A regression model is a statistical tool that allows us to explore the relationship between one or more explanatory variables (also called independent variables or predictors) and a response variable (also called dependent variable or outcome). Regression models can be used for various purposes, such as describing patterns, testing hypotheses, making predictions, or evaluating interventions.
There are different types of regression models depending on the nature and number of the explanatory and response variables, such as linear regression, logistic regression, multiple regression, etc. In this blog post, we will focus on linear regression, which is one of the most common and simple forms of regression.
Linear regression assumes that there is a linear relationship between one explanatory variable (X) and one response variable (Y), meaning that a change in X is associated with a proportional change in Y. The equation of this relationship can be written as:
Y = a + bX + e
where a is the intercept (the value of Y when X is zero), b is the slope (the amount of change in Y for a unit change in X), and e is the error term (the difference between the observed and predicted values of Y).
To illustrate how linear regression works, let’s use an example from the real world. Suppose we want to study how the height of a person affects their weight. We can collect data from a sample of people and plot their height (X) and weight (Y) on a scatterplot. Here is an example of such a plot:
We can see that there is a positive correlation between height and weight, meaning that taller people tend to weigh more than shorter people. But how can we quantify this relationship and make predictions based on it? This is where linear regression comes in.
Using linear regression, we can fit a straight line that best summarizes the pattern of the data points. This line is called the regression line or the line of best fit. The equation of this line can be estimated using a method called ordinary least squares (OLS), which minimizes the sum of squared errors (SSE), or the sum of squared differences between the observed and predicted values of Y. Here is an example of a regression line fitted to our data:
The equation of this line can be written as:
Y = 38.6 + 0.9X
This means that the intercept is 38.6 kg and the slope is 0.9 kg/cm. In other words, when height is zero (which is not possible in reality), weight is 38.6 kg, and for every centimeter increase in height, weight increases by 0.9 kg on average.
Using this equation, we can make predictions about the weight of a person given their height. For example, if we want to predict the weight of a person who is 170 cm tall, we can plug in X = 170 into the equation and get:
Y = 38.6 + 0.9 * 170
Y = 191.4
Therefore, we predict that a person who is 170 cm tall weighs 191.4 kg on average.
Of course, this prediction may not be very accurate for every individual, as there are other factors that affect weight besides height, such as age, gender, diet, exercise, etc. These factors are not accounted for by our simple linear regression model, and they contribute to the error term (e) in our equation. Therefore, we should not rely on our model for precise predictions, but rather use it as a general guide to understand the overall trend and direction of the relationship between height and weight.
Linear regression is a powerful and versatile tool that can help us analyze and interpret data from various fields and domains. However, it also has some limitations and assumptions that need to be checked before applying it to any data set. Some of these include:
- Linearity: The relationship between X and Y should be linear or approximately linear. If the relationship is nonlinear, such as quadratic or exponential, linear regression may not be appropriate or may need to be transformed.
- Independence: The observations should be independent of each other, meaning that they are not influenced by common factors or sources of variation. If there is dependence or correlation among the observations, such as in time series or clustered data, linear regression may not be valid or may need to be adjusted.
- Homoscedasticity: The variance of Y should be constant or similar across different values of X. If the variance of Y changes significantly with X, such as increasing or decreasing, linear regression may not fit well or may