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## Illustration of Regression Analysis

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**Illustration of Regression Analysis**This problem is the major problem for Chapter 4, "Multiple Regression Analysis," from the textbook. Illustration of Regression Analysis**Stage 1: Definition Of The Research Problem**In the first stage, we state the research problem, specify the variables to be used in the analysis, and specify the method for conducting the analysis: standard multiple regression, hierarchical regression, or stepwise regression. Relationship to be Analyzed HATCO management has long been interested in more accurately predicting the level of business obtained from its customers in the attempt to provide a better basis for production controls and marketing efforts. To this end, analysts at HATCO proposed that a multiple regression analysis should be attempted to predict the product usage levels of customers based on their perceptions of HATCO's performance. In addition to finding a way to accurately predict usage levels, the researchers were also interested in identifying the factors that led to increased product usage for application in differentiated marketing campaigns. (page 196) Specifying the Dependent and Independent Variables The dependent variable is Product Usage (x9). The independent variables are Delivery Speed (x1), Price Level (x2), Price Flexibility (x3), Manufacturer's Image (x4), Service (x5), Sales force Image (x6), and Product Quality (x7). (page 196) Method for including independent variables: standard, hierarchical, stepwise Since this is an exploratory analysis and we are interested in identifying the best subset of predictors, we will employ a stepwise regression. Illustration of Regression Analysis**Stage 2: Develop The Analysis Plan: Sample Size Issues**In stage 2, we examine sample size and measurement issues. Missingdata analysis If the significance level is set to 0.05, then with a sample size of 100, we can identify relationships that explain about 13% of the variance. (text, page 196), referencing the power table on page 165 of the text. Power to Detect Relationships: Page 165 of Text If the significance level is set to 0.05, then with a sample size of 100, we can identify relationships that explain about 13% of the variance. (text, page 196), referencing the power table on page 165 of the text. Minimum Sample Size Requirement: 15-20 Cases Per Independent Variable With 100 cases in the sample and 7 independent variables, we are very close to satisfying the 15 case per independent variable requirement. Illustration of Regression Analysis**Stage 2: Develop The Analysis Plan: Measurement Issues**In stage 2, we examine sample size and measurement issues. Incorporating Nonmetric Data with Dummy Variables All of the variables are metric, so no dummy coding is required. Representing Curvilinear Effects with Polynomials We do not have any evidence of curvilinear effects at this point in the analysis. Representing Interaction or Moderator Effects We do not have any evidence at this point in the analysis that we should add interaction or moderator variables. Illustration of Regression Analysis**Stage 3: Evaluate Underlying Assumptions**In this stage, we verify that all of the independent variables are metric or dummy-coded, and test for normality of the metric variables, linearity of the relationships between the dependent and the independent variables, and test for homogeneity of variance for the nonmetric independent variables. Illustration of Regression Analysis**Normality of metric variables**The null hypothesis in the K-S Lilliefors test of normality is that the data for the variable is normally distributed. The desirable outcome for this test is to fail to reject the null hypothesis. When the probability value in the Sig. column is less than 0.05, we conclude that the variable is not normally distributed. If a variable is not normally distributed, we can try three transformations (logarithmic, square root, and inverse) to see if we can induce the distribution of cases to fit a normal distribution. If one or more of the transformations induces normality, we have the option of substituting the transformed variable in the analysis to see if it improves the strength of the relationship. To reduce the tedium of the sequence of tests and computations that are required by an analysis of normality, I have produced an SPSS script that produces the tests and creates transformed variables where necessary. The use of this script is detailed below. The results of the tests of normality indicate that the following variables are normally distributed: X1 'Delivery Speed', X5 'Service', and X9 'Usage Level' (the dependent variable). X2 'Price Level' is induced to normality by a log and a square root transformation. X7 'Product Quality' is induced to normality by a log and a square root transformation. The other non-normal variables are not improved by a transformation. Note that this finding does not agree with the text, which finds that X2 'Price Level', X4 'Manufacturer Image', and X6 'Salesforce Image' are correctable with a log transformation. I have no explanation for the discrepancy. Illustration of Regression Analysis**Run the 'NormalityAssumptionAndTransformations' Script**Illustration of Regression Analysis**Complete the 'Test for Assumption of Normality' Dialog Box**First, move the variables that will be used in the regression to the 'Variable to Test:' list box. Second, click on the OK button to produce the output for the normality tests. Illustration of Regression Analysis**Output for a Variable that Passes the Test of Normality**If a variable passes the test of normality, the Sig value for the K-S Lilliefors test will be greater than the 0.05 alpha level. The table of output will be followed by the histogram and normality plot. Illustration of Regression Analysis**Output for a Variable that Fails the Test of Normality**If a variable has a Sig value of less than 0.05 for the K-S Lilliefors test, we conclude that the distribution of the variable is not normal. After the histogram and normality plot for this variable, the script will calculate the three transformations to correct for problems of normality, as shown below. Illustration of Regression Analysis**Output for a Variable that Fails the Test of Normality**The test of normality for the transformed variables is shown in a table following the output for the original form of the variable. Each row of the table contains one of the possible transformations. The formula used to compute the transformation in shown in square brackets at the end of the row label. If the distribution is negatively skewed, the reflection formulas for the transformations will be shown and used in the computations. In this example, two transformations induce normality, the log transformation and the square root transformation. Since log transformations are more commonly used, we might prefer that option in subsequent analyses. Illustration of Regression Analysis**Linearity between metric independent variables and dependent**variable Another script, 'LinearityAssumptionAndTransformations' tests for linearity of relationships between the dependent variable and each of the independent variables. Since there is no simple score that indicates whether or not a relationship is linear or nonlinear, a scatterplot matrix is created for the dependent variable, the independent variable, and transformations of the independent variable. The user can visually inspect the scatterplot matrix for evidence of nonlinearity. If nonlinearity is evident, but not corrected by a transformation of the independent variable, transformation of the dependent variable is available. More detailed information is available by requesting a correlation matrix for the variables included in the scatterplot matrix. If the scatterplot matrix does not provide sufficient detail, individual scatterplots overlaid with fit lines can be requested. When we run the script as described below, we do not find any nonlinear relationships between the dependent variables and the metric independent variables. Illustration of Regression Analysis**Run the 'LinearityAssumptionAndTransformations' Script**Illustration of Regression Analysis**Complete the 'Check for Linear Relationships' Dialog Box**First, move the variable that will be used for the dependent variable in the regression to the 'Dependent (Y) Variable:' list box. Second, move the variables that will be used as independent variables in the regression to the 'Independent (X) Variables: list box. Third, click on the OK button to produce the output for the linearity tests. Illustration of Regression Analysis**The Scatterplot Matrix**Each scatterplot matrix examines the linearity and impact of transformations for the dependent variable and one of the independent variables. Focusing our attention on the first column with a 'Y' in cell 1, we see the scatterplot for the independent variable and all of its transformations. In this scatterplot, we see in the second row of the first column that the relationship between X9 'Usage Level' and 'X1 Delivery Speed' is linear, so we ignore the remaining scatterplots in column 1. If the relationship were nonlinear, we would examine the scatterplots in column 1 to visually determine is one of them induced a linear relationship between the variables. Illustration of Regression Analysis**The Correlation Matrix**In the correlation matrix shown below the scatterplot matrix, we note that the correlation between the dependent and independent variable is strong (0.676) and that none of the correlations for the transformed versions of the independent variable offer a large increase in correlation over that for the original form of the variables. The correlation matrix confirms the linearity of the relationship. Illustration of Regression Analysis**Output Demonstrating Nonlinearity**None of the relationships in the HATCO data set demonstrate nonlinear relationships, so we cannot demonstrate what output looks like when nonlinearity is detected. The following example uses the World95.Sav data set and does demonstrate a nonlinear relationship between population increase and fertility. Visual inspection shows a nonlinear relationship in the second row the first column, which is corrected by the log transformation in row 3 and the inverse transformation in row 6. Illustration of Regression Analysis**Output Demonstrating Nonlinearity**The correlation matrix, shown below, indicates first of all that there is a very strong relationship between population increase and fertility, as we might expect from the nature of the variables. The strength of this relationship (0.840) is enhanced by both the log transformation (0.884) and the inverse transformation (0.886). The analysis would be strengthened by the use of the transformations, though a case could be made that the relationship is strongly linear without the transformation. Illustration of Regression Analysis**Constant variance across categories of nonmetric independent**variables While this particular analysis does not include any nonmetric variables, the nonmetric variable X8 'Firm Size' will be used in a subsequent analysis, so we will do the test for homogeneity of variance here. Another script, 'HomoscedasticityAssumptionAndTransformations' test for homogeneity of variance across groups designated by nonmetric independent variables. The script uses the One-Way ANOVA procedure to produce a Levene test of the homogeneity of variance. The null hypothesis of the Levene test is that the variance of all groups of the independent variable are equal. If the Sig value associated with the test is greater than the alpha level, we fail to reject the null hypothesis and conclude that the variance for all groups is equivalent. If the Sig value associated with the test is less than the alpha level, we reject the null hypothesis and conclude that the variance of at least one group is different. If we fail the homogeneity of variance test, we can attempt to correct the problem by applying the transformations for normality to the dependent variable specified in the test. The script computes the transformations and applies the Levene test to the transformed variables. If one of the transformed variables corrects the problem, we can consider substituting it for the original form of the variable. When we run the script for this problem, as described below, we find that the nonmetric X8 'Firm Size' variable passes the homogeneity of variance test so no transformation is required. Illustration of Regression Analysis**Run the 'HomoscedasticityAssumptionAndTransformations'**Script Illustration of Regression Analysis**Complete the Dialog Box**First, move the variable that will be used for the dependent variable in the regression to the 'Dependent (Y) Variable:' list box. Second, move the variables that will be used as independent variables in the regression to the 'Nonmetric Independent (X) Variables: list box. Third, click on the OK button to produce the output for the linearity tests. Illustration of Regression Analysis**Output for the Test of Homogeneity of Variances**The Sig value in the Test of Homogeneity of Variances test guides our conclusion about meeting this assumption. For the nonmetric variable X8 'Firm Size', the homogeneity of variance assumption is met when the X9 'Usage Level' is the dependent variable. Illustration of Regression Analysis**Output Failing to Pass the Homogeneity of Variance Test**The output shown below is from the 'GSS93 Subset.Sav' data set. It demonstrates what the script does when the relationship fails the Levene test. In this example, males and females did not have the same variance for the variable on spanking as a form of discipline, as indicated by the Sig value for the Levene Statistic being below 0.05. A log transformation of the spanking variable does correct the heterogeneity of variance problem, as shown in the second Levene test, which has a non-significant SIG value greater than 0.05. Illustration of Regression Analysis**Stage 4: Compute the Statistics And Test Model Fit:**Computations In this stage, we compute the actual statistics to be used in the analysis. Regression requires that we specify a variable selection method. The text specifies a stepwise regression procedure, which is appropriate to exploratory analyses where we are uncertain which variables are important predictors and we want the analysis to select the best subset of predictors. The objective of the stepwise regression procedure is to find the smallest subset of independent variables that have the strongest R Square relationship to the dependent variable. At each step, the independent variable that will be added is the one that contributes the largest, statistically significant increase to the R square measure of relationship between the dependent and the independent variables. At the conclusion of each step, the variables already entered into the equation are tested to see if the overall relationship would be stronger if one or more variables were removed. The significance levels for adding a variable are specified as PIN, the probability for inclusion and POUT, the criterion probability for taking a variable out of the regression equation. We will accept the SPSS default of 0.05 for PIN and 0.10 for POUT. The stepping procedure continues until none of the variables not included on a previous step have a significance for the increase in R square value that is less than the PIN value, or until the remaining variables have a tolerance less than that specified, or until all independent variables have been included. Illustration of Regression Analysis**Stage 4: Compute the Statistics And Test Model Fit:**Computations When SPSS is evaluating independent variables for possible inclusion in the regression equation, it also tests for multicollinearity among the independent variables. Multicollinearity exists when two or more independent variables are very highly intercorrelated, i.e, 0.90 or higher. Where there is high multicollinearity, it means that two independent variables are attempting to explain the same variance in the dependent variable. Extreme mutlicollinearity causes the mathematical computations to fail, producing incorrect answers. SPSS computes the tolerance for each variable as its test for multicollinearity. If the computed tolerance is less than the amount specified in the TOL option, the variable is excluded from consideration. The author's note that a TOL specification of 0.10 will eliminate variables correlated at the 0.95 level or higher (page 193). The author's also suggest that we set the tolerance higher than the SPSS default of 0.0001, but I cannot find a place in the menu commands where this specification can be made. If it is necessary to set the tolerance, the regression will have to be run using syntax commands. Illustration of Regression Analysis**Request the Regression Analysis**The first task in this stage is to request the initial regression model and all of the statistical output we require for the analysis. Illustration of Regression Analysis**Specify the Dependent and Independent Variables and the**Variable Selection Method Illustration of Regression Analysis**Specify the Statistics Options**Illustration of Regression Analysis**Specify the Plots to Include in the Output**Illustration of Regression Analysis**Specify Diagnostic Statistics to Save to the Data Set**Illustration of Regression Analysis**Complete the Regression Analysis Request**Illustration of Regression Analysis**Stage 4: Compute the Statistics And Test Model Fit: Model**Fit In this stage, we examine the relationships between our independent variables and the dependent variable. First, we look at the F test for R Square which represents the relationship between the dependent variable and the set of independent variables. This analysis tests the hypothesis that there is no relationship between the dependent variable and the set of independent variables, i.e. the null hypothesis is: R2 = 0. If we cannot reject this null hypothesis, then our analysis is concluded; there is no relationship between the dependent variable and the independent variables that we can interpret. If we reject the null hypothesis and conclude that there is a relationship between the dependent variable and the set of independent variables, then we examine the table of coefficients to identify which independent variables have a statistically significant individual relationship with the dependent variable. For each independent variable in the analysis, a t-test is computed that the slope of the regression line (B) between the independent variable and the dependent variable is not zero. The null hypothesis is that the slope is zero, i.e. B = 0, implying that the independent variable has no impact or relationship on scores on the dependent variable. This part of the analysis is more important in standard multiple regression where we enter all independent variables into the regression at one time, and hierarchical multiple regression where we specify the order of entry of independent variables than it is in stepwise multiple regression where the computer picks the order of entry and stops adding variables when some statistical limit is reached. In stepwise regression, we would expect all of the individual variables that passed the statistical entry for entry to have a significant individual relationship with the dependent variable. When we are determining which independent variables have a significant relationship with the dependent variable, we often are interested in the question of the relative importance of the predictor variables to predicting the dependent variable. To answer this question, we will examine the Beta coefficients, or standardized version of the coefficients of the individual independent variables. Illustration of Regression Analysis**Significance Test of the Coefficient of Determination R**Square Footnote C to the Model Summary table tells us that the regression model contained three independent variables when the stepwise criteria were satisfied: X5 Service, X3 Price Flexibility, and X6 Salesforce Image. The R² value for the three variable model is 0.768, which means that the three independent variables in the regression collectively explain 76.8% of the variance in the dependent variable. Using the descriptive framework described previously, we would describe this relationship as very strong. The Model Summary table also indicates that the increase in predictive power or accuracy at each step of the model's three steps was a statistically significant addition. The 'Sig. F Change' provides the probability that the increase in R² is greater than 0. Illustration of Regression Analysis**Significance Test of the Coefficient of Determination R**Square The 'Sig.' column of the ANOVA table for model 3 supports a conclusion that there is a statistically significant relationship between the dependent variable and the set of independent variables. Illustration of Regression Analysis**Significance Test of Individual Regression Coefficients**In addition to the issue of the statistical significance of the overall relationship between the dependent variable and the independent variables, SPSS provides us with the statistical tests of whether or not each of the individual regression coefficients are significantly different from 0, as shown in the table below: The t test for the B coefficient is a test of relationship between the dependent variable and a specific independent variable. The null hypothesis for this test is B=0, the slope of the regression line for the two variables in zero, making it flat or parallel to the x-axis. A flat line implies that for any value of x, we would predict the same value for y. Therefore, knowledge of x does not improve our ability to predict y, and there is no relationship. We also use the B coefficients to define the direction of the relationship between the independent and dependent variable. If the sign of the coefficient is positive (usually not shown the table), the relationship between the variables is direct; scores on the two variable change in the same direction. If the sign of the coefficient is negative, the relationship is inverse, meaning that increases in one variable correspond to decreases in the other variable. Another way to interpret the B coefficient is that it represents the amount of change in the dependent variable for a 1unit change in the independent variable. For example, it might tell us that starting salaries increase by $2,114 for each additional year of education. It can be problematic to compare B coefficients because they are often measured in units of different magnitudes. For example, a problem may involve GPA's measured from 0 to 4.0 and GRE scores measured from 400 to 1600. Computing standard scores for both reduces them to a common metric that supports direct comparison of their values. Beta is the standard score equivalent of the B coefficient, used to compare the relative importance of different variables toward the prediction of the dependent variable. We will discuss Beta coefficients in greater detail in stage 5 when we examine the importance of the predictors to use in our findings. Illustration of Regression Analysis**Stage 4: Compute the Statistics And Test Model Fit: Meeting**Assumptions Using output from the regression analysis to examine the conformity of the regression analysis to the regression assumptions is often referred to as "Residual Analysis" because if focuses on the component of the variance which our regression model cannot explain. Using the regression equation, we can estimate the value of the dependent variable for each case in our sample. This estimate will differ from the actual score for each case by an amount referred to as the residual. Residuals are a measure of unexplained variance or error that remains in the dependent variable that cannot be explained or predicted by the regression equation. Illustration of Regression Analysis**Linearity and Constant Variance for the Dependent**Variable:Residual Plot When we specified plots for regression output, we made a specific request for a plot of studentized residuals versus standardized predicted values. This plot is often referred to as the 'Residual Plot.' It is used to evaluate whether or not the derived regression model violates the assumption of linearity and constant variance in the dependence variable. If the plot shows a curvilinear pattern, it indicates a violation of the linearity assumption. If the spread of the residuals varies substantially across the range of the predicted values, it indicates a violation of the constant variance assumption. To correct for these violations of assumptions, we would employ transformations. If we do not see a pattern of nonlinearity or restricted spread to the residuals, the residual plot is termed a "Null Plot" to indicate that it does not show any violations of assumptions. The author's interpretation of the residual plot for this analysis is that the residuals fall in a generally random pattern similar to the null plot shown on page 174. Illustration of Regression Analysis**Normal Distribution of Residuals:Normality Plot of Residuals**To check for meeting the assumption that the residuals or error terms are normally distributed, we look at the Normal p-p plot of Regression Standardized Residual as shown below: Our criteria for normal distribution is the degree to which the plot for the actual values coincides with the green line of expected values. For this problem, the plot of residuals fits the expected pattern well enough to support a conclusion that the residuals are normally distributed. If a more exact computation is desired, we instruct SPSS to save the residuals in our data file and do a test of normality on the residual values using the Explore command. Illustration of Regression Analysis**Linearity of Independent Variables:Partial Plots**A partial regression plot is a scatterplot of the partial correlation of each independent with the dependent variable after removing the linear effects of the other independent variables in the model. The values plotted on this chart are two sets of residuals. The residuals from regressing the dependent variable on the other independent variables are plotted on the vertical axis. The residuals from regressing the particular predictor variable on all other independent variables are plotted on the horizontal axis. The partial regression, thus, shows the relationship between the dependent variable and a specific independent variable. We examine each plot to see if it shows a linear or nonlinear pattern. If the specific independent variable shows a linear relationship to the dependent variable, it meets the linearity assumption of multiple regression. If there is an obvious nonlinear pattern, we should consider a transformation of either the dependent or independent variable. I like to add a total fit line to the scatterplot to make it easier to interpret. We added the fit line to scatterplots in previous examples when we examined scatterplots for linearity. The partial regression plots for the three independent variables in the analysis are shown below. None of the plots demonstrates an obvious nonlinear pattern. Illustration of Regression Analysis**Linearity of Independent Variables:Partial Plots**Illustration of Regression Analysis**Independence of Residuals:Durbin-Watson Statistic**The next assumption is that the residuals are not correlated serially from one observation to the next. This means the size of the residual for one case has no impact on the size of the residual for the next case. While this is particularly a problem for time-series data, SPSS provides a simple statistical measure for serial correlation for all regression problems. The Durbin-Watson Statistic is used to test for the presence of serial correlation among the residuals. Unfortunately, SPSS does not print the probability for accepting or rejecting the presence of serial correlation, though probability tables for the statistic are available in other texts. The value of the Durbin-Watson statistic ranges from 0 to 4. As a general rule of thumb, the residuals are uncorrelated is the Durbin-Watson statistic is approximately 2. A value close to 0 indicates strong positive correlation, while a value of 4 indicates strong negative correlation. For our problem, the value of Durbin-Watson is 1.910, approximately equal to 2, indicating no serial correlation. Illustration of Regression Analysis**Identifying Dependent Variable Outliers:Casewise Plot of**Standardized Residuals As shown in the following table of Residual Statistics, all standardized residuals (Std. Residual) fell within +/- 3 standard deviations (actually, between -2.9 and 1.8). We do not have cases where the value of the dependent variable indicates an outlier. Illustration of Regression Analysis**Identifying Independent Variable Outliers - Mahalanobis**Distance While SPSS will save the Mahalanobis Distance score for each case to the data set, we must specifically request the probability to identify the outliers. Illustration of Regression Analysis**Identifying Independent Variable Outliers:Mahalanobis**Distance While SPSS will save the Mahalanobis Distance score for each case to the data set, we must specifically request the probability to identify the outliers. Illustration of Regression Analysis**Identifying Statistically Significant Mahalanobis Distance**Scores Illustration of Regression Analysis**Identifying Potential Outliers**The cases with a probability of Mahalanobis Distance smaller than 0.05 are shown in the half of the list titled 'Lowest.' As we can see, cases 96, 82, 42, and 5 are outliers by this criteria. Illustration of Regression Analysis**Identifying Influential Cases - Cook's Distance**In addition to the request for Mahalanobis distance score, we also requests that Cook's distance scores be saved to the data editor. Cook's distance identifies cases that are influential or have a large effect on the regression solution and may be distorting the solution for the remaining cases in the analysis. While we cannot associate a probability with Cook's distance, we can identify problematic cases that have a score larger than the criteria computed using the formula: 4/(n - k - 1), where n is the number of cases in the analysis and k is the number of independent variables. For this problem which has 100 subjects and 3 independent variables, the formula equates to: 4 / (100 - 3 - 1) = 0.042. To identify the influential cases with large Cook's distances, we sort the data set by the Cook's distance variable, 'coo_1' that SPSS created in the data set. Illustration of Regression Analysis**Sorting Cook's Distance Scores in Descending Order**Illustration of Regression Analysis**Cases with Large Cook's Distances**Illustration of Regression Analysis**Stage 5: Interpret The Findings - Regression Coefficients**Interpreting the regression coefficients enables us to make statements about the direction of the relationship between the dependent variable and each of the independent variables, the size of the contribution of each independent variable to the dependent variable, and the relative importance of the independent variables as predictors of the dependent variable. Illustration of Regression Analysis