# 2 Main workflow

## 2.1 Step 1: importation, check and normalization / transformation of data if needed

### 2.1.1 General format of imported data

#### 2.1.1.1 Importation of data from a unique text file

Whatever the type of data imported in DRomics (e.g. RNAseq, microarray, metabolomic data), data can be imported from a .txt file (e.g. “mydata.txt”) organized with **one row per item** (e.g. transcript, probe, metabolite) and **one column per sample**. In **an additional first row**, after a name for the item identifier (e.g. “id”), we must have the **tested doses or concentrations in a numeric format** for the corresponding sample (for example, if there are triplicates for each treatment, the first line could be “item”, 0, 0, 0, 0.1, 0.1, 0.1, etc.). An **additional first column** must give the **identifier of each item** (identifier of the probe, transcript, metabolite, …, or name of the endpoint for anchoring data), and the other columns give the responses of the item for each sample. This file will be imported within DRomics using an internal call to the function read.table() with its default field separator (sep argument).

Different examples of .txt files formatted for the DRomics workflow are available in the package, such as the one named “RNAseq_sample.txt”. You can have a look of how the data are coded in this file using the following code. **To use a local dataset formatted in the same way, such use a datafilename of type "yourchosenname.txt".**

```
# Import the text file just to see what will be automatically imported
system.file("extdata", "RNAseq_sample.txt", package = "DRomics")
datafilename <-# datafilename <- "yourchosenname.txt" # for a local file
# Have a look of what information is coded in this file
read.table(file = datafilename, header = FALSE)
d <-nrow(d)
```

`## [1] 1000`

`head(d)`

```
## V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
## 1 RefSeq 0 0 0.22 0.22 0.22 0.67 0.67 0.67 2
## 2 NM_144958 2072 2506 2519.00 2116.00 1999.00 2113.00 2219.00 2322.00 2359
## 3 NR_102758 0 0 0.00 0.00 0.00 0.00 0.00 0.00 0
## 4 NM_172405 198 265 250.00 245.00 212.00 206.00 227.00 246.00 265
## 5 NM_029777 18 29 25.00 19.00 19.00 13.00 22.00 19.00 19
## 6 NM_001130188 0 0 0.00 0.00 0.00 0.00 0.00 1.00 0
## V11 V12 V13 V14 V15
## 1 2 2 6 6 6
## 2 1932 1705 2110 2311 2140
## 3 0 0 0 0 0
## 4 205 175 288 315 242
## 5 26 16 26 32 33
## 6 0 0 1 0 1
```

#### 2.1.1.2 Importation of data as an R object

**Alternatively an R object of class data.frame can be directly given in input**, that corresponds to the output of read.table(file, header = FALSE) on a file described in the previous section.

You can see below an example of an RNAseq data set that is available in DRomics as an R object (named Zhou_kidney_pce) and which is an extended version (more rows) of the previous dataset coded in “RNAseq_sample.txt”.

```
# Load and look at the dataset directly coded as an R object
data(Zhou_kidney_pce)
nrow(Zhou_kidney_pce)
```

`## [1] 33395`

`head(Zhou_kidney_pce)`

```
## V1 V2 V3 V4 V5 V6 V7 V8 V9 V10
## 1 RefSeq 0 0 0.22 0.22 0.22 0.67 0.67 0.67 2
## 2 NM_144958 2072 2506 2519.00 2116.00 1999.00 2113.00 2219.00 2322.00 2359
## 3 NR_102758 0 0 0.00 0.00 0.00 0.00 0.00 0.00 0
## 4 NM_172405 198 265 250.00 245.00 212.00 206.00 227.00 246.00 265
## 5 NM_029777 18 29 25.00 19.00 19.00 13.00 22.00 19.00 19
## 6 NM_001130188 0 0 0.00 0.00 0.00 0.00 0.00 1.00 0
## V11 V12 V13 V14 V15
## 1 2 2 6 6 6
## 2 1932 1705 2110 2311 2140
## 3 0 0 0 0 0
## 4 205 175 288 315 242
## 5 26 16 26 32 33
## 6 0 0 1 0 1
```

If your data are already imported in R in a **different format as the one described above**, you can use the **formatdata4DRomics() function to build an R object directly useable by the DRomics workflow**. formatdata4DRomics() needs two arguments in input:

- the
**matrix of the data**with**one row for each item**and**one column for each sample**, - and the
**numeric vector coding for the dose**for each sample.

The names of the samples can be added in a third optional argument (see ?formatdata4DRomics for details). Below is an example using a RNAseq dataset of the package coded as an R object named zebraf.

```
# Load and look at the data as initially coded
data(zebraf)
str(zebraf)
```

```
## List of 3
## $ counts: int [1:1000, 1:16] 453 331 897 12 326 533 1948 904 583 154 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:1000] "ENSDARG00000102141" "ENSDARG00000102123" "ENSDARG00000114503" "ENSDARG00000115971" ...
## .. ..$ : chr [1:16] "I10_05mG_E5" "I10_05mG_E6" "I10_05mG_E7" "I10_C5" ...
## $ dose : num [1:16] 500 500 500 0 0 0 0 50000 50000 50000 ...
## $ batch : Factor w/ 2 levels "I10","I17": 1 1 1 1 1 1 1 2 2 2 ...
```

```
# Formatting of data for use in DRomics
#
formatdata4DRomics(signalmatrix = zebraf$counts,
data4DRomics <-dose = zebraf$dose)
# Look at the dataset coded as an R object
nrow(data4DRomics)
```

`## [1] 1001`

`head(data4DRomics)`

```
## V1 I10_05mG_E5 I10_05mG_E6 I10_05mG_E7 I10_C5 I10_C6 I10_C7
## 1 item 500 500 500 0 0 0
## 2 ENSDARG00000102141 453 656 590 636 529 453
## 3 ENSDARG00000102123 331 505 401 465 488 430
## 4 ENSDARG00000114503 897 1055 1073 1017 1022 992
## 5 ENSDARG00000115971 12 23 24 21 40 32
## 6 ENSDARG00000098311 326 544 416 484 330 297
## I10_C8 I17_50mG_E5 I17_50mG_E6 I17_50mG_E8 I17_5mG_E5 I17_5mG_E7 I17_5mG_E8
## 1 0 50000 50000 50000 5000 5000 5000
## 2 566 790 557 1005 688 775 425
## 3 582 498 427 815 450 675 307
## 4 1124 1153 900 2115 1064 1555 824
## 5 30 22 14 34 19 36 21
## 6 396 502 380 1065 522 688 332
## I17_C5 I17_C7 I17_C8
## 1 0 0 0
## 2 719 516 667
## 3 522 475 541
## 4 1449 1152 1341
## 5 44 27 22
## 6 666 471 506
```

Whatever the way you format your data, **we strongly recommend you to carefully look at the following sections** to **check that you use the good scale for you data**, which **depends on the type of measured signal ** (counts of reads, fluorescence signal, …).

### 2.1.2 What types of data can be analyzed using DRomics ?

#### 2.1.2.1 Description of the classical types of data handled by DRomics

DRomics offers the possibility to work on different types of omics data (see following subsections for their description) but also on continuous anchoring data. **When working on omics data, all the lines of the data frame** (except the first one coding for the doses or concentrations) **correspond to the same type of data** (e.g. raw counts for RNAseq data). **When working on anchoring data, the different lines** (except the first one coding for the doses or concentrations) **correspond to different endpoints that may correspond to different types of data** (e.g. biomass, length,..), but all are assumed continuous data compatible with a Gaussian (normal) error model (after transformation if needed, e.g. logarithmic transformation) for the selection and modeling steps (see the section on least squares if you need a reminder on the is condition).

**Three types of omics data** may be imported in DRomics using the following functions:

**RNAseqdata()**should be used to import**RNAseq as counts of reads**(for details look at the example with RNAseq data),**microarraydata()**should be used to import**single-channel microarray data in log2 scale**(for details look at the example with microarray data),**continuousomicdata()**should be used to import**other continuous omics data**such as metabolomics, proteomics,…,**in a scale that enables the use of a Gaussian error model**(for details look at the example with metabolomic omics data).

It is also possible to import in DRomics **continuous anchoring data** measured at the apical level, especially for **sake of comparison of benchmark doses** (see Step 4 for definition of BMD) estimated **at different levels of organization** but using **the same metrics**. Nevertheless, one should keep in mind that the DRomics workflow was optimized for an automatic analysis of high throughput omics data (especially implying a selection and modeling steps on high-dimensional data) and that **other tools may be better suited for the sole analysis of apical dose-response data** (for details look at the example with continuous apical data).

In Steps 1 and 2 **count data** are internally analysed using functions of the Bioconductor package DESeq2, continuous omics data (**microarray data and other continuous omics data**) are internally analysed using functions of the Bioconductor package limma and **continuous anchoring data** are internally analysed using the classical lm() function.

##### 2.1.2.1.1 An example with RNAseq data

For RNAseq data, **imperatively imported in raw counts** (if your counts come from Kallisto or Salmon put add the argument `round.counts = TRUE`

in order to round them), you have to choose the transformation method used to stabilize the variance (“rlog” or “vst”). In the example below “vst” was used to make this vignette quick to compile, but **“rlog” is recommended and chosen by default even if more computer intensive than “vst” except when the number of samples is very large (> 30)** (as encountered for *in situ* data for example: see ?RNAseqdata and the section dedicated to *in situ* data for details on this point). Whatever the chosen method, **data are automatically normalized with respect to library size and transformed in log2 scale**.

```
system.file("extdata", "RNAseq_sample.txt", package = "DRomics")
RNAseqfilename <-# RNAseqfilename <- "yourchosenname.txt" # for a local file
```

` RNAseqdata(RNAseqfilename, transfo.method = "vst")) (o.RNAseq <-`

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 0.22 0.67 2 6
## 2 3 3 3 3
## Number of items: 999
## Identifiers of the first 20 items:
## [1] "NM_144958" "NR_102758" "NM_172405" "NM_029777" "NM_001130188"
## [6] "NM_207141" "NM_001162368" "NM_008117" "NM_001168290" "NM_010910"
## [11] "NM_001004147" "NM_001146318" "NM_145597" "NM_001161797" "NM_021483"
## [16] "NR_002862" "NR_033520" "NM_134027" "NM_010381" "NM_019388"
## Data were normalized with respect to library size and tranformed using
## the following method: vst
```

`plot(o.RNAseq, cex.main = 0.8, col = "green")`

The plot of the output shows the distribution of the signal on all the contigs/genes, for each sample, before and after the normalization and transformation of data.

##### 2.1.2.1.2 An example with microarray data

For single-channel microarray data, **imperatively imported in log scale** (classical and recommended log2 scale), you can choose between array normalization methods (“cyclicloess”, “quantile”, “scale” or “none”). In the example below, “quantile” was used to make this vignette quick to compile, but **“cyclicloess” is recommended and chosen by default even if more computer intensive than the others** (see ?microarraydata for details).

```
system.file("extdata", "transcripto_sample.txt", package = "DRomics")
microarrayfilename <-# microarrayfilename <- "yourchosenname.txt" # for a local file
```

` microarraydata(microarrayfilename, norm.method = "quantile")) (o.microarray <-`

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 0.69 1.223 2.148 3.774 6.631
## 5 5 5 5 5 5
## Number of items: 1000
## Identifiers of the first 20 items:
## [1] "1" "2" "3" "4" "5.1" "5.2" "6.1" "6.2" "7.1" "7.2"
## [11] "8.1" "8.2" "9.1" "9.2" "10.1" "10.2" "11.1" "11.2" "12.1" "12.2"
## Data were normalized between arrays using the following method: quantile
```

`plot(o.microarray, cex.main = 0.8, col = "green")`

The plot of the output shows the distribution of the signal over all the probes, for each sample, before and after the normalization of data.

##### 2.1.2.1.3 An example with metabolomic data

**Neither normalization nor transformation** is provided in the function continuousomicdata(). The **pre-treatment of metabolomic data must be done before importation of data**, and **data must be imported in log scale**, so that they can be directly modelled using a **Gaussian (normal) error model**. This strong hypothesis is required both for selection of items and for dose-reponse modeling (see the section on least squares for a reminder if needed). In the context of a multi-omics approach we recommend the use of a log2 transformation, instead of the classical log10 for such data, so as to facilitate the comparison of results obtained with transcriptomics data generally handled in a log2 scale.

For instance, a basic procedure for the pre-treatment of metabolomic data could follow the three steps described thereafter: i) removing of metabolites for which the proportion of missing data (non detected) across all the samples is too high (more than 20 to 50 percents according to your tolerance level); ii) retrieving of missing values data using half minimum method (i.e. half of the minimum value found for a metabolite across all samples); iii) log-transformation of values. If a scaling to the total intensity (normalization by sum of signals in each sample) or another normalization is necessary and pertinent, we recommend to do it before those three previously described steps.

```
system.file("extdata", "metabolo_sample.txt", package = "DRomics")
metabolofilename <-# metabolofilename <- "yourchosenname.txt" # for a local file
```

` continuousomicdata(metabolofilename)) (o.metabolo <-`

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 0.69 1.1 1.79 2.92 4.78 7.76
## 10 6 2 2 2 6 2
## Number of items: 109
## Identifiers of the first 20 items:
##
## [1] "P_2" "P_4" "P_5" "P_6" "P_7" "P_10" "P_11" "P_12" "P_14" "P_16"
## [11] "P_19" "P_21" "P_22" "P_26" "P_32" "P_34" "P_35" "P_36" "P_37" "P_38"
```

`plot(o.metabolo, col = "green")`

The plot of the output shows the distribution of the signal over all the metabolites, for each sample.

The deprecated metabolomicdata() function was renamed continuousomicdata() in the recent versions of the package (while keeping the first name available) to **offer its use to other continuous omic data** such as **proteomics data** or **RT-qPCR data**. As for metabolomic data, the **pre-treatment** of other continuous omic data must be done **before importation**, and **data must be imported in a scale that enables the use of a Gaussian error model** as this strong hypothesis is required both for selection of items and for dose-response modeling.

##### 2.1.2.1.4 An example with continuous anchoring apical data

No transformation is provided in the function continuousanchoringdata(). **If needed the pre-treatment of data must be done before importation of data**, so that they can be directly modelled using a **Gaussian error model**. This strong hypothesis is required both for selection of responsive endpoints and for dose-reponse modeling (see the section on least squares for a reminder if needed).

```
system.file("extdata", "apical_anchoring.txt", package = "DRomics")
anchoringfilename <-# anchoringfilename <- "yourchosenname.txt" # for a local file
```

In the following example the argument backgrounddose is used to specify that doses below or equal to 0.1 are considered as 0 in the DRomics workflow. Specifying this argument is necessary when there is no dose at 0 in the data (see section on *in situ* data for details on this point).

` continuousanchoringdata(anchoringfilename, backgrounddose = 0.1)) (o.anchoring <-`

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 2.4 3.8 6.2 10.1 16.5 26.8 43.5 70.7
## 12 6 2 2 2 6 2 2 2
## Number of endpoints: 2
## Names of the endpoints:
## [1] "growth" "photosynthesis"
```

`plot(o.anchoring)`

For such data the plot() function simply provides a dose-response plot for each endpoint.

#### 2.1.2.2 Handling of data collected through specific designs

The DRomics workflow was first developed for data collected through a typical dose-response experiment, with a reasonable number of tested doses (or concentrations - at least 4 in addition to the control and ideally 6 to 8) and a small number of replicates per dose. Recently we made some modifications in the package to **make possible the use of designs with only 3 doses in addition to the control even if this type of design is not recommended for dose-response modeling**.

**We also extended our workflow to handle in situ (observational) data**, for which there is

**no replication, as the dose (or concentration) is not controlled**(see an example with in situ data for more details).

It is also now possible to handle **experimental data collected using a design with a batch effect** using DRomics together with functions from the Bioconductor package sva to correct for this batch effect before selection and modeling steps. We also developed the **PCAplot() function to help visualizing this batch effect** and the impact of the **batch effect correction (BEC)** on data (see an example with RNAseq data from an experiment with a batch effect for more details on how to handle such a case and see ?PCAplot for details on this specific function **that could also be used to identify potential outlier samples**).

##### 2.1.2.2.1 An example with *in situ* (observational) RNAseq data

One of the problem that may occur in particular with *in situ* data, is the absence of real control samples, corresponding to a strictly null exposure dose or concentration. **To prevent an hazardous calculation of the BMD (see Step 4 for definition of BMD) by extrapolation in such a case, one should use the argument backgrounddose to define the maximal measured dose that can be considered as a negligible dose.** All doses below or equal to the value given in backgrounddose will be fixed at 0, so as to be considered at the **background level of exposition**.

For *in situ* data (and more generally for data with a very large number of samples), the **use of the rlog transformation in RNAseqdata() is not recommended**, both for speed reason and because you are **more likely to encounter a problem with the rlog transformation in case of outliers** in such a case (see https://support.bioconductor.org/p/105334/ for an explanation of the author of DESeq2 and if you want to see an example of problems that may appear with outliers in that case, just force the `transfo.method`

to `"rlog"`

in the following example).

```
system.file("extdata", "insitu_RNAseq_sample.txt", package="DRomics")
datafilename <-# Importation of data specifying that observed doses below the background dose
# fixed here to 2e-2 will be considered as null dose to have a control
RNAseqdata(datafilename, backgrounddose = 2e-2, transfo.method = "vst")) (o.insitu <-
```

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 0.0205 0.0216 0.0248 0.0272 0.0322 0.0339 0.0387 0.0432 0.0463
## 25 1 1 1 1 1 1 1 1 1
## 0.04866 0.0528 0.0726 0.101 0.112 0.1122 0.167 2.089 2.474 2.892
## 1 1 3 1 1 1 1 1 1 1
## 2.899 2.904 3.008 3.16 3.199 3.251 3.323 3.483 3.604 4.484
## 1 1 1 1 1 1 1 1 1 1
## 4.917 4.924 5.509 8.53 9.16 9.38 9.5 9.83 10.35 11.06
## 1 1 1 1 1 1 1 1 1 1
## 11.34 11.94 12.11 12.39 13.51 14.8 16.26 16.89 18.2 18.28
## 1 1 1 1 1 1 1 1 1 1
## 19.14 20.62 20.7 21.13 23.0348 23.69 24.15 27.06 28.05 29.88
## 1 1 1 1 1 1 1 1 1 1
## 36.28
## 1
## Number of items: 1000
## Identifiers of the first 20 items:
## [1] "N00000000001_c0_g1" "N00000000002_c0_g1" "N00000000003_c0_g1"
## [4] "N00000000004_c0_g1" "N00000000005_c0_g1" "N00000000006_c0_g1"
## [7] "N00000000008_c0_g1" "N10000_c0_g1" "N10000_c0_g1.1"
## [10] "N10001_c0_g1" "N10008_c1_g1" "N10010_c0_g1"
## [13] "N10010_c1_g1" "N10011_c0_g1" "N10012_c0_g1"
## [16] "N100156_c0_g1" "N1001_c0_g1" "N10021_c0_g1"
## [19] "N10021_c0_g1.1" "N10021_c0_g1.2"
## Data were normalized with respect to library size and tranformed using
## the following method: vst
```

`plot(o.insitu)`

The plot of the output shows the distribution of the signal on all the contigs, for each sample (here the box plots are stuck to each other due to the large number of samples), before and after the normalization and transformation of data.

##### 2.1.2.2.2 An example with RNAseq data from an experiment with a batch effect

When omics data are collected through a **design with a known potential batch effect**, the **DRomics function PCAplot()** can be used as in the example below to **visualize the impact of this batch effect on the data**. If it seems necessary, **functions from specific packages** can then be used to perform **batch effect correction (BEC)**. We recommend the use of functions **ComBat()** and **ComBat_seq()** from the Bioconductor **sva** package for this purpose, respectively for **microarray** (or other continuous omic data) and **RNAseq data**.

As **sva** is a Bioconductor package, it must be installed in the same way as **DESeq2** and **limma** previously to be loaded. If needed look at the DRomics web page to get the good instruction to install Bioconductor packages: https://lbbe.univ-lyon1.fr/fr/dromics).

Below is an example using ComBat-seq() on RNAseq data with batch effect. As the sva package does not import the RNAseq data in the same format as DRomics, it is necessary to use the DRomics function formatdata4DRomics() to interoperate between ComBat-seq in DRomics functions (see the section on importation as an R object for details on this function or ?formatdata4DRomics).

```
# Load of data
data(zebraf)
str(zebraf)
```

```
## List of 3
## $ counts: int [1:1000, 1:16] 453 331 897 12 326 533 1948 904 583 154 ...
## ..- attr(*, "dimnames")=List of 2
## .. ..$ : chr [1:1000] "ENSDARG00000102141" "ENSDARG00000102123" "ENSDARG00000114503" "ENSDARG00000115971" ...
## .. ..$ : chr [1:16] "I10_05mG_E5" "I10_05mG_E6" "I10_05mG_E7" "I10_C5" ...
## $ dose : num [1:16] 500 500 500 0 0 0 0 50000 50000 50000 ...
## $ batch : Factor w/ 2 levels "I10","I17": 1 1 1 1 1 1 1 2 2 2 ...
```

```
# Look at the design of this dataset
xtabs(~ zebraf$dose + zebraf$batch)
```

```
## zebraf$batch
## zebraf$dose I10 I17
## 0 4 3
## 500 3 0
## 5000 0 3
## 50000 0 3
```

It appears in this design that the data were obtained using to batches, with only the controlled condition (null dose) appearing in both batches.

```
# Formating of data using the formatdata4DRomics() function
formatdata4DRomics(signalmatrix = zebraf$counts,
data4DRomics <-dose = zebraf$dose)
# Importation of data just to use DRomics functions
# As only raw data will be given to ComBat_seq after
RNAseqdata(data4DRomics)) (o <-
```

```
## Just wait, the transformation using regularized logarithm (rlog) may
## take a few minutes.
```

```
## Elements of the experimental design in order to check the coding of the data:
## Tested doses and number of replicates for each dose:
##
## 0 500 5000 50000
## 7 3 3 3
## Number of items: 1000
## Identifiers of the first 20 items:
## [1] "ENSDARG00000102141" "ENSDARG00000102123" "ENSDARG00000114503"
## [4] "ENSDARG00000115971" "ENSDARG00000098311" "ENSDARG00000104839"
## [7] "ENSDARG00000100143" "ENSDARG00000102474" "ENSDARG00000104049"
## [10] "ENSDARG00000102226" "ENSDARG00000103095" "ENSDARG00000102128"
## [13] "ENSDARG00000110470" "ENSDARG00000100422" "ENSDARG00000104632"
## [16] "ENSDARG00000100660" "ENSDARG00000113107" "ENSDARG00000099787"
## [19] "ENSDARG00000112451" "ENSDARG00000070546"
## Data were normalized with respect to library size and tranformed using
## the following method: rlog
```

```
# PCA plot to visualize the batch effect
PCAdataplot(o, batch = zebraf$batch)
```

The PCA plot shows an impact of the batch effect, that clearly appears on controls (red points) which were obtained on two different batches.

```
# Batch effect correction using ComBat_seq{sva}
require(sva)
ComBat_seq(as.matrix(o$raw.counts),
BECcounts <-batch = as.factor(zebraf$batch),
group = as.factor(o$dose))
```

```
# Formating of data after batch effect correction
formatdata4DRomics(signalmatrix = BECcounts,
BECdata4DRomics <-dose = o$dose)
RNAseqdata(BECdata4DRomics) o.BEC <-
```

```
## Just wait, the transformation using regularized logarithm (rlog) may
## take a few minutes.
```

```
# PCA plot after batch effect correction
PCAdataplot(o.BEC, batch = zebraf$batch)
```

The PCA plot after BEC shows the impact of the correction on the batch effect that is no more visible on controls (red points).

## 2.2 Step 2: selection of significantly responding items

For the second step of the workflow, the function **itemselect()** must be used simply **taking as input in a first argument the output of the function used in step 1** (output of RNAseqdata(), microarraydata(), continuousomicdata() or continuousanchoringdata()). Below is an example with microarray data.

` itemselect(o.microarray, select.method = "quadratic", FDR = 0.01)) (s_quad <-`

```
## Number of selected items using a quadratic trend test with an FDR of 0.01: 150
## Identifiers of the first 20 most responsive items:
## [1] "383.2" "384.2" "363.1" "383.1" "384.1" "363.2" "364.2" "364.1" "300.2"
## [10] "301.1" "300.1" "301.2" "263.2" "27.2" "25.1" "368.1" "351.1" "15"
## [19] "370" "350.2"
```

The **false discovery rate (FDR) corresponds to the expected proportion of items that will be falsely detected as responsive**. **With a very large data set** it is important to define a selection step based on an FDR not only **to reduce the number of items to be further processed**, but also **to remove too noisy dose-response signals that may impair the quality of the results**. We recommend to set a value between 0.001 and 0.1 depending of the initial number of items. When this number is very high (more than several tens of thousands), we recommend a FDR less than 0.05 (0.001 to 0.01) to increase the robustness of the results (Larras et al. 2018).

Concerning the method used for selection, **we recommend the default choice (“quadratic”) for a typical omics dose-response design (many doses/concentrations with few replicates per condition)**. It enables the **selection of both monotonic and biphasic dose-response relationships**. If you want to focus on monotonic dose-response relationships, the “linear” method could be chosen. For a design with a small number of doses/concentrations and many replicates (not optimal for dose-response modeling), the “ANOVA” method could be preferable. **For in situ data** (observational data without replicates due to uncontrolled dose),

**only trend tests will be proposed**as the use of an ANOVA test in absence of replicates for some conditions is not reasonable.

Each of the three methods proposed for this selection step is based on the use of a simple model (quadratic,linear or ANOVA-type) linking the signal to the dose in a rank scale. This model is internally fitted to data by an **empirical Bayesian approach** using the respective packages **DESeq2** and **limma** for **RNAseq data** and **microarray or continuous omics data**, and by classical **linear regression** using the lm() function for **continuous anchoring data**. The adjustment of p-values according to the specified FDR is performed in any case, even for continuous anchoring data, so as to ensure the unicity of the workflow independently of the type of data.

See ?itemselect for more details and Larras et al. 2018 comparison of the three proposed methods on an example.

## 2.3 Step 3: fit of dose-response models, choice of the best fit for each curve

### 2.3.1 Fit of the best model

For Step 3 the function drcfit() simply **takes as input in a first argument the output of itemselect()**. For each item selected in Step 2, the model that best fits the dose-response data is chosen among a **family of five simple models built to describe a wide variety of monotonic and biphasic dose-response curves (DRC)** (and exclusively monotonic and biphasic curves : it is why more flexible models such as polynomial third and fourth order classical polynomial models were deliberately not considered). For a complete description of those models see the last section of Step 3 or Larras et al. 2018.

The procedure used to **select the best fit** is based on an **information criterion** as described in Larras et al. 2018 and in ?drcfit. The **classical** and **former default option** of the **AIC** (Akaike criterion - default information criterion used in DRomics versions < 2.2-0) was **replaced by the default use of the AICc** (second-order Akaike criterion) in order to **prevent the overfitting** that may occur with dose-response designs with a small number of data points, as recommended and now classically done in regression (Hurvich and Tsai, 1989; Burnham and Anderson DR, 2004).

As the call to the drcfit() function may take time when the number of pre-selected items is large, by default a progressbar is provided. Some arguments of this function can be used to specify **parallel computing to accelerate the computation** (see ?drcfit for details).

` drcfit(s_quad, progressbar = FALSE)) (f <-`

```
## Results of the fitting using the AICc to select the best fit model
## 22 dose-response curves out of 150 previously selected were removed
## because no model could be fitted reliably.
## Distribution of the chosen models among the 128 fitted dose-response curves:
##
## Hill linear exponential Gauss-probit
## 1 30 39 47
## log-Gauss-probit
## 11
## Distribution of the trends (curve shapes) among the 128 fitted dose-response curves:
##
## U bell dec inc
## 19 38 37 34
```

In the following you can see the first lines of the output data frame on our example (see ?drcfit for a complete description of the columns of the output data frame.) This output data frame provides information for each item, such as **the best-fit model**, **its parameter values**, the **standard residual error (SDres)** (see the section on least squares for his definition), **coordinates of particular points**, and the **trend of the curve** (among increasing, decreasing, U-shaped, bell-shaped).

An extensive description of the outputs of the complete DRomics workflow is provided in the last section of the main workflow. Note that the number of items successfully fitted (output of Step 3) is often smaller that the number of items selected in Step 2, as for some of the selected items, all models may fail to converge or fail to significantly better describe the data than the constant model.

`head(f$fitres)`

```
## id irow adjpvalue model nbpar b c d e f SDres
## 1 383.2 725 2.08e-07 Gauss-probit 4 5.5836 8.58 8.58 1.70 3.62 0.157
## 2 384.2 727 2.08e-07 exponential 3 -0.0298 NA 12.24 1.76 NA 0.160
## 3 363.1 686 2.24e-07 exponential 3 -0.2058 NA 9.10 3.11 NA 0.218
## 4 383.1 724 2.24e-07 Gauss-probit 4 5.4879 8.75 8.75 1.72 3.58 0.169
## 5 384.1 726 3.41e-07 Gauss-probit 4 6.8453 7.26 7.26 1.77 5.01 0.158
## 6 363.2 687 7.01e-07 exponential 3 -0.1467 NA 9.10 2.77 NA 0.206
## typology trend y0 yatdosemax yrange maxychange xextrem yextrem
## 1 GP.bell bell 12.0 11.03 1.17 1.004 1.70 12.2
## 2 E.dec.concave dec 12.2 10.99 1.25 1.251 NA NA
## 3 E.dec.concave dec 9.1 7.58 1.53 1.527 NA NA
## 4 GP.bell bell 12.2 11.15 1.18 1.012 1.72 12.3
## 5 GP.bell bell 12.1 11.15 1.11 0.949 1.77 12.3
## 6 E.dec.concave dec 9.1 7.64 1.46 1.461 NA NA
```

### 2.3.2 Plot of fitted curves

By default the plot() function used on the output of the drcfit() function provides the first 20 fitted curves (or the ones you specify using the argument items) with observed points. **Fitted curves are represented in red, replicates are represented in open circles and means of replicates at each dose/concentration are represented by solid circles**. All the fitted curves may be saved in a pdf file using the plotfit2pdf() function (see ?drcfit).

`plot(f) `

The fitted curves may be **represented using a log scale** for dose/concentration, which will be more suited in case where the range of observed doses/concentrations is very wide, using argument `dose_log_transfo`

(see ?drcfit for details and examples).

Another **specific plot function named targetplot() can be used to plot targeted items, whether they were or not selected** in step 2 and fitted in step 3. See an example below and details in ?targetplot

```
c("88.1", "1", "3", "15")
targetitems <-targetplot(targetitems, f = f)
```

### 2.3.3 Plot of residuals

**To check the assumption of the Gaussian error model** (see the section on least squares), two types of residual plots can be used, `"dose_residuals"`

for plot of residuals against the observed doses/concentrations, or `"fitted_residuals"`

for plot of residuals against fitted values of the modeled signal. The residual plots for all items may also be saved in a pdf file using the plotfit2pdf() function (see ?drcfit).

`plot(f, plot.type = "dose_residuals")`

### 2.3.4 Description of the family of dose-response models fitted in DRomics

The best fit model is chosen among the five following models describing the observed signal \(y\) as a function of \(x\) the dose (or concentration):

- the
**linear model**: \[y = d + b \times x\] with**2 parameters**, \(b\) the slope and \(d\) the mean signal at control. - the
**exponential model**: \[y = d + b \times \left(exp\left(\frac{x}{e}\right)-1\right)\] with**3 parameters**,

\(b\) a shape parameter, \(d\) the mean signal at control, \(e\) a shape parameter.- When \(e>0\) the dose response curve - DRC - is increasing if \(b>0\) and decreasing if \(b<0\), with no asymptote for high doses.
- When \(e<0\) the DRC is increasing if \(b<0\) and decreasing if \(b>0\), with an asymptote at \(d-b\) for high doses.

- the
**Hill model**: \[y = c + \frac{d-c}{1+(\frac{x}{e})^b}\] with**4 parameters**,

\(b\) (\(>0\)) a shape parameter, \(c\) the asymtotic signal for high doses, \(d\) the mean signal at control, and \(e\) (\(>0\)) the dose at the inflection point of the sigmoid. - the
**Gauss-probit model**built as the sum of a Gauss and a probit part sharing the same parameters as defined below:\[y = f \times exp\left(-0.5 \left(\frac{x-e}{b}\right)^2\right) +d+(c-d) \times \Phi\left(\frac{x-e}{b}\right)\] with**5 parameters**,

\(b\) (\(>0\)) a shape parameter corresponding to the standard deviation of the Gauss part of the model, \(c\) the asymtotic signal for high doses, \(d\) the asymptotic signal on the left of the DRC (generally corresponding to a fictive negative dose), \(e\) (\(>0\)) a shape parameter corresponding to the mean of the Gauss part of the model, and \(f\) the amplitude and sign of the Gauss part of the model (the model is U-shaped if \(f<0\) and bell-shaped if \(f<0\)).

\(\Phi\) represents the cumulative distribution function (CDF) of the standard Gauss (also named normal or Gaussian) distribution.

This model encompasses**two simplifed versions with 4 parameters**, one**monotonic**(for \(f=0\)) and one with**symmetrical asymptotes**(for \(c=d\)). - the
**log-Gauss-probit model**, a variant of the previous one with on the log scale of the dose: \[y = f \times exp\left(-0.5\left(\frac{ln(x)-ln(e)}{b}\right)^2\right) +d+(c-d) \times \Phi\left(\frac{ln(x)-ln(e)}{b}\right)\]

with**5 parameters**,

\(b\) (\(>0\)) a shape parameter corresponding to the standard deviation of the Gauss part of the model, \(c\) the asymtotic signal for high dose, \(d\) the asymptotic signal on the left of the DRC, reached at the control (for \(ln(x) = ln(0) = -\infty\)), \(ln(e)\) (\(>0\)) a shape parameter corresponding to the mean of the Gauss part of the model, and \(f\) the amplitude and sign of the Gauss part of the model (the model is U-shaped if \(f<0\) and bell-shaped if \(f<0\)).

\(\Phi\) represents the cumulative distribution function (CDF) of the standard Gauss distribution.

As the previous one, this model encompasses**two simplifed versions with 4 parameters**, one**monotonic**(for \(f=0\)) and one with**symmetrical asymptotes**(for \(c=d\)).

This family of five models was built to be able to describe a wide range of monotonic and biphasic DRC. In the following plot were represented typologies of curves that can be described using those models, depending of the definition of their parameters.

### 2.3.5 Reminder on least squares regression

It is important when using DRomics to have in mind that the dose-response models are fitted using the **least squares regression**, assuming an **additive Gaussian (normal) error model** for the observed signal. It is why the scale under which you should import your data is very important: a log (or pseudo-log) transformation may be necessary to meet the use conditions of the model for some types of data.

Let us recall the formulation of the Gaussian model defining the signal (after transformation if needed) \(y\) as a function of the dose (or concentration) \(x\), \(f\) being one of the five models previously described, and \(\theta\) the vector of its parameters (of length 2 to 5).

\[y = f(x, \theta) + \epsilon\] with \[\epsilon \sim N(0, \sigma)\] \(N(0, \sigma)\) representing the Gaussian (normal) distribution of mean \(0\) and standard deviation (SD) \(\sigma\).

In this model, the **residual standard deviation \(\sigma\) is assumed constant**. It is the classical “homoscedasticity” hypothesis (see the following figure for an illustration). The examination of residuals (see the section on plot of residuals) is a good way to check that the error model is not strongly violated on your data.

## 2.4 Step 4: calculation of benchmark doses (BMD)

### 2.4.1 Calculation of BMD

The **two types of benchmark doses (BMD-zSD and BMD-xfold) proposed by the EFSA (2017) are systematically calculated** for each fitted dose-response curve using the function **bmdcalc()** with the output of the drcfit() function as a first argument, **but we strongly recommend the use of the first one (BMD-zSD)** for reasons explained in Larras et al. 2018 (see ?bmdcalc for details on this function).

` bmdcalc(f, z = 1, x = 10)) (r <-`

```
## 1 BMD-xfold values and 0 BMD-zSD values are not defined (coded NaN as
## the BMR stands outside the range of response values defined by the model).
## 61 BMD-xfold values and 0 BMD-zSD values could not be calculated (coded
## NA as the BMR stands within the range of response values defined by the
## model but outside the range of tested doses).
```

For the **recommended BMD-zSD**,the argument \(z\), by default at 1, is used to define the **BMD-zSD** as the dose at which the response is reaching the **BMR (benchmark response)** defined as \[BMR = y_0 \pm z \times SD\] with \(y_0\) the level at the control given by the dose-response fitted model and \(SD\) the residual standard deviation of the dose-response fitted model (also named \(\sigma\) in the previous mathematical definition of the Gaussian model).

For the less recommended BMD-xfold, the argument \(x\), by default at 10 (for 10%), is used to define the BMD-xfold as the dose at which the response is reaching the BMR defined as \(BMR = y_0 \pm \frac{x}{100} \times y_0\). So this second BMD version does not take into account the residual standard deviation, and is strongly dependent of the magnitude of \(y_0\), which may be a problem if the signal at the control is close to 0, which is not rare on omics data that are classically handled on log scale.

In the following you can see the first lines of the output data frame of the function bmdcalc() on our example. BMD values are coded `NA`

when the BMR stands within the range of response values defined by the model but outside the range of tested doses, and `NaN`

when the BMR stands outside the range of response values defined by the model due to asymptotes.

An extensive description of the outputs of the complete DRomics workflow is provided in the last section of the main workflow. You can also see ?bmdcalc for a complete description of the columns of the output data frame.

`head(r$res)`

```
## id irow adjpvalue model nbpar b c d e f SDres
## 1 383.2 725 2.08e-07 Gauss-probit 4 5.5836 8.58 8.58 1.70 3.62 0.157
## 2 384.2 727 2.08e-07 exponential 3 -0.0298 NA 12.24 1.76 NA 0.160
## 3 363.1 686 2.24e-07 exponential 3 -0.2058 NA 9.10 3.11 NA 0.218
## 4 383.1 724 2.24e-07 Gauss-probit 4 5.4879 8.75 8.75 1.72 3.58 0.169
## 5 384.1 726 3.41e-07 Gauss-probit 4 6.8453 7.26 7.26 1.77 5.01 0.158
## 6 363.2 687 7.01e-07 exponential 3 -0.1467 NA 9.10 2.77 NA 0.206
## typology trend y0 yatdosemax yrange maxychange xextrem yextrem BMD.zSD
## 1 GP.bell bell 12.0 11.03 1.17 1.004 1.70 12.2 1.33
## 2 E.dec.concave dec 12.2 10.99 1.25 1.251 NA NA 3.26
## 3 E.dec.concave dec 9.1 7.58 1.53 1.527 NA NA 2.25
## 4 GP.bell bell 12.2 11.15 1.18 1.012 1.72 12.3 1.52
## 5 GP.bell bell 12.1 11.15 1.11 0.949 1.77 12.3 1.41
## 6 E.dec.concave dec 9.1 7.64 1.46 1.461 NA NA 2.43
## BMR.zSD BMD.xfold BMR.xfold
## 1 12.19 NA 10.83
## 2 12.08 6.59 11.02
## 3 8.89 5.26 8.19
## 4 12.33 NA 10.95
## 5 12.26 NA 10.89
## 6 8.90 5.47 8.19
```

### 2.4.2 Plots of the BMD distribution

The default plot of the output of the bmdcalc() function provides the distribution of benchmark doses as an ECDF (Empirical Cumulative Density Function) plot for the chosen BMD (“zSD”" or “xfold”). See an example below.

`plot(r, BMDtype = "zSD", plottype = "ecdf") `

Different alternative plots are proposed (see ?bmdcalc for details) that can be obtained using the argument plottype to choose the type of plot (“ecdf”, “hist” or “density”) and the argument by to split the plot for example by “trend”.

On a BMD ECDF plot one can add a **color gradient for each item** coding for the **intensity of the signal** (after shift of the control signal at 0) as a function of the dose (see ?bmdplotwithgradient for details and an example below). It is generally necessary to use the argument line.size to manually adjust the width of lines in that plot as the default value does not always give a satisfactory resut. It is also recommended (but not mandatory and it is not the default option for this argument) to scale the signal as below using the argument `scaling`

if you want to focus on the shape of the dose-reponse curves and not on the amplitude of the signal change.

```
bmdplotwithgradient(r$res, BMDtype = "zSD",
facetby = "trend",
shapeby = "model",
line.size = 1.2,
scaling = TRUE) + labs(shape = "model")
```

### 2.4.3 Calculation of confidence intervals on the BMDs by bootstrap

**Confidence intervals on BMD values** can be calculated by **bootstrap**. As the call to this function may take much time, by default a progressbar is provided and some arguments can be used to specify parallel computing to accelerate the computation (see ?bmdboot for details).

In the example below, a small number of iterations was used just to make this vignette quick to compile, but **the default value of the argument niter (1000) should be considered as a minimal value to obtain stable results**.

` bmdboot(r, niter = 50, progressbar = FALSE)) (b <-`

```
## Bootstrap confidence interval computation failed on 16 items among 128
## due to lack of convergence of the model fit for a fraction of the
## bootstrapped samples greater than 0.5.
## For 6 BMD.zSD values and 69 BMD.xfold values among 128 at least one
## bound of the 95 percent confidence interval could not be computed due
## to some bootstrapped BMD values not reachable due to model asymptotes
## or reached outside the range of tested doses (bounds coded Inf)).
```

This function gives an output corresponding to the output of the bmdcalc() function completed with bounds of BMD confidence intervals (by default 95% confidence intervals) and the number of bootstrap iterations for which the model was successfully fitted to the data. An extensive description of the outputs of the complete DRomics workflow is provided in the last section of the main workflow.

`head(b$res)`

```
## id irow adjpvalue model nbpar b c d e f SDres
## 1 383.2 725 2.08e-07 Gauss-probit 4 5.5836 8.58 8.58 1.70 3.62 0.157
## 2 384.2 727 2.08e-07 exponential 3 -0.0298 NA 12.24 1.76 NA 0.160
## 3 363.1 686 2.24e-07 exponential 3 -0.2058 NA 9.10 3.11 NA 0.218
## 4 383.1 724 2.24e-07 Gauss-probit 4 5.4879 8.75 8.75 1.72 3.58 0.169
## 5 384.1 726 3.41e-07 Gauss-probit 4 6.8453 7.26 7.26 1.77 5.01 0.158
## 6 363.2 687 7.01e-07 exponential 3 -0.1467 NA 9.10 2.77 NA 0.206
## typology trend y0 yatdosemax yrange maxychange xextrem yextrem BMD.zSD
## 1 GP.bell bell 12.0 11.03 1.17 1.004 1.70 12.2 1.33
## 2 E.dec.concave dec 12.2 10.99 1.25 1.251 NA NA 3.26
## 3 E.dec.concave dec 9.1 7.58 1.53 1.527 NA NA 2.25
## 4 GP.bell bell 12.2 11.15 1.18 1.012 1.72 12.3 1.52
## 5 GP.bell bell 12.1 11.15 1.11 0.949 1.77 12.3 1.41
## 6 E.dec.concave dec 9.1 7.64 1.46 1.461 NA NA 2.43
## BMR.zSD BMD.xfold BMR.xfold BMD.zSD.lower BMD.zSD.upper BMD.xfold.lower
## 1 12.19 NA 10.83 0.489 3.93 Inf
## 2 12.08 6.59 11.02 1.724 4.58 6.40
## 3 8.89 5.26 8.19 1.366 3.52 4.61
## 4 12.33 NA 10.95 0.423 4.17 Inf
## 5 12.26 NA 10.89 0.480 4.27 Inf
## 6 8.90 5.47 8.19 1.334 3.31 4.75
## BMD.xfold.upper nboot.successful
## 1 Inf 36
## 2 Inf 47
## 3 5.96 50
## 4 Inf 36
## 5 Inf 25
## 6 6.00 50
```

The plot() function applied on the output of the bmdboot() function gives an ECDF plot of the chosen BMD with the confidence interval of each BMD (see ?bmdcalc for examples). By default BMDs with an infinite confidence interval bound are not plotted.

### 2.4.4 Plot of fitted curves with BMD values and confidence intervals

It is possible to add the output of bmdcalc() (or of bmdboot()) in the argument BMDoutput of the plot() function of drcfit(), in order to add BMD values (when defined) as a **vertical line on each fitted curve**, and **bounds of the confidence intervals** (when successfully calculated) as **two dashed lines**. **Horizontal dotted lines corresponding to the two BMR potential values will be also added**. See an example below.

```
# If you do not want to add the confidence intervals just replace b
# the output of bmdboot() by r the output of bmdcalc()
plot(f, BMDoutput = b)
```

All the fitted curves may also be saved in the same way in a pdf file using the plotfit2pdf() function (see ?drcfit).

### 2.4.5 Description of the outputs of the complete DRomics workflow

The **output of the complete DRomics workflow**, given in `b$res`

with `b`

being the output of bmdboot() is a **data frame** reporting the **results of the fit and BMD computation on each selected item** sorted in the ascending order of the adjusted p-values returned by the item selection step.

`str(b$res)`

```
## 'data.frame': 128 obs. of 28 variables:
## $ id : chr "383.2" "384.2" "363.1" "383.1" ...
## $ irow : int 725 727 686 724 726 687 689 688 568 569 ...
## $ adjpvalue : num 2.08e-07 2.08e-07 2.24e-07 2.24e-07 3.41e-07 ...
## $ model : Factor w/ 5 levels "Hill","linear",..: 4 3 3 4 4 3 3 3 3 3 ...
## $ nbpar : num 4 3 3 4 4 3 3 3 3 3 ...
## $ b : num 5.5836 -0.0298 -0.2058 5.4879 6.8453 ...
## $ c : num 8.58 NA NA 8.75 7.26 ...
## $ d : num 8.58 12.24 9.1 8.75 7.26 ...
## $ e : num 1.7 1.76 3.11 1.72 1.77 ...
## $ f : num 3.62 NA NA 3.58 5.01 ...
## $ SDres : num 0.157 0.16 0.218 0.169 0.158 ...
## $ typology : Factor w/ 12 levels "E.dec.concave",..: 6 1 1 6 6 1 1 1 2 2 ...
## $ trend : Factor w/ 4 levels "U","bell","dec",..: 2 3 3 2 2 3 3 3 3 3 ...
## $ y0 : num 12 12.2 9.1 12.2 12.1 ...
## $ yatdosemax : num 11.03 10.99 7.58 11.15 11.15 ...
## $ yrange : num 1.17 1.25 1.53 1.18 1.11 ...
## $ maxychange : num 1.004 1.251 1.527 1.012 0.949 ...
## $ xextrem : num 1.7 NA NA 1.72 1.77 ...
## $ yextrem : num 12.2 NA NA 12.3 12.3 ...
## $ BMD.zSD : num 1.33 3.26 2.25 1.52 1.41 ...
## $ BMR.zSD : num 12.19 12.08 8.89 12.33 12.26 ...
## $ BMD.xfold : num NA 6.59 5.26 NA NA ...
## $ BMR.xfold : num 10.83 11.02 8.19 10.95 10.89 ...
## $ BMD.zSD.lower : num 0.489 1.724 1.366 0.423 0.48 ...
## $ BMD.zSD.upper : num 3.93 4.58 3.52 4.17 4.27 ...
## $ BMD.xfold.lower : num Inf 6.4 4.61 Inf Inf ...
## $ BMD.xfold.upper : num Inf Inf 5.96 Inf Inf ...
## $ nboot.successful: num 36 47 50 36 25 50 50 50 50 50 ...
```

The columns of this data frame are:

- id: the item identifier
- irow: the row number in the initial dataset
- adjpvalue: the adjusted p-values returned by the item selection step
- model: the best model fitted
- nbpar: the number of parameters of this best model (that may be smaller than the maximal number of parameters of the model if a simplified version of it was chosen)
- b, c, d, e, and f, the model parameter values
**SDres**: the**residual standard deviation of the best model**- typology: the typology of the curve depending of the chosen model and of its parameter values with 16 classes,
- “H.inc” for increasing Hill curves
- “H.dec” for decreasing Hill curves
- “L.inc” for increasing linear curves
- “L.dec” for decreasing linear curves
- “E.inc.convex” for increasing convex exponential curves
- “E.dec.concave” for decreasing concave exponential curves
- “E.inc.concave” for increasing concave exponential curves
- “E.dec.convex” for decreasing convex exponential curves
- “GP.U” for U-shape Gauss-probit curves
- “GP.bell” for bell-shape Gauss-probit curves
- “GP.inc” for increasing Gauss-probit curves
- “GP.dec” for decreasing Gauss-probit curves
- “lGP.U” for U-shape log-Gauss-probit curves
- “lGP.bell” for bell-shape log-Gauss-probit curves
- “lGP.inc” for increasing log-Gauss-probit curves
- “lGP.decreasing” for decreasing log-Gauss-probit curves

**trend**: the rough trend of the curve defined with four classes,**U shape****bell shape****increasing**

**decreasing**

- y0: the y theoretical value at the control
- yatdosemax: the theoretical y value at the maximal dose
- yrange: the theoretical y range for x within the range of tested doses
- maxychange: the maximal absolute y change (up or down) from the control
- xextrem: for biphasic curves, x value at which their extremum is reached
- yextrem: the corresponding y value at this extremum
**BMD.zSD**: the BMD-zSD value- BMR.zSD: the corresponding BMR-zSD value
- BMD.xfold: the BMD-xfold value
- BMR.xfold: the corresponding BMR-xfold value
- nboot.successful: the number of bootstrap iterations for which the model was successfully fitted to the data

An incomplete version of this data frame is also given at the end of Step 3 (in `f$fitres`

with `f`

this output of drcfit()) and before bootstrap calculation on BMD values (in `r$res`

with `r`

is the output of bmdcalc()).