# Multiple Linear Regression – Definition and Calculations

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Several independent variables are always connected to the dependent variable (y) of concern. If this correlation can be calculated, it could allow us to develop more correct estimates of the dependent variable than would be achievable using linear regression.1

Multiple linear regression is used when there is more than one independent variable to predict.

## Multiple Linear Regression – In a Nutshell

• As its name implies, multiple linear regression is a statistical method that uses many key variables to foretell the result of a test statistic.
• Linear (OLS) regression employs a single explanatory variable, whereas multiple regression uses many variables.
• Inference in the fields of economics and finance relies heavily on MLR.

## Definition: Multiple linear regression

Statisticians use multiple linear regression to predict one variable’s value from several other variables. Multiple regression refers to an expansion of linear regression.

Dependent variables are the target of our prediction efforts, while independent or explanatory variables are the variables we utilize to estimate that target’s value.

A linear connection between the dependent variable and two or more independent variables is considered present for multiple regression.

Non-linearity occurs when the relationship between the dependent and independent variables is curvy rather than straight.

We are using two or more variables, both linear and non-linear regression to plot the relationship between the answer and those factors.

However, non-linear regression is notably hard to execute because it relies on predictions developed through trial and error.

Example:

You are a public health expert investigating the role of socioeconomic determinants in cardiovascular disease. You study 200 communities, collecting information on tobacco consumption, bicycle commuting rates, and the prevalence of cardiovascular illness in every location.

Because of their inherent structure, multiple linear regression may be used to examine the connection between your quantitative variables (two independent and one dependent).

## Multi-linear regression: Assumptions

The assumptions of multiple linear regression are identical to those of basic linear regression.

• Independence of Observation: All data points in the multiple linear regression set were acquired using statistically sound sampling techniques, and there are no ambiguous connections between the variables.
• Normality: The data are standard.
• Homogeneity of Variance: As the independent variable is adjusted, the extent of the error in our estimate remains constant.
• Linearity: The ideal line representing the relationship between the data points is straight, not a curve or other aggregating feature.

If you are using multiple linear regression, it’s a good idea to double-check for any potential correlations between the independent variables initially.2

If the relationship between two independent variables is too strong (r2 > 0.6), then only one should be included in the regression analysis.

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## Calculating a Multiple Linear Regression

The equation for multiple linear regression is as follows:  Dependent variable  Explanatory variables  Y-intercept (constant term)  Slope coefficients for each explanatory variable  The model’s error term The three things that multiple linear regression computes to determine the best-fit line for every independent variable are:

• The optimal regression coefficients minimize the total error in the equation.
• The model-wide t-statistic.
• If there is no correlation between the independent and dependent variables, the corresponding p-value indicates how probable it is that the t-statistic happened by chance.

## Multiple Linear Regression in R

Multiple linear regression may be done manually. However, nowadays, most researchers use statistical programs.

We will use R as our example program since it’s open-source, robust, and convenient.

### Multiple Linear Regression Results

The result of a regression model can look like this: In this example of multiple linear regression, Y is correlated with X1 ( ) and X2 ( ) to help explain the former.

Keeping everything else the same, we may read this model as follows:

• If X1 increases by one unit, Y increases by 3.2 units (if X1 increases by two units, Y increases by 6.4 units; All else equal.
• That’s to say; we see this correlation between X1 and X2 even when we remove X2.
• Similarly, with X1 held constant, a twofold reduction in Y is shown for every unit increase in X2. The y-intercept of 1.0 tells us that when X1 and X2 are both zero, Y equals 1.
• The residual (error term) is 0.21.

## Multiple linear regression: Reporting the results

Include the p-value, standard deviation of the estimate, and anticipated impact in your presentation.3 And do not forget to explain the significance of the regression coefficient to your audience by interpreting the data you provide.

Example:

We surveyed 500 communities and discovered a substantial correlation between the number of times people biked to work and the number of times they died from heart disease.

We discovered that the prevalence of heart disease decreased by 0.2% (0.0014) for every 1% rise in bicycling but increased by 0.178% (0.0035) for every 1% rise in smoking.

### Illustrating the results in a graph

Displaying your multiple linear regression findings on a graph might also be helpful. As the number of parameters in multiple linear regression exceeds what can be shown in a two-dimensional graphic, it is more difficult to implement than basic linear regression.

While one independent variable may be shown on the x-axis, there are other methods to present your findings, including the impact of several independent factors on the dependent variable.

Here, we have computed the projected values of the dependent variable (heart disease) throughout the whole range of actual values for the proportion of persons who bike to work. These projected values were determined by adjusting the independent variable for the impact of smoking at the lowest, medium, and highest recorded smoking rates.

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## FAQs

#### What is multiple linear regression?

A multiple linear regression model is a predictive method used to predict the linear connection between a dependent variable and one or more independent variables.

#### What is a major limitation of multiple regressions?

If the connection between the independent factors and the dependent variables is not linear, then multiple regression will not provide a satisfactory explanation.

#### How can one define a reliable multiple linear regression model?

The following are the presumptions upon which the multiple linear regression model rests: The correlation between the two sets of numbers is linear.

There is reasonable independence between the independent variables. The yi samples are taken at random from the whole population.

#### How is multiple linear regression used in your study design?

Researchers often use complex correlational research to investigate interrelationships among many study variables.

Advanced research findings may explore potential causal connections between variables using multiple regression methods.

## Sources

1 Gallo, Amy. “A Refresher on Regression Analysis.” Harvard Business Review. November 4, 2015. https://hbr.org/2015/11/a-refresher-on-regression-analysis.

2 CFI Team. “Regression Analysis.” CFI. November 24, 2022. https://corporatefinanceinstitute.com/resources/data-science/regression-analysis/.

3 Laerd Statistics. “Multiple Regression Analysis using SPSS Statistics.” Accessed February 21, 2023. https://statistics.laerd.com/spss-tutorials/multiple-regression-using-spss-statistics.php.

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