![multiple linear regression equation example multiple linear regression equation example](https://miro.medium.com/max/1280/1*oZ720COoRQhkBM3jimtOvQ.png)
![multiple linear regression equation example multiple linear regression equation example](https://sds-platform-private.s3-us-east-2.amazonaws.com/uploads/36_blog_image_1.png)
The "Coefficients" table presents the optimal weights in the regression model, as seen in the following. The output consists of a number of tables. In addition, under the "Save." option, both unstandardized predicted values and unstandardized residuals were selected. In the first analysis, Y 1 is the dependent variable and two independent variables are entered in the first block, X 1 and X 2. The multiple regression is done in SPSS by selecting Analyze/Regression/Linear. For that reason, computational procedures will be done entirely with a statistical package. They are messy and do not provide a great deal of insight into the mathematical "meanings" of the terms. The formulas to compute the regression weights with two independent variables are available from various sources (Pedhazur, 1997). Using the Graphs/Scatter/3-D commands in SPSS results in the following two graphs.
![multiple linear regression equation example multiple linear regression equation example](https://www.empirical-methods.hslu.ch/files/2017/02/equation-for-r2-multiple-regression.png)
Three-dimensional scatter plots also permit a graphical representation in the same information as the multiple scatter plots. These graphs may be examined for multivariate outliers that might not be found in the univariate view. The score on the review paper could not be accurately predicted with any of the other variables.Ī visual presentation of the scatter plots generating the correlation matrix can be generated using the Graphs/Scatter/Matrix commands in SPSS. Measures of intellectual ability and work ethic were not highly correlated. The measures of intellectual ability were correlated with one another. Interpreting the variables using the suggested meanings, success in graduate school could be predicted individually with measures of intellectual ability, spatial ability, and work ethic. In addition, X 1 is significantly correlated with X 3 and X 4, but not with X 2. In the case of the example data, it is noted that all X variables correlate significantly with Y 1, while none correlate significantly with Y 2. This can be done using a correlation matrix, generated using the Analyze/Correlate/Bivariate commands in SPSS. The second step is an analysis of bivariate relationships between variables. In the case of the example data, the following means and standard deviations were computed using SPSS by clicking Analyze/Summarize/Descriptives. In a multiple regression analysis, these score may have a large " influence" on the results of the analysis and are a cause for concern. Additional analysis recommendations include histograms of all variables with a view for outliers, or scores that fall outside the range of the majority of scores. The first step in the analysis of multivariate data is a table of means and standard deviations. X 3 - A second measure of intellectual ability. Y 1 - A measure of success in graduate school. If a student desires a more concrete description of this data file, meaning could be given the variables as follows: The example data can be obtained as a text file and as an SPSS data. The data used to illustrate the inner workings of multiple regression are presented below: The difference is that in simple linear regression only two weights, the intercept (b 0) and slope (b 1), were estimated, while in this case, three weights (b 0, b 1, and b 2) are estimated. In the same manner as in simple linear regression. The "b" values are called regression weights and are computed in a way that minimizes the sum of squared deviations Note that this transformation is similar to the linear transformation of two variables discussed in the previous chapter except that the w's have been replaced with b's and the X' i has been replaced with a Y' i. With two independent variables the prediction of Y is expressed by the following equation: The interpretation of the results of a multiple regression analysis is also more complex for the same reason. The computations are more complex, however, because the interrelationships among all the variables must be taken into account in the weights assigned to the variables. The predicted value of Y is a linear transformation of the X variables such that the sum of squared deviations of the observed and predicted Y is a minimum. Multiple regression is an extension of simple linear regression in which more than one independent variable (X) is used to predict a single dependent variable (Y). Multiple Regression with Two Predictor Variables