3166 Environmental Toxicology and Chemistry, Vol. 24, No. 12, pp. 3166–3172, 2005  2005 SETAC Printed in the USA 0730-7268/05 $12.00 + .00 Hazard/Risk Assessment IMPROVED EMPIRICAL MODELS DESCRIBING HORMESIS NINA CEDERGREEN,*² C HRISTIAN RITZ,‡ and JENS CARL STREIBIG² ²Department of Agricultural Sciences, The Royal Veterinary and Agricultural University, Højbakkega ˚rd Alle ´ 13, DK-2630 Taastrup, Denmark ‡Department of Natural Sciences, The Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK-1871 Frederiksberg C, Denmark ( Received 7 January 2005; Accepted 27 May 2005) Abstract—During the past two decades, the phenomenon of hormesis has gained increased recognition. To promote research in hormesis, a sound statistical quantification of important parameters, such as the level and significance of the increase in response and the range of concentration where it occurs, is strongly needed. Here, we present an improved statistical model to describe hormetic dose–response curves and test for the presence of hormesis. Using the delta method and freely available software, any percentage effect dose or concentration can be derived with its associated standard errors. Likewise, the maximal response can be extracted and the growth stimulation calculated. The new model was tested on macrophyte data from multiple-species experiments and on laboratory data of Lemna minor. For the 51 curves tested, significant hormesis was detected in 18 curves, and for another 17 curves, the hormesis model described that data better than the logistic model did. The increase in response ranged from 5 to 109%. The growth stimulation occurred at an average dose somewhere between zero and concentrations corresponding to approx- imately 20 to 25% of the median effective concentration (EC50). Testing the same data with the hormesis model proposed by Brain and Cousens in 1989, we found no significant hormesis. Consequently, the new model is shown to be far more robust than previous models, both in terms of variation in data and in terms of describing hormetic effects ranging from small effects of a 10% increase in response up to effects of an almost 100% increase in response. Keywords—Hormesis Hormoligosis Dose–response model Statistical model Delta method INTRODUCTION During the past decade or two, the phenomenon of hormesis has gained increased recognition. Evidence is accumulating that stimulatory responses to low levels of stress probably are the rule rather than the exception [1–3]. Stimulations have been shown for various population growth parameters, such as in- dividual growth, longevity, and number of eggs produced in insects and crustacean [1,4–6], but physiological and bio- chemical parameters, such as gene expression, enzyme activ- ity, and tumor formation, have shown hormetic behavior as well [7–10]. Hormetic responses are found in all organisms, from animals and plants to microbiota. These responses are induced by low doses of organic or inorganic chemicals as well as by low doses of radiation [1–3]. The recognition of the phenomenon of hormesis has led to the need for robust statistical models to test for hormesis and to assess the dose at which maximal hormetic response occurs. This need is ex- pressed not only by scientists who want the most appropriate model to describe their data but also by risk assessors who are concerned with the application of hazardous doses of chem- icals [11–13]. Traditionally, dose–response models are based on strictly monotone functions, such as the Gompertz and Weibull func- tions or, as shown here, the logistic function, assuming ap- proximately normally distributed data: d - c y = c + (1) 1 + exp{b[ln(x) - ln(e)]} In the logistic model, d denotes the untreated control, and c is the response at infinite dose. The parameter e is the dose at which the value of d - c is reduced by 50% (ED50), and b is proportional to the slope around ED50. These traditional * To whom correspondence may be addressed (ncf@kvl.dk). functions either are strictly decreasing from a maximal control response at zero dose to a lower limit at infinite dose or are strictly increasing from no effect at zero dose to maximum effect at infinite dose, depending on whether the response or the effect is being assessed. These functions therefore cannot be used to model dose responses that exhibit initial response stimulation or effect minimization. Over the years, several attempts have been made to introduce low-dose stimulatory effects in dose–response models. One of the earliest was that of Brain and Cousens [14], who introduced the following mod- ification of the four-parameter logistic function: r + fx y = d + (2) 1 + exp{b[ln(x) - ln(e)]} The modification is the function fx, whereas r denotes the response range. This model also can be expressed directly in terms of parameters for the untreated control d and the lower limit c using the terminology described by Streibig et al. [15]: d - c + fx y = c + (3) 1 + exp{b[ln(x) - ln(e)]} Compared to the logistic model, the parameters d and c retain their interpretation as the untreated control and lower limit of the dose–response curve, whereas the parameter e is no longer the ED50 level and b loses its interpretation as relative slope at x = ED50. The parameter f measures the rate of growth stimulation at doses close to zero [14]. The actual size of f may be difficult to relate to the extent of hormesis, apart from the fact that the hormetic effect increases with increasing val- ues of f, as long as f is positive. Letting f = 0 in Equation 3 reduces the model of Brain and Cousens to the four-parameter logistic model, providing a convenient way of testing for hor- mesis using either a t or F test. A serious drawback of the Brain and Cousens model is that