4  Model module

4.1 Available models

FXMATE uses the R package drc to model dose-response curves and effect concentrations and for consistency uses the package’s model names. The names are composed of a prefix (i.e., everything before the point) defining the model family and a suffix (i.e., everything after the point) defining the number of parameters. Currently 28 different models are implemented:

Family	Models
Hormesis	BC.4, BC.5, CRS.5a, CRS.5b, CRS.5c
Log-Logistic	LL.2, LL2.2, LL2.3, LL2.4, LL2.5, LL2.3u, LL.3, LL.4, LL.5, LL.3u
Log-Normal	LN.2, LN.3, LN.4, LN.3u
Logistic	L.3, L.4, L.5
Weibull (type I)	W1.2, W1.3, W1.4
Weibull (type II)	W2.2, W2.3, W2.4

If you want to learn more about these models, we recommend an article by Ritz (2010).

4.2 Model selection

How FXMATE guides your way to the best model

1. Only models that can be fit and are within the logical boundaries of the response variable (e.g., 0 and 1 for binary responses) are selectable. The way Model exclusion is handled can be selected via the dropdown menu in the sidebar. You can choose between Generous (i.e., exclusion of models with predictions outside of logical boundaries), Rigorous (i.e., additionally excludes models with parameters outside of logical boundaries), and None (i.e., no exclusion at all).
2. Since also logically possible models can drastically differ in their fit for the data, FXMATE further filters based on the rLik of the models (i.e., only models with an rLik > 0.05 are shown).
3. You can easily see which model is the best for each of the criteria (i.e., AIC , NW and S , in accordance with EFSA Supporting publication 2015:EN-924). Additionally, the Goodness of fit table features an EFSA protection class (based on the relationships of EC_X) and an FXSCORE which weighs the ranks of the models in terms of AIC (see Section 4.2.1), NW (see Section 4.2.2), and S (see Section 4.2.3) by 5, 3, and 1, respectively and scales the score between 0 (worst) and 100 (best). A model with decent AIC and NW can thus score higher than a model with great AIC but bad NW .
4. FXMATE is full of informative tooltips to guide you to the ideal modelling settings and also gives precise model advice next to the dose-response plot when necessary.

4.2.1 Akaike information criterion

The Akaike information criterion (AIC), formulated by Akaike (1998), is given by the equation:

\[ {\displaystyle \mathrm {AIC} \,=\,2k-2\ln({\hat {L}})} \]

where \(k\) is the number of parameters and \({\hat {L}}\) is the maximized value of the likelihood function. In general, this criterion has two decisive properties:

1. Lower values indicate better models

2. Models with more parameters are penalized. This means that if two models fit the data “equally well” the one with fewer parameters is recommended (i.e., Occam’s razor / principle of parsimony)

To compare models with each other, FXMATE calculates their relative likelihood (\(rLik\)):

\[ rLik = exp\left(\frac{{AIC}_{min} - {AIC}_i}{2}\right) \]

By default, FXMATE selects the model with the highest \(rLik\) and does not display models with \(rLik < 0.05\). This can be avoided by clicking the checkbox Show all valid models in the sidebar of the Model module.

4.2.2 Normalized width

The normalized width (\(NW\)) is defined as

\[NW = \frac{CI_{95\%}(EC_x)}{EC_x}\]

where \(EC_X\) is the concentration with an effect size \(X\) and \(CI_{95\%}(EC_x)\) is the width of the 95%-confidence-interval at \(EC_X\) (i.e., upper minus lower limit). For easy interpretation, FXMATE includes NW ratings in accordance with EFSA Supporting publication 2015:EN-924. However, note that these ratings refer to the NW of the EC₁₀ and only offer a vague reference for other EC_X.

Condition	Rating
NW < 0.2	Excellent
NW < 0.5	Good
NW < 1	Fair
NW < 2	Poor
NW ≥ 2	Bad

In the Goodness of fit panel, NW values refer to the \(EC_{50}\) and are averaged over group levels (if present).

4.2.3 Steepness

The steepness (\(S\)) of the dose-response model is defined as

\[ S = \frac{EC_i}{EC_{50}} \]

For easy interpretation, FXMATE includes steepness ratings in accordance with EFSA Supporting publication 2015:EN-924. However, note that these ratings refer to the S of the EC₁₀ and only offer a vague reference for other EC_X.

Condition	Rating
S < 0.33	Shallow
0.33 ≤ S ≤ 0.66	Medium
S > 0.66	Steep

In the Goodness of fit panel, S values refer to \(\frac{EC_{10}}{EC_{50}}\) and are averaged over group levels (if present).

4.3 Visualization and effects

Data are displayed as response vs. dose on a logarithmic axis. Plot elements are conveniently explained in the caption. The plot can be further customized in the Report module. To visually compare different candidate models, the user can add comparison models.

An important constraint of logarithmic axes is that they cannot display values ≤ 0. Hence, a value must be applied to visualize control treatments. FXMATE automatically calculates possible zero levels based on the test concentrations and their separation factor. The zero level should be adjusted by the user so that the left asymptote approaches horizontality.

To improve visualization, y-axes can be scaled Uniform or Variable (e.g., if response values drastically differ between groups). Furthermore, you can choose to plot confidence ribbons and CI limits that are not constrained by the range of the tested concentrations.

The desired effect levels to be predicted by the model can be added via a text input. This allows any arbitrary effect level or number of effect levels. However, this also requires that the user supplies the values in the proper format (i.e., separated by a comma and using “.” as decimal separator). For example, entering 10, 20, 22.87, 50 would yield the desired response levels, while 10 / 20 / 50 cannot be interpreted. Note that hormesis effects need to be specified as negative numbers and bound must be unselected in the Expert Options (Section 4.6).

The output table is displayed directly below the dose-response visualization.

The column

• EC shows the level of effect

• Estimate shows the median \(EC_X\)

• Std. Error shows the standard error of the \(EC_X\)

• LCL / UCL show the Lower and Upper Confidence Level of the \(EC_X\), respectively

• Normalized width / Rating show the normalized width at the respective effect level alongside its rating

• Prediction shows the response value predicted by the dose-response model at the respective effect level

• Steepness shows the steepness at the respective effect level (not possible if using EC_X Type “absolute”)

⚠ Proper interpretation of EC_X

Note that by default relative effect concentrations (in %) are calculated. This procedure uses the fitted (or fixed) values of the left and right asymptotes. For 50% effect size, this means that the estimated response value is directly in the middle of the horizontal axes of the left and right asymptotes.

Very important: Fitting asymptotes does not always make sense and can heavily influence your EC_X. For example, when the data doesn’t show a clear asymptote or there is a logical 100% effect size (e.g., growth rate of 0 in OECD Test No. 201: Freshwater Alga and Cyanobacteria, Growth Inhibition Test) it is often better to set a fixed asymptote or use a model with a fixed asymptote (e.g., models with two or three parameters). The model advice in FXMATE warns you if fitted or fixed asymptotes differ from what would be expected of an ideal test. See Section 4.6 for more details.

4.4 Compare effect concentrations

Assessing if differences between EC_X levels are statistically significant can inform about the confidence and protectiveness of your model. FXMATE calculates pairwise absolute differences between effect levels and their 95% CI assuming normality. Significance is assumed when this interval does not contain 0. For instance, in the visualization below, you can see that the \(EC_{50}\) is significantly different from both the \(EC_{10}\) and the \(EC_{20}\), while \(EC_{10}\) and \(EC_{20}\) do not differ significantly.

4.5 Guideline-specific considerations

Guideline	Response	Response Type	Considerations
OECD 201: Freshwater Alga and Cyanobacteria, Growth Inhibition Test	Yield	continuous	An ideal test would assume 0 yield at 100% effect. Thus, using a lower limit of 0 (or a three-parameter model) is reasonable if your data does not show a clear lower asymptote.
	Growth rate	continuous	An ideal test would assume 0 growth rate at 100% effect. Thus, using a lower limit of 0 (or a three-parameter model) is reasonable if your data does not show a clear lower asymptote.
OECD 202: Daphnia sp. Acute Immobilisation Test	Fraction Immobile	binomial	An ideal test would assume a response of 1 at 100% effect. Thus, using an upper limit of 1 is reasonable if your data does not show a clear upper asymptote.
OECD 203: Fish, Acute Toxicity Test	Dead	binomial	An ideal test would assume a response of 1 at 100% effect. Thus, using an upper limit of 1 is reasonable if your data does not show a clear upper asymptote.
OECD 211: Daphnia magna reproduction test	Total Offspring per Adult	Poisson	An ideal test would assume no offspring at 100% effect. Thus, using a lower limit of 0 (or a three-parameter model) is reasonable if your data does not show a clear lower asymptote. In case of mortality or when using a flowthrough-design, response values can become non-integer. In that case, choose a Response type of continuous in the Expert options
	Fraction Dead	binomial	An ideal test would assume a response of 1 at 100% effect. Thus, using an upper limit of 1 is reasonable if your data does not show a clear upper asymptote.

4.6 Expert options

Although in some cases required, Expert Options are only recommended for users with profound statistical expertise and thus hidden by default. Changing these inputs unreflectedly can result in erroneous models.

4.6.1 Model modes

4.6.1.1 Standard

The mode Standard is used for classical dose-response modelling. This means that the selected model is fit on the original data. Model parameters, \(EC_X\) values and confidence intervals are computed with the drc package.

4.6.1.2 Bootstraps

The mode Bootstraps creates 30, 100, 300, or 1000 bootstraps (i.e., sampling original data X times with replacement). For this, FXMATE stratifies by dose levels (and group levels, if present). The selected model is then fit on each bootstrap. Model predictions, \(EC_X\) estimates and model parameters are derived by averaging all bootstrapped models. Lower and upper 95%-confidence limits represent the 2.5% and 97.5% quantiles, respectively. The standard error is equal to the standard deviation when bootstrapping. Since computations can take a while we recommend a low number of bootstraps (e.g., 30) when testing candidate models and a high number of bootstraps (e.g., 1000) when the final model is selected.

4.6.1.3 Model Averaging

The mode Model Averaging fits all models above the selected rLik threshold on the original data and calculates a weighted average based on rLik . Averaged models often provide more robust estimates and more reliable confidence intervals. Since parameter composition and meaning differ between models, the averaged model itself does not have any fitted parameters and therefore FXMATE displays every parameter from all models included in the averaged model in the Model details panel. To derive \(EC_X\) values there are two methods of choice:

Buckland accounts for both model uncertainty and parameter uncertainty by calculating the standard error (\(SE_{avg}\)) of the averaged model as:

\[ SE_{avg} = \sqrt{\sum_{i=1}^n{\omega_i (SE_i^2 + (EC_{X_i} - \overline{EC_X})^2)}} \] where \(\omega_i\) is the weight, \(SE_i\) the standard error and \(EC_{X_i}\) the \(EC_X\) of model \(i\) and \(\overline{EC_X}\) the average \(EC_X\).

Kang applies weighted means to \(EC_X\), \(LCL\) and \(UCL\).

⚠ Careful when working with model averaging and fixed asymptotes

Sometimes it might be necessary to fix lower and upper asymptotes of the model. Often this leads to identical models in the goodness of fit table. For example, two-parameter (i.e., W2.2), three-parameter (i.e., W2.3), and four-parameter (i.e., W2.4) Weibull type II models are identical when lower and upper asymptotes are fixed to 0 and 1, respectively. Not removing two of these models from model averaging would thus lead to a triplication of the model weight and distort model averaging. FXMATE warns you via a popup and precisely informs which models are identical. It is recommended to remove the unwanted models until no repeated model is present anymore.

Note: FXMATE is deliberately liberal with these warnings. For example, models in the CRS family can be very similar but are not considered in this popup when their AIC differs. Hence, we suggest to think carefully about which models to include / exclude from averaging.

4.6.2 Transformations

If model residuals deviate strongly from normality (indicated by the Model diagnostics plot in the Residuals tab), it can be useful to transform response values. This can only be done with continuous data and has the implication that no model advice on asymptotes can be given, since the absolute values of the response are uninformative after transformation. Hence, we recommend to avoid transformations if possible. Still, FXMATE offers two options to transform continuous responses. Importantly, FXMATE automatically chooses \(\lambda\) based on maximum likelihood which avoids arbitrary decisions.

4.6.2.1 Modulus

This transformation is a generalization of power transformations like Box-Cox and allows positive and negative values. This has the benefit of not having to add an offset to the data.

\[ y^{(\lambda)} = sign(y) \frac{(|y|+1)^\lambda - 1}{\lambda} \]

4.6.2.2 Box-Cox

This transformation, developed by Box and Cox (1964), does not allow for negative values.

\[ y^{(\lambda)} = \frac{y^\lambda - 1}{\lambda} \]

Akaike, H., 1998. Information theory and an extension of the maximum likelihood principle, in: Selected Papers of Hirotugu Akaike. Springer, pp. 199–213.

Box, G.E., Cox, D.R., 1964. An analysis of transformations. Journal of the Royal Statistical Society Series B: Statistical Methodology 26, 211–243.

Ritz, C., 2010. Toward a unified approach to dose–response modeling in ecotoxicology. Environmental toxicology and chemistry 29, 220–229.