Chapter 15 Differences Between Groups and Relationships Among Variables

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Chapter 15Differences Between Groups and
Relationships Among Variables
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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

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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.

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EXHIBIT 15.1 Choosing the Right Statistic
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EXHIBIT 15.1 Choosing the Right Statistic (contd)
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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
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Cross-Tabulation Tables The ?2 Test for
Goodness-of-Fit

• Cross-Tabulation (Contingency) Table
• A joint frequency distribution of observations on
  two more variables.
• ?2 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.

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Example Papa Johns Restaurants
Univariate HypothesisPapa Johns 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.
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Chi-Square Test
?² 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
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Degrees of Freedom (d.f.)

• (R-1)(C-1)(2-1)(2-1)1

d.f.(R-1)(C-1)
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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.

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The t-Test for Comparing Two Means (contd)

• 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?

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The t-Test for Comparing Two Means (contd)

• 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.

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The t-Test for Comparing Two Means (contd)
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EXHIBIT 15.2 Independent Samples t-Test Results
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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.

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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 Night 6
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EXHIBIT 15.3 Illustration of ANOVA Logic
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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

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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

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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

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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

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EXHIBIT 15.4 Interpreting ANOVA
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Correlation Coefficient Analysis

• Correlation coefficient
• 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.

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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)

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Correlation, Covariance, and Causation

• When two variables covary, they display
  concomitant variation.
• This systematic covariation does not in and of
  itself establish causality.
• Roosters crow and the rising of the sun
• Rooster does not cause the sun to rise.

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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.

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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 a ßX )
• Y the continuous dependent variable
• X the independent variable
• a the Y intercept (regression line intercepts Y
  axis)
• ß the slope of the coefficient (rise over run)

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The Regression Equation

• Parameter Estimate Choices
• ß is indicative of the strength and direction of
  the relationship between the independent and
  dependent variable.
• a (Y intercept) is a fixed point that is
  considered a constant (how much Y can exist
  without X)
• Standardized Regression Coefficient (ß)
• 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).

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EXHIBIT 15.5 The Advantage of Standardized
Regression Weights
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The Regression Equation (contd)

• Parameter Estimate Choices (contd)
• Raw regression estimates (b1)
• Raw regression weights have the advantage of
  retaining the scale metricwhich 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 (ß1)
• Standardized regression estimates have the
  advantage of a constant scale.
• Standardized regression estimates should be used
  when the researcher is testing explanatory
  hypotheses.

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Multiple Regression Analysis

• 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.

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Multiple Regression Analysis (contd)

• 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 Competitors 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.

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Multiple Regression Analysis (contd)

• Statistical Results of the Multiple Regression
• Regression Equation

• Coefficient of multiple determination (R2) 0.845
• F-value 14.6 plt.05

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Multiple Regression Analysis (contd)

• 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.

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Multiple Regression Analysis (contd)

• 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.

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Multiple Regression Analysis (contd)

• 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).

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Multiple Regression Analysis (contd)

• 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

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F-test
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EXHIBIT 15.4Interpreting Multiple Regression
Results
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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.

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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.

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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 (ß)
• Multiple regression analysis
• Multivariate data analysis