morse tutorial

2018-06-12

The package morse is devoted to the analysis of data from standard toxicity tests. It provides a simple workflow to explore/visualize a dataset, and compute estimations of risk assessment indicators. This document illustrates a typical use of morse on survival and reproduction data, which can be followed step-by-step to analyze new datasets.

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 message

The 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     5

Step 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  0

Step 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 8.637e+00 3.977e+00 1.618e+01
## d 9.569e-01 9.081e-01 9.863e-01
## e 2.365e+02 2.103e+02 2.536e+02
## 
## Posteriors of the LCx (quantiles):
## 
##            50%      2.5%     97.5%
## LC10 1.834e+02 1.269e+02 2.144e+02
## LC20 2.015e+02 1.539e+02 2.265e+02
## LC30 2.145e+02 1.750e+02 2.355e+02
## LC40 2.257e+02 1.933e+02 2.438e+02
## LC50 2.365e+02 2.103e+02 2.536e+02

If 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.

Survival analysis using a toxicokinetic-toxicodynamic (TKTD) model: GUTS

The steps for a TKTD data analysis are absolutely analogous to what we described for the analysis at target time. Here the goal is to estimate the relationship between chemical compound concentration, time and survival rate using the GUTS models. GUTS, for General Unified Threshold models of Survival, is a TKTD models generalising most of existing mechanistic models for survival description. For details about GUTS models, see the vignette theory under modelling approaches, and the included references.

GUTS model with constant exposure concentrations

Here is a typical session to analyse concentration-dependent time-course data using the so-called “Stochastic Death” (SD) model:

# (1) load dataset
data(propiconazole)

# (2) check structure and integrity of the dataset
survDataCheck(propiconazole)
## No message
# (3) create a `survData` object
dat <- survData(propiconazole)

# (4) represent the number of survivors as a function of time
plot(dat, pool.replicate = FALSE)

# (5) check information on the experimental design
summary(dat)
## 
## Number of replicates per time and concentration: 
##        time
## conc    0 1 2 3 4
##   0     1 1 1 1 1
##   8.05  1 1 1 1 1
##   11.91 1 1 1 1 1
##   13.8  1 1 1 1 1
##   17.87 1 1 1 1 1
##   24.19 1 1 1 1 1
##   28.93 1 1 1 1 1
##   35.92 1 1 1 1 1
## 
## Number of survivors (sum of replicates) per time and concentration: 
##        0  1  2  3  4
## 0     20 19 19 19 19
## 8.05  20 20 20 20 19
## 11.91 20 19 19 19 17
## 13.8  20 19 19 18 16
## 17.87 21 21 20 16 16
## 24.19 20 17  6  2  1
## 28.93 20 11  4  0  0
## 35.92 20 11  1  0  0

GUTS model “SD”

To fit the Stochastic Death model, we have to specify the model_type as "SD":

# (6) fit the TKTD model SD
fit_cstSD <- survFit(dat, quiet = TRUE, model_type = "SD")

Then, the summary() function provides parameters estimates as medians and 95% credible intervals.

# (7) summary of parameters estimates
summary(fit_cstSD)
## Summary: 
## 
## Priors of the parameters (quantiles) (select with '$Qpriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 4.157e-02 2.771e-04 6.236e+00
##          hb 1.317e-02 2.708e-04 6.403e-01
##           z 1.700e+01 8.171e+00 3.539e+01
##          kk 5.727e-03 1.021e-05 3.212e+00
## 
## Posteriors of the parameters (quantiles) (select with '$Qposteriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 2.195e+00 1.591e+00 3.493e+00
##          hb 2.676e-02 1.260e-02 4.886e-02
##           z 1.688e+01 1.545e+01 1.873e+01
##          kk 1.230e-01 7.838e-02 1.896e-01

The plot() function provides a representation of the fitting for each replicates

plot(fit_cstSD)

Original data can be added by using the option adddata = TRUE

plot(fit_cstSD, adddata = TRUE)

A posterior predictive check is also possible using function ppc():

ppc(fit_cstSD)

GUTS model “IT”

The Individual Tolerance (IT) model is a variant of TKTD survival analysis. It can also be used with morse as demonstrated hereafter. For the IT model, we have to specify the model_type as "IT":

fit_cstIT <- survFit(dat, quiet = TRUE, model_type = "IT")

We can first get a summary of the estimated parameters:

summary(fit_cstIT)
## Summary: 
## 
## Priors of the parameters (quantiles) (select with '$Qpriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 4.157e-02 2.771e-04 6.236e+00
##          hb 1.317e-02 2.708e-04 6.403e-01
##       alpha 1.700e+01 8.171e+00 3.539e+01
##        beta 1.000e+00 1.259e-02 7.943e+01
## 
## Posteriors of the parameters (quantiles) (select with '$Qposteriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 7.198e-01 5.297e-01 9.359e-01
##          hb 1.576e-02 3.829e-03 3.767e-02
##       alpha 1.770e+01 1.513e+01 2.022e+01
##        beta 6.706e+00 4.882e+00 8.929e+00
# OR
fit_cstIT$estim.par
##   parameters     median         Q2.5       Q97.5
## 1         kd  0.7198470  0.529734985  0.93594269
## 2         hb  0.0157649  0.003828662  0.03767306
## 3      alpha 17.7038137 15.125450128 20.21705969
## 4       beta  6.7064622  4.882493589  8.92855078
plot(fit_cstIT, adddata= TRUE)

ppc(fit_cstIT)

GUTS model under time-variable exposure concentration

Here is a typical session fitting an SD or an IT model for a dataset under time-variable exposure scenario.

# (1) load dataset
data("propiconazole_pulse_exposure")

# (2) check structure and integrity of the dataset
survDataCheck(propiconazole_pulse_exposure)
## No message
# (3) create a `survData` object
dat <- survData(propiconazole_pulse_exposure)

# (4) represent the number of survivor as a function of time
plot(dat)

# (5) check information on the experimental design
summary(dat)
## 
## Occurence of 'replicate' for each 'time': 
##             time
## replicate    0 0.96 1 1.96 2 2.96 3 3.96 4 4.96 4.97 5 5.96 6 6.96 7 7.96
##   varA       1    1 1    1 1    1 1    1 1    1    1 1    1 1    1 1    1
##   varB       1    1 1    1 1    1 1    1 1    1    1 1    1 1    1 1    1
##   varC       1    1 1    1 1    1 1    1 1    1    1 1    1 1    1 1    1
##   varControl 1    0 1    0 1    0 1    0 1    0    0 1    0 1    0 1    0
##             time
## replicate    8 9 9.96 10
##   varA       1 1    1  1
##   varB       1 1    1  1
##   varC       1 1    1  1
##   varControl 1 1    0  1
## 
## Number of survivors per 'time' and 'replicate': 
##             0 0.96  1 1.96  2 2.96  3 3.96  4 4.96 4.97  5 5.96  6 6.96  7
## varA       70   NA 50   NA 49   NA 49   NA 45   NA   NA 45   NA 45   NA 42
## varB       70   NA 57   NA 53   NA 52   NA 50   NA   NA 46   NA 45   NA 44
## varC       70   NA 70   NA 69   NA 69   NA 68   NA   NA 66   NA 65   NA 64
## varControl 60   NA 59   NA 58   NA 58   NA 57   NA   NA 57   NA 56   NA 56
##            7.96  8  9 9.96 10
## varA         NA 38 37   NA 36
## varB         NA 40 38   NA 37
## varC         NA 60 55   NA 54
## varControl   NA 56 55   NA 54
## 
## Concentrations per 'time' and 'replicate': 
##                0  0.96    1 1.96  2 2.96     3  3.96    4 4.96 4.97  5
## varA       30.56 27.93 0.00 0.26 NA 0.21 27.69 26.49 0.00 0.18 0.18 NA
## varB       28.98 27.66 0.00 0.27 NA 0.26  0.26  0.26 0.26 0.25 0.25 NA
## varC        4.93  4.69 4.69 4.58 NA 4.58  4.58  4.54 4.54 4.58 4.71 NA
## varControl  0.00    NA 0.00   NA  0   NA  0.00    NA 0.00   NA   NA  0
##            5.96  6 6.96     7  7.96    8    9 9.96   10
## varA       0.18 NA 0.14  0.14  0.18 0.18 0.00 0.00 0.00
## varB       0.03 NA 0.00 26.98 26.28 0.00 0.12 0.12 0.12
## varC       4.71 NA 4.60  4.60  4.59 4.59 4.46 4.51 4.51
## varControl   NA  0   NA  0.00    NA 0.00 0.00   NA 0.00

GUTS model “SD”

# (6) fit the TKTD model SD
fit_varSD <- survFit(dat, quiet = TRUE, model_type = "SD")
## Warning: The estimation of the dominant rate constant (model parameter kd) lies 
##     outside the range used to define its prior distribution which indicates that this
##     rate is very high and difficult to estimate from this experiment !
## Warning: The estimation of Non Effect Concentration threshold (NEC) 
##       (model parameter z) lies outside the range of tested concentration 
##       and may be unreliable as the prior distribution on this parameter is
##       defined from this range !
# (7) summary of the fit object
summary(fit_varSD)
## Summary: 
## 
## Priors of the parameters (quantiles) (select with '$Qpriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 2.683e-02 1.119e-04 6.434e+00
##          hb 8.499e-03 1.094e-04 6.606e-01
##           z 9.154e-01 3.212e-02 2.608e+01
##          kk 5.080e-02 4.342e-06 5.942e+02
## 
## Posteriors of the parameters (quantiles) (select with '$Qposteriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 3.737e+00 1.610e+00 2.597e+01
##          hb 2.322e-02 1.632e-02 3.097e-02
##           z 2.111e+01 2.520e+00 3.101e+01
##          kk 8.861e-02 5.353e-03 6.953e-01
plot(fit_varSD)

ppc(fit_varSD)

GUTS model “IT”

# fit a TKTD model IT
fit_varIT <- survFit(dat, quiet = TRUE, model_type = "IT")
## Warning: The estimation of the dominant rate constant (model parameter kd) lies 
##     outside the range used to define its prior distribution which indicates that this
##     rate is very high and difficult to estimate from this experiment !
## Warning: The estimation of log-logistic median (model parameter alpha) 
##       lies outside the range of tested concentration and may be unreliable as 
##       the prior distribution on this parameter is defined from this range !
# (7) summary of the fit object
summary(fit_varIT)
## Summary: 
## 
## Priors of the parameters (quantiles) (select with '$Qpriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 2.683e-02 1.119e-04 6.434e+00
##          hb 8.499e-03 1.094e-04 6.606e-01
##       alpha 9.154e-01 3.212e-02 2.608e+01
##        beta 1.000e+00 1.259e-02 7.943e+01
## 
## Posteriors of the parameters (quantiles) (select with '$Qposteriors'):
## 
##  parameters    median      Q2.5     Q97.5
##          kd 4.333e-01 7.132e-04 1.844e+01
##          hb 2.420e-02 1.620e-02 3.310e-02
##       alpha 1.753e+01 2.531e-01 3.822e+01
##        beta 2.268e+00 4.836e-01 2.226e+01
plot(fit_varIT)

ppc(fit_varIT)

Prediction

GUTS models can be used to simulate the survival of the organisms under any exposure pattern, using the calibration done with function survFit() from observed data. The function for prediction is called predict() and returns an object of class survFitPredict.

# (1) upload or build a data frame with the exposure profile
# argument `replicate` is used to provide several profiles of exposure
data_4prediction <- data.frame(time = c(1:10, 1:10),
                               conc = c(c(0,0,40,0,0,0,40,0,0,0),
                                        c(21,19,18,23,20,14,25,8,13,5)),
                               replicate = c(rep("pulse", 10), rep("random", 10)))

# (2) Use the fit on constant exposure propiconazole with model SD (see previously)
predict_PRZ_cstSD_4pred <- predict(object = fit_cstSD, data_predict = data_4prediction)

From an object survFitPredict, results can ben plotted with function plot():

# (3) Plot the predicted survival rate under the new exposure profiles.
plot(predict_PRZ_cstSD_4pred)

Removing background mortality in prediction

While the model has been estimated using the background mortalityt parameter hb, it can be interesting to see the prediction without it. This is possible with the argument hb_value. If TRUE, the background mortality is taken into account, and if FALSE, the background mortality is set to \(0\) in the prediction.

# Use the same data set profile to predict without 'hb'
predict_PRZ_cstSD_4pred_hbOUT <- predict(object = fit_cstSD, data_predict = data_4prediction, hb_value = FALSE)
# Plot the prediction:
plot(predict_PRZ_cstSD_4pred_hbOUT)

Lethal concentration

Compared to the target time analysis, TKTD modelling allows to compute and plot the lethal concentration for any x percentage and at any time-point. The chosen time-point can be specified with time_LCx, by default the maximal time-point in the dataset is used.

# LC50 at the maximum time-point:
LCx_cstSD <- LCx(fit_cstSD, X = 50)
plot(LCx_cstSD)

# LC50 at time = 2
LCx(fit_cstSD, X = 50, time_LCx = 2) %>% plot()

## Note the use of the pipe operator, `%>%`, which is a powerful tool for clearly expressing a sequence of multiple operations.
## For more information on pipes, see: http://r4ds.had.co.nz/pipes.html

Warning messages are returned when the range of concentrations is not appropriated for one or more LCx calculation(s).

# LC50 at time = 15
LCx(fit_cstSD, X = 50, time_LCx = 15) %>% plot()

# LC50 at the maximum time-point:
LCx_cstIT <- LCx(fit_cstIT, X = 50)
plot(LCx_cstIT)

# LC50 at time = 2
LCx(fit_cstIT, X = 50, time_LCx = 2) %>% plot()

# LC30 at time = 15
LCx(fit_cstIT, X = 30, time_LCx = 15) %>% plot()

# LC50 at time = 4
LCx_varSD <- LCx(fit_varSD, X = 50, time_LCx = 4, conc_range = c(0,100))
plot(LCx_varSD)

# LC50 at time = 30
LCx(fit_varSD, X = 50, time_LCx = 30,  conc_range = c(0,100)) %>% plot()

# LC50 at time = 4
LCx(fit_varIT, X = 50, time_LCx = 4, conc_range = c(0,200)) %>% plot()
## Warning in pointsLCx(df_dose, X_prop): No 95%sup for survival rate of
## 0.453876328473077 in the range of concentrations under consideration: [ 0 ;
## 200 ]
## Warning: Removed 1 rows containing missing values (geom_point).

# LC50 at time = 30
LCx(fit_varIT, X = 50, time_LCx = 30, conc_range = c(0,100)) %>% plot()

Multiplication factors: ‘margin of safety’

Using prediction functions, GUTS models can be used to simulate the survival rate of organisms exposed to a given exposure pattern. In general, this realistic exposure profile does not result in any related mortality, but a critical question is to know how far the exposure profile is from adverse effect, that is a “margin of safety”.

This idea is then to multiply the concentration in the realistic exposure profile by a “multiplication factor”, denoted \(MF_x\), resulting in \(x\%\) (classically \(10\%\) or \(50\%\)) of additional death at a specified time (by default, at the end of the exposure period).

The multiplication factor \(MF_x\) then informs the “margin of safety” that could be used to assess if the risk should be considered as acceptable or not.

Computing an \(MF_x\) is easy with function MFx(). It only requires object survFit and the exposure profile, argument data_predict in the function. The chosen percentage of survival reduction is specified with argument X, the default is \(50\), and the chosen time-point can be specified with time_MFx, by default the maximal time-point in the dataset is used.

There is no explicit formulation of \(MF_x\) (at least for the GUTS-SD model), so the accuracy argument can be used to change the accuracy of the convergence level.

# (1) upload or build a data frame with the exposure profile
data_4MFx <- data.frame(time = 1:10,
                        conc = c(0,0.5,8,3,0,0,0.5,8,3.5,0))

# (2) Use the fit on constant exposure propiconazole with model SD (see previously)
MFx_PRZ_cstSD_4MFx <- MFx(object = fit_cstSD, data_predict = data_4MFx)
## q50 1 accuracy: 0.3901651  with multiplication factor: 1000 
## q50 2 accuracy: 0.3901651  with multiplication factor: 500 
## q50 3 accuracy: 0.3901651  with multiplication factor: 250 
## q50 4 accuracy: 0.3901651  with multiplication factor: 125 
## q50 5 accuracy: 0.3901651  with multiplication factor: 62.5 
## q50 6 accuracy: 0.3901651  with multiplication factor: 31.25 
## q50 7 accuracy: 0.3901651  with multiplication factor: 15.625 
## q50 8 accuracy: 0.3901329  with multiplication factor: 7.8125 
## q50 9 accuracy: 0.1957951  with multiplication factor: 3.90625 
## q50 10 accuracy: 0.3901651  with multiplication factor: 1.953125 
## q50 11 accuracy: 0.3124028  with multiplication factor: 2.929688 
## q50 12 accuracy: 0.0228327  with multiplication factor: 3.417969 
## q50 13 accuracy: 0.1009284  with multiplication factor: 3.662109 
## q50 14 accuracy: 0.04208264  with multiplication factor: 3.540039 
## q50 15 accuracy: 0.01037075  with multiplication factor: 3.479004 
## q50 16 accuracy: 0.00565318  with multiplication factor: 3.448486 
## qinf95 1 accuracy: 0.3901651  with multiplication factor: 1000 
## qinf95 2 accuracy: 0.3901651  with multiplication factor: 500 
## qinf95 3 accuracy: 0.3901651  with multiplication factor: 250 
## qinf95 4 accuracy: 0.3901651  with multiplication factor: 125 
## qinf95 5 accuracy: 0.3901651  with multiplication factor: 62.5 
## qinf95 6 accuracy: 0.3901651  with multiplication factor: 31.25 
## qinf95 7 accuracy: 0.3901651  with multiplication factor: 15.625 
## qinf95 8 accuracy: 0.3901634  with multiplication factor: 7.8125 
## qinf95 9 accuracy: 0.2457348  with multiplication factor: 3.90625 
## qinf95 10 accuracy: 0.2897097  with multiplication factor: 1.953125 
## qinf95 11 accuracy: 0.1959479  with multiplication factor: 2.929688 
## qinf95 12 accuracy: 0.05859171  with multiplication factor: 3.417969 
## qinf95 13 accuracy: 0.06811207  with multiplication factor: 3.173828 
## qinf95 14 accuracy: 0.003508197  with multiplication factor: 3.295898 
## qsup95 1 accuracy: 0.3901651  with multiplication factor: 1000 
## qsup95 2 accuracy: 0.3901651  with multiplication factor: 500 
## qsup95 3 accuracy: 0.3901651  with multiplication factor: 250 
## qsup95 4 accuracy: 0.3901651  with multiplication factor: 125 
## qsup95 5 accuracy: 0.3901651  with multiplication factor: 62.5 
## qsup95 6 accuracy: 0.3901651  with multiplication factor: 31.25 
## qsup95 7 accuracy: 0.3901651  with multiplication factor: 15.625 
## qsup95 8 accuracy: 0.3898172  with multiplication factor: 7.8125 
## qsup95 9 accuracy: 0.1360735  with multiplication factor: 3.90625 
## qsup95 10 accuracy: 0.4703236  with multiplication factor: 1.953125 
## qsup95 11 accuracy: 0.4183426  with multiplication factor: 2.929688 
## qsup95 12 accuracy: 0.1145479  with multiplication factor: 3.417969 
## qsup95 13 accuracy: 0.02938477  with multiplication factor: 3.662109 
## qsup95 14 accuracy: 0.03831341  with multiplication factor: 3.540039 
## qsup95 15 accuracy: 0.00335103  with multiplication factor: 3.601074

As the computing time can be long, the function prints the accuracy for each step of the tested multiplication factor, for the median and the 95% credible interval.

Then, we can plot the survival rate as a function of the tested multiplication factors. Note that it is a linear interpolation between tested multiplication factor (cross dots on the graph).

# (3) Plot the survival rate as function of the multiplication factors.
plot(MFx_PRZ_cstSD_4MFx)
## Warning in plot.MFx(MFx_PRZ_cstSD_4MFx): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

In this specific case, the x-axis needs to be log-scaled, what is possible by setting option log_scale = TRUE:

# (3 bis) Plot the survival rate as function of the multiplication factors in log-scale.
plot(MFx_PRZ_cstSD_4MFx, log_scale = TRUE)
## Warning in plot.MFx(MFx_PRZ_cstSD_4MFx, log_scale = TRUE): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

As indicated, the warning message just remind you how multiplication factors and linear interpolations between them have been computed to obtain the graph.

To compare the initial survival rate (corresponding to a multiplication factor set to 1) with the survival rate at the asked multiplication factor leading to a reduction of \(x\%\) of survival (provided with argument X), we can use option x_variable = "Time". The option x_variable = "Time" allows to vizualize differences in survival rate with and without the multiplication factor.

# (4) Plot the survival rate versus time. Control (MFx = 1) and estimated MFx.
plot(MFx_PRZ_cstSD_4MFx, x_variable =  "Time")

What is provided with the function plot() is direclty accessible within the object of class MFx. For instance, to have access to the median and \(95\%\) of returned MFx, we simply extract the element df_MFx which is the following data.frame:

MFx_PRZ_cstSD_4MFx$df_MFx
##         quantile      MFx
## 1         median 3.448486
## 2  quantile 2.5% 3.295898
## 3 quantile 97.5% 3.601074

Change the level of multiplication factor

Here is an other example with a 10 percent Multiplication Factor:

# (2 bis) fit on constant exposure propiconazole with model SD (see previously)
MFx_PRZ_cstSD_4MFx_x10 <- MFx(object = fit_cstSD, data_predict = data_4MFx, X = 10)
## q50 1 accuracy: 0.7022972  with multiplication factor: 1000 
## q50 2 accuracy: 0.7022972  with multiplication factor: 500 
## q50 3 accuracy: 0.7022972  with multiplication factor: 250 
## q50 4 accuracy: 0.7022972  with multiplication factor: 125 
## q50 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## q50 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## q50 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## q50 8 accuracy: 0.702265  with multiplication factor: 7.8125 
## q50 9 accuracy: 0.5079272  with multiplication factor: 3.90625 
## q50 10 accuracy: 0.07803303  with multiplication factor: 1.953125 
## q50 11 accuracy: 0.0002706852  with multiplication factor: 2.929688 
## qinf95 1 accuracy: 0.7022972  with multiplication factor: 1000 
## qinf95 2 accuracy: 0.7022972  with multiplication factor: 500 
## qinf95 3 accuracy: 0.7022972  with multiplication factor: 250 
## qinf95 4 accuracy: 0.7022972  with multiplication factor: 125 
## qinf95 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## qinf95 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## qinf95 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## qinf95 8 accuracy: 0.7022955  with multiplication factor: 7.8125 
## qinf95 9 accuracy: 0.5578669  with multiplication factor: 3.90625 
## qinf95 10 accuracy: 0.0224224  with multiplication factor: 1.953125 
## qinf95 11 accuracy: 0.0224224  with multiplication factor: 0.9765625 
## qinf95 12 accuracy: 0.0224224  with multiplication factor: 0.4882812 
## qinf95 13 accuracy: 0.0224224  with multiplication factor: 0.2441406 
## qinf95 14 accuracy: 0.0224224  with multiplication factor: 0.1220703 
## qinf95 15 accuracy: 0.0224224  with multiplication factor: 0.06103516 
## qinf95 16 accuracy: 0.0224224  with multiplication factor: 0.03051758 
## qinf95 17 accuracy: 0.0224224  with multiplication factor: 0.01525879 
## qinf95 18 accuracy: 0.0224224  with multiplication factor: 0.007629395 
## qinf95 19 accuracy: 0.0224224  with multiplication factor: 0.003814697 
## qinf95 20 accuracy: 0.0224224  with multiplication factor: 0.001907349 
## qinf95 21 accuracy: 0.0224224  with multiplication factor: 0.0009536743 
## qinf95 22 accuracy: 0.0224224  with multiplication factor: 0.0004768372 
## qinf95 23 accuracy: 0.0224224  with multiplication factor: 0.0002384186 
## qinf95 24 accuracy: 0.0224224  with multiplication factor: 0.0001192093 
## qinf95 25 accuracy: 0.0224224  with multiplication factor: 5.960464e-05 
## qinf95 26 accuracy: 0.0224224  with multiplication factor: 2.980232e-05 
## qinf95 27 accuracy: 0.0224224  with multiplication factor: 1.490116e-05 
## qinf95 28 accuracy: 0.0224224  with multiplication factor: 7.450581e-06 
## qinf95 29 accuracy: 0.0224224  with multiplication factor: 3.72529e-06 
## qinf95 30 accuracy: 0.0224224  with multiplication factor: 1.862645e-06 
## qinf95 31 accuracy: 0.0224224  with multiplication factor: 9.313226e-07 
## qinf95 32 accuracy: 0.0224224  with multiplication factor: 4.656613e-07 
## qinf95 33 accuracy: 0.0224224  with multiplication factor: 2.328306e-07 
## qinf95 34 accuracy: 0.0224224  with multiplication factor: 1.164153e-07 
## qinf95 35 accuracy: 0.0224224  with multiplication factor: 5.820766e-08 
## qinf95 36 accuracy: 0.0224224  with multiplication factor: 2.910383e-08 
## qinf95 37 accuracy: 0.0224224  with multiplication factor: 1.455192e-08 
## qinf95 38 accuracy: 0.0224224  with multiplication factor: 7.275958e-09 
## qinf95 39 accuracy: 0.0224224  with multiplication factor: 3.637979e-09 
## qinf95 40 accuracy: 0.0224224  with multiplication factor: 1.818989e-09 
## qinf95 41 accuracy: 0.0224224  with multiplication factor: 9.094947e-10 
## qinf95 42 accuracy: 0.0224224  with multiplication factor: 4.547474e-10 
## qinf95 43 accuracy: 0.0224224  with multiplication factor: 2.273737e-10 
## qinf95 44 accuracy: 0.0224224  with multiplication factor: 1.136868e-10 
## qinf95 45 accuracy: 0.0224224  with multiplication factor: 5.684342e-11 
## qinf95 46 accuracy: 0.0224224  with multiplication factor: 2.842171e-11 
## qinf95 47 accuracy: 0.0224224  with multiplication factor: 1.421085e-11 
## qinf95 48 accuracy: 0.0224224  with multiplication factor: 7.105427e-12 
## qinf95 49 accuracy: 0.0224224  with multiplication factor: 3.552714e-12 
## qinf95 50 accuracy: 0.0224224  with multiplication factor: 1.776357e-12 
## qinf95 51 accuracy: 0.0224224  with multiplication factor: 8.881784e-13 
## qinf95 52 accuracy: 0.0224224  with multiplication factor: 4.440892e-13 
## qinf95 53 accuracy: 0.0224224  with multiplication factor: 2.220446e-13 
## qinf95 54 accuracy: 0.0224224  with multiplication factor: 1.110223e-13 
## qinf95 55 accuracy: 0.0224224  with multiplication factor: 5.551115e-14 
## qinf95 56 accuracy: 0.0224224  with multiplication factor: 2.775558e-14 
## qinf95 57 accuracy: 0.0224224  with multiplication factor: 1.387779e-14 
## qinf95 58 accuracy: 0.0224224  with multiplication factor: 6.938894e-15 
## qinf95 59 accuracy: 0.0224224  with multiplication factor: 3.469447e-15 
## qinf95 60 accuracy: 0.0224224  with multiplication factor: 1.734723e-15 
## qinf95 61 accuracy: 0.0224224  with multiplication factor: 8.673617e-16 
## qinf95 62 accuracy: 0.0224224  with multiplication factor: 4.336809e-16 
## qinf95 63 accuracy: 0.0224224  with multiplication factor: 2.168404e-16 
## qinf95 64 accuracy: 0.0224224  with multiplication factor: 1.084202e-16 
## qinf95 65 accuracy: 0.0224224  with multiplication factor: 5.421011e-17 
## qinf95 66 accuracy: 0.0224224  with multiplication factor: 2.710505e-17 
## qinf95 67 accuracy: 0.0224224  with multiplication factor: 1.355253e-17 
## qinf95 68 accuracy: 0.0224224  with multiplication factor: 6.776264e-18 
## qinf95 69 accuracy: 0.0224224  with multiplication factor: 3.388132e-18 
## qinf95 70 accuracy: 0.0224224  with multiplication factor: 1.694066e-18 
## qinf95 71 accuracy: 0.0224224  with multiplication factor: 8.470329e-19 
## qinf95 72 accuracy: 0.0224224  with multiplication factor: 4.235165e-19 
## qinf95 73 accuracy: 0.0224224  with multiplication factor: 2.117582e-19 
## qinf95 74 accuracy: 0.0224224  with multiplication factor: 1.058791e-19 
## qinf95 75 accuracy: 0.0224224  with multiplication factor: 5.293956e-20 
## qinf95 76 accuracy: 0.0224224  with multiplication factor: 2.646978e-20 
## qinf95 77 accuracy: 0.0224224  with multiplication factor: 1.323489e-20 
## qinf95 78 accuracy: 0.0224224  with multiplication factor: 6.617445e-21 
## qinf95 79 accuracy: 0.0224224  with multiplication factor: 3.308722e-21 
## qinf95 80 accuracy: 0.0224224  with multiplication factor: 1.654361e-21 
## qinf95 81 accuracy: 0.0224224  with multiplication factor: 8.271806e-22 
## qinf95 82 accuracy: 0.0224224  with multiplication factor: 4.135903e-22 
## qinf95 83 accuracy: 0.0224224  with multiplication factor: 2.067952e-22 
## qinf95 84 accuracy: 0.0224224  with multiplication factor: 1.033976e-22 
## qinf95 85 accuracy: 0.0224224  with multiplication factor: 5.169879e-23 
## qinf95 86 accuracy: 0.0224224  with multiplication factor: 2.584939e-23 
## qinf95 87 accuracy: 0.0224224  with multiplication factor: 1.29247e-23 
## qinf95 88 accuracy: 0.0224224  with multiplication factor: 6.462349e-24 
## qinf95 89 accuracy: 0.0224224  with multiplication factor: 3.231174e-24 
## qinf95 90 accuracy: 0.0224224  with multiplication factor: 1.615587e-24 
## qinf95 91 accuracy: 0.0224224  with multiplication factor: 8.077936e-25 
## qinf95 92 accuracy: 0.0224224  with multiplication factor: 4.038968e-25 
## qinf95 93 accuracy: 0.0224224  with multiplication factor: 2.019484e-25 
## qinf95 94 accuracy: 0.0224224  with multiplication factor: 1.009742e-25 
## qinf95 95 accuracy: 0.0224224  with multiplication factor: 5.04871e-26 
## qinf95 96 accuracy: 0.0224224  with multiplication factor: 2.524355e-26 
## qinf95 97 accuracy: 0.0224224  with multiplication factor: 1.262177e-26 
## qinf95 98 accuracy: 0.0224224  with multiplication factor: 6.310887e-27 
## qinf95 99 accuracy: 0.0224224  with multiplication factor: 3.155444e-27 
## qinf95 100 accuracy: 0.0224224  with multiplication factor: 1.577722e-27
## Warning in binarySearch_MFx(object = object, spaghetti = spaghetti,
## mcmc_size = mcmc_size, : For qinf95 the number of iteration reached the
## threshold number of iteration.
## qsup95 1 accuracy: 0.7022972  with multiplication factor: 1000 
## qsup95 2 accuracy: 0.7022972  with multiplication factor: 500 
## qsup95 3 accuracy: 0.7022972  with multiplication factor: 250 
## qsup95 4 accuracy: 0.7022972  with multiplication factor: 125 
## qsup95 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## qsup95 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## qsup95 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## qsup95 8 accuracy: 0.7019493  with multiplication factor: 7.8125 
## qsup95 9 accuracy: 0.4482056  with multiplication factor: 3.90625 
## qsup95 10 accuracy: 0.1581915  with multiplication factor: 1.953125 
## qsup95 11 accuracy: 0.1062105  with multiplication factor: 2.929688 
## qsup95 12 accuracy: 0.1975842  with multiplication factor: 3.417969 
## qsup95 13 accuracy: 0.0292868  with multiplication factor: 3.173828 
## qsup95 14 accuracy: 0.04695287  with multiplication factor: 3.051758 
## qsup95 15 accuracy: 0.01055886  with multiplication factor: 3.112793 
## qsup95 16 accuracy: 0.009855705  with multiplication factor: 3.143311

The warning messages are just saying that the quantile at \(2.5\%\) was not possible to compute. You can see this in the object df_MFx included in MFx_PRZ_cstSD_4MFx_x10. The reason of this impossibility is obvious when you plot the multiplication factor-response curve:

# Plot with log scale
plot(MFx_PRZ_cstSD_4MFx_x10, log_scale = TRUE)
## Warning in plot.MFx(MFx_PRZ_cstSD_4MFx_x10, log_scale = TRUE): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

Then, you can reduce the threshold of iterations as:

# (2 ter) fit on constant exposure propiconazole with model SD (see previously)
MFx_PRZ_cstSD_4MFx_x10_thresh20 <- MFx(object = fit_cstSD, data_predict = data_4MFx, X = 10, threshold_iter = 20)
## q50 1 accuracy: 0.7022972  with multiplication factor: 1000 
## q50 2 accuracy: 0.7022972  with multiplication factor: 500 
## q50 3 accuracy: 0.7022972  with multiplication factor: 250 
## q50 4 accuracy: 0.7022972  with multiplication factor: 125 
## q50 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## q50 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## q50 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## q50 8 accuracy: 0.702265  with multiplication factor: 7.8125 
## q50 9 accuracy: 0.5079272  with multiplication factor: 3.90625 
## q50 10 accuracy: 0.07803303  with multiplication factor: 1.953125 
## q50 11 accuracy: 0.0002706852  with multiplication factor: 2.929688 
## qinf95 1 accuracy: 0.7022972  with multiplication factor: 1000 
## qinf95 2 accuracy: 0.7022972  with multiplication factor: 500 
## qinf95 3 accuracy: 0.7022972  with multiplication factor: 250 
## qinf95 4 accuracy: 0.7022972  with multiplication factor: 125 
## qinf95 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## qinf95 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## qinf95 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## qinf95 8 accuracy: 0.7022955  with multiplication factor: 7.8125 
## qinf95 9 accuracy: 0.5578669  with multiplication factor: 3.90625 
## qinf95 10 accuracy: 0.0224224  with multiplication factor: 1.953125 
## qinf95 11 accuracy: 0.0224224  with multiplication factor: 0.9765625 
## qinf95 12 accuracy: 0.0224224  with multiplication factor: 0.4882812 
## qinf95 13 accuracy: 0.0224224  with multiplication factor: 0.2441406 
## qinf95 14 accuracy: 0.0224224  with multiplication factor: 0.1220703 
## qinf95 15 accuracy: 0.0224224  with multiplication factor: 0.06103516 
## qinf95 16 accuracy: 0.0224224  with multiplication factor: 0.03051758 
## qinf95 17 accuracy: 0.0224224  with multiplication factor: 0.01525879 
## qinf95 18 accuracy: 0.0224224  with multiplication factor: 0.007629395 
## qinf95 19 accuracy: 0.0224224  with multiplication factor: 0.003814697 
## qinf95 20 accuracy: 0.0224224  with multiplication factor: 0.001907349
## Warning in binarySearch_MFx(object = object, spaghetti = spaghetti,
## mcmc_size = mcmc_size, : For qinf95 the number of iteration reached the
## threshold number of iteration.
## qsup95 1 accuracy: 0.7022972  with multiplication factor: 1000 
## qsup95 2 accuracy: 0.7022972  with multiplication factor: 500 
## qsup95 3 accuracy: 0.7022972  with multiplication factor: 250 
## qsup95 4 accuracy: 0.7022972  with multiplication factor: 125 
## qsup95 5 accuracy: 0.7022972  with multiplication factor: 62.5 
## qsup95 6 accuracy: 0.7022972  with multiplication factor: 31.25 
## qsup95 7 accuracy: 0.7022972  with multiplication factor: 15.625 
## qsup95 8 accuracy: 0.7019493  with multiplication factor: 7.8125 
## qsup95 9 accuracy: 0.4482056  with multiplication factor: 3.90625 
## qsup95 10 accuracy: 0.1581915  with multiplication factor: 1.953125 
## qsup95 11 accuracy: 0.1062105  with multiplication factor: 2.929688 
## qsup95 12 accuracy: 0.1975842  with multiplication factor: 3.417969 
## qsup95 13 accuracy: 0.0292868  with multiplication factor: 3.173828 
## qsup95 14 accuracy: 0.04695287  with multiplication factor: 3.051758 
## qsup95 15 accuracy: 0.01055886  with multiplication factor: 3.112793 
## qsup95 16 accuracy: 0.009855705  with multiplication factor: 3.143311
plot(MFx_PRZ_cstSD_4MFx_x10_thresh20, log_scale = TRUE) 
## Warning in plot.MFx(MFx_PRZ_cstSD_4MFx_x10_thresh20, log_scale = TRUE): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

After the plot() function, you have the following message: Warning message: Removed 1 rows containing missing values (geom_point). This message comes from the use of ggplot() function (see the ggplot2 package) as an echo of the warning message about the missing point at \(2.5\%\) that has not been computed.

Change the default time of multiplication factor assessment

Multiplication factor is also available for the GUTS IT model (option quiet = TRUE remove the output):

# (2) Use the fit on constant exposure propiconazole with model IT. No print of run messages.
MFx_PRZ_cstIT_4pred <- MFx(object = fit_cstIT, data_predict = data_4MFx, time_MFx = 4, quiet = TRUE)

# (3) Plot the survival rate versus multiplication factors.
plot(MFx_PRZ_cstIT_4pred, log_scale = TRUE)
## Warning in plot.MFx(MFx_PRZ_cstIT_4pred, log_scale = TRUE): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

This last example set a reduction of \(10\%\) of the survival rate, remove the background mortality by setting hb_value = FALSE and is computed at time time_MFx = 4.

# (2) Use the fit on constant exposure propiconazole with model IT. No print of run messages.
MFx_PRZ_cstIT_4pred <- MFx(object = fit_cstIT, X=10, hb_value = FALSE, data_predict = data_4MFx, time_MFx = 4, quiet = TRUE)

plot(MFx_PRZ_cstIT_4pred, log_scale = TRUE)
## Warning in plot.MFx(MFx_PRZ_cstIT_4pred, log_scale = TRUE): This is not an error message:
## Just take into account that MFx as been estimated with a binary
## search using the 'accuracy' argument. Cross point indicated the
## position of evaluated time series. To improve the shape of the curve, you 
## can use X = NULL, and computed time series around the median MFx, with the
##           vector `MFx_range`.

plot(MFx_PRZ_cstIT_4pred, x_variable =  "Time")

Compute a range of multiplication factors

Once we have obtained the desired multiplication factor inducing the \(x\%\) reduction of the survival rate, it can be relevant to explore the sentivity of this parameter by exploring survival rate over a range of multiplication factors. This is possible by setting argument X = NULL and providing a range of wanted multiplication factors, for instance MFx_range = c() in our first example.

# Use the fit on constant exposure propiconazole with model SD.
MFx_PRZ_cstSD_4pred_range <- MFx(object = fit_cstSD, data_predict = data_4MFx, X = NULL, MFx_range = 1:6)

The associated plot if given by:

# Plot survival rate versus the range of multiplication factor.
plot(MFx_PRZ_cstSD_4pred_range)

And the argument x_variable = "Time" returns all computed time series:

# Plot Survival rate as function of time.
plot(MFx_PRZ_cstSD_4pred_range, x_variable = "Time")

To select a specific time series, we can use the element ls_predict wich is a list of object of class survFitPredict to wich a plot is defined.

# Plot a specific time series.
plot(MFx_PRZ_cstSD_4pred_range$ls_predict[[4]])

Reproduction data analysis at target-time

The steps for reproduction data analysis are absolutely analogous to what we described for survival data. Here, the aim is to estimate the relationship between the chemical compound concentration and the reproduction rate per individual-day.

Here is a typical session:

# (1) load dataset
data(cadmium2)

# (2) check structure and integrity of the dataset
reproDataCheck(cadmium2)
## No message
# (3) create a `reproData` object
dat <- reproData(cadmium2)

# (4) represent the cumulated number of offspring as a function of time
plot(dat, concentration = 124, addlegend = TRUE, pool.replicate = FALSE)

# (5) represent the reproduction rate as a function of concentration
plotDoseResponse(dat, target.time = 28)

# (6) check information 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  0
## 
## Number of offspring (sum of replicates) per time and concentration: 
##     0    3    7   10   14   17   21   24   28   31   35   38   42   45
## 0   0 1659 1587 2082 1580 2400 2069 2316 1822 2860 2154 3200 1603 2490
## 53  0 1221 1567 1710 1773 1859 1602 1995 1800 2101 1494 2126  935 1629
## 78  0 1066 2023 1752 1629 1715 1719 1278 1717 1451 1826 1610 1097 1727
## 124 0  807 1917 1423  567  383  568  493  605  631  573  585  546  280
## 232 0  270 1153  252   30    0   37   28   46  119   19    9    0    0
## 284 0  146  275   18    1    0    0    0    0    0    0    0    0    0
##       49   52   56
## 0   1609 2149 2881
## 53  2108 1686 1628
## 78  2309 1954 1760
## 124  594  328  380
## 232    0    0    0
## 284    0    0    0
# (7) fit a concentration-effect model at target-time
fit <- reproFitTT(dat, stoc.part = "bestfit",
                  target.time = 21,
                  ecx = c(10, 20, 30, 40, 50),
                  quiet = TRUE)
summary(fit)
## Summary: 
## 
## The  loglogistic  model with a Gamma Poisson stochastic part was used !
## 
## Priors on parameters (quantiles):
## 
##             50%      2.5%     97.5%
## b     1.000e+00 1.259e-02 7.943e+01
## d     1.830e+01 1.554e+01 2.107e+01
## e     1.488e+02 7.902e+01 2.804e+02
## omega 1.000e+00 1.585e-04 6.310e+03
## 
## Posteriors of the parameters (quantiles):
## 
##             50%      2.5%     97.5%
## b     3.850e+00 2.851e+00 5.887e+00
## d     1.774e+01 1.547e+01 2.018e+01
## e     1.369e+02 1.140e+02 1.742e+02
## omega 1.440e+00 8.191e-01 2.858e+00
## 
## Posteriors of the ECx (quantiles):
## 
##            50%      2.5%     97.5%
## EC10 7.723e+01 5.459e+01 1.175e+02
## EC20 9.531e+01 7.174e+01 1.354e+02
## EC30 1.098e+02 8.628e+01 1.492e+02
## EC40 1.232e+02 9.979e+01 1.615e+02
## EC50 1.369e+02 1.140e+02 1.742e+02
plot(fit, log.scale = TRUE, adddata = TRUE,
     cicol = "orange",
     addlegend = TRUE)

ppc(fit)

As in the survival analysis, we assume that the reproduction rate per individual-day is a log-logistic function of the concentration. More details and parameter signification can be found in the modelling vignette.

Model comparison

For reproduction analyses, we compare one model which neglects the inter-individual variability (named “Poisson”) and another one which takes it into account (named “gamma Poisson”). You can choose either one or the other with the option stoc.part. Setting this option to "bestfit", you let reproFitTT() decides which models fits the data best. The corresponding choice can be seen by calling the summary function:

summary(fit)
## Summary: 
## 
## The  loglogistic  model with a Gamma Poisson stochastic part was used !
## 
## Priors on parameters (quantiles):
## 
##             50%      2.5%     97.5%
## b     1.000e+00 1.259e-02 7.943e+01
## d     1.830e+01 1.554e+01 2.107e+01
## e     1.488e+02 7.902e+01 2.804e+02
## omega 1.000e+00 1.585e-04 6.310e+03
## 
## Posteriors of the parameters (quantiles):
## 
##             50%      2.5%     97.5%
## b     3.850e+00 2.851e+00 5.887e+00
## d     1.774e+01 1.547e+01 2.018e+01
## e     1.369e+02 1.140e+02 1.742e+02
## omega 1.440e+00 8.191e-01 2.858e+00
## 
## Posteriors of the ECx (quantiles):
## 
##            50%      2.5%     97.5%
## EC10 7.723e+01 5.459e+01 1.175e+02
## EC20 9.531e+01 7.174e+01 1.354e+02
## EC30 1.098e+02 8.628e+01 1.492e+02
## EC40 1.232e+02 9.979e+01 1.615e+02
## EC50 1.369e+02 1.140e+02 1.742e+02

When the gamma Poisson model is selected, the summary shows an additional parameter called omega, which quantifies the inter-individual variability (the higher omega the higher the variability).

Reproduction data and survival functions

In morse, reproduction datasets are a special case of survival datasets: a reproduction dataset includes the same information as in a survival dataset plus the information on reproduction outputs. For that reason, the S3 class reproData inherits from the class survData, which means that any operation on a survData object is legal on a reproData object. In particular, in order to use the plot function related to the survival analysis on a reproData object, we can use survData as a conversion function first:

dat <- reproData(cadmium2)
plot(survData(dat))

  1. In Bayesian inference, the parameters of a model are estimated from the data starting from a so-called prior, which is a probability distribution representing an initial guess on the true parameters, before seing the data. The posterior distribution represents the uncertainty on the parameters after seeing the data and combining them with the prior. To obtain a point estimate of the parameters, it is typical to compute the mean or median of the posterior. We can quantify the uncertainty by reporting the standard deviation or an inter-quantile distance from this posterior distribution.↩