Analysis of Variance Correlation and Regression Analysis Chapter 16 & 17 Analysis of Variance Correlation and Regression Analysis Copyright © 2010 Pearson Education, Inc. 16-1

Chapter Outline Overview Relationship Among Techniques One-Way Analysis of Variance Statistics Associated with One-Way Analysis of Variance Conducting One-Way Analysis of Variance Identification of Dependent & Independent Variables Decomposition of the Total Variation Measurement of Effects Significance Testing Interpretation of Results

Chapter Outline 1) Overview 2) Product-Moment Correlation 3) Regression Analysis 4) Bivariate Regression 5) Statistics Associated with Bivariate Regression Analysis 6) Conducting Bivariate Regression Analysis i. Scatter Diagram ii. Bivariate Regression Model 7) Multiple Regression

Relationship Among Techniques Analysis of variance (ANOVA) is used as a test of means for two or more populations. The null hypothesis, typically, is that all means are equal. Analysis of variance must have a dependent variable that is metric (measured using an interval or ratio scale). There must also be one or more independent variables that are all categorical (nonmetric). Categorical independent variables are also called factors.

One-Way Analysis of Variance Marketing researchers are often interested in examining the differences in the mean values of the dependent variable for several categories of a single independent variable or factor. For example: Do the various segments differ in terms of their volume of product consumption? Do the brand evaluations of groups exposed to different commercials vary? What is the effect of consumers' familiarity with the store (measured as high, medium, and low) on preference for the store?

Statistics Associated with One-Way Analysis of Variance F statistic. The null hypothesis that the category means are equal in the population is tested by an F statistic based on the ratio of mean square related to X and mean square related to error. Mean square. This is the sum of squares divided by the appropriate degrees of freedom.

Conducting One-Way Analysis of Variance Interpret the Results If the null hypothesis of equal category means is not rejected, then the independent variable does not have a significant effect on the dependent variable. On the other hand, if the null hypothesis is rejected, then the effect of the independent variable is significant. A comparison of the category mean values will indicate the nature of the effect of the independent variable.

Illustrative Applications of One-Way Analysis of Variance We illustrate the concepts discussed in this chapter using the data presented in Table 16.2. The department store is attempting to determine the effect of in-store promotion (X) on sales (Y). For the purpose of illustrating hand calculations, the data of Table 16.2 are transformed in Table 16.3 to show the store sales (Yij) for each level of promotion. The null hypothesis is that the category means are equal: H0: µ1 = µ2 = µ3

Effect of Promotion and Clientele on Sales Table 16.2

One-Way ANOVA: Effect of In-Store Promotion on Store Sales Table 16.4 Cell means Level of Count Mean Promotion High (1) 10 8.300 Medium (2) 10 6.200 Low (3) 10 3.700 TOTAL 30 6.067 Source of Sum of df Mean F ratio F prob. Variation squares square Between groups 106.067 2 53.033 17.944 0.000 (Promotion) Within groups 79.800 27 2.956 (Error) TOTAL 185.867 29 6.409

Multivariate Analysis of Variance Multivariate analysis of variance (MANOVA) is similar to analysis of variance (ANOVA), except that instead of one metric dependent variable, we have two or more. In MANOVA, the null hypothesis is that the vectors of means on multiple dependent variables are equal across groups. Multivariate analysis of variance is appropriate when there are two or more dependent variables that are correlated.

SPSS Windows: One-Way ANOVA Select ANALYZE from the SPSS menu bar. Click COMPARE MEANS and then ONE-WAY ANOVA. Move “Sales [sales]” in to the DEPENDENT LIST box. Move “In-Store Promotion[promotion]” to the FACTOR box. Click OPTIONS. Click Descriptive. Click CONTINUE. Click OK.

Correlation The product moment correlation, r, summarizes the strength of association between two metric (interval or ratio scaled) variables, say X and Y. It is an index used to determine whether a linear or straight- line relationship exists between X and Y. As it was originally proposed by Karl Pearson, it is also known as the Pearson correlation coefficient. It is also referred to as simple correlation, bivariate correlation, or merely the correlation coefficient.

Product Moment Correlation From a sample of n observations, X and Y, the product moment correlation, r, can be calculated as: r = ( X i - ) Y S 1 n 2 D v s o f t h e u m a d b y g C O V x

Product Moment Correlation r varies between -1.0 and +1.0. The correlation coefficient between two variables will be the same regardless of their underlying units of measurement.

Explaining Attitude Toward the City of Residence Table 17.1

Product Moment Correlation The correlation coefficient may be calculated as follows: = (10 + 12 + 12 + 4 + 12 + 6 + 8 + 2 + 18 + 9 + 17 + 2)/12 = 9.333 X Y = (6 + 9 + 8 + 3 + 10 + 4 + 5 + 2 + 11 + 9 + 10 + 2)/12 = 6.583 ( i - ) S = 1 n = (10 -9.33)(6-6.58) + (12-9.33)(9-6.58) + (12-9.33)(8-6.58) + (4-9.33)(3-6.58) + (12-9.33)(10-6.58) + (6-9.33)(4-6.58) + (8-9.33)(5-6.58) + (2-9.33) (2-6.58) + (18-9.33)(11-6.58) + (9-9.33)(9-6.58) + (17-9.33)(10-6.58) + (2-9.33)(2-6.58) = -0.3886 + 6.4614 + 3.7914 + 19.0814 + 9.1314 + 8.5914 + 2.1014 + 33.5714 + 38.3214 - 0.7986 + 26.2314 + 33.5714 = 179.6668

Product Moment Correlation ( X i - ) 2 S = 1 n = (10-9.33)2 + (12-9.33)2 + (12-9.33)2 + (4-9.33)2 + (12-9.33)2 + (6-9.33)2 + (8-9.33)2 + (2-9.33)2 + (18-9.33)2 + (9-9.33)2 + (17-9.33)2 + (2-9.33)2 = 0.4489 + 7.1289 + 7.1289 + 28.4089 + 7.1289+ 11.0889 + 1.7689 + 53.7289 + 75.1689 + 0.1089 + 58.8289 + 53.7289 = 304.6668 Y = (6-6.58)2 + (9-6.58)2 + (8-6.58)2 + (3-6.58)2 + (10-6.58)2+ (4-6.58)2 + (5-6.58)2 + (2-6.58)2 + (11-6.58)2 + (9-6.58)2 + (10-6.58)2 + (2-6.58)2 = 0.3364 + 5.8564 + 2.0164 + 12.8164 + 11.6964 + 6.6564 + 2.4964 + 20.9764 + 19.5364 + 5.8564 + 11.6964 + 20.9764 = 120.9168 Thus, r 7 9 . 6 8 3 4 = 0.9361

Correlation Analysis Pearson Correlation Coefficient–statistical measure of the strength of a linear relationship between two metric variables Varies between – 1.00 and +1.00 The higher the correlation coefficient–the stronger the level of association Correlation coefficient can be either positive or negative

Strength of Correlation Coefficients

SPSS Pearson Correlation Example

Regression Analysis Regression analysis examines associative relationships between a metric dependent variable and one or more independent variables in the following ways: Determine whether the independent variables explain a significant variation in the dependent variable: whether a relationship exists. Determine how much of the variation in the dependent variable can be explained by the independent variables: strength of the relationship. Determine the structure or form of the relationship: the mathematical equation relating the independent and dependent variables. Predict the values of the dependent variable. Control for other independent variables when evaluating the contributions of a specific variable or set of variables. Regression analysis is concerned with the nature and degree of association between variables and does not imply or assume any causality.

Relationships between Variables Is there a relationship between the two variables we are interested in? How strong is the relationship? How can that relationship be best described?

Conducting Bivariate Regression Analysis Formulate the Bivariate Regression Model In the bivariate regression model, the general form of a straight line is: Y = X b + 1 where Y = dependent or criterion variable X = independent or predictor variable = intercept of the line = slope of the line The regression procedure adds an error term to account for the probabilistic or stochastic nature of the relationship: Yi = Xi + ei where ei is the error term associated with the i th observation.

Covariation and Variable Relationships First we should understand the covariation between variables Covariation is amount of change in one variable that is consistently related to the change in another variable A scatter diagram graphically plots the relative position of two variables using a horizontal and a vertical axis to represent the variable values

Plot of Attitude with Duration Fig. 17.3 4.5 2.25 6.75 11.25 9 13.5 3 6 15.75 18 Duration of Residence Attitude

Scatter Diagram Illustrates No Relationship

Positive Relationship between X and Y

Negative Relationship between X and Y

Curvilinear Relationship between X and Y

Straight Line Relationship in Regression

Formula for a Straight Line y = a + bX + ei y = the dependent variable a = the intercept b = the slope X = the independent variable used to predict y ei = the error for the prediction

Ordinary Least Squares (OLS) OLS is a statistical procedure that estimates regression equation coefficients which produce the lowest sum of squared differences between the actual and predicted values of the dependent variable

SPSS Results for Bivariate Regression

SPSS Results say... Percieved reasonableness of prices is positively related to overall customer satisfaction Th relationship is positive But weak! Prices and satisfaction is associated, but there are other factors as well!!

Multiple Regression Analysis Multiple regression analysis is a statistical technique which analyzes the linear relationship between a dependent variable and multiple independent variables by estimating coefficients for the equation for a straight line

Y = b + X . e Multiple Regression The general form of the multiple regression model is as follows: which is estimated by the following equation: = a + b1X1 + b2X2 + b3X3+ . . . + bkXk As before, the coefficient a represents the intercept, but the b's are now the partial regression coefficients. Y = b + 1 X 2 3 . k e

Statistics Associated with Multiple Regression Adjusted R2. R2, coefficient of multiple determination, is adjusted for the number of independent variables and the sample size to account for the diminishing returns. After the first few variables, the additional independent variables do not make much contribution. Coefficient of multiple determination. The strength of association in multiple regression is measured by the square of the multiple correlation coefficient, R2, which is also called the coefficient of multiple determination. F test. The F test is used to test the null hypothesis that the coefficient of multiple determination in the population, R2pop, is zero. This is equivalent to testing the null hypothesis. The test statistic has an F distribution with k and (n - k - 1) degrees of freedom.

VARIABLES IN THE EQUATION Multiple Regression Table 17.3 Multiple R 0.97210 R2 0.94498 Adjusted R2 0.93276 Standard Error 0.85974 ANALYSIS OF VARIANCE df Sum of Squares Mean Square Regression 2 114.26425 57.13213 Residual 9 6.65241 0.73916 F = 77.29364 Significance of F = 0.0000 VARIABLES IN THE EQUATION Variable b SEb Beta (ß) T Significance of T IMPORTANCE 0.28865 0.08608 0.31382 3.353 0.0085 DURATION 0.48108 0.05895 0.76363 8.160 0.0000 (Constant) 0.33732 0.56736 0.595 0.5668

SPSS Windows The CORRELATE program computes Pearson product moment correlations and partial correlations with significance levels. Univariate statistics, covariance, and cross-product deviations may also be requested. Significance levels are included in the output. To select these procedures using SPSS for Windows, click: Analyze>Correlate>Bivariate … Analyze>Correlate>Partial … Scatterplots can be obtained by clicking: Graphs>Scatter >Simple>Define … REGRESSION calculates bivariate and multiple regression equations, associated statistics, and plots. It allows for an easy examination of residuals. This procedure can be run by clicking: Analyze>Regression Linear …

SPSS Windows: Correlations Select ANALYZE from the SPSS menu bar. Click CORRELATE and then BIVARIATE. Move “Attitude[attitude]” into the VARIABLES box. Then move “Duration[duration]” into the VARIABLES box. Check PEARSON under CORRELATION COEFFICIENTS. Check ONE-TAILED under TEST OF SIGNIFICANCE. Check FLAG SIGNIFICANT CORRELATIONS. Click OK.

SPSS Windows: Bivariate Regression Select ANALYZE from the SPSS menu bar. Click REGRESSION and then LINEAR. Move “Attitude[attitude]” into the DEPENDENT box. Move “Duration[duration]” into the INDEPENDENT(S) box. Select ENTER in the METHOD box. Click on STATISTICS and check ESTIMATES under REGRESSION COEFFICIENTS. Check MODEL FIT. Click CONTINUE. Click OK.

Copyright © 2010 Pearson Education, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Copyright © 2010 Pearson Education, Inc.