Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall Basic Marketing Research: Using Microsoft Excel Data Analysis, 3 rd edition Alvin C. BurnsLouisiana State University Ronald F. BushUniversity of West Florida

Before we Discuss Relationships Between Two Variables… Every scale has unique descriptors, sometimes called levels or labels, which identify the different positions on that scale. The term “levels” implies that the scale is metric, whereas the term “labels” implies that the scale is categorical. This distinction is important in determining which statistical tool we will use to discover a relationship between two variables! Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 2

What Is a Relationship? A relationship is a consistent and systematic linkage between the levels (metric variables) or labels (categorical variables) for two variables. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 3

Finding Consistent and Systematic Relationships May be Useful! Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 4

Categorical Variables Relationships Relationships between two categorical variables with labels Example: McDonald’s knows breakfast customers typically purchase coffee while lunch customers typically purchase soft drinks Our labels are “breakfast” and “lunch” for choice of meal, and “coffee” and “soft drink” for choice of drink Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 5

Cross-Tabulation Analysis With cross-tabulation, the two categorical variables are arranged in a cross-tabulation table, defined as a table in which data are compared using a row-and-column format. The intersection of a row and column is called a cross-tabulation cell. A cross-tabulation analysis accounts for all of the relevant label-to-label relationships and it is the basis for the assessment of statistical significance of the relationships. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 6

A Cross-Tabulation Table Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 7

Chi-Square Analysis of a Cross-Tabulation Table Chi-square (x 2 ) analysis is the examination of frequencies for two categorical variables in a cross-tabulation table to determine whether the variables have a significant relationship. The Cross-tabulation table only depicts the two variables simultaneously; it is NOT a statistical test. Chi-square (x 2 ) analysis IS a statistical test…it will tell us if there is a “consistent, systematic” relationship…a “significant” association. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 8

Chi-Square Analysis of a Cross-Tabulation Table, Continued... The Chi-Square formula computes a value that is larger when there are greater differences between observed frequencies and those expected frequencies if the null hypothesis were true. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall 14- 9

How to Determine If You Have a Significant Relationship Using Chi-Square Analysis Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to Determine If You Have a Significant Relationship Using Chi-Square Analysis Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to present a Significant Cross- Tabulation Finding Convert your raw counts (observed frequencies) to percentages Examine Row or Column percentages The advantage of row or column percentages is that the will sum to 100%, and a graph will show the composition of each row or column Select either row OR column percentages based upon which one best “communicates” the relationship. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Presenting Significant Cross-Tabulation Relationships Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Using XL Data Analyst to Set Up a Cross- Tabulation Analysis Menu sequence: Relate-Crosstabs Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Using XL Data Analyst to Set Up a Cross- Tabulation Analysis, Continued... Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Correlation: Assessing Metric Variables Relationships The most intuitive relationship between two metric variables is a linear relationship. Knowledge of the amount of one variable will automatically yield knowledge of the amount of the other variable. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Straight-Line Relationship Illustrating the Intercept and the Slope Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Correlation Coefficients and Covariation.. when both Variables are Metric The correlation coefficient is an index number, constrained to fall between the range of -1.0 and +1.0, that communicates both the strength and the direction of the linear relationship between two metric variables. The strength of the linear relationship between two variables is communicated by the absolute size of the correlation coefficient, whereas its sign communicates the direction of the association. A plus sign means that the relationship is such that as one variable increases, the other variable increases. A negative sign means that as one variable increases, the other variable decreases. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Covariation Covariation is defined as the amount of change in variable systematically associated with a change in another variable. A scatter diagram, plots data pairs in an x- and y-axis graph… Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Scatter Diagrams Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Scatter Diagram Showing NO Association Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall There is no pattern…the dots are random!

Scatter Diagram of a Negative Relationship Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall A definite pattern, downward-sloping to the right, i.e. the relationship between dollars spent on training and employee turnover

Scatter Diagram of a Positive Relationship Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall A definite pattern; upward sloping and to the right, i.e. the relationship between the amount offered for bonuses for salespersons and sales per salesperson

Correlation Analysis Correlation analysis has the great ability of relating two variables that have different measurements because there is a standardization procedure in the computation of a correlation that eliminates the differences between the two measures involved. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to Interpret a Correlation Coefficient? First, you must assess the statistical significance of the correlation. If it is not significant, there is no relationship to interpret. If it is significant, you can take the second step, which is to interpret it by looking at the strength (closer to 1.00, the stronger) and the direction, positive or negative, given by the sign of the coefficient. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Statistical Significance of a Correlation, Continued... A correlation that is not statistically significant has no meaning at all because of the null hypothesis for a correlation which states that the population correlation coefficient is equal to zero. Most computer statistical programs will indicate the statistical significance level of the computed correlation coefficient. Your XL Data Analyst evaluates the significance and reports whether the correlation is significant at the 95% level of confidence Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Rules of Thumb for Correlation Strength If a correlation coefficient is statistically significant there are general “rules of thumb” concerning the strength of the relationship. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

The Pearson Product Moment Correlation Coefficient The Pearson product moment correlation measures the linear relationship between two metric-scaled variables. The correlation coefficient’s value comes to be restricted to -1.0 to The closer to 1.00, the higher the strength and the sign of the coefficient tells you the direction of the relationship… Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to Perform Correlation Analysis with the XL Data Analyst Menu sequence: Relate-Correlate Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to Perform Correlation Analysis with the XL Data Analyst, Continued... Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Regression Analysis Regression analysis is a predictive analysis technique in which two or more variables are used to predict the level of another by use of the straight-line formula: y= a + bx Bivariate regression analysis is a case in which only two variables are involved in the predictive model, and one is called dependent while the other is called independent Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Regression Analysis, Continued... Bivariate regression analysis, continued: The dependent variable is the one that is predicted, and it is customarily termed y in the regression straight-line equation. The independent variable is the one that is used to predict the dependent variable, and it is the x in the regression formula. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Computing the Intercept and Slope for the Bivariate Regression Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Testing for Statistical Significance of the Intercept and the Slope The values for a and b must be tested for statistical significance. To determine statistical significance, regression analysis requires that a t test be used for each parameter estimate. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Making a Prediction with Bivariate Regression Analysis Since a perfect correlation of +1.0 or -1.0 is almost never is found, we know that our regression prediction will only be an estimate. Regression analysis provides for a standard error of the estimate, which is a measure of the accuracy of the predictions of the regression equation. Residuals are the differences between each predicted y value for each x value in the data set compared to the actual x value. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Making a Prediction with Bivariate Regression Analysis, Continued... The prediction process is accomplished by applying the following equation: Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Making a Prediction with Bivariate Regression Analysis, Continued... Let us use the regression equation to make a prediction about the dollar amount of grocery purchases that would be associated with a certain family size… Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Making a Prediction with Bivariate Regression Analysis, Continued... The interpretation of these three numbers is as follows: For a typical family represented by the sample, the expected average weekly grocery purchases amount to $175. The 95% confidence interval reveals that the sales figure should fall between $136 and $214 (rounded values). The R-squared value is the squared correlation coefficient between the independent and dependent variable ranges from 0 to 1, and the closer it is found to 1, the stronger is the linear relationship and the more precise will be the predictions. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Multiple Regression Analysis Multiple regression analysis is an expansion of bivariate regression analysis such that more than one independent variable is used in the regression equation. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Multiple Regression Example Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Multiple Regression Multiple regression is a very powerful tool, because it tells us: a.which factors predict the dependent variable; b.which way (the sign) each factor influences the dependent variable, and; c.how much (the size of b i ) each factor influences it Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Multiple Regression Multiple R, also called the coefficient of determination, is a handy measure of strength of the overall linear relationship. Multiple R ranges from 0 to +1.0 and represents the amount of the dependent variable “explained,” or accounted for, by the combined independent variables. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Three Uses of Multiple Regressions Bivariate regression is used only for prediction, whereas multiple regression can be used for: Prediction Understanding As a screening device Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Three Uses of Multiple Regressions, Continued... As a tool for understanding, multiple regressions may use standard beta coefficients allowing for directly comparing the importance of each independent variable. Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Three Uses of Multiple Regressions, Continued... Using multiple regression analysis as a screening device means it allows a researcher to “narrow down” the many independent variables thought to predict the dependent variable to a smaller, more manageable set. When regression is used as a screening device for understanding, the items to report are: Dependent variable Statistically significant independent variables Signs of beta coefficients Standardized beta coefficients for the significant variables Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Using XL Data Analyst to Perform Regression Analysis Menu sequence: Relate-Regression Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Using XL Data Analyst to Perform Regression Analysis, Continued... Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

How to Present Regression Analysis Findings To properly present regression results the following are recommended: Dependent variable Statistically significant independent variables Intercept Beta coefficients, including signs Adjusted R 2 Standard error of the estimate Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

Flow Chart on Relationship Analyses Copyright ©2010 Pearson Education, Inc. publishing as Prentice Hall

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