# Statistics - Dummy (Coding|Variable) - One-hot-encoding (OHE)

### Table of Contents

## 1 - About

Dummy coding is:

- a classic way to transform nominal into numerical values.
- a system to code categorical predictors in a regression analysis

A system to code categorical predictors in a regression analysis in the context of the general linear model.

We can't put categorical predictors such as character variable, or a string variable into a regression analysis function. We need to make it a numeric variable in some way. That's where dummy coding comes in.

It allows to look at categorical predictors in the same model as continuous predictors and put them together in moderation analyses.

Featurizing via a one-hot-encoding representation lead to a very large feature vector. To reduce the dimensionality of the feature space, feature hashing is generally used.

## 2 - Articles Related

## 3 - Scheme

### 3.1 - Pairwise

Four levels of my (independent|grouping) variable:

- Blue
- Green
- Red
- Yellow

Groups | Dummy Code 1 Variable 1 | Dummy Code 2 Variable 2 | Dummy Code 3 Variable 3 |
---|---|---|---|

Blue | 0 | 0 | 0 |

Green | 1 | 0 | 0 |

Red | 0 | 1 | 0 |

Yellow | 0 | 0 | 1 |

The number of dummy code (dummy variable) is the number of value minus 1. Blue is the reference group and get 0 across the board.

The regression constant in a multiple regression, it's the predicted score on the outcome variable, when all others variables are zero which means that it will be the predicted score for the reference group.

The regression constant will be the mean for **Blue** because it's the reference group.

In this scheme, you will compare the reference group with the others group. If you want to make a comparison between two groups that are not a reference group, you will have to make one of them the reference group.

These is pairwise comparisons approach and is for this reason generally not used. A one-way ANOVA would be done instead where, you can code a categorical variable in a multiple regression analysis.

### 3.2 - Effects coding

In Effects coding schemes, the regression constant is the predictive score for all individuals, across all groups, not just the predictive score for the reference group.

#### 3.2.1 - unweighed

Instead of giving O across the board, the reference group get minus one.

Groups | Dummy Code 1 | Dummy Code 2 | Dummy Code 3 |
---|---|---|---|

Blue | -1 | -1 | -1 |

Green | 1 | 0 | 0 |

Red | 0 | 1 | 0 |

Yellow | 0 | 0 | 1 |

The codes (variables) are now effects codes instead of dummy codes.

The regression constant will now represent the average across all groups. It will not be exactly the mean for the entire group because it's unweighed as it doesn't take into account the fact that there are different number of individuals into each group.

If you wanted to know the difference between the overall mean and the reference group, you would have to re-code this with a difference reference group.

#### 3.2.2 - weighted

If you want to get the regression constant to be the exact mean of all individuals, you have to weight the score by the number of individuals in each group.

Groups | Dummy Code 1 | Dummy Code 2 | Dummy Code 3 |
---|---|---|---|

Blue | <math>- \frac{\displaystyle N_{Blue}}{\displaystyle N_{Total}}</math> | <math>- \frac{\displaystyle N_{Blue}}{\displaystyle N_{Total}}</math> | <math>- \frac{\displaystyle N_{Blue}}{\displaystyle N_{Total}}</math> |

Green | <math>\frac{\displaystyle N_{Green}}{\displaystyle N_{Total}}</math> | 0 | 0 |

Red | 0 | <math>\frac{\displaystyle N_{Red}}{\displaystyle N_{Total}}</math> | 0 |

Yellow | 0 | 0 | <math>\frac{\displaystyle N_{Yellow}}{\displaystyle N_{Total}}</math> |

## 4 - Example

Dictionary with the mapping (featureId, Value) = NewFeatureId

### 4.1 - Before Mapping

Feature 0: Color

- Red
- Blue
- Purple

Feature 1: Status

- Not started
- In Progress
- Final

### 4.2 - After mapping

- New Feature 0 = (Color (Feature 0), Red)
- New Feature 1 = (Color (Feature 0), Blue)
- New Feature 2 = (Color (Feature 0), Purple)
- New Feature 3 = (Status (Feature 1), Not started)
- New Feature 4 = (Status (Feature 1), In Progress)
- New Feature 5 = (Status (Feature 1), Final)