Assumptions of Linear Regression - Multivariate Normality

Introduction

Linear regression is a widely used statistical method for modelling the relationship between a dependent variable and one or more independent variables. It is based on the linear relationship between the variables and is widely used in various fields, including economics, psychology, and engineering. However, certain assumptions must be met for the results of linear regression analysis to be meaningful and accurate. One of these assumptions is the assumption of multivariate normality.

Multivariate normality assumes that the residuals, or the difference between the observed and predicted values, are normally distributed. This assumption is important because it allows for various statistical tests and inference methods, such as hypothesis tests and confidence intervals, that rely on the normality of the residuals. This assumption is necessary for the results of linear regression analysis to be accurate and accurate.

Assumptions of Linear Regression - Multivariate Normality

Multivariate Normality

Analyses using linear regression determine whether one or more predictor variables adequately account for the dependent (or criterion) variable. Five major hypotheses underlie the regression −

• Linear relationship

• Multivariate normality

• No or little multicollinearity

• No autocorrelation

• Homoscedasticity

Linear regression is one of the most widely used statistical techniques for modelling the relationship between a dependent variable and one or more independent variables. It is a popular method for modelling continuous and numerical outcomes and is particularly useful for identifying the strength and direction of the relationship between variables. However, for linear regression to be an effective tool for data analysis, it is important to understand and respect its underlying assumptions.

One of the most critical assumptions of linear regression is multivariate normality. This refers to the idea that the model's error terms, or residuals, should be normally distributed. In other words, the residuals should have a mean of zero and be distributed in a bell-shaped curve. This assumption is important because it allows us to use various statistical tests and confidence intervals to make inferences about the model and its parameters.

Multivariate normality is a central component of the classical linear regression framework. It is necessary to validate many statistical results and inferences made using the model. In particular, the central limit theorem states that the sum of many independent random variables approaches a normal distribution and applies to the residuals in linear regression. This means that, as the number of observations increases, the residuals will become more and more normally distributed, even if the individual observations are not themselves normally distributed.

There are several ways to assess the assumption of multivariate normality in a linear regression model. One common method is to plot a histogram of the residuals and visually inspect the distribution for evidence of normality. A normal probability plot can also be used to graphically assess the residuals' normality. Another method is to perform a normality test, such as the Shapiro-Wilk test or the Anderson-Darling test, to test the hypothesis that the residuals are normally distributed formally.

If the assumption of multivariate normality is not met, the analysis has several potential implications. One of the most serious consequences is that the estimates of the standard errors and confidence intervals for the model's parameters may need to be corrected. This, in turn, can affect the results of hypothesis tests and lead to incorrect inferences about the relationship between the dependent and independent variables. Furthermore, the validity of other statistical results, such as the F-test for the overall significance of the model, may also be compromised.

There are several ways to address the violation of the assumption of multivariate normality. One option is to transform the dependent variable to make the residuals more normally distributed. For example, transforming the dependent variable to its logarithmic scale or a power function can often lead to a more normal distribution of the residuals. Other techniques, such as transforming the independent variables or using a different model altogether, such as a nonlinear regression model or a robust regression model, can also be used to address the violation of the assumption of multivariate normality.

Specifications with Real-World Entities

It is important to note that the assumption of multivariate normality is only sometimes met in real-world data sets, particularly in the case of smaller sample sizes. In these cases, it is crucial to consider alternative methods for modelling the data that are not dependent on the assumption of normality. For example, robust regression methods, such as the M-estimators, are designed to be more resistant to outliers and deviations from normality and can be used to fit a regression model in cases where the residuals are not normally distributed.

It is also important to consider the underlying relationships between the dependent and independent variables in a linear regression model. In some cases, transforming the variables or using nonlinear regression methods may be necessary to model the relationship between variables accurately. For example, if the relationship between the dependent and independent variables is nonlinear, a polynomial or spline regression model may be more appropriate.

Example and Equations

An example of the assumption of multivariate normality in linear regression can be seen in a study investigating the relationship between income and years of education. The dependent variable, income, is continuous and numerical, while the independent variable, years of education, is also continuous. To model the relationship between these variables, a linear regression model is fit using a sample of data collected from individuals in the population.

One of the key assumptions of linear regression is that the residuals, or the difference between the observed and predicted values, should be normally distributed. To assess this assumption, a histogram of the residuals can be plotted and visually inspected for evidence of normality. A normal probability plot can also be used to graphically assess the residuals' normality. If the residuals are not normally distributed, alternative methods for modeling the data, such as robust regression or generalized linear models, should be considered.

In this example, let's assume that the residuals from the linear regression model are found to be not normally distributed. One potential solution is to transform the dependent variable, income, by taking the logarithm of the values. This transformation can often result in a more normal distribution of the residuals. A new linear regression model can then be fitted using the logarithmic transformation of the dependent variable, and the residuals can be assessed for normality once again. If the residuals still need to be distributed, alternative data modelling methods should be considered.

This example highlights the importance of understanding and respecting the assumptions of linear regression to obtain accurate results and make valid inferences about the relationship between variables. By considering alternative methods for modelling the data and addressing any violations of the assumptions, researchers can ensure that the results of their analysis are valid and meaningful.

Equations

The equation for a simple linear regression model with a single independent variable is given −

Y = β0 + β1X + ε

Y is the dependent variable, X is the independent variable, β0 is the intercept, β1 is the slope or regression coefficient, and ε is the error term.

The goal of linear regression is to estimate the values of β0 and β1 that minimize the sum of the squared residuals, defined as −

RSS = Σ(Yi - Ŷi)^2

Where Yi is the observed value of the dependent variable, Ŷi is the predicted value of the dependent variable, and the sum is taken over all observations.

The estimates of β0 and β1 can be obtained using the least squares method, which minimizes the RSS. The estimated values of β0 and β1 can then be used to make predictions about the dependent variable based on the values of the independent variable.

In the case of multiple independent variables, the equation for a multiple linear regression model is given −

Y = β0 + β1X1 + β2X2 + ... + βkXk + ε

Where X1, X2, ... Xk are the independent variables, β0 is the intercept, β1, β2, ... βk are the regression coefficients, and ε is the error term. The estimates of β0, β1, β2, ... βk can be obtained using the method of least squares, as described above.

Conclusion

The assumption of multivariate normality is an important component of linear regression analysis and must be carefully considered to obtain meaningful and accurate results. In cases where the residuals are not normally distributed, alternative methods for modelling the data, such as robust regression or generalized linear models, should be considered. By understanding the limitations of linear regression and considering alternative methods, researchers can make more informed decisions about their data and better understand the relationships between variables. Overall, the assumption of multivariate normality is a crucial component of linear regression analysis and should be carefully considered to ensure the validity of the results.