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STANDARDIZED VS UNSTANDARDIZED REGRESSION COEFFICIENT

This article was published as a part of the Data Science Blogathon.

Introduction

Some time back, I was making the predictive model using Linear Regression and I found a variable whose unstandardized coefficient of regression(beta or estimate) close to zero but after some analysis, I find that it is statistically significant (means p-value <0.05). We know that if a variable is significant for a particular model it means its coefficient value is significant and different from zero. So, the question that arises is “Why coefficient value is close to zero but that variable is significant for our predictive model?”.

The answer to this question lies in the difference between the standardized and unstandardized coefficients of regression. So, in this article, we will see the basic concepts behind these coefficients and how they are different from each other with their advantages and disadvantages.

The concept of standardization or standard coefficients come into the picture when independent variables or predictor for a particular model are expressed in different units. For Example, let say we have three independent features namely height, age, and weight. Her height is in inches, her weight in kilograms, and her age in years. If we want to rank these predictors based on the unstandardized coefficient (which directly comes when we train a regression model), it would not be a fair comparison since the units for all the predictors are different.

Unstandardized Regression Coefficients 

1. What are unstandardized regression coefficients?

Unstandardized coefficients are those which are produced by the linear regression model after its training using the independent variables which are measured in their original scales i.e, in the same units in which we are taken the dataset from the source to train the model.

– Unstandardized coefficient should not be used to drop or rank predictors (aka independent variables) as it does not eliminate the unit of measurement.

For Example, let’s take a hypothetical example where we want to predict the income(in rupees) of a person based on its age (in years), height(in cm), and weight(in kg). So, here inputs for our regression model are age, height, and weight, and output is income. Then,

Income(rupees)=a0+a1*age(years)+a2*height(cm)+a3*weight(kg)+e                (eqn-1)

 

2. How to interpret the unstandardized regression coefficients?

These are used to interpret the effect of each independent variable on the outcome(response/output). Their interpretation is straightforward and intuitive.

– All other variables held constant, a shift of 1 unit in Xi(predictors) implies there is an average change of ai units in Y(outcome).

In the above example, if a1=0.3, a2=0.2, and a3=0.4 (and assume all are statistically significant), then we interpret these coefficients as :

Getting 1 year older is associated with an increase of 0.3 in income, assuming other variables are constant (means there is no change in height and weight).

Similarly, we can interpret the coefficient for other independent variables as well.

It represents the amount by which dependent variable changes if we change independent variable by one unit keeping other independent variables constant.

3. Limitations of the unstandardized regression coefficients

– Unstandardized coefficients are great for interpreting the relationship between an independent variable X and an outcome Y. However, they are not useful for comparing the effect of an independent variable with another one in the model.

– For Example, which variable has a larger impact on Income, Age or Height, or weight?

We can try to answer this question by looking at equation-1 and again assume that a1=0.3, a2=0.2, and a3=0.4, we conclude that :

“An increase of 20 cm in height has the same effect on the weight increases 10 times”

Still, this does not answer the question of which variable affects more Income.

Specifically, the statement that “the effect of the increase of weight by 10 times = the effect of the increase in height of 20 cm ” is meaningless without specifying how hard it is to increase height by 20 cm, specifically for someone who’s not familiar with this scale.

So, at last, we conclude that a direct comparison of the regression coefficients for any of the pair of independent variables is not making sense or not useful as these independent variables are on the different scales (age in years, weight in kg, and height in cm).

It turns out that the effects of these variables can be compared by using the standardized version of their coefficients. And that’s what we’re going to discuss next.

Standardized regression coefficients

1. What are standardized regression coefficients?

The standardized coefficients of regression are obtained by training(or running) a linear regression model on the standardized form of the variables.

The standardized variables are calculated by subtracting the mean and dividing by the standard deviation for each observation, i.e. calculating the Z-score. It would make mean 0 and standard deviation 1. Then, they don’t represent their original scales since they have no unit.

For each observation “j” of the variable X, we calculate the z-score using the formula:

z-score formula

2.  Which variables we have to standardize for finding the standardized regression coefficients i.e, both predictor and response or either one of them?

Yes, we standardize both the dependent(response) and the independent(predictor) variables before running the linear regression model (as this is the widely accepted practice when we want to find the standardized form of the variables).

3. How to interpret the standardized regression coefficients?

The interpretation of standardized regression coefficients is non-intuitive compared to their unstandardized versions:

A change of 1 standard deviation in X is associated with a change of β standard deviations of Y.

Note:

– If there is a categorical variable in place of a numerical variable in our analysis, then its standardized coefficient cannot be interpreted as it does not make sense to change X by 1 standard deviation. In general, this is not a problem for our model since these coefficients are not meant to be interpreted individually, but to be compared to one another in order to get a sense of the importance of each variable in the linear regression model.

The standardized coefficient is measured in units of standard deviation. A beta value of 2.25 indicates that a change of one standard deviation in the independent variable results in a 2.25 standard deviations increase in the dependent variable.

4. What is the real use of standardized coefficients?

They are mainly used to rank predictors (or independent or explanatory variables) as it eliminate the units of measurement of independent and dependent variables). We can rank independent variables with an absolute value of standardized coefficients. The most important variable will have the maximum absolute value of the standardized coefficient.

For example:

Y = β0 + β1 X1 + β2 X2 + ε

If the standardized coefficients β1 = 0.5 and β2 = 1, we can conclude that:

X2 is twice as important as X1 in predicting Y, assuming that both X1 and X2 follow roughly the same distribution and their standard deviations are not that different.

5. Limitations of Standardized regression coefficients

The standardized coefficients are misleading if the variables in the model have different standard deviations means all variables are having different distributions.

Take a look at the following linear regression equation:

Income($) = β0 + β1 Age(years) + β2 Experience(years) + ε

Because our independent variables Age and Experience are on the same scale (years) and if it is reasonable to assume that their standard deviations differ a lot, then in this case:

– Their unstandardized coefficients should be used to compare their importance/influence in the model.

– Standardized these variables would, in fact, cause them to be on a different scale (different standard deviations or follows different distribution)

Calculation of Standardized Coefficients

1. For Linear Regression(Another approach as we see one approach in the above part of the article)

The standardized coefficient is found by multiplying the unstandardized coefficient by the ratio of the standard deviations of the independent variable and dependent variable.

STANDARDIZED vs UNSTANDARDIZED for linear regression formula

2.  For Logistic Regression

STANDARDIZED UNSTANDARDIZED logistic regression

 

End Notes

This article covered some basic thing but necessary concepts when we work on a real-life project in Machine Learning and Artificial Intelligence. I hope you understood the concepts explained in this article very well. In this article in the last part, we see only the formulation related to the concepts but we do not go into much depth about the Mathematics behind them, that part we will discuss in some other article.

If you have any questions, let me know in the comments section!

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