Survival data analysis at target time (TT)
The following example shows all the steps to perform survival analysis on standard toxicity test data and to produce estimated values of the \(LC_x\). We will use a dataset of the library named cadmium2, which contains both survival and reproduction data from a chronic laboratory toxicity test. In this experiment, snails were exposed to six concentrations of a metal contaminant (cadmium) during 56 days.
Step 1: check the structure and the dataset integrity
The data from a survival toxicity test should be gathered in a data.frame with a specific layout. This is documented in the paragraph on survData in the reference manual, and you can also inspect one of the datasets provided in the package (e.g., cadmium2). First, we load the dataset and use the function survDataCheck to check that it has the expected layout:
data(cadmium2)
survDataCheck(cadmium2)## No messageThe output ## No message just informs that the dataset is well-formed.
Step 2: create a survData object
The class survData corresponds to survival data and is the basic layout used for the subsequent operations. Note that if the call to survDataCheck reports no error (i.e., ## No message), it is guaranteed that survData will not fail.
dat <- survData(cadmium2)
head(dat)## # A tibble: 6 x 6
## conc time Nsurv Nrepro replicate Ninit
## <dbl> <int> <int> <int> <dbl> <int>
## 1 0 0 5 0 1 5
## 2 0 3 5 262 1 5
## 3 0 7 5 343 1 5
## 4 0 10 5 459 1 5
## 5 0 14 5 328 1 5
## 6 0 17 5 742 1 5Step 3: visualize your dataset
The function plot can be used to plot the number of surviving individuals as a function of time for all concentrations and replicates.
plot(dat, pool.replicate = FALSE)
Two graphical styles are available, "generic" for standard R plots or "ggplot" to call package ggplot2 (default). If argument pool.replicate is TRUE, datapoints at a given time-point and a given concentration are pooled and only the mean number of survivors is plotted. To observe the full dataset, we set this option to FALSE.
By fixing the concentration at a (tested) value, we can visualize one subplot in particular:
plot(dat, concentration = 124, addlegend = TRUE,
pool.replicate = FALSE, style ="generic")
We can also plot the survival rate, at a given time-point, as a function of concentration, with binomial confidence intervals around the data. This is achieved by using function plotDoseResponse and by fixing the option target.time (default is the end of the experiment).
plotDoseResponse(dat, target.time = 21, addlegend = TRUE)
Function summary provides some descriptive statistics on the experimental design.
summary(dat)##
## Number of replicates per time and concentration:
## time
## conc 0 3 7 10 14 17 21 24 28 31 35 38 42 45 49 52 56
## 0 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 53 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 78 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 124 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 232 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
## 284 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
##
## Number of survivors (sum of replicates) per time and concentration:
## 0 3 7 10 14 17 21 24 28 31 35 38 42 45 49 52 56
## 0 30 30 30 30 29 29 29 29 29 28 28 28 28 28 28 28 28
## 53 30 30 29 29 29 29 29 29 29 29 28 28 28 28 28 28 28
## 78 30 30 30 30 30 30 29 29 29 29 29 29 29 29 29 27 27
## 124 30 30 30 30 30 29 28 28 27 26 25 23 21 18 11 11 9
## 232 30 30 30 22 18 18 17 14 13 12 8 4 3 1 0 0 0
## 284 30 30 15 7 4 4 4 2 2 1 1 1 1 1 1 0 0Step 4: fit an exposure-response model to the survival data at target time
Now we are ready to fit a probabilistic model to the survival data, in order to describe the relationship between the concentration in chemical compound and survival rate at the target time. Our model assumes this latter is a log-logistic function of the former, from which the package delivers estimates of the parameters. Once we have estimated the parameters, we can then calculate the \(LC_x\) values for any \(x\). All this work is performed by the survFitTT function, which requires a survData object as input and the levels of \(LC_x\) we want:
fit <- survFitTT(dat,
target.time = 21,
lcx = c(10, 20, 30, 40, 50))The returned value is an object of class survFitTT providing the estimated parameters as a posterior1 distribution, which quantifies the uncertainty on their true value. For the parameters of the models, as well as for the \(LC_x\) values, we report the median (as the point estimated value) and the 2.5 % and 97.5 % quantiles of the posterior (as a measure of uncertainty, a.k.a. credible intervals). They can be obtained by using the summary method:
summary(fit)## Summary:
##
## The loglogisticbinom_3 model with a binomial stochastic part was used !
##
## Priors on parameters (quantiles):
##
## 50% 2.5% 97.5%
## b 1.000e+00 1.259e-02 7.943e+01
## d 5.000e-01 2.500e-02 9.750e-01
## e 1.227e+02 5.390e+01 2.793e+02
##
## Posteriors of the parameters (quantiles):
##
## 50% 2.5% 97.5%
## b 8.580e+00 4.023e+00 1.625e+01
## d 9.570e-01 9.099e-01 9.862e-01
## e 2.364e+02 2.098e+02 2.533e+02
##
## Posteriors of the LCx (quantiles):
##
## 50% 2.5% 97.5%
## LC10 1.831e+02 1.267e+02 2.139e+02
## LC20 2.012e+02 1.538e+02 2.261e+02
## LC30 2.142e+02 1.747e+02 2.350e+02
## LC40 2.255e+02 1.928e+02 2.436e+02
## LC50 2.364e+02 2.098e+02 2.533e+02If the inference went well, it is expected that the difference between quantiles in the posterior will be reduced compared to the prior, meaning that the data were helpful to reduce the uncertainty on the true value of the parameters. This simple check can be performed using the summary function.
The fit can also be plotted:
plot(fit, log.scale = TRUE, adddata = TRUE, addlegend = TRUE)
This representation shows the estimated relationship between concentration of chemical compound and survival rate (orange curve). It is computed by choosing for each parameter the median value of its posterior. To assess the uncertainty on this estimation, we compute many such curves by sampling the parameters in the posterior distribution. This gives rise to the grey band, showing for any given concentration an interval (called credible interval) containing the survival rate 95% of the time in the posterior distribution. The experimental data points are represented in black and correspond to the observed survival rate when pooling all replicates. The black error bars correspond to a 95% confidence interval, which is another, more straightforward way to bound the most probable value of the survival rate for a tested concentration. In favorable situations, we expect that the credible interval around the estimated curve and the confidence interval around the experimental data largely overlap.
A similar plot is obtained with the style "generic":
plot(fit, log.scale = TRUE, style = "generic", adddata = TRUE, addlegend = TRUE)
Note that survFitTT will warn you if the estimated \(LC_{x}\) lie outside the range of tested concentrations, as in the following example:
data("cadmium1")
doubtful_fit <- survFitTT(survData(cadmium1),
target.time = 21,
lcx = c(10, 20, 30, 40, 50))## Warning: The LC50 estimation (model parameter e) lies outside the range of
## tested concentrations and may be unreliable as the prior distribution on
## this parameter is defined from this range !plot(doubtful_fit, log.scale = TRUE, style = "ggplot", adddata = TRUE,
addlegend = TRUE)
In this example, the experimental design does not include sufficiently high concentrations, and we are missing measurements that would have a major influence on the final estimation. For this reason this result should be considered unreliable.
Step 5: validate the model with a posterior predictive check
The fit can be further validated using so-called posterior predictive checks: the idea is to plot the observed values against the corresponding estimated predictions, along with their 95% credible interval. If the fit is correct, we expect to see 95% of the data inside the intervals.
ppc(fit)
In this plot, each black dot represents an observation made at a given concentration, and the corresponding number of survivors at target time is given by the value on the x-axis. Using the concentration and the fitted model, we can produce the corresponding prediction of the expected number of survivors at that concentration. This prediction is given by the y-axis. Ideally observations and predictions should coincide, so we’d expect to see the black dots on the points of coordinate \(Y = X\). Our model provides a tolerable variation around the predited mean value as an interval where we expect 95% of the dots to be in average. The intervals are represented in green if they overlap with the line \(Y=X\), and in red otherwise.
