1. Chapter 15 Differences Between Groups and Relationships Among Variables

2. LEARNING OUTCOMES After studying this chapter, you should be able to
Understand what multivariate statistical analysis involves and know the two types of multivariate analysis
Interpret results from multiple regression analysis.
Interpret results from multivariate analysis of variance (MANOVA)
Interpret basic exploratory factor analysis results

3. What Is the Appropriate Test of Difference?
Test of Differences
An investigation of a hypothesis that two (or more) groups differ with respect to measures on a variable.
Behavior, characteristics, beliefs, opinions, emotions, or attitudes
Bivariate Tests of Differences
Involve only two variables: a variable that acts like a dependent variable and a variable that acts as a classification variable.
Differences in mean scores between groups or in comparing how two groups’ scores are distributed across possible response categories.

4. EXHIBIT 15.1 Choosing the Right Statistic

5. EXHIBIT 15.1 Choosing the Right Statistic (cont’d)

6. Common Bivariate Tests
Type of Measurement
Differences between two independent groups
Differences among three or more independent groups
Interval and ratio
Independent groups:
t-test or Z-test
One-way ANOVA
Ordinal
Mann-Whitney U-test
Wilcoxon test
Kruskal-Wallis test
Nominal
Z-test (two proportions)
Chi-square test
Chi-square test

7. Cross-Tabulation Tables: The χ2 Test for Goodness-of-Fit
Cross-Tabulation (Contingency) Table
A joint frequency distribution of observations on two more variables.
χ2 Distribution
Provides a means for testing the statistical significance of a contingency table.
Involves comparing observed frequencies (Oi) with expected frequencies (Ei) in each cell of the table.
Captures the goodness- (or closeness-) of-fit of the observed distribution with the expected distribution.

8. Example: Papa John’s Restaurants
Univariate Hypothesis: Papa John’s restaurants are more likely to be located in a stand-alone location or in a shopping center.
Bivariate Hypothesis: Stand-alone locations are more likely to be profitable than are shopping center locations.

9. Chi-Square Test χ² = chi-square statistic
Oi = observed frequency in the ith cell
Ei = expected frequency on the ith cell
Ri = total observed frequency in the ith row
Cj = total observed frequency in the jth column
n = sample size

10. Degrees of Freedom (d.f.)
(R-1)(C-1)=(2-1)(2-1)=1
d.f.=(R-1)(C-1)

11. The t-Test for Comparing Two Means
Independent Samples t-Test
A test for hypotheses stating that the mean scores for some interval- or ratio-scaled variable grouped based on some less than interval classificatory variable.

12. The t-Test for Comparing Two Means (cont’d)
Determining when an independent samples t-test is appropriate:
Is the dependent variable interval or ratio?
Can the dependent variable scores be grouped based upon some categorical variable?
Does the grouping result in scores drawn from independent samples?
Are two groups involved in the research question?

13. The t-Test for Comparing Two Means (cont’d)
Pooled Estimate of the Standard Error
An estimate of the standard error for a t-test of independent means that assumes the variances of both groups are equal.

14. The t-Test for Comparing Two Means (cont’d)

15. EXHIBIT 15.2 Independent Samples t-Test Results

16. What Is ANOVA? Analysis of Variance (ANOVA)
An analysis involving the investigation of the effects of one treatment variable on an interval-scaled dependent variable
A hypothesis-testing technique to determine whether statistically significant differences in means occur between two or more groups.
A method of comparing variances to make inferences about the means.
ANOVA tests whether “grouping” observations explains variance in the dependent variable.

17. Simple Illustration of ANOVA
How much coffee respondents report drinking each day based on which shift they work (GY stands for Graveyard shift).
Day 1
Day 3
Day 4
Day 0
Day 2
GY 7
GY 2
GY 1
GY 6
Night 6
Night 8
Night 3
Night 7

18. EXHIBIT 15.3 Illustration of ANOVA Logic

19. Partitioning Variance in ANOVA
Total Variability
Grand mean
The mean of a variable over all observations.
SST
The total observed variation across all groups and individual observations
SST = Total of (observed value-grand mean)2

20. Partitioning Variance in ANOVA
Between-groups Variance
The sum of differences between the group mean and the grand mean summed over all groups for a given set of observations.
SSB
Systematic variation of scores between groups due to manipulation of an experimental variable or group classifications of a measured independent variable or between-group variance.
SSB = Total of ngroup(Group Mean − Grand Mean)2

21. Partitioning Variance in ANOVA
Within-group Error or Variance
The sum of the differences between observed values and the group mean for a given set of observations; also known as total error variance.
SSE
Variation of scores due to random error or within-group variance due to individual differences from the group mean.
This is the error of prediction.
SSE = Total of (Observed Mean − Group Mean)2

22. The F-Test
F-Test
Is used to determine whether there is more variability in the scores of one sample than in the scores of another sample.
Variance components are used to compute f-ratios
SSE, SSB, SST

23. EXHIBIT 15.4 Interpreting ANOVA

24. Correlation Coefficient Analysis
A statistical measure of the covariation, or association, between two at-least interval variables.
Covariance
Extent to which two variables are associated systematically with each other.

25. Simple Correlation Coefficient
Correlation coefficient (r)
Ranges from +1 to -1
Perfect positive linear relationship = +1
Perfect negative (inverse) linear relationship = -1
No correlation = 0
Correlation coefficient for two variables (X,Y)

26. Correlation, Covariance, and Causation
When two variables covary, they display concomitant variation.
This systematic covariation does not in and of itself establish causality.
Rooster’s crow and the rising of the sun
Rooster does not cause the sun to rise.

27. Coefficient of Determination
Coefficient of Determination (R2)
A measure obtained by squaring the correlation coefficient; the proportion of the total variance of a variable accounted for by another value of another variable.
Measures that part of the total variance of Y that is accounted for by knowing the value of X.

28. Regression Analysis Simple (Bivariate) Linear Regression
A measure of linear association that investigates straight-line relationships between a continuous dependent variable and an independent variable that is usually continuous, but can be a categorical dummy variable.
The Regression Equation (Y = α + βX )
Y = the continuous dependent variable
X = the independent variable
α = the Y intercept (regression line intercepts Y axis)
β = the slope of the coefficient (rise over run)

29. The Regression Equation
Parameter Estimate Choices
β is indicative of the strength and direction of the relationship between the independent and dependent variable.
α (Y intercept) is a fixed point that is considered a constant (how much Y can exist without X)
Standardized Regression Coefficient (β)
Estimated coefficient of the strength of relationship between the independent and dependent variables.
Expressed on a standardized scale where higher absolute values indicate stronger relationships (range is from -1 to 1).

30. EXHIBIT 15.5 The Advantage of Standardized Regression Weights

31. The Regression Equation (cont’d)
Parameter Estimate Choices (cont’d)
Raw regression estimates (b1)
Raw regression weights have the advantage of retaining the scale metric—which is also their key disadvantage.
If the purpose of the regression analysis is forecasting, then raw parameter estimates must be used.
This is another way of saying when the researcher is interested only in prediction.
Standardized regression estimates (β1)
Standardized regression estimates have the advantage of a constant scale.
Standardized regression estimates should be used when the researcher is testing explanatory hypotheses.

32. Multiple Regression Analysis
An analysis of association in which the effects of two or more independent variables on a single, interval-scaled dependent variable are investigated simultaneously.
Dummy variable
The way a dichotomous (two group) independent variable is represented in regression analysis by assigning a 0 to one group and a 1 to the other.

33. Multiple Regression Analysis (cont’d)
A Simple Example
Assume that a toy manufacturer wishes to explain store sales (dependent variable) using a sample of stores from Canada and Europe.
Several hypotheses are offered:
H1: Competitor’s sales are related negatively to sales.
H2: Sales are higher in communities with a sales office than when no sales office is present.
H3: Grammar school enrollment in a community is related positively to sales.

34. Multiple Regression Analysis (cont’d)
Statistical Results of the Multiple Regression
Regression Equation:
Coefficient of multiple determination (R2): 0.845
F-value= 14.6; p<.05

35. Multiple Regression Analysis (cont’d)
Regression Coefficients in Multiple Regression
Partial correlation
The correlation between two variables after taking into account the fact that they are correlated with other variables too.
R2 in Multiple Regression
The coefficient of multiple determination in multiple regression indicates the percentage of variation in Y explained by all independent variables.

36. Multiple Regression Analysis (cont’d)
Coefficients of Partial Regression bn
Independent variables correlated with one another
The percentage of variance in the dependent variable that is explained by a single independent variable, holding other independent variables constant R2
The percentage of variance in the dependent variable that is explained by the variation in the independent variables.

37. Multiple Regression Analysis (cont’d)
Statistical Significance in Multiple Regression
F-test
Tests statistical significance by comparing the variation explained by the regression equation to the residual error variation.
Allows for testing of the relative magnitudes of the sum of squares due to the regression (SSR) and the error sum of squares (SSE).

38. Multiple Regression Analysis (cont’d)
Degrees of Freedom (d.f.)
k = number of independent variables
n = number of observations or respondents
Calculating Degrees of Freedom (d.f.)
d.f. for the numerator = k
d.f. for the denominator = n - k - 1

39. F-test

40. EXHIBIT 15.4 Interpreting Multiple Regression Results

41. Steps in Interpreting a Multiple Regression Model
Examine the model F-test.
Examine the individual statistical tests for each parameter estimate.
Examine the model R2.
Examine collinearity diagnostics.

42. Other Multivariate Techniques
Multivariate Data Analysis
A group of statistical techniques allowing for the simultaneous analysis of three or more variables.
Multivariate Techniques
Exploratory factor analysis
Confirmatory factor analysis
Multivariate analysis of variance (MANOVA)
Multiple discriminant analysis
Cluster analysis.

43. Key Terms and Concepts Cross-tabulation (contingency table)
Univariate analysis
Bivariate analysis
Analysis of variance (ANOVA)
Grand mean
Between-groups variance
Within-group error or variance
F-test
Within-group variation
Between-group variance
Total variability (SST)
Correlation coefficient
Coefficient of determination (r2)
Simple linear regression
Standardized regression coefficient (β)
Multiple regression analysis
Multivariate data analysis