Article pubs.acs.org/est
Are Chlorophyll a−Total Phosphorus Correlations Useful for Inference and Prediction? Craig A. Stow*,† and YoonKyung Cha‡ †
National Oceanic and Atmospheric Administration Great Lakes Environmental Research Laboratory, 4840 South State Road, Ann Arbor, Michigan 48108, United States ‡ Cooperative Institute for Limnology and Ecosystems Research, University of Michigan, Ann Arbor, Michigan 48108, United States ABSTRACT: Correlations between chlorophyll a and total phosphorus in freshwater ecosystems were first documented in the 1960s and have been used since then to infer phosphorus limitation, build simple models, and develop management targets. Often these correlations are considered indicative of a cause−effect relationship. However, many scientists regard the use of these associations for modeling and inference to be misleading due to their potentially spurious nature. Using data from Saginaw Bay, Lake Huron, we examine the relationship among chlorophyll a, total phosphorus, and algal biomass measurements. We apply graphical models and recently developed “structure learning” principles that use conditional dependencies to help identify causal relationships among observational data. The spurious relationship suspected by some is not supported by our data, whereas a direct relationship between chlorophyll a and total phosphorus is always supported, and an additional indirect relationship with an algal biomass intermediary is plausible under some circumstances. Thus, we conclude that these correlations are useful for simple model building but encourage the use of modern statistical methods to avoid common model-assumption violations.
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INTRODUCTION Empirical relationships between chlorophyll a and total phosphorus have been used for years to examine factors associated with aquatic ecosystem productivity and estimate simple models and have been proposed as a basis to develop numerical nutrient criteria.1 In early work, Sakamoto2 plotted data from several lakes to reveal an approximately linear relationship on a log−log scale between average chlorophyll concentration and average total phosphorus concentration. He inferred ranges of nutrient limitation based on the nitrogen/ phosphorus ratios of the lakes and the position of each lake relative to a line drawn among them on the plot. Similarly, Brydges3 showed a relationship between chlorophyll a and total phosphorus measurements in Lake Erie and inferred that phosphorus reductions would be accompanied by lower algal levels. Edmondson4 demonstrated a relationship between yearly values of chlorophyll a and total phosphorus in Lake Washington, while Megard5 showed a chlorophyll a−total phosphorus relationship among various locations in a lake with distinct sub-basins. Subsequently, Dillon and Rigler6 used these data, and measurements from other lakes, to produce the linear regression model:
example on which most ensuing models have been based. The apparent validity of this simple modeling approach for predictive purposes was further supported when Scavia and Chapra,7 using a more complex process-based model for Lake Ontario, demonstrated their model predictions were similar to those obtained from the empirically based Dillon and Rigler model. Subsequently, Nicholls and Dillon8 documented notable differences among published chlorophyll a−total phosphorus relationships, which they attributed to high intrinsic variability in cellular chlorophyll content, causing them to question the suitability of chlorophyll a as a surrogate for algal biomass. Although they did not explicitly examine the relationship between algal biovolume and chlorophyll a, they did compare plots of chlorophyll a versus total phosphorus and phytoplankton volume versus total phosphorus and concluded that cell volume provided a better relationship with total phosphorus than did chlorophyll a. Thus, they encouraged the use of biovolume rather than chlorophyll a as a measure of algal content. On the basis of this recommendation, Canfield et al.9 analyzed relationships among algal biomass, chlorophyll a, and total phosphorus in cross-sectional data from Florida lakes. They found the relationship between chlorophyll a and biomass to be approximately linear on a log−log scale, albeit noisy. In contrast to Nicholls and Dillon,8 Canfield et al.9 concluded that
log10[Chla] = − 1.136 + 1.449· log10[P] + ε
where Chla is the average summer chlorophyll a concentration (μg/L), P is the average spring total phosphorus concentration (μg/L), and ε is the model error term. They found that this model compared favorably with the data presented by Sakamoto,2 and their approach has become the canonical © 2013 American Chemical Society
Received: Revised: Accepted: Published: 3768
December 6, 2012 March 8, 2013 March 15, 2013 March 15, 2013 dx.doi.org/10.1021/es304997p | Environ. Sci. Technol. 2013, 47, 3768−3773
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Figure 1. Summary of structure learning procedures to examine total phosphorus (TP)−chlorophyll a (Chla)−algal biovolume relationships compatible with observed data. Each scenario was tested with differing numbers of bins. The symbols * and # indicate that the direction of the causal relationships TP → Chla and biovolume → Chla, respectively, were defined by the user.
“correlated” are nearly synonyms, correlation being a measure of the tendency of quantities to covary. However, the core of this rationale is that the correlation is spurious; phosphorus and chlorophyll a have a common antecedent, the phytoplankton in which they are bound. This reasoning is consistent with the adage: “correlation does not imply causation”; however, Shipley24 points out that a relationship between variables that is not coincidental does in fact imply an unresolved causal structure. Semantics may also underlie this apparent heresy; in logical reasoning, the word “imply” indicates sufficiency, whereas more casually “imply” means to suggest. Shipley means the latter; thus, the causal structure implied by the correlation between chlorophyll a and total phosphorus merits exploration. Because the chlorophyll a−total phosphorus correlation has been so widely observed, we start with the premise that it is not a coincidence. Therefore, the correlation can arise from three fundamental relationships: (1) total phosphorus causes chlorophyll a, (2) chlorophyll a causes total phosphorus, and (3) something else causes both chlorophyll a and total phosphorus. These possibilities are not mutually exclusive and do not exclude the possibility of indirect effects, with intervening factors occurring within the causal pathways. To clarify these associations, we present three graphical models depicting the alternative causal pathways that would produce a correlation between chlorophyll a and total phosphorus (Figure 1). Graphical models provide a compact representation of assumptions about relationships among variables and depict their joint probability structure. Further, these diagrams facilitate inference about the values of particular variables from observed values of other variables, using Bayes’s theorem.25 Figure 1a shows the spurious relationship constituting the argument that the inclusion of chlorophyll a and total phosphorus in algal cells produces the observed correlation between them. A third variable, phytoplankton biomass (or
chlorophyll a provided a better relationship with nutrient content than did algal biomass. Later studies have confirmed noisy and variable relationships between chlorophyll a and algal biomass (or biovolume), dependent on a number of factors,10−12 and other modeling efforts have also indicated a high residual variance in the relationship between algal biovolume and total phosphorus.13,14 Thus, because all of these relationships have a relatively high intrinsic variance and due to the relative ease of analyzing chlorophyll a versus quantifying algal biomass, investigators continue to report relationships between chlorophyll a and total phosphorus.15−18 While the limitations of applying models based on multilake data to predict outcomes in individual lakes have been acknowledged,19,20 generally, models based on empirical relationships have been regarded as predictive, although Reckhow21 cautioned that, without additional evidence, they should not be regarded as causal. Nonetheless, the chlorophyll a−total phosphorus correlation is often assumed to represent a cause−effect relationship.22 However, because chlorophyll a and total phosphorus are both bound within algal cells, some limnologists regard them as measurements of essentially the same thing. This outlook was captured by Lewis and Wurtsbaugh23 who asserted that the correlation is a tautology because, as “essential components of phytoplankton biomass”, “measurements of phosphorus and chlorophyll that are taken in a lake over the same span of time are not independent variables; there must always be a correlation between the two variables”. Because this perspective has implications for the veracity of considerable historical work, as well for the continued use of this relationship for model building and management decision making, it invites further scrutiny. From a probabilistic standpoint, the argument stated by Lewis and Wurtsbaugh23 is itself a tautology; variables that are not independent must be correlated. The tautology may be largely semantics; in probabilistic terms, “dependent” and 3769
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process of learning the model structure. For example, BN software programs require users to discretize continuous variables into multinomial forms; the location and number of bins created to discretize the variables can influence the resultant model structure. Alameddine et al.28 recommended creating bins of equal frequency (similar number of observations in each bin) over equal interval (similar range covered by each bin) for environmental data; because many environmental measurements have a lower bound of zero, they are generally right-skewed and often include outliers. Under these circumstances, equal interval binning results in too few data points in certain bins and a failure to reveal significant causal relationships between variables. Additionally, decreasing the significance level used for statistical testing results in fewer arrows in the model structure. Changing the number of bins also can influence the model structure identified by a structure learning algorithm. Alameddine et al.28 found that increasing the number of bins results in fewer arrows in the model structure and adds ambiguity (the arrow direction is indeterminate) to conditional dependence relationships. When structures are ambiguous, the necessary path condition (NPC) algorithm29 allows users to specify the direction of an arrow to yield plausible structures that are compatible with data, conditional on the user-specified arrow. To explore the likely causal relationships in our data, we used the Bayesian structure learning approach implemented in Hugin,30 a software that retrieves model structures compatible with observational data. We used the NPC algorithm with equal frequency binning while fixing the significance level at 0.05 and varying the number of bins from two to four (Table 1). Four was the maximum number of bins that provided informative probabilities, given the number of observations and conditional relationships among the variables.
biovolume), appears as the direct cause of both total phosphorus and chlorophyll a. Notably, the causal structure simulated by Lewis and Wurtsbaugh23 (their Figure 1) differs from this depiction. Phytoplankton biomass is not involved in their simulation; instead, total phosphorus values are stochastically generated as a function of initial chlorophyll a values, consistent with a depiction of chlorophyll concentration causing total phosphorus concentration (Figure 1b). Neither of these configurations portrays the concept that phosphorus is a necessary, often limiting, nutrient that causes algal growth, resulting in chlorophyll a concentrations that are directly proportional to algal biomass. This possibility, an expansion of the direct relationship between total phosphorus and chlorophyll a (Figure 1c), describes an indirect relationship with algal biomass as an intermediate variable (Figure 1d) and is also consistent with the argument that both chlorophyll a and total phosphorus are phytoplankton biomass components. We examine all these possibilities using principles developed by Pearl25 and Shipley24 to evaluate which structures are best supported by our data. Differentiating the plausible from the implausible structures provides further evidence toward resolving the question of the causal structure underlying the chlorophyll a−total phosphorus correlation and whether or not this relationship is a sound basis for prediction and inference.
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METHODS Data. Our data consist of chlorophyll a, total phosphorus, and total dissolved phosphorus measurements taken on 174 surface grab samples collected by the National Oceanic and Atmospheric Administration’s Great Lakes Environmental Research Laboratory (NOAA GLERL) from 16 monitoring locations in Saginaw Bay, Lake Huron. Samples were collected from May through October in 2008−2010 and analyzed in accordance with documented protocols.26 Algal biovolume was measured in 55 of the samples that were collected in 2009 and 2010. To calculate the ratio of bound phosphorus to the total phosphorus, we subtracted the total dissolved phosphorus concentration from the total phosphorus concentration and divided this difference by the total phosphorus concentration. Bayesian Network Structure Learning. Graphical models (Figure 1) can be effective tools to assess causal relationships among variables using observational data. A Bayesian network (BN) is a directed acyclic graphical model (a network of nodes and arrows in which there are no cycles) that identifies conditional dependencies to infer causal relationships.25,27 Pearl25 develops a rigorous mathematical formulation of causality based on graphical models in which any joint probability distribution of multiple variables can be represented as a directed acyclic graph. There may be more than one graph that is compatible with the observed joint distribution. If so, then these graphs are termed “observationally equivalent” and cannot be further distinguished based on uncontrolled observational data alone. Often, however, observational data can be used to eliminate models that are incompatible with the distribution implied by those data. In some cases, it is even possible to establish unambiguous causal dependencies (definitively directed arrows) among variables from patterns of probabilistic dependence. Numerous structure learning algorithms, based on the principles elucidated by Pearl,25 are available to examine potential causal pathways among observational data. Alameddine et al.28 discuss the sensitivity of structure learning algorithms to the parameters that must be set up in the
Table 1. Ranges and Cutoffs of Algal Biovolume (106 μm3/ L), Chla (μg/L), and TP (μg/L) Depending on the Number of Bins Used in the Structure Learning Analysisa minimum cutoff 2 bins 3 bins 4 bins maximum sample average a
biovolume
Chla
TP
38.6 448.9 327.5/661.9 219.4/448.9/876.3 2167.3 626.9
0.2 5.7 1.7/7.3 1.4/5.5/8.8 47.8 7.1
0.6 12.4 7.7/15.2 5.8/11.7/17.6 77.2 15.2
Overall sample averages are indicated in the bottom line.
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RESULTS The proportion of bound total phosphorus ranges 0.076− 0.999, with a sample mean and median of 0.736 and 0.777, respectively, and increases with chlorophyll a concentration (Figure 2). Algal biovolume measurements were available for 55 of the 174 samples; those 55 samples exhibit a pattern consistent with the larger set of observations. These results demonstrate that the notion that chlorophyll a and total phosphorus are “the same thing” may be supported only at relatively high chlorophyll a concentrations. A wide range of trophic conditions is reflected in these data; sample sites encompass a gradient that includes eutrophy near the mouth of the Saginaw River to oligotrophy near the main body of Lake Huron (Table 1). Biovolume, chlorophyll a, and total phosphorus are all positively correlated, although the 3770
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for a relationship that would be consistent with the argument that chlorophyll a and total phosphorus are essential algal components as well as a causal relationship originating with phosphorus. Interestingly, though, in some binning situations, a model with an indirect relationship between phosphorus and chlorophyll a, with biovolume intervening, and a direct relationship from total phosphorus to chlorophyll a was demonstrated by the data (Figure 1e). This model was not one we had anticipated but could plausibly arise because water clarity and consequently light penetration decrease at higher total phosphorus concentrations. Lower light availability has been shown to result in a community-level shift to algae with more chlorophyll a per unit biovolume so that the resident algae are more able to capture the available light.11 If this occurs, then there would be an indirect relationship, mediated by light, between total phosphorus and chlorophyll a that is independent of and in addition to the relationship between total phosphorus and biovolume. This possibility could also explain why the chlorophyll a−total phosphorus relationship is better determined than the chlorophyll a−biovolume relationship in some data sets.
Figure 2. Chlorophyll a vs the ratio of bound phosphorus to the total phosphorus. Filled circles depict observations for which algal biomass measurements were available, and empty circles depict observations for which biomass was not measured.
relationships are noisy and somewhat nonlinear (Figure 3a). Re-expression in the log metric linearizes the relationships, as the near coincidence of the locally weighted scatterplot smoothing (LOWESS) and least-squares regression lines indicates, but there is still scatter about the fitted lines (Figure 3b). Thus, these data exhibit patterns generally consistent with those that have been previously reported in the referred literature. In the structure learning exercise, a link between total phosphorus and chlorophyll a was always supported (Figure 1b,c); consequently, the spurious relationship with algal biovolume as the common antecedent for both total phosphorus and chlorophyll a was not corroborated by our data (Figure 1a). While the link between total phosphorus and chlorophyll a was always supported, with only two variables, the direction of the arrow cannot be discerned; thus, models 1b and 1c are observationally equivalent. Although these two possibilities cannot be differentiated by this approach, we cannot envision a causal mechanism in which chlorophyll a either directly or indirectly influences lake phosphorus content. The consistent support for a direct link between total phosphorus and chlorophyll a also means that model 1d was not confirmed by our data. This model was our a priori choice
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DISCUSSION
Our analysis does not support the presence of a lurking spurious relationship with algal biomass as the latent variable causing total phosphorus and chlorophyll a to covary (Figure 1). Thus, on the basis of our data, the argument that chlorophyll a and total phosphorus are correlated only via their contemporaneous algal cell residency seems unfounded. We cannot rule out the possibility of a spurious relationship with some other factor generating the observed association, but it is unclear what the alternative antecedent could be. Although our results do not prove causality, there is no scientific doubt that there is a causal relationship between phosphorus and algal biomass,31,32 and total phosphorus and chlorophyll a are well established surrogates for these entities. Thus, we posit that chlorophyll a−total phosphorus correlations are a reasonable basis to estimate predictive models that are consistent with a causal interpretation.
Figure 3. Scatterplot matrix and histograms of algal biovolume (106 μm3/L), Chla (μg/L), and TP (μg/L). A histogram of each variable is displayed on the diagonal panels. On each of the remaining plots, the relationship between each pair among the three variables is displayed. Empty circles denote the observations for the pair, and the curved line is derived from LOWESS. 3771
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on common characteristics such as landscape setting, ecoregion, or political boundaries. Soranno et al.48 point out that, for developing predictive models, it is better to group lakes based on the similarity of their responses to changes in the predictor variable rather than creating groups of lakes with similar features. Groups of lakes with similar features may encompass a narrow range of phosphorus or chlorophyll concentrations and produce a model with poorly identified intercept and slope parameters and high predictive uncertainty. Additionally, groups composed of lakes that differ in their responses to stressor-induced changes will generate models that reflect the average response of the lakes and perform badly for any specific lake. During the seminal work of the 1960s and 1970s and into the 1980s, computational limitations meant that estimating even a simple linear regression model could be tedious. However, modern methods make it straightforward to develop models accommodating previously intractable problems. Bayesian hierarchical models42,49 allow considerable flexibility in choosing groups because they allow pooling or partial pooling of information among lakes or groups of lakes. Similarly, software, such as WinBUGS50 and R make it possible to construct models that incorporate sample average data based on different numbers of observations or that have differing variances. Careful testing and application of models developed using these contemporary approaches should support improved inference and prediction into the future.
Further, our data demonstrate the presence of a considerable proportion of extracellular phosphorus, indicating that total phosphorus and chlorophyll a are not approximately redundant measures, except perhaps at relatively high chlorophyll a concentrations (Figure 2). Because total dissolved phosphorus does not measure sediment-associated, extra-cellular phosphorus, it is likely that these proportions somewhat overestimate the actual algal-bound phosphorus. Overestimation in these data is particularly likely at high chlorophyll a concentrations, which are from samples in or near the Saginaw River plume, where suspended sediment concentrations are relatively high. Thus, the high bound to total phosphorus ratio at high chlorophyll a concentrations may occur in part because extracellular phosphorus is strongly associated with nonalgal particulate matter, not because all of the phosphorus is contained within algal cells. Numerous factors have been hypothesized to influence the pattern and scatter of chlorophyll a−total phosphorus correlations including physical processes,33 zooplankton community composition,34−36 dreissenid mussels,37−39 nitrogen/phosphorus ratios,40,41 and landscape features.42 Consequently, in addition to having utility as predictive tools, the rigorous estimation of model parameters and their uncertainty and comparisons of alternative model forms43,44 provides a context for informative inference about the processes affecting lake primary productivity. However, there are methodological issues in the model estimation process that can strongly influence parameter and uncertainty estimation that have not been systematically addressed in the literature. For example, it is common to estimate chlorophyll a−total phosphorus models using data that have been averaged over time and/or space.45 Often, sample averages are used to aggregate multiple observations from single lakes, sub-basins, temporal periods, or some combination thereof. Implicit in this practice is the idea that these sample averages represent an underlying, but unknown, (and often unspecified), “true” population mean. However, using sample averages as surrogates for population means is inconsistent with two underlying regression model assumptions. The first assumption violation is that the predictor variable, in this case total phosphorus, is observed without error. Violating this assumption causes the slope estimator to be biased toward zero.46 In other words, it is likely that the regression slope estimate will be too low. The second violation is that the variance of the response variable, chlorophyll a, is the same throughout the range of the data. Using sample averages, particularly averages calculated from differing numbers of individual observations, violates this assumption and influences both the model error and parameter variance estimates. Thus, comparisons among models that are based on these variance estimates, including practices such as significance testing, can result in tenuous decisions. An additional problem occurs when sample averages are log-transformed. Because the log of a sample average is not mathematically equivalent, the underlying mean of the observations in the log metric,47 it becomes unclear exactly what the log-transformed sample average represents. While there is ample theory to address these problems, they are typically ignored, in part because they are not widely recognized by practitioners but also because until recently these complexities were not easy to accommodate using available statistical software. Another important consideration is the process used to classify lakes for model estimation. Often chlorophyll a−total phosphorus models are developed by first grouping lakes based
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS Pat Soranno, Ken Reckhow, Steve Carpenter, and Tara Stow reviewed early versions of this manuscript. This research was sponsored by a grant from the National Oceanic and Atmospheric Administration Center for Sponsored Coastal Ocean Research. GLERL contribution number 1655.
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