Multiple Linear Regression Analysis consists of more than just fitting a linear line through a cloud of data points. It consists of 3 stages – (1) analyzing the correlation and directionality of the data, (2) estimating the model, i.e., fitting the line, and (3) evaluating the validity and usefulness of the model.
Firstly, the scatter plots should be checked for directionality and correlation of data. Typically you would look at an individual scatter plot for every independent variable in the analysis.
In the two examples shown here the first scatter plot indicates a positive relationship between the two variables. The data is fit to run a multiple linear regression analysis.
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The second scatter plot seems to have an arch-shape this indicates that a regression line might not be the best way to explain the data, even if a correlation analysis establishes a positive link between the two variables. However, most often data contains quite a large amount of variability (just as in the third scatter plot example) in these cases it is up for decision how to best proceed with the data.
The second step of multiple linear regression is to formulate the model, i.e. that variable X1, X2, and X3 have a causal influence on variable Y and that their relationship is linear.
The third step of regression analysis is to fit the regression line.
Mathematically least square estimation is used to minimize the unexplained residual. The basic idea behind this concept is illustrated in the following graph.
In our example we want to model the relationship between age, job experience, and tenure on one hand and job satisfaction on the other hand. The research team has gathered several observations of self-reported job satisfaction and experience, as well as age and tenure of the participant.
When we fit a line through the scatter plot (for simplicity only one dimension is shown here), the regression line represents the estimated job satisfaction for a given combination of the input factors. However in most cases the real observation might not fall exactly on the regression line.
Because we try to explain the scatter plot with a linear equation of
for i = 1…n. The deviation between the regression line and the single data point is variation that our model can not explain. This unexplained variation is also called the residual ei.
The method of least squares is used to minimize the residual.
The multiple linear regression’s variance is estimated by
where p is the number of independent variables and n the sample size.
The result of this equation could for instance be yi = 1 + 0.1 * xi1+ 0.3 * xi2 – 0.1 * xi3+ 1.52 * xi4. This means that for additional unit x1 (ceteris paribus) we would expect an increase of 0.1 in y, and for every additional unit x4 (c.p.) we expect 1.52 units of y.
Now that we got our multiple linear regression equation we evaluate the validity and usefulness of the equation.
The key measure to the validity of the estimated linear line is R². R² = total variance / explained variance. The following graph illustrates the key concepts to calculate R². In our example the R² is approximately 0.6, this means that 60% of the total variance is explained with the relationship between age and satisfaction.
As you can easily see the number of observations and of course the number of independent variables increases the R². However, over fitting occurs easily with multiple linear regression, over fitting happens at the point when the multiple linear regression model becomes inefficient. To identify whether the multiple linear regression model is fitted efficiently a corrected R² is calculated (it is sometimes called adjusted R²), which is defined
where J is the number of independent variables and N the sample size. As you can see the larger the sample size the smaller the effect of an additional independent variable in the model.
In our example R²c = 0.6 – 4(1-0.6)/95-4-1 = 0.6 – 1.6/90 = 0.582. Thus we find the multiple linear regression model quite well fitted with 4 independent variables and a sample size of 95.
The last step for the multiple linear regression analysis is the test of significance. Multiple linear regression uses two tests to test whether the found model and the estimated coefficients can be found in the general population the sample was drawn from. Firstly, the F-test tests the overall model. The null hypothesis is that the independent variables have no influence on the dependent variable. In other words the F-tests of the multiple linear regression tests whether the R²=0. Secondly, multiple t-tests analyze the significance of each individual coefficient and the intercept. The t-test has the null hypothesis that the coefficient/intercept is zero.
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