1. Discrete distributions | 1.1. Base R distribution | 1.1.1. Geometric distribution | 1.1.2. Negative binomial distribution | 1.1.3. Poisson distribution | 1.1.4. Binomial distribution | 1.2. logarithmic distribution | 1.3. Zero truncated distributions | 1.4. Zero modified distributions | 1.5. Poisson inverse Gaussian distribution | 2. Continuous distributions | 2.1. Normal distribution | 2.2. Lognormal distribution | 2.3. Beta distribution (of the first kind) | 2.4. Other continuous distribution in actuar | 2.4.1. Log-gamma | 2.4.2. Gumbel | 2.4.3. Inverse Gaussian distribution | 2.4.4. Generalized beta | 2.5. Feller-Pareto family | The gradient with respect to $\theta, \alpha, \gamma, \tau$ is\begin{equation}\nabla\mathcal L(\mu, \theta, \alpha, \gamma, \tau) | 2.5.1. Transformed beta | 2.5.2. Generalized Pareto | 2.5.3. Burr | The survival function is$$1-F(x) = (1+(x/\theta)^\gamma)^{-\alpha}.$$Using the median $q_2$, we have$$\log(1/2) = - \alpha \log(1+(q_2/\theta)^\gamma).$$The initial value is\begin{equation}\alpha | 2.5.4. Loglogistic | 2.5.5. Paralogistic | 2.5.6. Inverse Burr | 2.5.7. Inverse paralogistic | 2.5.8. Inverse pareto | 2.5.9. Pareto IV | The first and third quartiles $q_1$ and $q_3$ verify$$((\frac34)^{-1/\alpha}-1)^{1/\gamma} = \frac{q_1-\mu}{\theta},((\frac14)^{-1/\alpha}-1)^{1/\gamma} = \frac{q_3-\mu}{\theta}.$$Hence we get two useful relations\begin{equation}\gamma | \frac{\log\left(\frac{(\frac43)^{1/\alpha}-1}{(4)^{1/\alpha}-1}\right)}{\log\left(\frac{q_1-\mu}{q_3-\mu}\right)},(#eq:pareto4gammarelation)\end{equation}\begin{equation}\theta | 2.5.10. Pareto III | Pareto III corresponds to Pareto IV with $\alpha=1$.\begin{equation}\gamma | \begin{equation}\theta | 2.5.11. Pareto II | 2.5.12. Pareto I | 2.5.13. Pareto | Pareto corresponds to Pareto IV with $\gamma=1$, $\mu=0$.\begin{equation}\theta | 2.6. Transformed gamma family | 2.6.1. Transformed gamma distribution | 2.6.2. gamma distribution | 2.6.3. Weibull distribution | 2.6.4. Exponential distribution | 2.7. Inverse transformed gamma family | 2.7.1. Inverse transformed gamma distribution | 2.7.2. Inverse gamma distribution | 2.7.3. Inverse Weibull distribution | 2.7.4. Inverse exponential | 3. Bibliography | 3.1. General books | 3.2. Books dedicated to a distribution family | 3.3. Books with applications

Prerequisites | Use case | Install and load kissDE | Quick start | kissDE's workflow | Input data | Condition vector | User's own data (without KisSplice): table of counts format | Input table from KisSplice output | Input table from KisSplice2refgenome output | Quality Control | Differential analysis | Output results | Final table | f/PSI table | kissDE's theory | Normalization | Estimation of dispersion | Pre-test filtering | Model fitting | Likelihood ratio test | Flagging low counts | Magnitude of the effect | Case studies | Application of kissDE to alternative splicing | Dataset | Load data | Quality control | Differential analysis | Export results | Explore results | One command to rule them all | Application of kissDE to SNPs/SNVs | Time / Requirements | Session info

1. Questions regarding distributions | 1.1. How do I know the root name of a distribution? | 1.2. How do I find "non standard" distributions? | 1.3. How do I set (or find) initial values for non standard distributions? | 1.4. Is it possible to fit a distribution with at least 3 parameters? | 1.5. Why there are differences between MLE and MME for the lognormal distribution? | 1.6. Can I fit a distribution with positive support when data contains negative values? | 1.7. Can I fit a finite-support distribution when data is outside that support? | 1.8. Can I fit truncated distributions? | 1.9. Can I fit truncated inflated distributions? | 1.10. Can I fit a uniform distribution? | 1.11. Can I fit a beta distribution with the same shape parameter? | 1.12. How to estimate support parameter? the case of the four-parameter beta | 2. Questions regarding goodness-of-fit tests and statistics, Cullen-Frey graph | 2.1. Where can we find the results of goodness-of-fit tests ? | 2.2. Is it reasonable to use goodness-of-fit tests to validate the fit of a distribution ? | 2.2.1. Should I reject a distribution because a goodness-of-fit test rejects it ? | 2.2.2. Should I accept a distribution because goodness-of-fit tests do not reject it ? | 2.3. Why all goodness-of-fit tests are not available for every distribution ? | 2.4. How can we use goodness-of-fit statistics to compare the fit of different distributions on a same data set ? | 2.5. Can we use a test to compare the fit of two distributions on a same data set ? | 2.6. Can we get goodness-of-fit statistics for a fit on censored data ? | 2.7. Why Cullen-Frey graph may be misleading? | 3. Questions regarding optimization procedures | 3.1. How to choose optimization method? | 3.2. The optimization algorithm stops with error code, e.g., 100. What shall I do? | 3.3 Why distribution with a log argument may converge better? | 3.4. What to do when there is a scaling issue? | 3.5. How do I set bounds on parameters when optimizing? | 3.5.1. Setting bounds for scale parameters | 3.5.2. Setting bounds for shape parameters | 3.5.3. Setting bounds for probability parameters | 3.5.4. Setting bounds for boundary parameters | 3.5.5. Setting linear inequality bounds | 3.6. How does quantile matching estimation work for discrete distributions? | 3.7. Why setting a parameter to the true value does not lead to the expected result for other parameters? | 4. Questions regarding uncertainty | 4.1. Can we compute marginal confidence intervals on parameter estimates from their reported standard error ? | 4.2. How can we compute confidence intervals on quantiles from the fit of a distribution ? | 4.3. How can we compute confidence intervals on any function of the parameters of the fitted distribution ? | 4.4. How do we choose the bootstrap number? | 5. How to personalize plots | 5.1. Can I personalize the default plot given for an object of class fitdist or fitdistcens? | 5.2. How to personalize goodness-of-fit plots ? | 5.3. Is it possible to obtain ggplot2 plots ? | 5.4. Is it possible to add the names of the observations in a goodness-of-fit plot, e.g. the names of the species in the plot of the Species Sensitivity Distribution (SSD) classically used in ecotoxicology ? | 6. Questions regarding (left, right and/or interval) censored data | 6.1. How to code censored data in fitdistrplus ? | 6.2. How do I prepare the input of fitdistcens() with Surv2fitdistcens()? | 6.3. How to represent an empirical distribution from censored data ? | 6.4. How to assess the goodness-of-fit of a distribution fitted on censored data ?

1. Introduction | 2. Fitting distributions to continuous non-censored data | 2.1. Choice of candidate distributions | 2.2. Fit of distributions by maximum likelihood estimation | 2.3. Uncertainty in parameter estimates | 3. Advanced topics | 3.1. Alternative methods for parameter estimation | 3.3. Fitting distributions to other types of data | 4. Conclusion | Acknowledgments | References

1. Quick overview of main optimization methods | 1.1. Derivative-free optimization methods | 1.2. Hessian-free optimization methods | 1.2.1. Computing the direction $d_k$ | 1.2.2. Computing the stepsize $t_k$ | 1.3. Benchmark | 2. Numerical illustration with the beta distribution | 2.1. Log-likelihood function and its gradient for beta distribution | 2.1.1. Theoretical value | 2.1.2. R implementation | 2.2. Random generation of a sample | 2.3 Fit Beta distribution | 2.4. Results of the numerical investigation | 3. Numerical illustration with the negative binomial distribution | 3.1. Log-likelihood function and its gradient for negative binomial distribution | 3.1.1. Theoretical value | 3.1.2. R implementation | 3.2. Random generation of a sample | 3.3. Fit a negative binomial distribution | 3.4. Results of the numerical investigation | 4. Conclusion

Introduction | Main workflow | Step 1: importation, check and normalization / transformation of data if needed | General format of imported data | Importation of data from a unique text file | Importation of data as an R object | What types of data can be analyzed using DRomics ? | Description of the classical types of data handled by DRomics | An example with RNAseq data | An example with microarray data | An example with metabolomic data | An example with continuous anchoring apical data | Handling of data collected through specific designs | An example with in situ (observational) RNAseq data | An example with RNAseq data from an experiment with a batch effect | Step 2: selection of significantly responding items | Step 3: fit of dose-response models, choice of the best fit for each curve | Fit of the best model | Plot of fitted curves | Plot of residuals | Description of the family of dose-response models fitted in DRomics | Reminder on least squares regression | Step 4: calculation of benchmark doses (BMD) | Calculation of BMD | Plots of the BMD distribution | Calculation of confidence intervals on the BMDs by bootstrap | Filtering BMDs according to estimation quality | Plot of fitted curves with BMD values and confidence intervals | Plot of all the fitted curves in one figure with points at BMD-BMR values | Description of the outputs of the complete DRomics workflow | Help for biological interpretation of DRomics outputs | Interpretation of DRomics results in a simple case with only one data set obtained in one experimental condition | Augmentation of the data frame of DRomics results with biological annotation | Various plots of results by biological group | BMD ECDF plots split by group defined from biological annotation | Sensitivity plot of biological groups | Trend plot per biological group | Plot of dose-response curves per biological group | Comparison of DRomics results obtained at different experimental levels, for example in a multi-omics approach | Augmentation of the data frames of DRomics results with biological annotation | Binding of the data frames corresponding the results at each experimental level | Comparison of results obtained at the different experimental levels using basic R functions | Comparison of results obtained at the different experimental levels using DRomics functions | ECDF plot of BMD values per group and experimental level using DRomics functions | Plot of the trend repartition per group and experimental level | Sensitivity plot per group and experimental level | Selection of groups on which to focus using the selectgroups() function | BMD ECDF plot with color gradient split by group and experimental level | Plot of the dose-response curves for a selection of groups | References

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A Simple Example: Male Gammarus Single | Data | Inference | Results | TK parameters | Male Gammarus Merged | Male Gammarus seanine with growth | Oncorhynchus | Chironomus benzo-a-pyrene | Prediction