Bivariate Analysis Differences Between Sample Groups

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Bivariate AnalysisDifferences Between Sample
Groups

• Chapter 16

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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

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Application
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
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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 observed
• Measurement of the strength of the association by
  means of an agreement index

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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

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Testing of Hypotheses

• When applying the SE formulas, the following
  conditions must be met
• Samples must be independent
• Individual items in samples must be drawn in a
  random manner
• The population being sampled must be normally
  distributed (or sample size sufficiently large)
• For small samples, the population variances must
  be equal
• The data must be at least intervally scaled

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Testing of Hypotheses (cont.)

• Steps
• Specify the null hypothesis
• Establish the level of statistical significance
• a 0.05
• a) Calculate the Z-value
• Means
• Proportions

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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. (tables)
• 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.

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Testing the Means of Two Groups The Independent
Samples t-Test

• When testing variances in large samples
• Pooled variance estimate
• When testing for the same population proportion
  in two populations
• Testing the difference in means between two small
  samples

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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 effects.

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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

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One-way (single factor) ANOVA
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One-way (single factor) ANOVA
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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 Bonnferonis test, Duncans multiple range
  tests, Scheffes test, etc.
• These test statistics control the probability
  that a Type I error will occur when a series of
  statistical test are conducted.

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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)

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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)