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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 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 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 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 equal 5. The data must be at least intervally scaled Testing of Hypotheses (cont.) Steps: 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. (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. 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 samples 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. 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 conducted. 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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posted: | 3/12/2008 |

language: | English |

pages: | 18 |

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