Regression analysis

Regression Analysis Examples

Regression analysis has practical applications in numerous fields. Some notable examples include:

• Economy and finance
  • sales prediction :
    utilizando variables como la publicidad, el precio de los productos y las condiciones del mercado, se puede predecir el volumen de ventas futuras;
  • investment analysis : using logistic regression, the probability of success of certain financial instruments is evaluated based on economic indicators.
• Health Sciences
  • effect of medical treatments : analyzing how variables such as the dose of a medication or the age of patients affect the results of a treatment;
  • Risk prediction : Using Cox regression models to estimate the risk of mortality in survival studies.
• Marketing
  • customer segmentation : using polynomial regression to identify purchasing patterns in different customer groups;
  • advertising campaign optimization : determining the impact of each advertising channel on conversions.
• social sciences
  • inequality analysis : exploring how variables such as educational level and geographic location affect average income;
  • Prediction of electoral behavior : using variables such as age, education and political background.
• Engineering and natural sciences
  • machinery failure forecasting : analyzing how factors such as temperature and load affect the life of mechanical components;
  • climate change modeling – using regression models to study how greenhouse gas emissions affect global temperature.

• Main models
  • linear model : assumes a linear relationship between the variables;
  • nonlinear model : captures more complex relationships.
• Types of regression
  • simple linear regression : relationship between two variables;
  • multiple linear regression – includes several independent variables.
• Calculation and estimation
  • regression coefficients : determine the weight of each independent variable;
  • standard error : measures the precision of the estimated coefficients;
  • ordinary least squares (OLS) : Method for minimizing the differences between observed and predicted values.

Diagnosis and validation of the model

A regression model must be evaluated to ensure that it meets the necessary statistical assumptions. This validation includes key tests and diagnostics.

Residue analysis

Residuals, which are the differences between observed and predicted values, provide essential information about model fit. A residual analysis verifies whether they are normally distributed and whether they meet the condition of homoscedasticity (constant variance). If the residuals show patterns, it may be necessary to adjust the model or transform the variables.

Assumption testing

The Durbin-Watson test and some similar tests detect autocorrelation in the residuals, while others such as the Shapiro-Wilk test verify normality. If the assumptions are not met, techniques such as logarithmic transformations or the use of alternative models, such as quantile regression, may be useful.

Cross validation

To evaluate the predictive ability of the model, cross validation is used, which divides the data into subsets to train and test the model. This avoids overfitting problems and guarantees better generalization.

Data transformations and adjustments

To ensure the validity and accuracy of models, it is often necessary to transform data or adjust statistical assumptions.

• Homoscedasticity and heteroskedasticity
  • homoscedasticity : the variability of the residuals is expected to be constant;
  • heteroscedasticity : If the variability is not constant, it can affect the validity of the model.
• Common transformations
  • logarithm – used to stabilize variance and convert exponential relationships to linear ones;
  • exponential : inverse of logarithm, used in certain types of data;
  • square root – reduces the variability of highly dispersed data;
  • Box-Cox - Automatically determines the most appropriate transformation.
• Purposes of transformations
  • normalize data distribution;
  • improve the interpretation of the coefficients;
  • fit the data to the statistical assumptions of the model.

Model optimization and variable selection

Building an effective model requires optimizing its structure and carefully choosing the variables included.

Variable selection methods

There are several methods to select relevant variables. The backward elimination method progressively eliminates the least significant ones, while the forward selection method adds variables one by one. For its part, the stepwise method combines both strategies to find the optimal model.

Regularization and advanced methods

To handle large data sets and avoid overfitting, techniques such as ridge regression , lasso regression and elastic net are used. These incorporate penalties that limit the complexity of the model, prioritizing the most important variables.

Data transformations and assumptions

When data does not meet the basic assumptions of a regression model, transformations are crucial tools to address these problems.

Types of transformations

Transformations such as logarithm, exponential, and square root can stabilize variance, improve linearity, and handle nonlinear relationships. The Box-Cox method, in particular, is a versatile technique that automatically selects the most appropriate transformation.

Heteroscedasticity and correction

Heteroscedasticity, or non-constant variation in errors, can be detected using Levene's test , among others. In case of heteroscedasticity, robust transformations or models can improve the reliability of the results.

Assumptions and visualization

Finally, verifying model assumptions using residual plots, histograms, and scatterplots ensures that interpretations are valid. This approach also encourages a deeper understanding of the data.