Lære Backward Difference Coding in Practice | Advanced Categorical Coding Schemes

Backward Difference coding is a powerful technique for representing categorical variables in regression models.

Purpose: compare the mean of each category to the mean of all previous categories;
Contrast: not limited to a single reference or the overall mean;
Best for: ordinal or nominal categories where the order carries meaningful information.

When you use Backward Difference coding:

Each coefficient in your regression model shows the difference in the mean response between the current category and the average of all categories that come before it;
This approach helps you interpret the incremental effect of moving from one category to the next in sequence;
The last category is left out and serves as the implicit reference; all differences are calculated relative to this cumulative structure.

Example context

Suppose you have a categorical feature called Education with three levels:

High School;
Bachelors;
Masters.

The following code demonstrates how to encode this feature using Backward Difference coding with Python, pandas, and scikit-learn.


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import pandas as pd
from sklearn.preprocessing import OneHotEncoder

# Sample data
df = pd.DataFrame({
    "Education": ["High School", "Bachelors", "Masters", "Bachelors", "Masters", "High School"]
})

# Define category order for Backward Difference coding
categories = [["High School", "Bachelors", "Masters"]]

# Custom function for Backward Difference coding
def backward_difference_coding(series, categories):
    n = len(categories[0])
    codes = []
    for val in series:
        row = []
        for i in range(n - 1):
            if categories[0].index(val) <= i:
                row.append(0)
            elif categories[0].index(val) == i + 1:
                row.append(1)
            else:
                row.append(-1)
        codes.append(row)
    col_names = []
    for i in range(n - 1):
        col_names.append(f"BD_{categories[0][i+1]}_vs_prev")
    return pd.DataFrame(codes, columns=col_names)

# Apply Backward Difference coding
bd_encoded = backward_difference_coding(df["Education"], categories)
result = pd.concat([df, bd_encoded], axis=1)
print(result)

Backward Difference coding implementation

The code sample above demonstrates how to perform Backward Difference coding for a categorical variable using Python. Here's how each part of the implementation works:

Custom function definition: The function backward_difference_coding takes a pandas Series and a list of ordered category labels. The order in this list defines the sequence for Backward Difference comparisons;
Category order and dimensions: The variable n stores the number of unique categories. For each value in the series, you create a new row of coded values with n - 1 columns, matching the number of comparisons required by the scheme;
Row-wise encoding logic:
- For each category value, the function checks its position in the order;
- For each column (comparison), if the value’s index is less than or equal to the current comparison, assign 0;
- If the value’s index is exactly one after the current comparison, assign 1;
- If the value’s index is greater than one after the current comparison, assign -1;
- This logic ensures that each row captures the difference between the current category and the mean of all previous categories.
Column naming: Each column is labeled to indicate which category is being compared to all those before it. For example, BD_Bachelors_vs_prev shows the difference between "Bachelors" and "High School";
Output DataFrame: The resulting DataFrame has the original category and two new columns (since there are three categories). Each row’s values in these columns represent the Backward Difference coded representation for that observation.

Output interpretation

A 1 in a column means the current category is the next in sequence compared to previous categories;
A -1 means the current category is after the previous comparison group;
A 0 means the current category is before or equal to the current comparison.

This encoding allows you to fit regression models where each coefficient directly reflects the difference between a category and the mean of all categories that come before it in the specified order.

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