Commission Regulation (EC) No 761/2009Show full title

Regression analysis

Models

A regression analysis aims at quantitatively describing the concentration-response curve in the form of a mathematical regression function Y = f (C) or more frequently F (Z) where Z = log C. Used inversely C = f^–1 (Y) allows the calculation of EC_x figures including the EC₅₀, EC₁₀ and EC₂₀, and their 95 % confidence limits. Several simple mathematical functional forms have proved to successfully describe concentration-response relationships obtained in algal growth inhibition tests. Functions include, for instance, the logistic equation, the non-symmetrical Weibul equation and the log normal distribution function, which are all sigmoid curves asymptotically approaching one for C → 0, and zero for C → infinity.

The use of continuous threshold function models (e.g. the Koyman model ‘for inhibition of population growth’ Kooijman et al. 1996) is a recently proposed or alternative to asymptotic models. This model assumes no effects at concentrations below a certain threshold, EC0+, that is estimated by extrapolation of the response concentration relationship to intercept the concentration axis using a simple continuous function that is not differentiable in the starting point.

Note that the analysis can be a simple minimisation of sums of residual squares (assuming constant variance) or weighted squares if variance heterogeneity is compensated.

Procedure

The procedure can be outlined as follows: select an appropriate functional equation, Y = f (C), and fit it to the data by non-linear regression. Preferably use the measurements from each individual flask rather than the mean values of the replicates, in order to extract as much information from the data as possible. If the variance is high, on the other hand, practical experience suggests that the mean values of the replicates may provide a more robust mathematical estimation, less influenced by systematic errors in the data, than with each individual data point retained.

Plot the fitted curve and the measured data and examine whether the curve fit is appropriate. Analysis of residuals may be a particularly helpful tool for this purpose. If the chosen functional relationship to fit the concentration response does not describe the whole curve or some essential part of it, such as the response at low concentrations well, choose another curve fit option — e.g. a non-symmetrical curve like the Weibul function, instead of a symmetrical one. Negative inhibitions may be a problem with, for instance, the log-normal distribution function, likewise demanding an alternative regression function. It is not recommended to assign a zero or a small positive value to such negative values because this distorts the error distribution. It may be appropriate to make separate curve fits on parts of the curve such as the low inhibition part to estimate EC_{low x} figures. Calculate from the fitted equation (by ‘inverse estimation’, C = f^–1 (Y), characteristic point estimates EC_x’s, and report as a minimum the EC₅₀ and one or two EC_{low x} estimates. Experience from practical testing has shown that the precision of the algal test normally allows a reasonably accurate estimation at the 10 % inhibition level if data points are sufficient — unless stimulation occurs at low concentrations as a confounding factor. The precision of an EC₂₀ estimate is often considerably better than that of an EC₁₀, because the EC₂₀ is usually positioned on the approximately linear part of the central concentration response curve. Sometimes EC₁₀ can be difficult to interpret because of growth stimulation. So, while the EC₁₀ is normally obtainable with a sufficient accuracy, it is also recommended to report always the EC₂₀.

Weighting factors

The experimental variance is not generally constant and typically includes a proportional component, a weighted regression is therefore advantageously carried out routinely. Weighting factors for such an analysis are normally assumed inversely proportional to the variance:

W_i = 1/Var(r_i)

Many regression programs allow the option of weighted regression analysis with weighting factors listed in a table. Conveniently, weighting factors should be normalised by multiplying them by n/Σ w_i (n is the number of data points) so that their sum equals one.

Normalising responses

Normalising by the mean control response gives some principle problems and gives rise to a rather complicated variance structure. Dividing the responses by the mean control response for obtaining the percentage of inhibition, one introduces an additional error caused by the error on the control mean. Unless this error is negligibly small, weighting factors in the regression and confidence limits must be corrected for the covariance with the control (17). Note that high precision on the estimated mean control response is important in order to minimise the overall variance for the relative response. This variance is as follows:

(subscript i refers to concentration level i and subscript 0 to the controls)

Y_i = Relative response = r_i/r₀ = 1 — I = f (C_i)

with a variance:

Var (Y_i) = Var (r_i/r₀) ≅ (∂Y_i / ∂ r_i)²·Var(r_i) + (∂ Y_i/ ∂ r₀)²·Var (r₀)

and since

(∂ Y_i/ ∂ r_i) = 1/r₀ and (∂ Y_i / ∂ r₀) = r_i/r₀ ²

with normally distributed data and m_i and m₀ replicates:

Var(r_i) = σ²/m_i

the total variance of the relative response, Y_i thus becomes:

Var(Y_i) = σ²/(r₀ ² m_i) + r_i ²·σ²/r₀ ⁴ m₀

The error on the control mean is inversely proportional to the square root of the number of control replicates averaged, and sometimes it can be justified to include historical data and in this way greatly reduce the error. An alternative procedure is not to normalise the data and fit the absolute responses, including the control response data, but introducing the control response value as an additional parameter to be fitted by non-linear regression. With a usual 2 parameter regression equation, this method necessitates the fitting of 3 parameters, and therefore demands more data points than non-linear regression on data that are normalised using a pre-set control response.

Inverse confidence intervals

The calculation of non-linear regression confidence intervals by inverse estimation is rather complex and not an available standard option in ordinary statistical computer program packages. Approximate confidence limits may be obtained with standard non-linear regression programs with re-parameterisation (Bruce and Versteeg, 1992), which involves rewriting the mathematical equation with the desired point estimates, e.g. the EC₁₀ and the EC₅₀ as the parameters to be estimated. (Let the function be I = f (α, β, concentration) and utilise the definition relationships f (α, β, EC₁₀) = 0,1 and f (α, β, EC₅₀) = 0,5 to substitute f (α, β, concentration) with an equivalent function g (EC₁₀, EC₅₀, concentration).

A more direct calculation (Andersen et al, 1998) is performed by retaining the original equation and using a Taylor expansion around the means of r_i and r₀.

Recently ‘boot strap methods’ have become popular. Such methods use the measured data and a random number generator directed frequent re-sampling to estimate an empirical variance distribution.