5  Other EC_X module

5.1 General

This module of FXMATE let’s you determine \(LC_{50}\) / \(EC_{50}\) values for quantal datasets with little or no partial responses. Quantal data arises from binary (0 or 1) data. A classical example is mortality data, where animals can only be either dead or alive. A partial response would refer, in this context, to a treatment where not all animals share the same condition (i.e., either all are alive or all are dead).

Note that methods in this module require a positive effect direction. To this end, the app tries to capture the effect direction of your data and, if judged as negative, reverses your data (i.e., 1 - response). You can overrule the app’s judgement in the Effect direction dropdown menu in the sidebar.

5.2 Methods

5.2.1 Trimmed Spearman-Karber

This is a non-parametric method to determine \(LC_{50}\) / \(EC_{50}\) values, which means - contrasting to the approaches in the Model module - no specific model needs to be assumed. The method was developed by Hamilton et al. (1977).

It requires zero mortality in the control, which makes it necessary to apply a correction if this is not the case. In such situations, FXMATE automatically applies Abbott’s formula (Abbott, 1987). The user should be aware that this can introduce large errors into the data. Next, the effect direction needs to be monotonically increasing. If this is not the case, an amalgamation algorithm is used to satisfy this prerequisite. Both Abbott’s correction and amalgamation are realized by functions from the ecotoxicology package.

Then \(alpha\), a proportion, of the lowest and highest responses are removed. \(alpha\) is automatically calculated in FXMATE as the maximum of 10%, \(100p_1\), and \(100(1 – p_k)\) as suggested by Hamilton et al. (1977). \(p_1\) and \(p_k\) are the adjusted proportions observed in the lowest and highest exposure treatments, respectively.

Using this \(alpha\), the Trimmed Spearman-Karber method is then applied as realized by the ecotoxicology package. Note that the source code of the respective function was altered to allow FXMATE style figures.

The app will provide you with a plot displaying inter alia original data, trim boundaries, the \(LC_{50}\) / \(EC_{50}\) value, and the associated 95% confidence interval, which are also numerically presented in a table.

5.2.2 Binary method

This is the simplest method to estimate \(LC_{50}\) / \(EC_{50}\) values. It requires at least one treatment with 0% response, at least one treatment with 100% response, no partial responses, and all treatments showing 100% response need to be test doses/concentrations higher than for all treatments showing 0% response.

Under these conditions, the \(LC_{50}\) / \(EC_{50}\) value is the geometric mean of the two neighboring concentrations where the step from 0% to 100% response takes place.

Note that for this method fiducial intervals are estimated, which are slightly different in their interpretation compared to confidence intervals. Moreover, LCL and UCL can only take on values of test doses/concentrations. FXMATE provides you with fiducial bounds (i.e., LCL and UCL ) that are associated with the least wide interval with \(\geq\) 95% confidence. The exact level of confidence is provided in the output table.

The Binary method was realized by adpating code provided by Green et al. (2018).

Abbott, W., 1987. A method of computing the effectiveness of an insecticide.

Green, J.W., Springer, T.A., Holbech, H., 2018. Statistical analysis of ecotoxicity studies. John Wiley & Sons.

Hamilton, M.A., Russo, R.C., Thurston, R.V., 1977. Trimmed spearman-karber method for estimating median lethal concentrations in toxicity bioassays. Environmental science & technology 11, 714–719.