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.