Environmental Toxicology and Chemistry, Vol. 34, No. 4, pp. 816–820, 2015 # 2015 SETAC Printed in the USA
Metal Mixture Modeling Evaluation APPLICATION OF A GENERALIZED LINEAR MIXED MODEL TO ANALYZE MIXTURE TOXICITY: SURVIVAL OF BROWN TROUT AFFECTED BY COPPER AND ZINC YUICHI IWASAKI*yz and STEPHEN F. BRINKMANx yDepartment of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado, USA zResearch Center for Life and Environmental Sciences, Toyo University, Oura, Gunma, Japan xColorado Parks and Wildlife, Fort Collins, Colorado, USA (Submitted 18 August 2014; Returned for Revision 13 November 2014; Accepted 16 December 2014) Abstract: Increased concerns about the toxicity of chemical mixtures have led to greater emphasis on analyzing the interactions among the mixture components based on observed effects. The authors applied a generalized linear mixed model (GLMM) to analyze survival of brown trout (Salmo trutta) acutely exposed to metal mixtures that contained copper and zinc. Compared with dominant conventional approaches based on an assumption of concentration addition and the concentration of a chemical that causes x% effect (ECx), the GLMM approach has 2 major advantages. First, binary response variables such as survival can be modeled without any transformations, and thus sample size can be taken into consideration. Second, the importance of the chemical interaction can be tested in a simple statistical manner. Through this application, the authors investigated whether the estimated concentration of the 2 metals binding to humic acid, which is assumed to be a proxy of nonspecific biotic ligand sites, provided a better prediction of survival effects than dissolved and free-ion concentrations of metals. The results suggest that the estimated concentration of metals binding to humic acid is a better predictor of survival effects, and thus the metal competition at the ligands could be an important mechanism responsible for effects of metal mixtures. Application of the GLMM (and the generalized linear model) presents an alternative or complementary approach to analyzing mixture toxicity. Environ Toxicol Chem 2015;34:816–820. # 2015 SETAC Keywords: Windermere humic aqueous model (WHAM) humic acid distribution Random effect
Concentration addition
Response addition
Binomial
Survival response is binary (i.e., dead or alive) and is commonly investigated in acute toxicity tests with fish or invertebrates (e.g., Daphnia magna). The effect on survival rate traditionally has been analyzed using nonlinear regression models that assume the normal (Gaussian) function as the error distribution [10,11]. However, such models have some statistical drawbacks (see Kerr and Meador [12] for more details). For example, the calculation of the proportion (the number of living organisms divided by the total number of organisms tested) loses information on the sample size (the number of organisms added in each chamber); 90 surviving organisms out of 100 organisms and 9 out of 10 are equal (0.9) in terms of proportion. In contrast, the generalized linear model (GLM) with a binomial distribution can analyze survival data while retaining the sample size information (using raw binary data instead of aggregated proportions) and thus can be used in an ecotoxicology context [12]. Moreover, the GLM has an appealing feature in its application to analysis of the mixture toxicity. Suppose we have toxicity data on survival of a species after exposure to each of 2 chemicals (M1, M2) and their mixtures. The expected survival rate (S) can be modeled using the binomial GLM with a logit link
INTRODUCTION
Predicting and analyzing the toxicity of chemical mixtures is a central but challenging topic in environmental toxicology [1,2]. Understanding critical toxic mechanisms and developing theoretically sound approaches for the prediction of mixture toxicity are ideal goals but are difficult to achieve. To date, 2 classical but sometimes arbitrarily applied methods, concentration addition (i.e., similar joint action) and response addition (i.e., independent joint action), have been adopted to categorize the major types of chemical interactions and to predict mixture toxicity [3]. In addition, based on toxicity tests with single chemicals and their mixtures, the extent of the observed mixture toxicity is typically categorized as additive, more than additive (synergistic), and less than additive (antagonistic) [4–6]. In testing chemical interactions, the concentration addition approach, which generally assumes similar modes of action and often requires parallel concentration–response relationships among the mixture components, is predominantly used [5–7]. A common procedure is to quantify concentration–response relationships for individual chemicals; estimate the concentration of each chemical that causes x% effect (ECx; often 50% is used); calculate the sum of toxic units by adding up the ratios of each chemical concentration in the mixture divided by its corresponding ECx; and, based on the sum of toxic units, compare predicted and observed effects to evaluate chemical interactions (see Laetz et al. [8] and Faust et al. [9] for more detailed examples).
S¼
1 1 þ expððb0 þ b1 M1 þ b2 M2 þ b3 M1 M2 ÞÞ
ð1Þ
and this equation can be expressed as
S 1S ¼ b0 þ b1 M1 þ b2 M2 þ b3 M1 M2
logit ðSÞ ¼ log All Supplemental Data may be found in the online version of this article. * Address correspondence to [email protected] Published online 18 December 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etc.2862
ð2Þ
where b0, b1, b2, and b3 are regression coefficients to be estimated. The term b3M1M2 corresponds to the influence of the chemical interaction, and the importance can be evaluated by 816
Generalized linear mixed model to analyze mixture toxicity
simply determining the statistical significance of b3 or by performing model selection based on information criteria such as Akaike’s information criterion (AIC; [13]). In addition to within-test variability, nonsimultaneous toxicity testing of the individual chemicals and their mixtures can increase variability (e.g., changes in initial conditions of test organisms) and result in invalid data interpretations [14] if not addressed properly. In the framework of GLM, the additional variability that affects the intercept of the concentration– response curve can conventionally be handled by including dummy variables in the model. For example, if toxicity tests were performed in n different time periods, n–1 dummy variables would be needed. However, because such additional variability is not usually the main research focus but instead is considered a nuisance factor, it is preferable to use a smaller number of variables to simplify the data handling and the resulting model. In this regard, a simple extension of the GLM called the generalized linear mixed model (GLMM) would be effective, because the GLMM requires only an additional parameter (called a random effect; see Methods for further explanation). The GLMMs rarely have been used in ecotoxicological studies [15], but are frequently used in ecology [16]. Simple introductions to GLMMs are available elsewhere [17]. The present study demonstrates an application of the GLMM to analyze survival data from toxicity tests using metal mixtures. We used available data on survival responses of brown trout (Salmo trutta) to copper and zinc [18]. Through this application, we test whether the concentration of metals binding to humic acid is a better predictor of metal effects on survival of brown trout than 2 other measures: dissolved and free-ion concentrations of metals. The concentration of metals binding to humic acid, estimated using a chemical speciation model (Windermere humic aqueous model [WHAM]), has been recently proposed as a new predictor of metal exposure [19]. This approach (referred to in the present study as the WHAM-HA approach, a term also used by Iwasaki et al. [20]) uses humic acid as a proxy of organism binding and has been applied to responses of river macroinvertebrates to metals and protons in field and microcosm settings [19–21] as well as to copper toxicity to duckweed (Lemna minor [22]). Evidence is accumulating for the use of the WHAM-HA approach to model the effects of metals on aquatic organisms [20,22–24], and the present study provides additional support for this approach. METHODS
Toxicity data
The data used in the present study were from acute toxicity tests conducted with brown trout exposed to zinc, copper, and their mixtures [18]. For those tests, brown trout eggs were obtained from the Colorado Division of Wildlife’s Fish Research Hatchery in Bellevue, Colorado (USA), and swimup-stage fry hatched from those eggs were used for the tests (6–9 wk old: average length and weight were 35.4 mm and 0.408 g, respectively). A total of 8 flowthrough toxicity tests were conducted at 5 different times (January to March 2002), including 4 single-metal tests and 4 mixture tests in which the target mass ratios of zinc to copper were 4:1, 8:1, 16:1, and 32:1 (Supplemental Data, Figure S1). These target ratios were selected to bracket the expected zinc:copper individual-metal median lethal concentration (LC50) ratio of 12:1. The concentration ranges of dissolved zinc and copper were
Metal Mixture Modeling Evaluation APPLICATION OF A GENERALIZED LINEAR MIXED MODEL TO ANALYZE MIXTURE TOXICITY: SURVIVAL OF BROWN TROUT AFFECTED BY COPPER AND ZINC YUICHI IWASAKI*yz and STEPHEN F. BRINKMANx yDepartment of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado, USA zResearch Center for Life and Environmental Sciences, Toyo University, Oura, Gunma, Japan xColorado Parks and Wildlife, Fort Collins, Colorado, USA (Submitted 18 August 2014; Returned for Revision 13 November 2014; Accepted 16 December 2014) Abstract: Increased concerns about the toxicity of chemical mixtures have led to greater emphasis on analyzing the interactions among the mixture components based on observed effects. The authors applied a generalized linear mixed model (GLMM) to analyze survival of brown trout (Salmo trutta) acutely exposed to metal mixtures that contained copper and zinc. Compared with dominant conventional approaches based on an assumption of concentration addition and the concentration of a chemical that causes x% effect (ECx), the GLMM approach has 2 major advantages. First, binary response variables such as survival can be modeled without any transformations, and thus sample size can be taken into consideration. Second, the importance of the chemical interaction can be tested in a simple statistical manner. Through this application, the authors investigated whether the estimated concentration of the 2 metals binding to humic acid, which is assumed to be a proxy of nonspecific biotic ligand sites, provided a better prediction of survival effects than dissolved and free-ion concentrations of metals. The results suggest that the estimated concentration of metals binding to humic acid is a better predictor of survival effects, and thus the metal competition at the ligands could be an important mechanism responsible for effects of metal mixtures. Application of the GLMM (and the generalized linear model) presents an alternative or complementary approach to analyzing mixture toxicity. Environ Toxicol Chem 2015;34:816–820. # 2015 SETAC Keywords: Windermere humic aqueous model (WHAM) humic acid distribution Random effect
Concentration addition
Response addition
Binomial
Survival response is binary (i.e., dead or alive) and is commonly investigated in acute toxicity tests with fish or invertebrates (e.g., Daphnia magna). The effect on survival rate traditionally has been analyzed using nonlinear regression models that assume the normal (Gaussian) function as the error distribution [10,11]. However, such models have some statistical drawbacks (see Kerr and Meador [12] for more details). For example, the calculation of the proportion (the number of living organisms divided by the total number of organisms tested) loses information on the sample size (the number of organisms added in each chamber); 90 surviving organisms out of 100 organisms and 9 out of 10 are equal (0.9) in terms of proportion. In contrast, the generalized linear model (GLM) with a binomial distribution can analyze survival data while retaining the sample size information (using raw binary data instead of aggregated proportions) and thus can be used in an ecotoxicology context [12]. Moreover, the GLM has an appealing feature in its application to analysis of the mixture toxicity. Suppose we have toxicity data on survival of a species after exposure to each of 2 chemicals (M1, M2) and their mixtures. The expected survival rate (S) can be modeled using the binomial GLM with a logit link
INTRODUCTION
Predicting and analyzing the toxicity of chemical mixtures is a central but challenging topic in environmental toxicology [1,2]. Understanding critical toxic mechanisms and developing theoretically sound approaches for the prediction of mixture toxicity are ideal goals but are difficult to achieve. To date, 2 classical but sometimes arbitrarily applied methods, concentration addition (i.e., similar joint action) and response addition (i.e., independent joint action), have been adopted to categorize the major types of chemical interactions and to predict mixture toxicity [3]. In addition, based on toxicity tests with single chemicals and their mixtures, the extent of the observed mixture toxicity is typically categorized as additive, more than additive (synergistic), and less than additive (antagonistic) [4–6]. In testing chemical interactions, the concentration addition approach, which generally assumes similar modes of action and often requires parallel concentration–response relationships among the mixture components, is predominantly used [5–7]. A common procedure is to quantify concentration–response relationships for individual chemicals; estimate the concentration of each chemical that causes x% effect (ECx; often 50% is used); calculate the sum of toxic units by adding up the ratios of each chemical concentration in the mixture divided by its corresponding ECx; and, based on the sum of toxic units, compare predicted and observed effects to evaluate chemical interactions (see Laetz et al. [8] and Faust et al. [9] for more detailed examples).
S¼
1 1 þ expððb0 þ b1 M1 þ b2 M2 þ b3 M1 M2 ÞÞ
ð1Þ
and this equation can be expressed as
S 1S ¼ b0 þ b1 M1 þ b2 M2 þ b3 M1 M2
logit ðSÞ ¼ log All Supplemental Data may be found in the online version of this article. * Address correspondence to [email protected] Published online 18 December 2015 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etc.2862
ð2Þ
where b0, b1, b2, and b3 are regression coefficients to be estimated. The term b3M1M2 corresponds to the influence of the chemical interaction, and the importance can be evaluated by 816
Generalized linear mixed model to analyze mixture toxicity
simply determining the statistical significance of b3 or by performing model selection based on information criteria such as Akaike’s information criterion (AIC; [13]). In addition to within-test variability, nonsimultaneous toxicity testing of the individual chemicals and their mixtures can increase variability (e.g., changes in initial conditions of test organisms) and result in invalid data interpretations [14] if not addressed properly. In the framework of GLM, the additional variability that affects the intercept of the concentration– response curve can conventionally be handled by including dummy variables in the model. For example, if toxicity tests were performed in n different time periods, n–1 dummy variables would be needed. However, because such additional variability is not usually the main research focus but instead is considered a nuisance factor, it is preferable to use a smaller number of variables to simplify the data handling and the resulting model. In this regard, a simple extension of the GLM called the generalized linear mixed model (GLMM) would be effective, because the GLMM requires only an additional parameter (called a random effect; see Methods for further explanation). The GLMMs rarely have been used in ecotoxicological studies [15], but are frequently used in ecology [16]. Simple introductions to GLMMs are available elsewhere [17]. The present study demonstrates an application of the GLMM to analyze survival data from toxicity tests using metal mixtures. We used available data on survival responses of brown trout (Salmo trutta) to copper and zinc [18]. Through this application, we test whether the concentration of metals binding to humic acid is a better predictor of metal effects on survival of brown trout than 2 other measures: dissolved and free-ion concentrations of metals. The concentration of metals binding to humic acid, estimated using a chemical speciation model (Windermere humic aqueous model [WHAM]), has been recently proposed as a new predictor of metal exposure [19]. This approach (referred to in the present study as the WHAM-HA approach, a term also used by Iwasaki et al. [20]) uses humic acid as a proxy of organism binding and has been applied to responses of river macroinvertebrates to metals and protons in field and microcosm settings [19–21] as well as to copper toxicity to duckweed (Lemna minor [22]). Evidence is accumulating for the use of the WHAM-HA approach to model the effects of metals on aquatic organisms [20,22–24], and the present study provides additional support for this approach. METHODS
Toxicity data
The data used in the present study were from acute toxicity tests conducted with brown trout exposed to zinc, copper, and their mixtures [18]. For those tests, brown trout eggs were obtained from the Colorado Division of Wildlife’s Fish Research Hatchery in Bellevue, Colorado (USA), and swimup-stage fry hatched from those eggs were used for the tests (6–9 wk old: average length and weight were 35.4 mm and 0.408 g, respectively). A total of 8 flowthrough toxicity tests were conducted at 5 different times (January to March 2002), including 4 single-metal tests and 4 mixture tests in which the target mass ratios of zinc to copper were 4:1, 8:1, 16:1, and 32:1 (Supplemental Data, Figure S1). These target ratios were selected to bracket the expected zinc:copper individual-metal median lethal concentration (LC50) ratio of 12:1. The concentration ranges of dissolved zinc and copper were
