CRITERIA FOR USE. A GRAPHICAL EXPLANATION OF BI-VARIATE (2 VARIABLE) REGRESSION ANALYSISSys
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1 Multiple Regression Analysis 1 CRITERIA FOR USE Multiple regression analysis is used to test the effects of n independent (predictor) variables on a single dependent (criterion) variable. Regression tests the deviation about the means, and all variables must be at least interval scaled. Computationally, regression analysis may be conducted using either a raw data matrix (respondents by variables) or a correlation matrix. Regression analysis measures the degree of influence of the independent variables on a dependent variable. In the case of simple bivariate regression where there is a single independent variable, the dependent variable could be predicted from the independent variable by the simple equation: y = a + bx {where a is constant} This could be extended to a multi-variable concept as follows: y = a + b 1 x 1 + b 2 x 2 + b 3 x b n x n It should be noted that whether it be for a single variable or for multiple variables, the relationship predicted is always linear. A GRAPHICAL EXPLANATION OF BI-VARIATE (2 VARIABLE) REGRESSION ANALYSISSys A simple approach to approximate a regression equation for a single variable is to plot the relationship between the variables. The task requires that we first plot the dependent variable against the independent variable. This type of plotting is called the scatter diagram. Next, identify the straight line that best represents the trend through the mid-point of the data. This line must be the one with the `best fit. The regression analysis line identifies the trend or relationship between the independent and dependent variables. The relationship, once identified, is used to predict the various values of the dependent variable given specific values of the independent variable. This predicted relationship is always in the form of a linear trend. The table below identifies a set of values for an independent (X) and dependent (Y) variable that are shown in the X-Y scatterplot. The scatter plot of the variables is given below: 1 November 1, 2011 Version: This tutorial is edited from the BIOMED statistical package regression analysis program, as developed under a National Science Foundation grant. Multiple Regression Analysis 1
2 LINEAR REGRESSION MODEL Regression analysis is utilized to develop an accurate mathematical formulation of the regression analysis. The line of best fit is defined as a line for which minimizes the sum of squares of deviation of the various data points from the line. The regression line is also referred to as the least squares line. In case of a multi-variable regression, the analysis is a sequence of multiple linear regression equations that are developed in a stepwise manner. At each step of the sequence, one variable is added to the regression equation. The variable added is the one that makes the greatest reduction in the error sum of squares of the sample data. Equivalently it is the variable that when added, provides the greatest increase in the F value. Variables not having a significant correlation with the dependent variable, are those whose addition does not increase the F value and are not featured in the regression equation. MATHEMATICAL COMPUTATION OF THE REGRESSION COEFFICIENTS WITH ONE INDEPENDENT VARIABLE The Mathematical Computation of the Regression Coefficients for the case of a single independent variable is given below: The slope (regression coefficient) for the line of least squares is given by b, where Multiple Regression Analysis 2
3 The intercept of the line is given by a, where a = y bx. The mathematical formula used for this computation is as follows: THE RESIDUAL The residual is defined as the difference between the actual and predicted values of the dependent variable. The standard error of the estimate is the standard deviation of the residuals. The standard error of the estimate can be calculated as follows: A NUMERICAL EXAMPLE: ONE DEPENDENT VARIABLE Multiple Regression Analysis 3
4 VARIABLE LABELS V1 AGE V2 WEIGHT V3 VARIABLE 3 V4 HEIGHT V5 STATUS Multiple Regression Analysis 4
5 V6 DEPENDENT CONTINUES THE OUTPUT FILE The output for most regression analysis programs contain the following: Stepwise input of independent variables r2 value Standard Regression Coefficients Unstandardized Regression Coefficients Sum of squares, mean squares, F-ratio F value to enter a variable in the equation F value to remove a variable from the equation Summary table of stepwise analysis We re Here to Help! Qualtrics.com provides the most advanced online survey building, data collection (via panels or corporate / personal contacts), real-time view of survey results, and advanced dashboard reporting tools. If you are interested in learning more about how the Qualtrics professional services team can help you with a conjoint analysis research project, contact us at research@qualtrics.com. Multiple Regression Analysis 5
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