Target Audience: New Graduates/Analysts
One-hot coding and effect coding are two common methods used to encode categorical variables in statistical analysis. One-hot coding involves creating binary columns for each category in the variable. Each column represents a category, and a value of 1 indicates the presence of that category, while 0 indicates the absence. This method results in a binary representation of the categories. Effect coding is a contrast coding method that compares each level of a categorical variable to the grand median (in AVM) of all levels. It uses one less variable than one-hot coding and allows for comparison between categories.
Example 1
Let's say we have a categorical variable, "Neighborhood," that describes three neighborhoods: "Downtown," "Suburb," and "Rural" in a city. We want to encode it using one-hot coding and effect coding.
One-Hot Coding:
·
"Downtown"
would be encoded as [1, 0, 0]
·
"Suburb"
would be encoded as [0, 1, 0]
·
"Rural"
would be encoded as [0, 0, 1]
Effect
Coding:
·
"Downtown"
would be encoded as [-2, 1, 1]
- "Suburb"
would be encoded as [1, -2, 1]
- "Rural"
would be encoded as [1, 1, -2]
In effect coding, the coefficients sum
to zero, with one level as the reference category (-2) and the other as
comparisons (1).
In summary, one-hot coding creates
binary columns for each category, while effect coding compares each category to
the grand median. Both methods have their uses in different statistical
analyses and modeling techniques.
While one-hot coding and effect coding are two common methods used
to encode categorical variables in statistical analysis, they are not the only
methods available. Here are a few other common methods:
1. Label
Encoding: Label encoding assigns a unique numerical value to each category in
the variable. This method is simple and efficient but may not be suitable for
models that assume ordinal relationships between categories.
2. Binary
Encoding: Binary encoding converts each category into binary digits and
represents them in a new set of columns. This method reduces the number of
columns compared to one-hot encoding while preserving the information about the
categories.
3. Helmert Coding: Helmert coding compares each categorical variable level with the subsequent levels' median. This method is useful for capturing changes or trends in the categories.
4. Sum
Coding: Sum coding compares each level of a categorical variable with the overall
median of all levels. This method provides information about differences
between each level and the overall median.
Each encoding method has its advantages and drawbacks, and the
choice of method depends on the nature of the categorical variable and the
specific requirements of the analysis or model being used.
Continuing with our "Neighborhood" categorical variable
in the housing market for the other encoding methods:
1.
Label
Encoding: If we use label encoding for the "Neighborhood" variable:
- "Downtown"
could be encoded as 1
- "Suburb"
could be encoded as 2
- "Rural"
could be encoded as 3
2.
Binary
Encoding: If we use binary encoding for the "Neighborhood" variable:
- "Downtown"
could be encoded as [0, 0]
- "Suburb"
could be encoded as [0, 1]
- "Rural"
could be encoded as [1, 0]
3.
Helmert
Coding: If we use Helmert coding for the "Neighborhood" variable:
- "Downtown"
would be encoded as [1, -1, 0]
- "Suburb"
would be encoded as [1, 1, -2]
- "Rural"
would be encoded as [1, 1, 1]
4.
Sum
Coding: If we use sum coding for the "Neighborhood" variable:
- "Downtown"
would be encoded as [1, -1, 0]
- "Suburb"
would be encoded as [0, 1, -1]
- "Rural"
would be encoded as [0, 0, 1]
These examples demonstrate how
different encoding methods can represent the same categorical variable in
various numerical formats, each with its own characteristics and implications
for analysis or modeling in the housing market context.
One-Hot Coding
When dealing with a categorical variable with more than two levels
(e.g., four categories: Mid-density Non-HOA, High-density Non-HOA, PUD, and
MPUD), AVM analysts can use a technique called one-hot encoding to represent
these categories in a regression model.
One-hot encoding involves creating
binary dummy variables for each category of the categorical variable. Each dummy
variable represents one category and takes on values of 0 or 1.
Here's how they can encode the above
categorical variable with four categories for a regression model:
1.
Mid-density
Non-HOA: This category would be represented by a dummy variable that takes on
the value of 1 if the observation falls into this category and 0 otherwise.
2.
High-density
Non-HOA: Another dummy variable would be created for this category with the
same logic as above.
3.
PUD:
They would assign a dummy variable for this category.
4.
MPUD:
Similarly, they would create a dummy variable for this category.
Using one-hot encoding in this
manner, they can incorporate a categorical variable with multiple levels into
their regression model, allowing them to analyze the impact of different categories
on the dependent variable while maintaining the regression model's linearity assumption.
Important Note
AVM analysts will create four
dummy variables to represent each category using one-hot encoding for a
categorical variable with four distinct categories (Mid-density Non-HOA,
High-density Non-HOA, PUD, and MPUD). Each dummy variable would take on 1 if the observation belongs to that particular category and 0 if it
does not. This approach allows them to include all four variable categories in
the regression model and analyze their effects on the dependent variable
separately.
Effect
Coding
Example
2
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Effect coding a categorical variable like the
TOWNSHIP variable in a regression model can be statistically valid and useful
for comparing the mean (or median) response across different categorical variable levels to an overall mean (or median).
By effect coding the TOWNSHIP variable, the model examines how
each township's median SP/SF deviates from the countywide median. This
approach allows for the evaluation of the average effect of each township on
the dependent variable, Adjusted Sale Price while accounting for the
countywide effect.
Median vs. Median in
Effect Coding
Even if the data series does not contain many outliers, using the
median instead of the median can still be preferable in certain
instances, especially in an AVM. The median is a robust measure of central
tendency less affected by extreme values than the median. By using the
median to code a categorical variable like the TOWNSHIP variable in a regression model, analysts ensure that the central tendency measure
chosen is not unduly influenced by any outliers in the data.
In cases where the data is skewed or if there are concerns about
outliers impacting the average, the median can provide a more stable estimate
of the central tendency and better reflect the typical value within each
township. Therefore, even if the data series does not contain many outliers,
selecting the median over the median can still be suitable for creating more robust and reliable effect coding for the regression model.
Case for Effect Coding
Effect coding can be employed to represent a categorical variable with four levels (Mid-density Non-HOA, High-density Non-HOA, PUD, and MPUD) as a single variable in a regression model for sale price while preserving linearity. This technique captures
the combined influence of all four categories on the dependent variable by
considering their deviations from the overall median. By incorporating these
effect-coded variables into the model, analysts can analyze the impact of the
categorical variable while maintaining a linear relationship with the sale price.
Example 3
Let's consider a numerical example to demonstrate effect coding
for a categorical variable with four categories: Mid-density Non-HOA,
High-density Non-HOA, PUD, and MPUD, with the following data:
·
Mid-density
Non-HOA: Median Sale Price = $300,000
·
High-density
Non-HOA: Median Sale Price = $320,000
·
PUD:
Median Sale Price = $350,000
·
MPUD:
Median Sale Price = $380,000
·
Overall
Median Sale Price = $337,500
To
apply effect coding, the differences between each category median and the
overall median have to be calculated:
1.
Mid-density
Non-HOA Effect: $300,000 (category median) - $337,500 (overall median) =
-$37,500
2.
High-density
Non-HOA Effect: $320,000 (category median) - $337,500 (overall median) =
-$17,500
3.
PUD
Effect: $350,000 (category median) - $337,500 (overall median) = $12,500
4.
MPUD
Effect: $380,000 (category median) - $337,500 (overall median) = $42,500
Effect-coded variables for the
four categories using these calculated values have been created. For each
observation, the effect-coded variable will reflect the difference between the
category median and the overall median of Sale Price. In the regression model, analysts
would include these effect-coded variables and other independent
variables to analyze the impact of the categorical variable on the dependent
variable (Adjusted Sale Price) while maintaining linearity.
Important to Note
Creating the effect-coded variable using a normalized variable such as SP/SF instead of the sale price itself can be a reasonable approach, especially if analysts are interested in
capturing the impact of the categorical variable on the price per square foot
specifically.
When using normalized variables,
they are standardizing the metric or unit of measurement, which can help improve the interpretability and comparability of coefficients
in the regression model.
In this case, creating effect-coded variables based on the SP/SF (a normalized measure) for the four
categories can help them assess how each category influences the price per
square foot compared to the overall price per square foot median.
However, it is important to keep in mind that when using
normalized variables, the interpretation of the coefficients in the regression
model will be based on the price per square foot, not the absolute sale price.
Additionally, including the living area as a separate independent variable in
the regression equation will allow them to analyze its individual effect on the
dependent variable while assessing the impact of the categorical variable on
the price per square foot.
Conclusion
One-hot and effect coding are valuable techniques
for transforming categorical variables into a format usable by automated
valuation models (AVMs). Here's a breakdown of their strengths and weaknesses
to help you choose the best approach for your AVM:
One-Hot Coding:
·
Strengths:
Simple to implement, easy to interpret coefficients, avoid multicollinearity.
·
Weaknesses:
Increases model complexity with many categories, can be data-hungry for large
datasets.
Effect Coding:
·
Strengths: It is more efficient for models with many categories, reduces multicollinearity, and allows
for comparisons between specific categories.
· Weaknesses: Interpretation of coefficients can be less intuitive and requires choosing a reference category that may influence results.
Choosing the Right Approach:
·
One-hot coding is a
good choice for AVMs with a small number of categorical variables or when a
clear interpretation of individual category coefficients is crucial.
·
Effect coding is
preferable for AVMs with many categories or when you're interested
in comparing specific categories (e.g., pool vs. no pool).
The optimal approach depends on
the specific characteristics of the AVM and the intended research questions. To
determine the most suitable method, it is recommended that the model be run with both coding approaches and that the results be compared. This comparison will
reveal which method yields the most interpretable and accurate data assessment.
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