Improving confidence limits for concentration-response models with quantal data
In ecotoxicology statistics there are three main types of data:
- Continuous
- Discrete
- Quantal / binary
Continuous data can take any numerical value over a range (e.g. weight), while discrete data take a finite number of values (e.g. colour). Quantal data, the focus of this post, can take only one of two values (e.g. alive/dead or mobile/immobile) and are often recorded as the number of affected individuals out of the total in each experimental unit.
Key Takeaways
- Quantal data represent binary outcomes and are commonly analysed using concentration–response models.
- The probit model is widely used because it constrains predictions between 0 and 100% and enforces monotonicity.
- Including a background (natural response) parameter, as recommended by the OECD, properly accounts for control mortality and uncertainty.
The previous post described the advantages of modelling the concentration–response relationship; here we describe one OECD-approved approach to modelling this relationship for quantal data.
Read our NOEC and LOEC blog here
The probit model for quantal data
A popular choice for modelling quantal concentration–response data in ecotoxicology is the probit model. This model is restricted to values between 0 and 1 (or 0–100%) and is monotone (non-decreasing or non-increasing, depending on the response).
The standard probit model has two parameters, similar to a linear model:
- a = intercept
- b = slope
Unlike the linear model, the probit model applies the cumulative distribution function (Φ) of the normal distribution to the linear predictor.
The log of the concentration (or the concentration itself) is typically used, implying a log-normal tolerance distribution for test organisms—an assumption supported by long-standing empirical experience.
Accounting for background response
The standard probit model assumes zero probability of response in the control (zero concentration) group. In ecotoxicology this is often unrealistic, as some natural response may occur.
Abbott’s correction adjusts for control response but assumes that the background rate is known with certainty. This is generally undesirable because the background response is estimated from the data and therefore uncertain.
The OECD instead recommends including a background (natural response) parameter directly in the model. This parameter (c) is estimated alongside the intercept and slope, using the control data explicitly. Because the model is nonlinear, all parameters influence one another.
Implications for ECX estimation
The ECX is the concentration associated with an x% response. When a background response is included, ECX refers to an x% response among the fraction of the population that did not respond at zero concentration.
Compared with Abbott’s correction, estimating the background response as part of the model typically leads to wider confidence intervals for ECX values. This reflects appropriate uncertainty in the background response and results in more reliable inference.
Although confidence-interval estimation for ECX values becomes slightly more complex when a background parameter is included, this added complexity leads to statistically sounder conclusions.
Reference
Organisation for Economic Co-operation and Development (OECD). 2006. Current Approaches in the Statistical Analysis of Ecotoxicity Data: A Guidance to Application. OECD Series on Testing and Assessment No. 54, ENV/JM/MONO(2006)18.
