What a standard analysis is?

A standard analysis for survival toxicity tests at final time consists of fitting a standard concentration-response curve to survival data at a given fixed time point; by default, MOSAIC\(_{surv}\) uses the final time of your experiment.

For such a standard analysis, we expect you have survival data at constant exposure concentration, at least at one time point (which will be considered as the final time). Example data sets are available for your convenience, either to try the module or to see how data must be formatted.

The mean survival rate at final time is related to the contaminant concentration \(c\) with a three-parameters log-logistic relationship: \[\begin{equation} f(c) = \frac{d}{1+\left(\frac{c}{e} \right)^b} \end{equation}\] where parameters are all positive: \(d\) corresponds to the survival rate under control conditions, \(e\) corresponds to the \(LC_{50}\) and parameter \(b\) is related to the effect intensity of the contaminant (‘slope’ of the curve).

Assuming that deaths of two animals are two independent events and given an initial number \(N_{0i}\) of animals at the \(i^\textrm{th}\) concentration \(c_i\), the number \(N_i\) of surviving animals at concentration \(c_i\) follows a binomial distribution: \[\begin{equation} N_i \sim \mathcal{B}\left(N_{0i},f(c_i) \right) \end{equation}\]

Model parameters \(d\), \(e\) and \(b\) are estimated using Bayesian inference, where the joint posterior distribution is computed from the likelihood of the observed data combined with prior distributions on the parameters. All details on priors can be found here or in the original research paper [2]. The \(50^{\textrm{th}}\), \(2.5^{\textrm{th}}\) and \(97.5^{\textrm{th}}\) quantiles are extracted from the marginal posterior distributions for each parameter, to serve as median and 95% credible intervals, respectively.

\(LC_x\) values, whatever \(x\), are then obtained from model parameters with their uncertainty, as probability distributions from which \(50^{\textrm{th}}\), \(2.5^{\textrm{th}}\) and \(97.5^{\textrm{th}}\) quantiles are extracted to serve as median and 95% credible intervals, respectively.

The different steps in using MOSAIC\(_{surv}\) to analyse your data are explained below and illustrated with a survival data set of a chronic toxicity test with snails exposed to six concentrations of cadmium during 56 days [3, 4]. Six concentrations were tested, with six replicates per concentration. Each replicate contained five organisms. Survival was monitored at 17 time points. Raw data can be downloaded here.

Step 1: Data uploading

When using MOSAIC\(_{surv}\), the first step is to upload input data.

Step 2: Results and interpretation

After clicking on Run, you immediately see a summary of the experimental design corresponding to the data you uploaded.

You also get plots of the survival raw data:

• The number of survivors over time for each replicate and each concentration.

• The observed fraction of surviving animals at final time at each tested contaminant concentration (black dots) together with the 95% binomial confidence intervals (black vertical segments).

Then you get the fitting results:

• The fitting plot which shows the mean survival rate as a function of the contaminant concentration (orange plain line) as well as the 95% credible band around this mean (light gray zone delimited by orange dotted lines). Black dots correspond to observed data while black vertical segments stand for the between-replicate variability (95% binomial confidence intervals).

• Model parameter estimates: we report the median (as the estimated value) and the 2.5% and 97.5% quantiles of the posterior (as a measure of uncertainty, a.k.a. 95% credible intervals).

        50%      2.5%     97.5%
b 1.135e+01 5.382e+00 7.026e+01
d 9.227e-01 8.522e-01 9.711e-01
e 1.175e+02 1.043e+02 1.247e+02

Take care about the value of parameter \(e\) corresponding to the \(LC_{50}\) estimate. If bounds are higher than the highest tested concentration, the value has to be considered with cautious.

• \(LC_x\) estimates: we report the median (as the estimated value) and the 2.5% and 97.5% quantiles (as a measure of uncertainty, a.k.a. 95% credible intervals) of several \(LC_x\) estimates, \(x = 1,...,80\).

     median  Q2.5 Q97.5
LC1    77.7  46.7   115
LC5    89.9  63.3   118
LC10   96.1  72.3   119
LC15  100.0  78.5   120
LC20  103.0  83.4   120
LC25  106.0  87.5   121
LC30  109.0  91.3   122
LC40  113.0  98.0   123
LC50  117.0 104.0   125
LC60  121.0 110.0   130
LC70  125.0 116.0   138
LC80  130.0 123.0   151

As already stated before, the \(LC_{50}\) estimate equals parameter \(e\) estimate.

• The posterior predictive check (PPC): we plot the observed data against the corresponding predicted value from the fitted model as a 95% credible interval. If the fit is correct, we expect to see 95% of the observed values fall inside the credible intervals.

Each black dot corresponds to one observation from the data set. The corresponding observed value can be read on the x-axis, while the y-axis reports the value predicted by the fitted model, as well as its 95% credible interval. The interval is colored in green if it contains the observed value, in red otherwise.

• R script: a gateway to perform further calculations directly within the R software [5].

library(morse)
data("cadmium2")
dsurv <- survData(cadmium2)
plot(dsurv, style = "ggplot")
plotDoseResponse(dsurv, addlegend = FALSE, style = "generic")
fit <- survFitTT(dsurv, lcx = c(1,5,10,15,20,25,30,40,50,60,70,80), quiet = TRUE)
plot(fit, adddata = TRUE, style = "generic")
summary(fit, quiet = TRUE)$Qpost
signif(fit$estim.LCx, digits = 3)
ppc(fit, style = "generic")

References

[1] Baudrot V, Charles S, Delignette-Muller ML, Duchemin W, Kon-Kam-King G, Lopes C, Ruiz P and Veber P (2018). morse: Modelling Tools for Reproduction and Survival Data in Ecotoxicology. R package version 3.1.0. https://CRAN.R-project.org/package=morse.

[2] Forfait-Dubuc C, Charles S, Billoir E, Delignette-Muller M. 2012. Survival data analyses in ecotoxicology: critical effect concentrations, methods and models. What should we use? Ecotoxicology. 12:1072–1083.

[3] Ducrot V, Askem C, Azam D, Brettschneider D, Brown R, Charles S, Coke M, Collinet M, Delignette-Muller ML, Forfait-Dubuc C, et al. 2014. Development and validation of an OECD reproductive toxicity test guideline with the pond snail Lymnaea stagnalis (Mollusca, Gastropoda). Regul. Toxicol. Pharmacol. 70:605–614.

[4] Charles S, Ducrot V, Azam D, Benstead R, Brettschneider D, De Schamphelaere K, Filipe Goncalves S, Green JW, Holbech H, Hutchinson TH, et al. 2016. Optimizing the design of a reproduction toxicity test with the pond snail Lymnaea stagnalis. Regul. Toxicol. Pharmacol. 81:47–56.

[5] R Core Team (2018). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.