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					Bivariate Analysis
Differences Between Sample Groups

   Chapter 16
    Bivariate Cross-Tabulation

• Bivariate statistical analysis: the analysis of
  relationships (e.g., differences) between two variables
• Chapter overview:
   – T-test of differences in means of two independent samples
   – One-way analysis of variance (ANOVA) for k groups
   – Two-way ANOVA of differences in between two variables and
     within the k groups defining each of the variables

Research Question: Is occupational status associated with the loyalty status?
Means of analysis: Chi-square; compute theoretical frequencies for each cell on
the null hypothesis that loyalty is statistically independent of occupation.

                                     Degrees of freedom: (R-1)(C-1). Significance level: 0.05
           Computer Programs
           for Cross-Tabulation
Most programs provide:
  – Computations of row and column percentages
  – Introduction of a third variable to describe association between
    a pair of variables
  – Determination of a statistical significance of the association
  – Measurement of the strength of the association by means of
    an agreement index
           Bivariate Analysis:
  Differences in Means and Proportions
• Standard Error of Differences
  – SE of Difference in Means:
     • If the population stdev are not
       known, they must be estimated.

  – SE of Difference in Proportions:
        Testing of Hypotheses
When applying the SE formulas, the following conditions
   must be met:
1.   Samples must be independent
2.   Individual items in samples must be drawn in a
     random manner
3.   The population being sampled must be normally
     distributed (or sample size sufficiently large)
4.   For small samples, the population variances must be
5.   The data must be at least intervally scaled
 Testing of Hypotheses (cont.)
1. Specify the null hypothesis

2.       Establish the level of statistical significance
                  α = 0.05
3.       a) Calculate the Z-value
     •      Means:

     •      Proportions:
 Testing of Hypotheses (cont.)
3. b) For unknown population variance and small
   samples, the Student t distribution must be used.

4. Determine the probability of the observed difference of
   the two sample statistics having occurred by chance.
5. If the probability of the observed difference is greater
   than the alpha risk, accept the null hypothesis; if the
   opposite, reject the null hypothesis.
    Testing the Means of Two Groups:
     The Independent Samples t-Test
•     When testing variances in large samples:

•     Pooled variance estimate:

     1. When testing for the same population proportion in
        two populations
     2. Testing the difference in means between two small
Testing of group means: ANOVA
 t-Test: tests differences between two group means
 ANOVA: tests the overall difference in k group means,
  where the k groups are thought as levels of a treatment
  or control variable(s) or factor(s).
   – The variables influencing the results are called experimental
     (control) factors.
       • Control factors in agriculture: seed type, fertilizer type, fertilizer
         dosage, temperature, moisture, etc.
   – ANOVA tests the statistical significance of differences in mean
     responses given the introduction of one or more treatment
           ANOVA Methodology
• ANOVA designs:
    – Total sum of squares
    – Between-treatment sum of squares
    – Within-treatment sum of squares
• Compares the between-treatment-groups sum of squares with the
  within-treatment-group sum of squares  F statistic:

• F statistic indicates the strength of the grouping factor; the larger
  the ratio of between to within, the more inclined to reject Ho.
• If the variance of the error distribution is large relative to
  differences among treatments, the true effects may be swamped
  Accept Ho when it is false
One-way (single factor) ANOVA
One-way (single factor) ANOVA
            Follow-up Tests
       of Treatment Differences
• F-ratio only provides information that differences exist.
  Then which treatments differ?
• To find out, perform a follow-up analysis: series of
  independent sample t-tests.
   – Ex: Bonnferoni’s test, Duncan’s multiple range tests, Scheffe’s test, etc.

• These test statistics control the probability that a Type I
  error will occur when a series of statistical test are
N-Way (Factorial) ANOVA Designs
Factorial experiment: an equal number of observations
  is made of all combinations involving at least two levels
  of at least two variables.
        • Enables researchers to study possible interactions among the
          variables of interest.
        • These Interactions can be ordinal and disordinal.

    Note: Response increments
    differ, line segments are not
    parallel. (differential effect)
     Nonparametric Analysis

• Other tests:
  – Wilcoxon Rank Sum
  – Mann-Whitney
  – Kolmogorov-Smirnov

• Indexes of Agreement:
  – Chi-square
  – 2x2 Case (phi correlation coefficient)
  – RxC Case (contingency coefficient)

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