Multiple Linear Regression using SPSS:
Learn with Easy Examples
Multiple linear regression is a statistical method used to model the linear relationship between a response variable and several predictor variables. It is a useful technique for understanding the relationship between variables in a data set and making predictions about the response variable. In this article, we will cover the basics of multiple linear regression and how to perform this analysis using SPSS (Statistical Package for the Social Sciences).
What is Multiple Linear Regression?
Multiple linear regression is a type of regression analysis in which two or more independent variables are used to predict the value of a dependent variable. The dependent variable is referred to as the response variable and the independent variables are referred to as predictor variables. The relationship between the predictor variables and the response variable is modelled as a linear equation.
Equation for Multiple Linear Regression
In multiple linear regression, the relationship between the dependent variable and independent variables is expressed as an equation in the form of:
y = b0 + b1x1 + b2x2 + … + bnxn
where:
- y is the dependent variable
- b0 is the intercept or constant
- b1 to bn are the coefficients or slopes
- x1 to xn are the independent variables
Why Use Multiple Linear Regression?
The goal of multiple linear regression is to estimate the values of the coefficients that provide the best fit to the data. Multiple linear regression is useful in several scenarios, including:
- To understand the relationship between several predictor variables and a response variable
- To make predictions about the response variable based on the values of the predictor variables
- To compare the relative importance of different predictor variables on the response variable
Multiple Linear Regression Example 1
Let’s take an example to understand MLR better. Suppose you are interested in understanding the factors that influence the salary of employees in a company. You have data on the salaries of employees along with their age, years of experience, and education level. You want to build a model that can predict the salary of employees based on these factors.
Here, the dependent variable is the salary, and the independent variables are age, years of experience, and education level. The MLR equation for this example would be:
salary = b0 + b1(age) + b2(years of experience) + b3(education level)
You would use the data to estimate the values of the coefficients b0, b1, b2, and b3 that provide the best fit to the data. Once you have the coefficients, you can use the equation to predict the salary of employees based on their age, years of experience, and education level.
Multiple Linear Regression Example 2
Another example where MLR can be applied is in the field of finance. Suppose you are interested in predicting the stock price of a company based on various financial indicators such as revenue, profit margin, debt-to-equity ratio, and market capitalization. The MLR equation for this example would be:
stock price = b0 + b1(revenue) + b2(profit margin) + b3(debt-to-equity ratio) + b4(market capitalization)
Again, the goal is to estimate the values of the coefficients that provide the best fit to the data and use the equation to predict the stock price based on the financial indicators.
Steps to Perform Multiple Linear Regression using SPSS
- Load the data into SPSS
- Select “Analyse” from the menu bar, then “Regression,” and finally “Linear.”
- Select the dependent variable (response variable) and the independent variables (predictor variables) by clicking and dragging them into the “Dependent” and “Independent(s)” boxes.
- Select the “Options” button and choose the desired options for the analysis.
- Select the “OK” button to run the analysis.
Multiple Linear Regression in SPSS: Output Interpretation
The output from multiple linear regression in SPSS includes several tables, including a model summary table, a coefficients table, and a residuals table.
Model Summary Table
The model summary table provides information about the overall fit of the regression model, including the R-squared value, the adjusted R-squared value, and the F-statistic. The R-squared value represents the proportion of variation in the response variable that is explained by the predictor variables. The adjusted R-squared value takes into account the number of predictor variables in the model and adjusts the R-squared value accordingly. The F-statistic represents the overall significance of the regression model.
Coefficients Table
The coefficients table provides information about each predictor variable, including the coefficient estimate, the standard error, the t-statistic, and the significance level. The coefficient estimate represents the change in the response variable for a one-unit change in the predictor variable, holding all other predictor variables constant. The standard error represents the standard deviation of the estimate of the coefficient. The t-statistic represents the test statistic for testing the hypothesis that the coefficient is equal to zero. The significance level represents the probability of observing a t-statistic as extreme as the one computed, if the null hypothesis that the coefficient is equal to zero is true.
Residuals Table
The residuals table provides information about the residuals, which are the differences between the observed and predicted values of the response variable. The residuals can be plotted in a scatterplot to examine the distribution of the residuals and identify any patterns or outliers. If the residuals are evenly distributed and do not show any patterns, it indicates that the linear regression model is a good fit for the data. However, if the residuals are not evenly distributed and show patterns, it indicates that the model is not a good fit for the data and may need to be improved.
One important aspect of the residuals table is the residual standard error, which measures the average amount that the residuals deviate from the mean. A smaller residual standard error indicates a better fit of the model to the data. Additionally, the residuals table also includes the residual degrees of freedom, which is the number of observations in the data minus the number of parameters in the model.
Another important aspect of the residuals table is the residual sum of squares, which measures the total variance of the residuals. The residual sum of squares is used to determine the goodness of fit of the model and can be compared to the total sum of squares to determine the proportion of variance in the response variable that is explained by the model.
Finally, the residuals table also includes the residual mean square, which is the mean of the residuals squared. The residual mean square can be used to calculate the F-statistic, which measures the ratio of the explained variance to the residual variance. The F-statistic is used to determine if the overall linear regression model is significant and if any individual predictor variables are significant in explaining the response variable.
The residuals table provides important information about the fit of the regression model. The residuals for each observation are displayed in the table along with other important statistics such as the mean, standard deviation, minimum and maximum values, and the number of missing values.
The residuals can be used to assess the assumptions of linearity, independence, and homoscedasticity. If the residuals are normally distributed and have constant variance, this indicates that the assumptions of linearity and homoscedasticity are satisfied. If the residuals are independent, this indicates that the independence assumption is satisfied.
To further assess the goodness of fit of the regression model, a scatterplot of residuals versus fitted values can be created. The scatterplot should show a random pattern, indicating that the residuals are not systematically related to the fitted values.
In SPSS, the residuals table can be obtained by selecting the “Save” option in the “Linear Regression” dialog box and checking the “Residuals” box. The residuals table can then be viewed in the “Data View” window.
Multiple Linear Regression using SPSS: Conclusion
In conclusion, the residuals table and scatterplot of residuals versus fitted values are important tools for evaluating the fit of the regression model and ensuring that the assumptions of the model are satisfied. By using these tools, PhD students can ensure that their data analysis is reliable and accurate.
