cvms

R Package: Cross-validate one or multiple gaussian or binomial regression models at once. Perform repeated cross-validation. Returns results in a tibble for easy comparison, reporting and further analysis.

https://github.com/ludvigolsen/cvms

Keywords from Contributors

Repository

R Package: Cross-validate one or multiple gaussian or binomial regression models at once. Perform repeated cross-validation. Returns results in a tibble for easy comparison, reporting and further analysis.

README.Rmd

---
output: github_document
---



```{r, echo = FALSE}
if (requireNamespace("knitr", quietly = TRUE)){
  knitr::opts_chunk$set(
    collapse = TRUE,
    comment = "#>",
    fig.path = "man/figures/README-",
    dpi = 92,
    fig.retina = 2
  )
}

# Get minimum R requirement 
dep <- as.vector(read.dcf('DESCRIPTION')[, 'Depends'])
rvers <- substring(dep, 7, nchar(dep)-1)
# m <- regexpr('R *\\\\(>= \\\\d+.\\\\d+.\\\\d+\\\\)', dep)
# rm <- regmatches(dep, m)
# rvers <- gsub('.*(\\\\d+.\\\\d+.\\\\d+).*', '\\\\1', dep)

# Function for TOC
# https://gist.github.com/gadenbuie/c83e078bf8c81b035e32c3fc0cf04ee8


```
# cvms 
**Cross-Validation for Model Selection**  
**Authors:** [Ludvig R. Olsen](https://www.ludvigolsen.dk/) ( r-pkgs@ludvigolsen.dk ), Hugh Benjamin Zachariae 

**License:** [MIT](https://opensource.org/license/mit) 

**Started:** October 2016 

[![CRAN_Status_Badge](https://www.r-pkg.org/badges/version/cvms)](https://cran.r-project.org/package=cvms)
[![metacran downloads](https://cranlogs.r-pkg.org/badges/cvms)](https://cran.r-project.org/package=cvms)
[![minimal R version](https://img.shields.io/badge/R%3E%3D-`r rvers`-6666ff.svg)](https://cran.r-project.org/)
[![Codecov test coverage](https://codecov.io/gh/ludvigolsen/cvms/branch/master/graph/badge.svg)](https://app.codecov.io/gh/ludvigolsen/cvms?branch=master)
[![GitHub Actions CI status](https://github.com/ludvigolsen/cvms/actions/workflows/R-check.yaml/badge.svg?branch=master)](https://github.com/ludvigolsen/cvms/actions/workflows/R-check.yaml?branch=master)
[![AppVeyor build status](https://ci.appveyor.com/api/projects/status/github/LudvigOlsen/cvms?branch=master&svg=true)](https://ci.appveyor.com/project/LudvigOlsen/cvms)
[![DOI](https://zenodo.org/badge/71063931.svg)](https://zenodo.org/badge/latestdoi/71063931)


## Overview

R package for model evaluation and comparison.

* **Cross-validate** one or multiple regression or classification models with relevant evaluation metrics in a **tidy** format. 
* **Validate** the best model on a test set and compare it to a **baseline** evaluation.
* Perform **hyperparameter tuning** with *grid search*. 
* **Evaluate** predictions from an external model. 
* Extract the observations that were the **most challenging** to predict.

Currently supports regression (`'gaussian'`), binary classification (`'binomial'`), and (some functions only) multiclass classification (`'multinomial'`). Many of the functions allow **parallelization**, e.g. through the `doParallel` package.

**NEW**: Our new [application](https://huggingface.co/spaces/ludvigolsen/plot_confusion_matrix) for plotting confusion matrices with `plot_confusion_matrix()` without any code is now available on [Huggingface Spaces](https://huggingface.co/spaces/ludvigolsen/plot_confusion_matrix).

### Main functions

| Function              | Description                                                        |
|:----------------------|:-------------------------------------------------------------------|
|`cross_validate()`     |Cross-validate linear models with `lm()`/`lmer()`/`glm()`/`glmer()` |
|`cross_validate_fn()`  |Cross-validate a custom model function                              |
|`validate()`           |Validate linear models with (`lm`/`lmer`/`glm`/`glmer`)             |
|`validate_fn()`        |Validate a custom model function                                    |
|`evaluate()`           |Evaluate predictions with a large set of metrics                    |
|`baseline()`
`baseline_gaussian()`
`baseline_binomial()`
`baseline_multinomial()`           |Perform baseline evaluations of a dataset |


### Evaluation utilities

| Function              | Description                                                        |
|:----------------------|:-------------------------------------------------------------------|
|`confusion_matrix()`   |Create a confusion matrix from predictions and targets |
|`evaluate_residuals()` |Evaluate residuals from a regression task |
|`most_challenging()`   |Find the observations that were the most challenging to predict |
|`summarize_metrics()`  |Summarize numeric columns with a set of descriptors |


### Formula utilities

| Function                | Description                                                        |
|:------------------------|:-------------------------------------------------------------------|
|`combine_predictors()`   |Generate model formulas from a list of predictors |
|`reconstruct_formulas()` |Extract formulas from output tibble |
|`simplify_formula()`     |Remove inline functions with more from a formula object |


### Plotting utilities

| Function                 | Description                                                        |
|:-------------------------|:-------------------------------------------------------------------|
|`plot_confusion_matrix()` |Plot a confusion matrix (see also our [no-code application](https://huggingface.co/spaces/ludvigolsen/plot_confusion_matrix)) |
|`plot_metric_density()`   |Create a density plot for a metric column |
|`font()` |Set font settings for plotting functions (currently only `plot_confusion_matrix()`) |
|`sum_tile_settings()` |Set settings for sum tiles in `plot_confusion_matrix()` |


### Custom functions

| Function                  | Description                                             |
|:--------------------------|:--------------------------------------------------------|
|`model_functions()`        |Example model functions for `cross_validate_fn()`        |
|`predict_functions()`      |Example predict functions for `cross_validate_fn()`      |
|`preprocess_functions()`   |Example preprocess functions for `cross_validate_fn()`   |
|`update_hyperparameters()` |Manage hyperparameters in custom model functions         |


### Other utilities

| Function              | Description                                                        |
|:----------------------|:-------------------------------------------------------------------|
|`select_metrics()`     |Select the metric columns from the output |
|`select_definitions()` |Select the model-defining columns from the output |
|`gaussian_metrics()`
`binomial_metrics()`
`multinomial_metrics()` |Create list of metrics for the common `metrics` argument |
|`multiclass_probability_tibble()` |Generate a multiclass probability tibble |


### Datasets

| Name                      | Description                                                        |
|:--------------------------|:-------------------------------------------------------------------|
|`participant.scores`       |Made-up experiment data with 10 participants and two diagnoses |
|`wines`                    |A list of wine varieties in an approximately Zipfian distribution |
|`musicians`                |Made-up data on 60 musicians in 4 groups for multiclass classification |
| `predicted.musicians`     |Predictions by 3 classifiers of the 4 classes in the `musicians` dataset |
|`precomputed.formulas`     |Fixed effect combinations for model formulas with/without two- and three-way interactions |
|`compatible.formula.terms` |162,660 pairs of compatible terms for building model formulas with up to 15 fixed effects |

## Table of Contents

```{r toc, echo=FALSE}
cvms:::render_toc("README.Rmd")
```

## Important News

>
> Check `NEWS.md` for the full list of changes.
>

* Version `1.2.0` contained **multiple breaking changes**. Please see `NEWS.md`. (18th of October 2020)


## Installation

CRAN:

> `install.packages("cvms")`

Development version:  

> `install.packages("devtools")`  
>
> `devtools::install_github("LudvigOlsen/groupdata2")`  
>
> `devtools::install_github("LudvigOlsen/cvms")`

## Vignettes  

`cvms` contains a number of vignettes with relevant use cases and descriptions:  
  
> `vignette(package = "cvms")` # for an overview   

# Examples

## Attach packages

```{r warning=FALSE, message=FALSE}
library(cvms)
library(groupdata2)   # fold() partition()
library(knitr)        # kable()
library(dplyr)        # %>% arrange()

```

## Load data

The dataset `participant.scores` comes with `cvms`:

```{r}
data <- participant.scores
```

## Fold data

Create a grouping factor for subsetting of folds using `groupdata2::fold()`. Order the dataset by the folds:

```{r}
# Set seed for reproducibility
set.seed(7)

# Fold data 
data <- fold(
  data = data, k = 4,
  cat_col = 'diagnosis',
  id_col = 'participant') %>% 
  arrange(.folds)

# Show first 15 rows of data
data %>% head(15) %>% kable()
```

## Cross-validate a single model

### Gaussian

```{r warning=FALSE, message=FALSE}
CV1 <- cross_validate(
  data = data,
  formulas = "score ~ diagnosis",
  fold_cols = '.folds',
  family = 'gaussian',
  REML = FALSE
)

# Show results
CV1

# Let's take a closer look at the different parts of the output 

# Metrics and formulas
CV1 %>% select_metrics() %>% kable()

# Just the formulas
CV1 %>% select_definitions() %>% kable()

# Nested predictions 
# Note that [[1]] picks predictions for the first row
CV1$Predictions[[1]] %>% head() %>% kable()

# Nested results from the different folds
CV1$Results[[1]] %>% kable()

# Nested model coefficients
# Note that you have the full p-values, 
# but kable() only shows a certain number of digits
CV1$Coefficients[[1]] %>% kable()

# Additional information about the model
# and the training process
CV1 %>% select(14:19, 21) %>% kable()

CV1$Process[[1]]

```

### Binomial

```{r fig.width = 4.25, fig.height = 4.25, fig.align='center'}
CV2 <- cross_validate(
  data = data,
  formulas = "diagnosis~score",
  fold_cols = '.folds',
  family = 'binomial'
)

# Show results
CV2

# Let's take a closer look at the different parts of the output 
# We won't repeat the parts too similar to those in Gaussian

# Metrics
CV2 %>% select(1:9) %>% kable(digits = 5)
CV2 %>% select(10:15) %>% kable()

# Confusion matrix
CV2$`Confusion Matrix`[[1]] %>% kable()

# Plot confusion matrix
plot_confusion_matrix(CV2$`Confusion Matrix`[[1]], add_sums = TRUE)

```


## Cross-validate multiple models

### Create model formulas

```{r}
model_formulas <- c("score ~ diagnosis", "score ~ age")
mixed_model_formulas <- c("score ~ diagnosis + (1|session)",
                          "score ~ age + (1|session)")
```

### Cross-validate fixed effects models

```{r}
CV3 <- cross_validate(
  data = data,
  formulas = model_formulas,
  fold_cols = '.folds',
  family = 'gaussian',
  REML = FALSE
)

# Show results
CV3
```

### Cross-validate mixed effects models

```{r}
CV4 <- cross_validate(
  data = data,
  formulas = mixed_model_formulas,
  fold_cols = '.folds',
  family = 'gaussian',
  REML = FALSE
)

# Show results
CV4
```

## Repeated cross-validation

Instead of only dividing our data into folds once, we can do it multiple times and average the results. As the models can be ranked differently with different splits, this is generally preferable.

Let's first add some extra fold columns. We will use the `num_fold_cols` argument to add 3 *unique* fold columns. We tell `fold()` to *keep* the existing fold column and simply add three extra columns. We could also choose to remove the existing fold column, if, for instance, we were changing the number of folds (`k`). Note, that the original fold column will be renamed to `".folds_1"`. 

```{r}
# Set seed for reproducibility
set.seed(2)

# Ungroup data
# Ootherwise we would create folds within the existing folds
data <- dplyr::ungroup(data)

# Fold data 
data <- fold(
  data = data, 
  k = 4,
  cat_col = 'diagnosis',
  id_col = 'participant',
  num_fold_cols = 3,
  handle_existing_fold_cols = "keep"
)

# Show first 15 rows of data
data %>% head(10) %>% kable()

```

Now, let's cross-validate the four fold columns. We use `paste0()` to create the four column names:

```{r}
CV5 <- cross_validate(
  data = data,
  formulas = c("diagnosis ~ score",
               "diagnosis ~ score + age"),
  fold_cols = paste0(".folds_", 1:4),
  family = 'binomial'
)

# Show results
CV5

# Subset of the results per fold for the first model
CV5$Results[[1]] %>% select(1:8) %>% kable()
```

## Cross-validating custom model functions

`cross_validate_fn()` allows us to cross-validate a custom model function, like a support vector machine or a neural network. It works with regression (`gaussian`), binary classification (`binomial`), and multiclass classification (`multinomial`). 

It is required to pass a model function and a predict function. Further, it is possible to pass a preprocessing function and a list of hyperparameter values to test with grid search. You can check the requirements for these functions at `?cross_validate_fn`.

### SVM

Let's cross-validate a support-vector machine using the `svm()` function from the `e1071` package.
First, we will create a model function. You can do anything you want inside it, as long as it takes the arguments `train_data`, `formula`, and `hyperparameters` and returns a fitted model object:

```{r eval=requireNamespace("e1071")}
# Create model function
#
# train_data : tibble with the training data
# formula : a formula object
# hyperparameters : a named list of hyparameters

svm_model_fn <- function(train_data, formula, hyperparameters){
  
  # Note that `formula` must be passed first
  # when calling svm(), otherwise it fails
  e1071::svm(
    formula = formula,
    data = train_data,
    kernel = "linear",
    type = "C-classification",
    probability = TRUE
  )
}
```

We also need a predict function. This will usually wrap the `stats::predict()` function. The point is to ensure that the predictions have the correct format. In this case, we want a single column with the probability of the positive class. 
Note, that you do not need to use the `formula`, `hyperparameters`, and `train_data` arguments within your function. These are there for the few cases, where they are needed.

```{r}
# Create predict function
#
# test_data : tibble with the test data
# model : fitted model object
# formula : a formula object
# hyperparameters : a named list of hyparameters
# train_data : tibble with the training data

svm_predict_fn <- function(test_data, model, formula, hyperparameters, train_data){
  
  # Predict the test set with the model
  predictions <- stats::predict(
    object = model,
    newdata = test_data,
    allow.new.levels = TRUE,
    probability = TRUE
  )

  # Extract the probabilities
  # Usually the predict function will just 
  # output the probabilities directly
  probabilities <- dplyr::as_tibble(
    attr(predictions, "probabilities")
  )

  # Return second column
  # with probabilities of positive class
  probabilities[[2]]
}
```

With these functions defined, we can cross-validate the support-vector machine:

```{r eval=requireNamespace("e1071")}
# Cross-validate svm_model_fn
CV6 <- cross_validate_fn(
  data = data,
  model_fn = svm_model_fn,
  predict_fn = svm_predict_fn,
  formulas = c("diagnosis ~ score", "diagnosis ~ age"),
  fold_cols = '.folds_1',
  type = 'binomial'
)

CV6
```

### Naïve Bayes

Let's try with a naïve Bayes classifier as well. First, we will define the model function:

```{r eval=requireNamespace("e1071")}
# Create model function
#
# train_data : tibble with the training data
# formula : a formula object
# hyperparameters : a named list of hyparameters

nb_model_fn <- function(train_data, formula, hyperparameters){
  e1071::naiveBayes(
    formula = formula,
    data = train_data
  )
}
```

And the predict function:

```{r}
# Create predict function
#
# test_data : tibble with the test data
# model : fitted model object
# formula : a formula object
# hyperparameters : a named list of hyparameters
# train_data : tibble with the training data

nb_predict_fn <- function(test_data, model, formula, hyperparameters, train_data){
  stats::predict(
    object = model,
    newdata = test_data,
    type = "raw",
    allow.new.levels = TRUE)[, 2]
}
```

With both functions specified, we are ready to cross-validate our naïve Bayes classifier:

```{r eval=requireNamespace("e1071")}
CV7 <- cross_validate_fn(
  data = data,
  model_fn = nb_model_fn,
  predict_fn = nb_predict_fn,
  formulas = c("diagnosis ~ score", "diagnosis ~ age"),
  type = 'binomial',
  fold_cols = '.folds_1'
)

CV7
```

## Extracting the most challenging observations

If we wish to investigate why some observations are harder to predict than others, we should start by identifying the most challenging observations. This can be done with `most_challenging()`. 

Let's first extract the predictions from some of the cross-validation results:

```{r eval=requireNamespace("e1071")}
glm_predictions <- dplyr::bind_rows(CV5$Predictions, .id = "Model")
svm_predictions <- dplyr::bind_rows(CV6$Predictions, .id = "Model")
nb_predictions <- dplyr::bind_rows(CV7$Predictions, .id = "Model")
predictions <- dplyr::bind_rows(
  glm_predictions, 
  svm_predictions, 
  nb_predictions, 
  .id = "Architecture"
)
predictions[["Target"]] <- as.character(predictions[["Target"]])

predictions
```

Now, let's find the *overall most difficult to predict* observations. `most_challenging()` calculates the `Accuracy`, `MAE`, and `Cross-Entropy` for each prediction. We can then extract the observations with the ~20% highest `MAE` scores. Note that `most_challenging()` works with grouped data frames as well.

```{r eval=requireNamespace("e1071")}
challenging <- most_challenging(
  data = predictions,
  prediction_cols = "Prediction",
  type = "binomial",
  threshold = 0.20,
  threshold_is = "percentage"
)

challenging
```

We can then extract the difficult observations from the dataset. First, we add an index to the dataset. Then, we perform a right-join, to only get the rows that are in the `challenging` data frame.

```{r eval=requireNamespace("e1071")}
# Index with values 1:30
data[["Observation"]] <- seq_len(nrow(data))

# Add information to the challenging observations
challenging <- data %>% 
  # Remove fold columns for clarity
  dplyr::select(-c(.folds_1, .folds_2, .folds_3, .folds_4)) %>% 
  # Add the scores
  dplyr::right_join(challenging, by = "Observation")

challenging %>% kable()
```

Note: You may have to scroll to the right in the table.

## Evaluating predictions

We can also evaluate predictions from a model trained outside `cvms`. This works with regression (`'gaussian'`), binary classification (`'binomial'`), and multiclass classification (`'multinomial'`). 

### Gaussian evaluation

Extract the targets and predictions from the first cross-validation we performed and evaluate it with `evaluate()`. We group the data frame by the `Fold` column to evaluate each fold separately:

```{r}
# Extract the predictions from the first cross-validation
predictions <- CV1$Predictions[[1]] 
predictions %>% head(6) %>% kable()

# Evaluate the predictions per fold
predictions %>% 
  group_by(Fold) %>% 
  evaluate(
    target_col = "Target",
    prediction_cols = "Prediction",
    type = "gaussian"
  )
```

### Binomial evaluation

We can do the same for the predictions from the second, binomial cross-validation:

```{r}
# Extract the predictions from the second cross-validation
predictions <- CV2$Predictions[[1]] 
predictions %>% head(6) %>% kable()

# Evaluate the predictions per fold
predictions %>% 
  group_by(Fold) %>% 
  evaluate(
    target_col = "Target",
    prediction_cols = "Prediction",
    type = "binomial"
  )
```

### Multinomial evaluation

We will use the `multiclass_probability_tibble()` helper to generate a data frame with predicted probabilities for three classes, along with the predicted class and the target class. Then, we will 1) evaluate the three probability columns against the targets (preferable format), and 2) evaluate the predicted classes against the targets:

```{r}
# Create dataset for multinomial evaluation
multiclass_data <- multiclass_probability_tibble(
  num_classes = 3, # Here, number of predictors
  num_observations = 30,
  apply_softmax = TRUE,
  add_predicted_classes = TRUE,
  add_targets = TRUE) 

multiclass_data

# Evaluate probabilities
# One prediction column *per class*
ev <- evaluate(
  data = multiclass_data,
  target_col = "Target",
  prediction_cols = paste0("class_", 1:3),
  type = "multinomial"
)

ev

# The one-vs-all evaluations
ev$`Class Level Results`[[1]]

# Evaluate the predicted classes
# One prediction column with the class names
evaluate(
  data = multiclass_data,
  target_col = "Target",
  prediction_cols = "Predicted Class",
  type = "multinomial"
)
```

## Baseline evaluations

While it's common to find the chance-level baseline analytically (in classification tasks), it's often possible to get a better evaluation than that by chance. Hence, it is useful to check the range of our metrics when randomly guessing the probabilities. 

Usually, we use `baseline()` on our test set at the start of our modeling process, so we know what level of performance we should beat.

Note: Where `baseline()` works with all three families (`gaussian`, `binomial` and `multinomial`), 
each family also has a wrapper function (e.g. `baseline_gaussian()`) that is easier to use. We use those here.

Start by partitioning the dataset:

```{r}
# Set seed for reproducibility
set.seed(1)

# Partition the dataset 
partitions <- groupdata2::partition(
  participant.scores,
  p = 0.7,
  cat_col = 'diagnosis',
  id_col = 'participant',
  list_out = TRUE
)

train_set <- partitions[[1]]
test_set <- partitions[[2]]
```


### Binomial baseline

Approach: `n` random sets of predictions are evaluated against the dependent variable in the test set. We also evaluate a set of all `0`s and a set of all `1`s.

Create the baseline evaluations:

```{r}
# Perform binomial baseline evaluation
# Note: It's worth enabling parallelization (see ?baseline examples)
binomial_baseline <- baseline_binomial(
  test_data = test_set, 
  dependent_col = "diagnosis",
  n = 100
)

binomial_baseline$summarized_metrics

```

On average, we can expect an `F1` score of approximately `0.481`. The maximum `F1` score achieved by randomly guessing was `0.833` though. That's likely because of the small size of the test set, but it illustrates how such information could be useful in a real-life scenario.

The `All_1` row shows us that we can achieve an `F1` score of `0.667` by always predicting `1`. Some model architectures, like neural networks, have a tendency to always predict the majority class. Such a model is quite useless of course, why it is good to be aware of the performance it could achieve. We could also check the confusion matrix for such a pattern.

```{r}
binomial_baseline$random_evaluations
```

We can plot the distribution of `F1` scores from the random evaluations:

```{r fig.width = 6, fig.height = 4.5, fig.align='center'}
# First, remove the NAs from the F1 column
random_evaluations <- binomial_baseline$random_evaluations
random_evaluations <- random_evaluations[!is.na(random_evaluations$F1),]

# Create density plot for F1
plot_metric_density(baseline = random_evaluations, 
                    metric = "F1", xlim = c(0, 1))
```


### Multinomial baseline

Approach: Creates one-vs-all (binomial) baseline evaluations for `n` sets of random predictions against the dependent variable, along with sets of `all class x,y,z,...` predictions.

Create the baseline evaluations:

```{r warning=FALSE}

multiclass_baseline <- baseline_multinomial(
  test_data = multiclass_data, 
  dependent_col = "Target",
  n = 100
)

# Summarized metrics
multiclass_baseline$summarized_metrics

```

The `CL_` measures describe the `Class Level Results` (aka. one-vs-all evaluations). One of the classes have a maximum `Balanced Accuracy` score of `0.770`, while the maximum `Balanced Accuracy` in the random evaluations is `0.664`.

```{r}

# Summarized class level results for class 1
multiclass_baseline$summarized_class_level_results %>% 
  dplyr::filter(Class == "class_1") %>%
  tidyr::unnest(Results)

# Random evaluations
# Note, that the class level results for each repetition
# are available as well
multiclass_baseline$random_evaluations

```

### Gaussian baseline

Approach: The baseline model `(y ~ 1)`, where `1` is simply the intercept (i.e. mean of `y`), is fitted on `n` random subsets of the training set and evaluated on the test set. We also perform an evaluation of the model fitted on the entire training set.

We usually wish to establish whether our predictors add anything useful to our model. We should thus at least do better than a model without any predictors. 

Create the baseline evaluations:

```{r}
gaussian_baseline <- baseline_gaussian(
  test_data = test_set,
  train_data = train_set,
  dependent_col = "score",
  n = 100
)

gaussian_baseline$summarized_metrics
```

The `All_rows` row tells us the performance when fitting the intercept model on the full training set. It is quite close to the mean of the random evaluations.

```{r}
gaussian_baseline$random_evaluations
```

Plot the density plot for `RMSE`:

```{r fig.width = 6, fig.height = 3.5, fig.align='center'}
plot_metric_density(baseline = gaussian_baseline$random_evaluations,
                    metric = "RMSE")
```

In this instance, the `All_rows` row might have been enough, as the subsets mainly add higher `RMSE` scores.

## Generate model formulas

Instead of manually typing all possible model formulas for a set of fixed effects (including the possible interactions), `combine_predictors()` can do it for you (with some constraints). 

When including interactions, >200k formulas have been precomputed for up to 8 fixed effects, with a maximum interaction size of 3, and a maximum of 5 fixed effects per formula. It's possible to further limit the generated formulas.

We can also append a random effects structure to the generated formulas.

```{r}
combine_predictors(
  dependent = "y",
  fixed_effects = c("a", "b", "c"),
  random_effects = "(1|d)"
)
```

If two or more fixed effects should not be in the same formula, like an effect and its log-transformed version, we can provide them as sublists.

```{r}
combine_predictors(
  dependent = "y",
  fixed_effects = list("a", list("b", "log_b")),
  random_effects = "(1|d)"
)
```

Owner

GitHub Events

Committers

Last synced: about 1 year ago

Issues and Pull Requests

Last synced: 10 months ago

Packages