, and identified in each census (Hubbell and Foster 1983; Condit et al. 2012). Counting as individuals every genet, meaning genetic individuals with one or more stems, we have tagged a total of 423,617 individual trees. Though every stem is measured, here we only make use of the largest stem per individual, and growth rates are included only when the same stem was measured at the same height in consecutive censuses. Trees are counted dead when all stems died.

### Changing methods

Our goal is to document long-term trends in forest demography, and it is thus crucial we use comparable methods across all eight censuses. Starting in 1990, we settled on consistent methods and published them in detail (Condit 1998b). During the first two censuses, however, refinements were made where needed. We address here three concerns where improvements meant changing methods.

#### DBH precision in saplings

Stems < 55 mm dbh were measured with 5-mm precision in 1982 and 1985, and 1-mm precision since (all larger stems were always measured with 1-mm precision). Condit et al. (1993b) discuss in detail growth estimates associated with a shift in precision. Here we bypass the problem by using only stems 60–79 mm dbh to analyze sapling growth, because they were always measured in the same way.

#### DBH around buttresses

In 1982, dbh was recorded at 1.3 m above the ground even in trees with large buttresses, but starting in 1985, all measurements were above buttresses where the bole was regular. The height at which trees were measured in 1982 and 1985 was always recorded, and we omit growth records where the point-of-measure changed between censuses. Here, we minimize the impact of excluding many buttressed trees in 1982–1985 by using only trees 30–50 cm dbh across all census intervals to analyze canopy-level growth.

#### Census dates

Since 1985, the 50-ha census proceeded identically, always starting in the southwest and proceeding east, and always finishing in 10–11 months. As a result, the census interval for individuals never differed much from 5 years. But a much longer census was unavoidable in 1982 when all trees were tagged. One full pass lasted from March, 1981, until August, 1982, then, since a 50-ha census was much larger than any prior and we were concerned about missing trees, a second pass was done between August, 1982, and March, 1983. A few new individuals were added (< 5% of the total). This meant the 1982–1985 time interval for individual trees was as little as 2.5 years and as long as 4 years. It was unfortunate that the unusual drought of 1983 fell during this first census interval, but that could not be planned. We have carefully examined notes from the first census to be certain that we have the date on which each tree was observed alive and measured, and all growth rates and death rates are calculated from individual census dates.

### Climate and dry season

The Smithsonian Environmental Sciences Program maintains a weather station on Barro Colorado Island, and the Panama Canal Authority collects rainfall data throughout the canal watershed. With the two sources combined, we have daily records on Barro Colorado since 1929 (Engelbrecht et al. 2007). Potential evapotranspiration (PET) was estimated from a pan on the island since 1994, and the daily average was taken from 1994–2007 and assumed to hold every year. Throughout the entire period, daily water deficit was measured as observed rainfall minus the average potential evapotranspiration (PET) for the same date. A cumulative deficit was calculated as the sum of daily deficits over any set of consecutive days. Starting on 30 September each year (well before the dry season), the set of consecutive days for which the cumulative deficit reached its most extreme was identified, and that deficit defined the year’s dry season (Engelbrecht et al. 2007). Daily minimum and maximum temperatures are available from the Barro Colorado weather station since 1970.

### Demographic analyses

We wish to test how death rates, growth rates, and population sizes, for the entire stand and for every species, changed through time. For growth and death, we utilize the seven consecutive census intervals, and for population size, the eight censuses. A simple trend through time would be described using linear regression, where the demographic variable is the response and time the predictor. We wished, however, to expand the hypothesis to consider whether demographic traits followed two different trends, shifting in slope after the drought. To this end we employed piecewise regression, a tool producing an objective test of whether a linear regression slope shifts somewhere on the *x*-axis. The method simultaneously identifies the two trend slopes and the position on the *x*-axis where the shift occurs, that is, it identifies a model consisting of two lines connecting at any interior value of *x* that best fits the observations. Statistical tests are possible by calculating credible intervals on all slope parameters. When a credible interval did not overlap zero, we conclude there was a significant change through time, and when credible intervals from two phases did not overlap, we conclude that the two-phase model is a better fit to the data then a simple linear regression.

Piecewise regression with two phases is a four-parameter model describing the response of an independent variable *y*as a function of a single predictor *x*,

$$ \hat{y} = \left\{\begin{array}{ll} y_{b}+a_{1}(x-x_{b}) & x<x_{b} \\ y_{b}+a_{2}(x-x_{b}) & x \ge x_{b}, \end{array}\right. $$

(1)

where \(\hat {y}\) means the predicted *y*, *a*
_{1} is the slope of the response in phase 1, *a*
_{2} is the slope of the response in phase 2, *x*
_{
b
} is the breakpoint, the *x* at which the slope changes, and *y*
_{
b
} is \(\hat {y}\) at the breakpoint. In our model, *x* is time, and we defined it as the calendar year minus 1997 so that it was centered near zero. The time assigned an individual in one census interval was the mid-point between the two dates the tree was measured, expressed in years; this includes dead trees, which had an exact census date. For the purpose of graphing, observed or modeled demographic rates in a census interval were plotted at the mean of the mid-points for all individuals in the analysis.

#### Demographic measures

The response variables were demographic rates and abundances. Mortality was defined as the mortality rate constant, *m*, which is the negative of the logarithm of annual survival,

$$ m= \frac{\log(N_{0}-S_{t})}{t}, $$

(2)

where *S*
_{
t
} is the number of survivors at time *t* in a cohort that began (at *t*=0) with *N*
_{0} trees (Condit et al. 1999). The time interval *t* was calculated as the mean number of days between censuses of the *N*
_{0} trees, converted to a year. Growth rates were defined for each individual tree as dbh increment, or

$$ g= \frac{d_{t}-d_{0}}{t}, $$

(3)

where *d*
_{
t
} is dbh at time *t* (Condit et al. 1999). Abundance was defined simply as the total number of individual trees in 50 ha.

#### Data transformation

Modeling *g*is difficult statistically because of two associated features. First, the distribution of *g* is extremely right skewed, which routinely calls for logarithmic transformation. Unfortunately, however, we often observe *g*<0, caused by small errors in dbh but also falling bark or other stem damage. We know that error can account for many negative growth rates, because we did double-blind remeasurements of 4070 randomly selected trees. The SD of repeated measurements, *σ*
_{
e
}, was ∼1 mm in saplings, greater than mean sapling growth rate (*σ*
_{
e
} is described in more detail below), and it is thus not surprising that 21.1% of 66191 sapling measurements analyzed had *g*≤0. The simplest approach is to convert the non-positive rates to a number below the minimum observable growth of 1 mm in 5 years (Condit et al. 1993a, 2006). But with 21% of rates so converted, the resulting distribution of transformed *g* is highly sensitive to that minimum. Choosing 0.01, 0.05, or 0.1 mm ·*y*
^{−1} alters the skewness of log-transformed growth.

An alternative transformation for right-skewed distributions is by a power < 1, that is

$$ g_{t}= g^{\lambda}, $$

(4)

where *g*
_{
t
} is a transformed growth rate and *λ*<1 (Tukey 1957). The advantage of this method is that it works smoothly for negative growth rates as

$$ g_{t}(\lambda)= \left\{\begin{array}{ll} g^{\lambda} & g \ge 0\\ -\{(-g)^{\lambda}\} & g<0. \end{array}\right. $$

(5)

This is known as the modulus transformation (John and Draper 1980). It reigns in negative growth outliers in the same way that it does for positive growth. We found that in the range *λ*∈(0.3,0.6), transformed growth rates have low skewness, and median and mean are close. For any sample of growth increments, a *λ* can be located that minimizes skewness, but we sought one value that worked reasonably for all species and dbh categories. The main purpose is to reign in the big outliers that can cause peculiar model results, and *λ*=0.4 was satisfactory for both saplings and large trees.

All modeling was thus done with *g*
_{
t
}(0.4), but we prefer presenting the original growth rates *g* via back-transformation. Arithmetic means do not back-transform, that is

$$ \hat{g} \ne \widehat{g_{t}}^{1/\lambda} $$

(6)

where \(\hat g\) is the arithmetic mean. The same problem holds for log-transformations. But medians back-transform exactly, so the model results represent median dbh increments in mm ·*y*
^{−1} per dbh category. Due to high skewness, median growth is well below mean growth, even at a fixed dbh (Kenfack et al. 2014), and the median is arguably a better statistic to present, in the same way that summaries of income are often presented as medians, not means.

The modulus transformation allows negative growth rates to be included, but there were extreme errors, both negative and positive, that were excluded. Any growth rate amounting to a dbh increment > 75 mm ·*y*
^{−1} was discarded, since this was the highest rate observed with confidence in *Ochroma pyrimidale, Trema*spp*.,* and *Cecropia* spp., the fastest growing species in the forest. Negative outliers were also eliminated based on the measurement error we estimated from the 4070 double-blind remeasures, *σ*
_{
e
}, modeled as a function of dbh *d*: *σ*
_{
e
}=0.006214*d*+0.9036. Any stem whose later measurement fell <4*σ*
_{
e
} below the earlier measurement was considered an extreme error. These extremes happen either when a decimal is misplaced, or when the wrong tree is measured, the latter plausible when two stems are in contact. Details on the procedures for estimating error and filtering extreme growth rates are given in Condit (1998b) and Condit et al. (2004).

Population sizes were modeled after log-transformation. The mortality rate parameter is already a logarithm, and thus was not further transformed.

#### Diameter categories

To eliminate dbh as a predictor of mortality and growth and to assess how understory and canopy trees responded differently, all tests were done separately on several dbh categories. Since growth rate increases substantially with dbh, we used three narrow dbh categories for growth analyses: [6,8), [10,15), and [30,50) cm. Mortality rate varies only slightly with dbh, so two wider categories were employed: [1,10) and ≥10 cm. Abundance was also considered in the latter two categories. The dbh category of a tree was redefined at every census, meaning trees changed categories if they grew enough, and trees entering the census for the first time were always included. For mortality and growth analyses, dbh category was defined based on the earlier of the two censuses on which the calculation was based.

#### Species included

Over eight censuses, a total of 321 taxa were consistently identified. Species were included in analyses, however, only if they had a demographic measure in every census or census interval, meaning rare species were omitted when they were absent in one or more censuses. For some analyses, we report observed or fitted demographic rates in the more common species, those averaging at least 50 individuals alive over the eight censuses in a single dbh interval, and some graphs highlight the most abundant species in any category.

### Modeling demography through time

#### Multi-level model

The piecewise regression model was run in an hierarchical, or multi-level, framework: species within the community. The parameter estimates (see Eq. 1) for all the species were thus assumed to form a multi-dimensional Gaussian distribution, ie a hyper-distribution. Define *θ*
_{
k
}=(*a*
_{1},*a*
_{2},*x*
_{
b
},*y*
_{
b
}) as the set of four piecewise-regression parameters for species *k*, then our assumption is that \(\boldsymbol \theta _{k} \sim \text {Norm} \left (\text {Mean} = \hat {\boldsymbol \theta }, \text {SD}= \boldsymbol \sigma _{\theta } \right)\). The multivariate mean \(\hat {\boldsymbol \theta }\) and standard deviation *σ*
_{
θ
} are called hyperparameters. They describe the community-wide mean and SD of species responses. In the terminology of random-effect models, the hyperparameters are the fixed effects and species parameters are random effects (Gelman and Hill 2007).

We have used hierarchical models in analyzing species demography in the 50-ha plot for many years (Condit et al. 2006; Rüger et al. 2009, 2011a, b). It works well because the large number of rare species cannot be analyzed independently, but the hierarchical approach allows them to be included. The multi-level approach combines species responses, so each species is supported by the others, but still allows the response of species to differ. Gelman and Hill (2007) describe this as partial pooling, where complete pooling means analyzing the community as a whole, effectively ignoring species, while the opposite, no pooling at all, would be analyzing all species separately, requiring many rare species to be omitted. The latter was our approach in analyzing demography in papers prior to 2005 (Hubbell and Foster 1990; Condit et al. 1992a, 1995, 1996a, c, 2004). In a sense, in the multi-level model, the entire community serves as a prior for the response of any one species. For rare species, that prior is important and can dominate the result. But for species with ample data, the prior exerts little pull.

#### Model fitting

One model covered one demographic rate (mortality, growth, or abundance) across all species in a single dbh category. The following parameters and equations refer to a single size class, and subscripts identifying dbh category are supressed for easier presentation.

In order to fit parameters in a Bayesian framework, likelihood functions describing the probability of all observations given the model and any set of parameters must be defined. Write the piecewise regression function for species *k* compactly as *f*, so a predicted demographic parameter is

$$ \widehat{y_{k}} = f\left(x,\boldsymbol\theta_{k}\right), $$

(7)

with *θ*
_{
k
} the set of four regression parameters for species *k* and *x* is time (of a census or mid-point between consecutive censuses). The prediction, \(\widehat {y_{k}}\), refers to either the mortality parameter *m*, transformed growth *g*
_{
t
}, or logarithm of abundance.

For the growth and abundance models, the likelihood function for species *k* is

$$ \begin{aligned} L_{k} &= \sum_{i} \log \left[ \text{Norm} \left(y_{i},\text{Mean}=\widehat{y_{k}},\text{SD}=\epsilon \right) \right]\\ &\quad+ \log \left[ \text{Prob} \left(\widehat{y_{k}} \right) \right], \end{aligned} $$

(8)

where Norm means the Gaussian probability of one observation *y*
_{
i
} given the predicted \(\widehat {y_{k}}\) and a residual error, *ε*. For growth, the observation is a single tree of species *k* in one census interval and the summation is thus across all individuals in a census then across censuses. For abundance, the observation is the population size of species *k* in one census and the summation is across censuses. The residual requires another parameter, but just one more because we assume it is identical for all species, standard in multi-level models (Gelman and Hill 2007). The term \(\text {Prob} \big (\widehat {y_{i}} \big)\) is the hyper-probability of the predicted \(\widehat {y_{k}}\), defined below (Eq. 10).

The likelihood function for mortality differs because it is a binomial process, so

$$ \begin{aligned} L_{k} &= \log \Big[ \text{Binom} \big(S_{k},\text{Mean}=e^{-\widehat{y_{k}} t},\text{Size}=N_{k} \big) \Big]\\ &\quad+ \log \Big[ \text{Prob} \big(\widehat{y_{k}} \big) \Big], \end{aligned} $$

(9)

where *S*
_{
k
} is the number of survivors observed of *N*
_{
k
} individuals over a census interval of *t* years, in species *k*. Binom is the binomial probability of observing *S*
_{
k
} out of *N*
_{
k
} given mean survival \(e^{-\widehat {y_{k}} t}\). The residual *ε* is no longer needed.

Whether mortality, growth, or population size, the total log-likelihood of observations is the sum of all the species’ log-likelihoods, \(L = \sum _{k} L_{k}\). To accommodate the hierarchical aspect, another likelihood function must be written for the hyperparameters,

$${} \begin{aligned} H &= \log \left[\text{Prob} \left(\widehat{y_{k}} \right) \right]\\ &= \sum_{k} \left[ \log \left(\text{Norm} \left(\widehat{y_{k}},\text{Mean}=\hat{\boldsymbol\theta},\text{SD}=\boldsymbol\sigma_{\theta} \right) \right.\right]. \end{aligned} $$

(10)

Equation 10 covers the four regression parameters, because \(\hat {\boldsymbol \theta }\) is the vector of four means and *σ*
_{
θ
} the vector of four standard deviations (we ignored covariance among the parameters). The full likelihood of an entire set of model parameters combines the likelihood of the observations and the likelihood of the hyperparameters,

$$ \Theta \left(\boldsymbol\theta_{k},\hat{\boldsymbol\theta},\boldsymbol\sigma_{\theta},\epsilon \right)=H+L. $$

(11)

With *S* species, there are 4*S*+9 parameters needed for the full likelihood *Θ* in growth and abundance models, and 4*S*+8 (without *ε*) for mortality. Priors for all parameters were assumed to be non-informative. Every valid value was equally likely.

#### Parameter fitting

We used a Bayesian parameter-fitting method, sampling the posterior distributions by repeated Metropolis updates based on the full likelihood function (Eq. 11), generating Monte-Carlo Markov chains of each parameter. The MCMC chains represent full posterior distributions of all parameters and of other estimators derived from parameters. Chains were run 12000 steps for each model, examined visually for mixing, and the initial 2000 steps discarded as burn-in. We report the mean of post-burn-in chains as best estimates, and quartiles 0.025 and 0.975 for 95% credible intervals. Hypotheses were tested by checking whether credible intervals of one slope parameter overlapped zero, or whether credible intervals of two slope parameters for one species overlapped each other.

### Drought impact score

As in Newbery and Lingenfelder (2009), we estimated a drought impact score for every species using its estimated growth and mortality rates. Fitted parameters of the piecewise regression model were plugged in Eq. 1 for every species and every diameter category to estimate the demographic rates in the first census interval (*x*=−14, which is year 1983 and during the drought) and the third interval (*x*=−5, after the drought). The difference in the estimated rates, early minus late, indicates drought impact. With mortality in two dbh categories and growth in three, there were five measures per species. We checked correlation among the measures, and then tested whether drought impact scores were significant predictors of population performance both during and after the drought, also estimated from piecewise regression.