Interactive Transformation Models

Welcome!

This application is designed to help you understand transformation models as defined by Hothorn, Möst & Bühlmann, 2018.

The purpose of a transformation model is to predict the distribution of a response $Y$.
Increasingly complex transformation models for a continuous response are presented on their respective page:

The unconditional case, where the distribution of $Y$ is modeled without involving any predictor.
The linear case, where predictors $X$ are introduced in the model as a linear shift term with a fixed effect.
The stratified linear case, where each stratum gets a different transformation but the regession coefficients of the shift term stay the same.
The conditional case, where the distribution of $Y$ is modeled fully interacting with $X$. The predictors no longer have a fixed effect, but rather an effect that can vary depending on the response.

Additionaly, transformation models for other types of response variables are presented:

Categorical response variable (unconditional case).
Count response variable.

Features

I recommend opening the application in full-screen.

On each model's page, a transformation model is fitted to a pre-loaded dataset. You can display the dataset and information about it at the bottom of the page.

In the left-side menu, the model status displays the current parameters. Below, you can change some parameters of the model, which is instantly fitted again. This might take a few seconds.

In the centre of the page, a summary of the currently fitted model and several plots are shown. The plots are updated at the same time as the model.

On the last page, you can build a transformation model adapted to your specific needs. Describe your data, and obtain R code ready to be copied into your environment. This feature is limited at the moment.

Acknowledgements

This application was created as a Master Thesis project in Applied Information and Data Science at Lucerne University of Applied Sciences and Arts.

I would like to express my sincere gratitude to the people who contributed to its developement:

Dr. Luisa Barbanti for her kind and insightful guidance throughout this project.
Nisia Trisconi for co-supervising the thesis and testing the application.
Dr. Torsten Hothorn for providing ideas of what to implement in the application, and testing it.
Dr. Sandra Siegfried for testing the application.
Dr. Balint Tamasi for testing the application.
Dr. Lucas Kook for providing the code of the categorical plots originally found in Kook et al., 2020.

Copyrights and Reproducibility

This work is licensed under CC BY-NC-SA 4.0

Main packages versions:

mlt (T. Hothorn, 2025): 1.7-1
tram (T. Hothorn, L. Barbanti, S. Siegfried, L. Kook, 2025): 1.2-5
cotram (S. Siegfried, L. Barbanti, T. Hothorn, 2025): 0.5-3
shiny (W. Chang, J. Cheng, JJ. Allaire, C. Sievert, B. Schloerke, G. Aden-Buie, Y. Xie, J. Allen, J. McPherson, A. Dipert, B.Borges, 2025): 1.11.1
ggplot2 (H. Wickham, W. Chang, L. Henry, T. Pedersen, K. Takahashi, C. Wilke, K. Woo, H. Yutani, D. Dunnington, T. van den Brand, 2025): 4.0.0

GitHub directory: https://github.com/jugwen/interactive-transformation-models

Contact: Gwen Junod - gwen.junod@gmail.com

Unconditional Transformation Model

In the unconditional case, the distribution is defined by a transformation function $h(y)$ and a distribution function $F_Z$, so we can write $\mathbb{P}(Y \leq y) = F_Z(h(y))$.

The transformation function $h(y)$ is parameterised using a basis function $a$, so we can write $h(y) = a(y)^\top \theta$ where $\theta$ denotes parameters to be estimated. So, $$ \mathbb{P}(Y \leq y) = F_Z(h(y)) = F_Z(a(y)^\top \theta) $$ To specify a transformation model, we must define the basis function $a$, choose the link function $F_Z$, and estimate the parameter vector $\theta$.

$a$ is a vector of Bernstein polynomials of order $M$ that must be defined on an interval corresponding to the range of $Y$. To do so, a numeric variable representing $Y$ is created. The author of the mlt package recommends choosing $M$ between 5 and 10.
$F_Z$ is a link function that defines the distribution to transfom $Y$ to. It can be chosen freely and influences the interpretation of the regression coefficients.
$\theta$ is the vector of parameters estimated by the model depending on $a$ and $F_Z$.

The resulting transformation model can be fitted to the data.

Interactive Model

A case of unconditional transformation model, more precisely a continuous model for a continuous response, is fitted to the Old Faithful dataset with the mlt package.

In this model, waiting is defined as the response variable.

Bernstein Basis
If you change the Bernstein Basis to a lower order, the first mode is less or even not represented in the PDF plot. You could set the order to 1 and increase it 1 by 1 with the arrow. The first mode slowly forms as the model captures more complexity. If you keep increasing the order, the modeled density stays stable until around 15.

Distribution
The distribution parameter defines $F_Z$. It is the distribution we want to transform $Y$ to. For unconditional transformation models, that choice is not really important since there are no coefficients to interpret. However, the Bernstein basis order must be large enough so that the model captures enough complexity to estimate a correct shape.

Numeric Variable
A transformation model is parametrised without seeing the data. Instead, a numeric variable representing the response variable is defined for the Bernstein basis. This is the only reference to the data before the fitting.
The support argument represents the range of the observed response, so from the smallest to the largest response value in the dataset. The bounds argument specifies the range of all possible values for the response variable. Theoretically and generally, a duration such as the waiting time variable can only be positive (there is no such thing as a negative duration), and can be infinitely large (there is no such thing as a too long duration).

Model Status

Bernstein Basis

Order ($M$):

Must be an integer >= 1

Model Options

Distribution ($F_Z$):

Numeric Variable

The support range must be contained into the bounds limits

Support Min:

Support Max:

Lower Bound:

-Inf Numeric input

Enter value:

Upper Bound:

Inf Numeric input

Enter value:

Show Dataset

Fitted Model Summary

Baseline Transformation Function

Baseline transformation $h(y)$ estimated by the model. It is the transformation applied to the response variable to make it behave like the chosen distribution $F_Z$.

Probability Density Function

Likelihood for the response variable to have a certain value.

Cumulative Distribution Function

Area under the PDF curve at x.

Dataset

Old Faithful Geyser Data — contains 272 observations on 2 variables of the Old Faithful geyser in Yellowstone National Park.

eruptions — duration of an eruption in mins
waiting — time until the next eruption in mins

Summary

Categorical Transformation Model

Categorical transformation models deal with ordered discrete response variables. Here, only the unconditional case is presented.

Explanations about how such a model is implemented should be added.

Model

A case of unconditional categorical transformation model is fitted with the tram package.

In this model, rating is defined as the response variable.

Link Function
That part of the model is not interactive on this page. It is set to Logistic, and since it is an unconditional model, that choice is of little importance.

Model Status

Show Dataset

Fitted Model Summary

Density and Distribution Functions

Plot A is the PDF, showing the probability for a response to belong to each category. For example, there is a 36% chance that a rating is 3.

Plot B is the CDF, so the probability for a response to belong to the observed category or any category below. For example, there is a 90% chance that a rating is 4 or below. The height of a step corresponds to the chance of belonging to that category, as depicted in the PDF. For example, the PDF shows that there is a 31% chance that a rating is 2. In the CDF, the corresponding step from 0.07 to 0.38 is 0.31.

Plots C and D show the PDF and CDF of the latent variable $Z$, which is an unobserved continuous variable used in the computation of the model. The transformation function maps the discrete response variable to $Z$. More precise information should be added here.
These two plots can be read in parallel to A and B, because they depict the same relationship but from a different point of view. For example, the area under the curve in C corresponds to the probability in A, so we know that the area between $h_1$ and $h_2$ is 0.31 (value of category 2 in A).

Transformation Function Mapping the Discrete Response Variable to the Latent Variable $Z$

This is another representation of the relationship between $Y$ and $Z$. Plot C is the PDF again, equivalent to plot A above.
The density of $Y$ (plot C) is mapped to the density of $Z$ (plot A, equivalent to plot C above) through the transformation step function $h$ (plot B).

Dataset

Bitterness of wine — dataframe containing 72 observations on 6 variables of a tasting experiment on the bitterness of wine.

response — scorings of wine bitterness on a 0-100 continuous scale
rating — ordered factor with 5 levels; a grouped version of response
temp — temperature during production as a factor with two levels
contact — contact between juice and skins during production as a factor with two levels
bottle — factor with eight levels
judge — factor with nine levels

Summary

Count Transformation Model

Count transformation models are specifically designed for count response variables.

They are expressed by $F_{Y|X=x}(y | x) = \mathbb{P}(Y \leq y | x) = F(h(\lfloor y \rfloor) - x^\top \beta)$, with $F$ being the link function and $y$ being rounded to the nearest integer.

Interactive Model

A case of count transformation model is fitted with the cotram package.

In this model, DVC is defined as the response variable depending on all the other variables.

Bernstein Basis
The Bernstein basis is interactive in this model.

Link Function
The choice of the link function is important because it defines the scale on which to interpret regression coefficients. Still, we can choose any $F$ that interests us.

Log-first
When it is set to TRUE, the model transforms the response with $log(y+1)$ before the Berstein basis is applied. That changes the interpretation scale of the coefficients from the response scale to the log scale, meaning that the coefficients have a multiplicative effect.

Model Status

Bernstein Basis

Order ($M$):

Must be an integer >= 1

Model Options

Link function ($F$):

Log-first:

TRUE FALSE

Show Dataset

Fitted Model Summary

When the link function is cloglog, the linear predictor is interpreted as discrete hazard ratio. We can interpret the sign of the coefficients: if it is positive, there is a higher risk of having a collision compared to the baseline. For example, the baseline for weekday is Monday. All other weekdays, there is a higher risk of having a collision. However, in the weekend, there is a smaller risk of having a collision.

Explanations for the other link functions have yet to be implemented.

Hazard Ratio for the Year 2011

Evolution of the collision risk across a year, estimated for each day, with fixed effects.

The changes in the hazard ratio are relative to the baseline of January 1st, so a higher ratio means a higher risk of collision. For example, we see that there is a peak of collision risk in May with about 12.5 times more risk to have a collision than on January 1st.

Baseline Transformation Function

Baseline transformation $h(y)$ estimated by the model. It is the transformation applied to the response variable to make it behave like the chosen distribution $F$.

Probability Density Function by Year

The PDF and CDF plots represent the isolated year effect on the collision count.

Cumulative Distribution Function by Year

Area under the PDF curve at x.

Dataset

Deer-Vehicle Collisions preprocessed according to the cotram package vignette code (DVC-data and DVC-setup chunks) — time series containing 3'652 observations on 25 variables of collisions between roe deer and vehicles between 2002 and 2011 in Bavaria, Germany.

day — date
DVC — number of deer-vehicle collisions that day
weekday — day of the week
year — year
time — days since beginning (01-01-2002)
tvar1 - tvar20 — sine-cosine transformed times (allow modelling of periodic (yearly) effects)

tvar variables rounded to the fifth decimal in this view.

Summary

Build a Transformation Model

On this page, you can generate a made-to-measure transformation model. Define the parameters of the model in the left menu, generate the model's code, and copy it into your R environment. At the moment, this feature offers only limited options.

Response Variable

Describe your response variable $Y$.

Type:

Name:

Support is the range of Y in your dataset.

Support Min:

Support Max:

Bounds is the range of all possible values for Y.

Lower Bound:

-Inf

Numeric Value

Lower Bound Value:

Upper Bound:

Inf

Numeric Value

Upper Bound Value:

Distribution

Decide the distribution $F_Z$ you want to transform $Y$ to. This will influence the interpretation of coefficients.

Covariates

Enter all the predictors you want to include and name them as in your dataset.

Model Structure

Define how the covariates interact.

Variables interaction:

Shifting terms only (linear effects)

Stratified by one variable

Conditional (all variables interact)

Model Structure

Define how the covariates enter the model.

Variables interaction:

Shifting terms only

Stratified by one variable

Bernstein Basis

Choose how much complexity the model can capture. Usually, an order between 5 and 10 is a good compromise between flexibility and computing time.

Order:

To create a model equivalent to a normal linear regression model, set to 1.

Your R Code

Copy the code below and replace the placeholder names:

Placeholders

my_data : Your data frame name

Welcome!

Features

Acknowledgements

Copyrights and Reproducibility

Unconditional Transformation Model

Interactive Model

Model Status

Bernstein Basis

Model Options

Numeric Variable

Fitted Model Summary

Baseline Transformation Function

Probability Density Function

Cumulative Distribution Function

Dataset

Summary

Linear Transformation Model

Interactive Model

Model Status

Bernstein Basis

Model Options

Predictors

Fitted Model Summary

Baseline Transformation Function

Probability Density Function

Cumulative Distribution Function

Quantile Distribution - Fitted Model

Quantile Distribution - Normal Linear Model

Dataset

Summary

Stratified Linear Transformation Model

Interactive Model

Model Status

Bernstein Basis

Predictors

Fitted Model Summary

Baseline Transformation Function

Probability Density Function

Cumulative Distribution Function

Quantile Distribution for each stratum

Dataset

Summary

Conditional Transformation Model

Interactive Model

Model Status

Bernstein Basis

Predictors

Fitted Model Summary

Dataset

Summary

Categorical Transformation Model

Model

Model Status

Fitted Model Summary

Density and Distribution Functions

Transformation Function Mapping the Discrete Response Variable to the Latent Variable \(Z\)

Dataset

Summary

Count Transformation Model

Interactive Model

Model Status

Bernstein Basis

Model Options

Fitted Model Summary

Hazard Ratio for the Year 2011

Baseline Transformation Function

Probability Density Function by Year

Cumulative Distribution Function by Year

Dataset

Summary

Build a Transformation Model

Response Variable

Distribution

Covariates

Model Structure

Model Structure

Bernstein Basis

Your R Code

Placeholders