Interpreting Parameters and Scalability

In our previous discussions, we delved into the mechanics of running linear regression and utilizing it for prediction. We also explored the various metrics used to evaluate the performance of linear regression models. However, I realized we were missing a crucial aspect: understanding the parameters and interpreting the coefficients. Thus, this article serves as an extension to the last two articles on my LinkedIn page, aiming to fill this gap.

Y (dependent variable) = B0 + B1(X1) + B2(X2) ….Bn(Xn) + E (random error)

Let's understand this equation:

• B0, termed as the intercept, provides insights into the magnitude and direction of Y when all the X variables are set to zero.
• B1, B2, ..., Bn represent coefficients, illustrating the rate of change in Y for a one-unit change in each corresponding X variable.

E, or epsilon, or random error, denotes the random error component, which captures the variability in Y that cannot be explained by the linear relationship with the X variables.

Understanding coefficients in a linear regression model is crucial. A positive coefficient means a positive relationship between variables, while a negative one shows the opposite. The intercept (B0) gives the starting point when predictors are zero. Coefficients (B1, B2, ..., Bn) tell us how much the dependent variable changes when the predictor changes by one unit, helping us see which predictors matter more.

Let's Look at an example-

To help understand this further, let's explore our favorite example involving ice cream sales. Imagine we're analyzing the relationship between ice cream sales and various factors like temperature and marketing expenses. In this scenario, the coefficients (B1, B2, ...) would quantify the impact of each factor on ice cream sales, while the intercept (B0) would provide insights into the baseline sales when all other factors are absent. I used a random data in Python and generated the summary for the model, here are my results (modified to show important parameters.)

               OLS Regression Results                            
====================================================================
                coeff     std err      tstat     P>|t|      

const          0.1622      0.095      1.712      0.090     
temp.          1.9336      0.090     21.540      0.000     
market_exp.    2.9906      0.087     34.426      0.000                   

Now let's interpret the betas:

• Intercept (const): With a coefficient of 0.1622, it indicates the expected value of the dependent variable when all other predictors are zero.
• Temperature (temp.): Showing a coefficient of 1.9336, it implies that for each one-unit rise in temperature, the dependent variable is expected to increase by 1.9336 units, with other variables held constant.
• Marketing Expenditure (market_exp.): With a coefficient of 2.9906, it signifies that for every one-unit increase in marketing expenditure, the dependent variable is anticipated to increase by 2.9906 units, all other variables being constant.

Now, the question arises - Does altering the scale of our variables impact the regression coefficients and their interpretation?

The answer is yes.

When we change the scale of our variables, it affects the coefficients in the regression model. This means that a one-unit change in a variable might now have a bigger or smaller impact on the outcome, depending on the scale. Let's understand better with example, we convert temperature from Celsius to Fahrenheit in a linear regression model, the coefficient representing the temperature's effect on ice cream sales changes. For instance, if a 1-degree Celsius increase in temperature corresponds to a 50-unit increase in sales, in Fahrenheit, it would equate to roughly a 90-unit increase. This demonstrates how scaling affects the interpretation of coefficients in regression analysis.

In a nutshell, linear regression is helpful for predicting outcomes, but we need to be able to interpret its parameters and scalability to get the most from our data. Understanding these concepts lets us make better decisions and extract valuable insights. Let's continue exploring data science together to unlock the secrets in our data.