Statisticians use regression analysis to ascertain the value of an independent variable based on another variable, thereby predicting the outcome or trend of particular events or situations. Numerous industries rely on this methodology, such as fitness product manufacturers who use regression analysis to evaluate the correlation between pricing and promotional tactics and the volume of sales.

Regression analysis is a dynamic instrument for identifying and investigating correlations among variables and can be executed through various techniques. Each technique of regression analysis comes with its pros and cons, and not all of these techniques are suitable for all types of problem statements. In this article, we will delve into the mathematical principles behind some of the most frequently used regression methods.

## What are the practical applications of statistical regression analysis?

Undoubtedly, regression analysis is a powerful tool for business analysts and data professionals hoping to comprehend their data’s significance. Regression analysis can elucidate the impact of a solitary independent variable on each of the dependent variables, allowing analysts to identify the relevant factors while discarding the irrelevant ones. This can aid business analysis methodologies, leading to successful and desired outcomes.

**Note:** It is crucial to have a clear understanding of variables’ meanings before including them in a model. Numerous elements can impact an organisation’s effectiveness and success.

## Different Types of Regression Methods

Every regression analysis method comes with its own set of advantages and disadvantages. See the list below for some typical ones.

### The Regression Line

Linear regression is a modelling method whereby statisticians establish a linear relationship between an independent variable set (X) and a dependent variable set (Y). It is important to note that this methodology is only valid when the variables exhibit a linear relationship, as indicated by its name.

Independent variables are selected by the researcher and not influenced by the other variables in the experiment. For example, in the case of gym supplements, sales are the dependent variable, while pricing and advertising are independent variables.

#### Assumptions of Linear Regression

- The independent and dependent variables must be linearly related.
- None of the variables must be dependent or related to any other variable.
- Outliers must be excluded before fitting a regression line.
- Multicollinearity is not acceptable.

#### Polynomial Regression

You may have observed that in the equations above (Y = m*x+c), the independent variable is multiplied by a factor of one. When the independent variable’s power is greater than one, the regression type in use is known as polynomial regression (Y = m*x2+c).

When the data’s degree is greater than one, a smooth curve connecting all data points is required, rather than the typical straight line. This curve will best represent the context in which the data is presented.

#### Important Information

- Regularly plotting variable relationships is crucial to ensure that the curve is appropriately fitted and avoids overfitting or underfitting due to excessively high polynomial degrees. This helps to reduce the risk of inaccurate outcomes and guarantee the best possible results.
- Higher-degree polynomial extrapolation can result in misleading conclusions, so it’s important to pay attention to the curve’s shape as you approach the extrema.
- Logarithmic Regression Analysis
- Logistic regression is a statistical method used to assess the probability of an event occurring, typically when the dependent variable can be split into two distinct categories, such as a binary variable (e.g. 0 or 1, yes or no, cat or dog). The result of the analysis is a probability, which will be a value between 0 and 1.
- Logistic regression is frequently applied in classification tasks.
- Applying a non-linear log transformation to the estimated odds ratio enables the avoidance of the linear relationship assumption between the dependent and independent variables, which is fundamental to linear regression.
- Multinomial logistic regression is employed when the final outcome comprises multiple categories.
- Like linear regression, nonlinear regression does not allow for multicollinearity.

### Steep Incline Linear Regression

Before diving into ridge regression, let’s examine regularization, a technique for making models more efficient in handling unfamiliar data by removing less significant features. Check out our guide for creating your own SEN model using spaCy.

Two types of regularization techniques are ridge regression and lasso regularization.

In actual data, it’s unrealistic to find variables that are entirely unrelated to one another. As a result, when dealing with actual data, multicollinearity should always be considered. Least squares approach is often ineffective because of the large variations between the observed and actual values. Ridge regression can help to avoid overfitting by punishing models with high variability and forcing their beta coefficients to zero.

The objective in linear regression is to minimize the cost function to develop a successful model. For this purpose, the cost function should include ‘lambda’ and ‘slope’. This will decrease the bias and standard deviation, resulting in a highly accurate model.

### Lasso Regression Example

LASSO (Least Absolute Shrinkage and Selection Operator) regression reduces the variability of linear regression models, similar to Ridge regression. Furthermore, it can aid in selecting the appropriate features by imposing a penalty function that employs absolute values rather than the squared values utilized in Ridge regression. Check out our blog post on the importance of financial planning.

As previously stated, the ridge regression’s best fitting line was approaching zero. The overall slope was not linear and instead tended towards a decreasing trend. However, in Lasso regression, the slope appeared to reach zero. Whenever the slope value is low in this type of regression, specific features are removed, indicating that these characteristics are not essential in determining the optimal fitting line. As a result, it becomes simpler for us to choose the most relevant features.

## Choosing the Suitable Model for Regression Analysis

Keep in mind that the regression models discussed are not exhaustive. Selecting the most appropriate model from such a vast range of possibilities can be a difficult task. It is critical to take into account the data’s dimensionality and any other pertinent information while making this decision. By prioritizing these elements, we can ensure that the best regression model is selected.

Consider the following factors when selecting the optimal regression model:

- Prior to creating a dependable prediction model, conducting exploratory data analysis is crucial. This first step in the process reveals the relationship between various factors, allowing for a more informed selection of the appropriate model.
- Comparing a given model to other models using various statistical metrics such as R Square, Adjusted R Square, Area Under the Curve (AUC), and Receiver Operating Characteristic (ROC) Curve allows us to assess its adequacy. These metrics assist us in determining how well the model fits the data and, as a result, the reliability of the model results.
- Cross-validation is an effective technique for determining the accuracy of a model. The dataset is divided into two parts using this method: a training set and a validation set. This approach helps us determine whether the model is overly restrictive or overly lenient with the data. It also enables us to identify any potential flaws in our model more easily and make any necessary adjustments.
- Feature selection methods, such as lasso and ridge regression, can be extremely helpful when working with datasets with a large number of features or variables that display multicollinearity. These techniques simplify the model and improve its performance by eliminating irrelevant or redundant features. Check out our blog post on exploring the information universe.

Regression analysis offers two significant advantages. Firstly, it identifies the connection between input and output variables, providing insights into how alterations in the independent variable can impact the dependent variable. Secondly, regression analysis helps determine the independent variable’s impact on the dependent variable.

Before attempting to apply any of the regression methods mentioned, it is crucial to consider the data’s characteristics. The nature of the variables, such as whether they are continuous or discrete, can assist in selecting the appropriate technique. Although all of the regression techniques discussed are based on the same fundamental principles, their complexities increase with the number of variables or the strength of the independent variable.