Chapter 12 & 13 - Analysis of Variance & Sign Test

Chapter 12 - Analysis of Variance

Page 1

• Title: Analysis of Variance

• Presented by: Prof. Faggella

Page 2 - Opening Activity

• Conducted a study on moose populations in a burned area.

• Four habitat types were identified on a map.

• Researchers expect moose observations to be proportional to the habitat acreage.

• Statistical test required to assess consistency with expectations.

Page 3 - Data from the Study

• Habitat Type & Data:

  • Type 1: 34% of total acreage, 25 moose observed

  • Type 2: 10% of total acreage, 22 moose observed

  • Type 3: 10% of total acreage, 30 moose observed

  • Type 4: 46% of total acreage, 40 moose observed

• Total Observations: 117 moose

Page 4 - One Way ANOVA Introduction

• Definition of ANOVA: Analysis of Variance, akin to chi-square tests for multiple means.

• Tests equality of population means.

• Utilizes F-distribution:

  • Skewed right.

  • Positive values only.

  • Shape is determined by two degrees of freedom.

Page 5 - When to Use ANOVA

• Applicable when data is categorized into one treatment group.

Page 6 - Hypotheses and Assumptions in ANOVA

• Hypotheses:

  • Null: All means are equal.

• Assumptions:

  • Populations are approximately normal.

  • Equal variances among populations.

  • Random samples.

  • Independent samples.

Page 7 - Example of One Way ANOVA

• Case Study: Turkey farmer tests three types of feeds.

• Weights of turkeys (in pounds):

  • Feed A: [12.3, 11.4, 13.4, 13.4, 12.0]

  • Feed B: [12.1, 13.4, 12.8, 12.5, 14.2]

  • Feed C: [11.5, 12.1, 13.6, 11.8, 14.0]

• Significance Level: α = 0.05

• Conduct pairwise tests if differences are found.

Page 8 - F Statistic in ANOVA

• F Statistic Definition: Ratio of variances.

• Degrees of Freedom:

  • Numerator: k - 1, where k is the number of samples.

  • Denominator: k(n - 1), where n is the sample size.

Page 9 - Example Dataset Calculation

• Sample datasets:

  • Sample 1: [7, 1, 3]

  • Sample 2: [5, 6, 6]

  • Sample 3: [6, 5, 7]

• Summary Stats:

  • Means and sample sizes computed.

• F-test statistic calculation:

  • F = 0.1428

  • P-value = 0.8688

Page 10 - Two Way ANOVA Overview

• Major change: Data presented in a table with rows and columns.

• Examines interaction effects between factors drawn from rows and columns.

Page 11 - Requirements for Two Way ANOVA

• Assumes normality, equal variance, simple random sampling, independence, and balanced design.

Page 12 - Two Way ANOVA Hypotheses

• Null Hypothesis: No interaction between two factors.

• Alternative Hypothesis: Interaction exists between two factors.

• If no interaction is found, separate one-way ANOVA tests can be performed.

Page 13 - Testing Procedure for Two Way ANOVA

• Test for interaction using the formula: F = MS(interaction) / MS(error)

• Conclude interaction if P-value is less than or equal to 0.05.

Page 14 - Example of Two Way ANOVA

• Detailed analysis and results expected but not elaborated in this section.

Page 15 - Chapter 13 - Nonparametric Tests

• Introduction to nonparametric statistical methods presented by Prof. Faggella.

Page 16 - Nonparametric Test Definition

• Nonparametric tests do not rely on population distribution assumptions.

Page 17 - Advantages of Nonparametric Tests

• Less rigid requirements allow wider application.

• Can be used with a variety of data types, including ranks and categorical data.

Page 18 - Disadvantages of Nonparametric Tests

• May waste information by simplifying quantifiable data into qualitative form.

• Generally less efficient than parametric tests, requiring stronger evidence to reject null hypotheses.

Page 19 - Overview of the Sign Test

• Nonparametric test utilizing positive/negative signs to evaluate claims about sample data.

Page 20 - Sign Test Process

• Analyzes frequency of signs to assess significance.

• A flowchart exists to guide through the procedures.

Page 21 - Types of Sign Tests

• Applicable to claims involving:

  • Matched pairs

  • Nominal data

  • Assessing the median of a single population.

Page 22 - Sign Test Example

• Case Study of Male Weights: Test for claims about equal medians.

Page 23 - Sign Test Procedure

• Example illustrated: Test statistic of x = 3, failing to reject H0 of no difference.

Page 24 - Sign Test Example Study Plan

• Final notes on structuring case studies for application of nonparametric testing.