Regression analysis is a statistical method used to determine the value of a dependent variable based on the value of another variable. It is widely used to predict the impact or trend of certain events or occurrences. For example, a supplement manufacturer in the fitness industry may utilise regression analysis to determine the effect of pricing and promotional strategies on sales.

Regression analysis is a powerful tool for identifying relationships between variables, and can be conducted in a variety of ways. Each method of regression analysis carries its own advantages and disadvantages, and not all of them are suitable for all types of problem statements. In this article, we will explore the mathematical principles behind some of the most commonly used regression techniques.

## In what ways are approaches for statistical regression analysis useful?

It is clear that regression analysis can be an effective tool for business analysts and data professionals to better understand the significance of their data points. This type of analysis helps to demonstrate how a modification of a single independent variable will affect the dependent variables, enabling analysts to identify the relevant factors while discarding those that are not. This in turn can aid in the application of sound business analysis methodologies to realise the desired outcomes.

**Note:** Prior to including any variable in a model, it is essential to have a clear understanding of its meaning. The efficacy of an organisation may be contingent upon a range of elements.

## Distinctive Regression Methods

Each method of regression analysis has its own advantages and disadvantages. Typical ones are listed below.

### Regression line

Linear regression is a modelling technique used in statistics to establish a linear relationship between a set of independent variables (X) and a set of dependent variables (Y). It is important to note that this technique can only be used when there is a linear connection between the variables, as suggested by its name.

Independent variables are those that are selected by the researcher, without being influenced by the other variables in the experiment. In the example of the gym supplement, sales would be the dependent variable, while pricing and the influence of advertising would be the independent factors.

#### Linear Regression Assumptions

- There should be a linear relationship between the independent and dependent variables.
- It’s important that none of the variables are reliant on or related to any others.
- For a regression line to be properly fitted, outliers must first be eliminated.
- It is unacceptable to have multicollinearity.

#### Regression using Polynomials

It is probable that you have noticed that the independent variable in the equations above (Y = m*x+c) was multiplied by a factor of one. When the power of the independent variable exceeds one, the form of regression used is known as polynomial regression (Y = m*x2+c).

Given that the degree of the data is greater than one, it is necessary to construct a smooth curve that connects all of the points in the data, instead of the traditional straight line. This curve will provide the best fit representation of the data in the given context.

#### The Facts You Need to Know

- It is vital to regularly plot the relationships between variables to ensure that the curve is properly fitted and not overfitted, as overfitting or underfitting may occur if the polynomial degree is too high. Doing so will help to minimise the risk of inaccurate results and ensure the best possible outcome.
- Extrapolating higher-degree polynomials may lead to inaccurate findings, so keep an eye on the shape of the curve as you approach the extrema.
- Regression analysis with logs
- Logistic regression is a statistical methodology which is commonly used to evaluate the likelihood of an event occurring. This is typically done when the dependent variable can be reduced to two distinct categories, such as a binary variable (e.g. 0 or 1, yes or no, cat or dog). As the result of the analysis is a probability, the outcome will be a number between 0 and 1.
- Classification issues are a common application domain for logistic regression.
- By applying a non-linear log transformation to the projected odds ratio, it becomes possible to avoid the requirement of a linear relationship between the dependent and independent variables that is typical of linear regression.
- Multinomial logistic regression is used when there are several categories in the final result.
- Similarly to linear regression, multicollinearity is not permitted in nonlinear regression.

### Linear Regression with a Steep Incline

Let us explore regularisation, a method of making models more effective in dealing with unseen data by removing less important characteristics, before we delve into ridge regression.

Ridge regression and lasso regularisation are two kinds of regularisation methods.

It is impossible to come across a situation in which all of the variables are unrelated to one another in the real world. As a result, multicollinearity is always a factor to consider when dealing with actual data. Unfortunately, this renders the least squares approach less effective, as the observed value is often far from the actual value due to the high variations. To prevent overfitting, ridge regression can be employed, as it penalises models with high variability by making their beta coefficients equal to zero.

In linear regression, we aim to minimise the cost function in order to achieve a successful model. To accomplish this, the cost function must include the terms ‘lambda’ and ‘slope’. This will reduce the bias and standard deviation, thus resulting in a model with maximum accuracy.

### An Example of Lasso Regression

The Least Absolute Shrinkage and Selection Operator (LASSO) regression is similar to Ridge regression in that it serves to reduce the variance of linear regression models. Additionally, it can be utilised to assist in feature selection by introducing a penalty function that is expressed in terms of absolute values, as opposed to the squared values used in Ridge regression.

As previously mentioned, the best fitting line in the ridge regression was getting close to zero (0). The slope as a whole was not linear, but instead, was decreasing. However, in lasso regression, it seemed to be heading towards zero. In this type of regression, features are eliminated when the slope value is low, thus indicating that these attributes are not significant in determining the ideal fitting line. Consequently, this makes it easier for us to select the most relevant features.

## Selecting the Appropriate Model for Regression Analysis

It is important to note that the regression models presented are not exhaustive. Selecting the most suitable model from such a wide range of options can be a challenging task. When making this decision, it is imperative to take into consideration the number of dimensions of the data as well as any other essential information. Prioritising these aspects will ensure that the most appropriate regression model is chosen.

Several considerations are listed below to help you choose the best regression model:

- It is essential to conduct exploratory data analysis prior to constructing a reliable prediction model. This is the initial step in the process and provides insight into the connection between the various factors. Consequently, a more informed decision can be made about the model to use.
- It is possible to assess the adequacy of a given model by comparing it to other models using a range of statistical metrics, such as R Square, Adjusted R Square, Area Under the Curve (AUC), and Receiver Operating Characteristic (ROC) Curve. These metrics enable us to determine how well the model fits the data and thus how reliable the model results are.
- Cross-validation is a useful technique for evaluating the accuracy of a model. It involves splitting the dataset into two parts: a training set and a validation set. This allows us to gain an understanding of whether the model is overly restrictive or overly liberal when it comes to the data. By using this technique, we can better identify potential flaws in our model and make necessary adjustments accordingly.
- The use of feature selection methods such as lasso regression and ridge regression can be beneficial for data sets with a high number of features or in cases where the variables present multicollinearity. These methods can help to reduce the complexity of the model and optimise the performance of the model by removing unnecessary or redundant features from the data set.

Regression analysis provides two key benefits. Firstly, it is able to identify the relationship between input and output variables, thus providing insights into how changes in the independent variable can affect the dependent variable. Secondly, regression analysis can also indicate the significance of the impact of the independent variable on the dependent variable.

It is essential to consider the characteristics of the data before attempting to apply any of the regression methods mentioned. The nature of the variables, such as whether they are continuous or discrete, can be used as a guideline for choosing the suitable technique. All the regression techniques discussed are based on the same fundamental principles, but their complexities rise when the number of variables or the strength of the independent variable increases.