Analysis of Multiple Regression - Essay Example

Multiple Regression Multiple Regression is a tool which involves a single dependent variable and two or more independent variables. The general form of the multiple regression is Y= a +b1X1 +b2X2+b3X3...+bnXn + e which can be estimated by Y= a +b1X1 +b2X2+b3X3...+bnXn, where a is the intercept and bi's are the partial regression coefficients. The various statistics associated with Multiple Regression are Regression Coefficient: The estimated parameter of b Standard Error of Estimate: Is the standard deviation of the actual values of Y from the predicted values of Y Standard Error The standard deviation of b ie SE(b) Coefficient of Multiple Determination: R2 which gives the % of the variance in the dependent variable that is explained by the variation in the independent variables. It measures the strength of association and varies between 0 and 1. It signifies the proportion of the total variation in Y that is accounted for by the variation in X Adjusted R square is the coefficient of multiple determination is adjusted for the number of independent variables and the sample size to account for diminishing returns After first few variables the additional independent variables do not make much contribution to the dependent variable. t- statistic is used to test the null hypothesis that no linear relationship exists between X and Y with Ho = b1 = 0 The test statistic t is given by t = b / SE(b) which follows t distribution with n-2 df. F test is used to test the null hypothesis that the coefficient of multiple determination in the population is zero ie Ho = b1=b2= = bn = 0 Partial Regression Coefficient: It denotes the change in the predicted value of Y per unit change in X1 when the other independent variables are held constant Working of Multiple Regression Once the dependent and independent variable are selected, the method for variables to enter can be selected or to be removed using any of the methods say stepwise, Remove, Backward or Forward. When clicked on the statistics, the estimate confidence intervals and model fit are selected and in the residuals Durbin-Watson is selected. The significance of each of these is as follows. Confidence Intervals: It produces confidence intervals for each of the unstandardized regression coefficients. They help in assessing the likely value of the regression coefficients in the population. Model Fit: It provides a statistical test of the models ability to predict the outcome variable and also the value of R, R2 and adjusted R2. Estimates: They give the estimated coefficients of the regression mode. The test statistics and their significances are also obtained for each regression. Here T-test is used to see whether each b differences significantly from zero. Durbin Watson: This test statistic tests the assumption of independent errors. If the value is different from value Z, then it is cause of concern. When this data is run on SPSS taking Satisfaction as dependent variable and functional, Epistemic, Social, Emotional, Conditional as independent variables, we get the following outputs. The first output is the Descriptive statistics. Descriptive Statistics Mean Std. Deviation N Satisfaction 5.1369 1.18900 389 Functional 5.4989 .91570 389 Epistemic 5.3492 .95148 389 Social 5.3209 1.27152 389 Emotional 5.4961 1.04150 389 Conditional 3.4679 1.01706 389 This table gives the mean and standard deviation of each of the variables. This is useful for summary of data. Correlations Satisfaction Functional Epistemic Social Emotional Conditional Conditional Pearson Correlation Satisfaction 1.000 .555 .618 .339 .565 .004 Functional .555 1.000 .509 .221 .520 .011 Epistemic .618 .509 1.000 .328 .531 .109 Social .339 .221 .328 1.000 .338 .148 Emotional .565 .520 .531 .338 1.000 .154 Conditional .004 .011 .109 .148 .154 1.000 Sig. (1-tailed) Satisfaction . .000 .000 .000 .000 .471 Functional .000 . .000 .000 .000 .415 Epistemic .000 .000 . .000 .000 .016 Social .000 .000 .000 . .000 .002 Emotional .000 .000 .000 .000 . .001 Conditional .471 .415 .016 .002 .001 . N Satisfaction 389 389 389 389 389 389 Functional 389 389 389 389 389 389 Epistemic 389 389 389 389 389 389 Social 389 389 389 389 389 389 Emotional 389 389 389 389 389 389 Conditional 389 389 389 389 389 389 The correlation matrix gives the Pearson correlation coefficient between every pair of variables. It also gives the one significance of each correlation. Here we observe that the correlation is significant with p Read More