Marketing Research Data Analysis

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MarketingResearchCh12.pdf

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Chapter 12

Examining Relationships in

Quantitative Research

Learning Objectives

• Understand and evaluate the types of relationships between variables

• Explain the concepts of association and co- variation

• Discuss the differences between Pearson correlation and Spearman correlation

Learning Objectives (continued)

• Explain the concept of statistical significance versus practical significance

• Understand when and how to use regression analysis

Examining Relationships between Variables

• Concepts used

– Presence

– Direction

– Strength of association

– Type

• Linear relationship: Association between two variables whereby the strength and nature of the relationship remains the same over the range of both variables

• Curvilinear relationship: Relationship between two variables whereby the strength and/or direction of the relationship changes over the range of both variables

Covariation and Variable Relationships

• Covariation: Amount of change in one variable that is consistently related to the change in another variable of interest

• Scatter diagram: Graphic plot of the relative position of two variables using a horizontal and a vertical axis to represent their values

– Possible relationships - Positive, negative, curvilinear, and non-existent

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

• 0 - No association between two variables

• – 1.00 or 1.00 - Perfect link between two variables

• Correlation coefficient can be either positive or negative, depending on the direction of the relationship

Exhibit 12.5 - Rules of Thumb about the Strength of Correlation Coefficients

Range of Coefficient Description of Strength

±.81 to ±1.00 Very Strong

±.61 to ±.80 Strong

±.41 to ±.60 Moderate

±.21 to ±.40 Weak

±.00 to ±.20 Weak to No Relationship

Assumptions Involved in Calculating Pearson’s Correlation Coefficient

• Variables are to be measured using interval- or ratio-scaled measures

• Nature of the relationship to be measured is linear

– Straight line describes the relationship between the variables of interest

• Variables to be analyzed have a normally distributed population

Exhibit 12.6 - SPSS Pearson Correlation Example for Santa Fe Grill Customers

Descriptive Statistics

Mean Std. Deviation N

X22-- Satisfaction 4.54 1.002 253

X24-- Likely to

Recommend

3.61 .960 253

Exhibit 12.6 - SPSS Pearson Correlation Example for Santa Fe Grill Customers

(continued)

Correlations

X22 -- Satisfaction X24 -- Likely to

Recommend

X22 -- Satisfaction Pearson Correlation 1 .776**

Sig. (2-tailed) .000

N 253 253

X24-- Likely to

Recommend Pearson Correlation .776** 1

Sig. (2-tailed) .000

N 253 253

**. Correlation is significant at the 0.01 level (2-tailed).

Substantive Significance of the Correlation Coefficient

• Coefficient of determination (r2): Number measuring the proportion of variation in one variable accounted for by another

– Can be represented as a percentage

– Ranges from 0.0 to 1.00

– Larger the size of the coefficient of determination, stronger the linear relationship between the two variables being examined

Influence of Measurement Scales on Correlation Analysis

• Spearman rank order correlation coefficient: Statistical measure of the linear association between two variables where both have been measured using ordinal (rank order) scales

Exhibit 12.7 - SPSS Spearman Rank Order Correlation

Correlations

X27-- Food Quality X29-- Service

Spearman's rho X27 -- Food Quality Correlation Coefficient 1.000 -.130**

Sig. (2-tailed) .009

N 405 405

X29 -- Service Correlation Coefficient -.130** 1.000

Sig. (2-tailed) .009

N 405 405

** Correlation is significant at the 0.01 level (2-tailed).

Regression Analysis

• Bivariate regression analysis

– Statistical technique

– Analyzes the linear relationship between two variables by estimating coefficients for an equation for a straight line

• One variable is designated as a dependent variable

• Other variable is called an independent or predictor variable

Assumptions Behind Regression Analysis

• Linear relationship

– Describes the relationship between two variables

• Dependent and independent variables

– Labeling does not infer that one variable influences the behavior of another

Assumptions Behind Regression Analysis (continued)

• Use of a simple regression model assumes that:

– Variables of interest are measured on interval or ratio scales

– Variables come from a normal population

– Error terms associated with making predictions are normally and independently distributed

Exhibit 12.9 - The Straight Line Relationship in Regression

Fundamentals of Regression Analysis

• General formula for a straight line

Y = a + bX + ei – Where,

• Y - Dependent variable

• a - Intercept (point where the straight line intersects the Y-axis when X = 0)

• b - Slope (the change in Y for every 1 unit change in X)

• X - Independent variable used to predict Y

• ei - Error for the prediction

Fundamentals of Regression Analysis (continued)

Least squares procedure

• Determines the best-fitting line by minimizing the vertical distances of all the points from the line

Unexplained variance

• Amount of variation in the dependent variable that cannot be accounted for by the combination of independent variables

Exhibit 12.10 - Fitting the Regression Line Using the Least Squares Procedure

Developing and Estimating the Regression Coefficients

Ordinary least squares

• Estimates regression equation coefficients that produce the lowest sum of squared differences between the actual and predicted values of the dependent variable

Regression coefficient

• Indicator of the importance of an independent variable in predicting a dependent variable

• Large coefficients are good predictors, and small coefficients are weak predictors

Exhibit 12.11 - SPSS Results for Bivariate Regression

Model summary

Model R R Square Adjusted R Square Std. Error of the Estimate

1 .479a .230 .227 .881

a. Predictors: (Constant), X16 -- Reasonable Prices

Exhibit 12.11 - SPSS Results for Bivariate Regression (continued 1)

ANOVAb

Model Model Sum of Squares df Mean Square F Sig.

1 Regression 58.127 1 58.127 74.939 .000a

Residual 194.688 251 .776

Total 252.814 252

a. Predictors: (Constant), X16 -- Reasonable Prices

b. Dependent Variable: X22 -- Satisfaction

Exhibit 12.11 - SPSS Results for Bivariate Regression (continued 2)

Coefficientsa

Model Model Unstandardized

Coefficients

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

Model Model B Std. Error Beta t Sig.

1 (Constant) 2.991 .188 15.951 .000

X16 --

Reasonable

Prices

.347 .040 .479 8.657 .000

a. Dependent Variable: X22 -- Satisfaction

Significance of Regression Coefficients

• Regression coefficients help addressing questions regarding relationships

– Is there a relationship between the dependent and independent variable?

• How strong is the relationship?

Multiple Regression Analysis

• Analyzes the linear relationship between a dependent variable and multiple independent variables

• Beta coefficient: Estimated regression coefficient recalculated to have a mean of 0 and a standard deviation of 1

– Enables independent variables with different measurement units to be directly compared on the association with the dependent variable

Examining the Statistical Significance of Each Coefficient

• Each regression coefficient is divided by its standard error to produce a t statistic

– t statistic is compared against the critical value to determine whether the null hypothesis can be rejected

• Model F statistic: Compares the amount of variation in the dependent measure associated with the independent variables to the error variance

Substantive Significance

• The multiple r2 describes the strength of the relationship between all the independent variables and the dependent variable

• Evaluating the results of a regression analysis

– Assess the statistical significance of the overall regression model using the F statistic

– Evaluate the obtained r2

– Examine the individual regression coefficients and their t statistics

– Examine the beta coefficients

Multiple Regression Assumptions

• Linear relationship

• Homoskedasticity: Pattern of the covariation is constant around the regression line

• Normal distribution

Multiple Regression Assumptions (continued)

• Heteroskedasticity: Pattern of covariation around the regression line is not constant

– Indicates that the shape of variable distribution is equal both above and below the mean

• Normal curve: Indicates the shape of the distribution of a variable is equal both above and below the mean

Exhibit 12.14 - SPSS Results for Multiple Regression

Model Summary

Model R R Square Adjusted R

Square

Std. Error of

the Estimate

1 .646a .417 .410 .770

a. Predictors: (Constant), X20 -- Proper Food Temperature, X15 -- Fresh Food,

X18 -- Excellent Food Taste

Exhibit 12.14 - SPSS Results for Multiple Regression (continued 1)

ANOVAb

Model Model Sum of Squares df Mean Square F Sig.

1 Regression 105.342 3 35.114 59.288

.000a

Residual 147.472 249 .592

Total 252.814 252

a. Predictors: (Constant), X20 -- Proper Food Temperature, X15 -- Fresh Food, X18 -Excellent

Food Taste

b. DependentVariable:X22- Satisfaction

Exhibit 12.14 - SPSS Results for Multiple Regression (continued 2)

Coefficients3

Model Model Unstandardized

Coefficients

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

Model Model B Std. Error Beta t Sig.

1 (Constant) 2.144 .269 7.984 .000

X15--Fresh Food .660 .068 .767 9.642 .000

X18 -- Excelent

Food Taste

-.304 .095 -.267 -3.202 .002

X20-- Proper Food

Temperature

.090 .069 .096 1.312 .191

a. Dependent Variable: X22-- Satisfaction

Multicollinearity

• Several independent variables are highly correlated

– May result in difficulty in estimating independent regression coefficients for the correlated variables

Marketing Research in Action Developing a Customer Satisfaction Program

• Will the results of this regression model be useful to the QualKote plant manager? How?

• Which independent variables are helpful in predicting A36–Customer Satisfaction?

• How would the manager interpret the mean values for the variables reported in Exhibit 12.16?

• What other regression models might be examined with the questions from the survey?