Topics in Quantitative Methods

Algebra Review Guide

Algebra Review Guide
Linear Equations, Formulas and Inequalities
Practice Problems
Practice Problems
Practice Problems
Practice Problems
Terminology
Operation With Fractions
Simplify
Simplifying Fractions
Multiply
Multiplying Fractions
Divide
Dividing Fractions
Add and Subtract
Adding/Subtracting Fractions
Rules of Exponents
Rules of Radicals
Plus/Minus Sign
Data Transformation
Summation
Acknowledgements

Basic Principles

Basic Principles
Introduction
Scientific Method
Four Basic Steps
Iterative Process
Statistical Science
Usage Example
Experimental Design
Observational Experiments
Designed Experiments
Study Questions 1
Key Characteristics
Separating Effects
Study Questions 2
Increasing Precision
Randomization
Design Control
Blocking
Study Questions 3
Types of Data
Quantitative vs. Qualitative
Types of Data
Discussion Topic
Measures of Center and Dispersion
Nominal and Ordinal Scales
Continuous Scales
Significant Digits
Study Questions 5
Calculation Statistics
Standard of Measurement
Populations and Samples
Parameters
Population and Sample Mean
Median and Mode
Study Questions 6
Variance
Standard Deviation
Coefficient of Variation
Study Questions 7
Excel Exercises
Introduction
Try This! Excel Exercises
Ex. 1: Excel Layout
Ex. 1: Data Columns
Ex. 1: Column F Formula
Ex. 1: Column G Formula
Ex. 1: Completed Data Table
Ex. 2: Sorting Steps
Ex. 3: Filtering Data
Ex. 4: Evaluating Data With a Pivot Table
Ex. 5: Graphing Data in Excel
Ex. 6: Calculating Measures of Dispersion with a Pivot Table
Ex. 7: Using Excel Functions to Calculate Statistics
Ex. 8: Calculate Descriptive Statistics
Summary
Reflection
Basic Principles

Categorical Data: Binary

Categorical Data - Binary
Introduction
Definition of Binomial
Binomial - Definition
Another Binomial Situation
Study Questions 1
Study Questions 2
Study Questions 3
Assumption of Independence
Discussion
The Binomial Probability Function
Calculate Probabilities
Probability Results
Study Questions 4 and 5
Try This! Probability Exercises
Ex. 1: Binomial Probabilities (1)
Ex. 1: Binomial Probabilities (2)
Ex. 2: Table of Probabilities
Ex. 3: Cumulative Binomial Probability
Ex. 4: Probability Computations
The Mean and Variance
Mean and Variance
Standard Deviations
Study Questions 6 and 7
Estimating Trial Numbers
The Normal Approximation
Approximate the Binomial
Normal Approximation
Study Questions 8
Conclusions from the Example
Study Questions 9
Computing a Probability
Sample Exercises
Ex. 5: Compute the Binomial Probability
Ex. 6: Compute the Binomial Probability
Study Questions 10 and 11
Reason to Use Normal Approximation
Confidence Intervals
Sample Proportion
Study Questions 12 and 13
Confidence Interval Exercise
Ex. 7: Confidence Interval for p
Testing Hypotheses
Testing for a Proportion
Ex. 8: Test the Hypothesis for p
Ex. 9: Test the Hypothesis for p
Comparing Proportions
Summary
Reflection
Categorical Data - Binary

Categorical Data: Multivariate

Categorical Data - Multivariate
Introduction
Chi-Square Testing
Purpose
Evaluation
Formula
Evaluating Hypotheses
Yates' Correction
Testing Proportions
Hypotheses
Calculation
Calculating a Chi-Square Test (1)
Testing a 9:3:3:1 genetic ratio
Observations vs. Expectations
Contingency Tables
Description
Expected Values
Degrees of Freedom
Test for Independence
Two-way Example
Blackleg Example
Calculating Differences
Testing for Independence of Data
Testing for Independence of Data (2)
Testing for Independence of Data (3)
Two-way Contingency Tables
Test for Heterogeneity
Description
Pooling Data
Chi-Square Values
Testing for Hetero/Homogeneity
Summary
Reflection
Categorical Data - Multivariate

Central Limit Theorem, Confidence Intervals, and Hypothesis Tests

Central Limit Theorem, Confidence Intervals, and Hypothesis Tests
Introduction
Distribution of Sample Averages
Averages of Samples
Study Questions 1
Extreme Sample Mean Values
Probability Distribution
Standard Error of the Mean
Study Questions 2
Central Limit Theorem
Normal Distribution
Study Questions 3
Data Distributions
Ex. 1: Evaluating a Distribution
Calculations
Create Histogram
Finished Histogram
Probability Plot
Create Probability Plot
Finished Probability Plot
Evaluating the Distribution
Ex. 2: Calculate the Z-Statistic
Using Excel Formulas
Further Application
Confidence Interval for a Mean
Computing a Confidence Interval
Estimating a True Mean
Study Questions 4
Null Hypothesis
Testing a Hypothesis
Errors Can Occur
Examining the Hypothesis
Study Questions 5
Experiment Conclusions
Testing a Hypothesis
Soil Sampling Example
Sampling Exercise Application
Sampling Conclusions
Hypothesis Testing
Summary
Reflection
Central Limit Theorem, Confidence Intervals, and Hypothesis Tests

Continuous Data

Continuous Data
Introduction
The t-Distribution
When the Variance is Unknown
A Sample t-Value Table
Study Questions 1
Sample Means in Values of t
t-Value Scenarios
Study Questions 2 and 3
Confidence Limits
Confidence Intervals
Try This! Calculation Exercise
Ex. 1: Calculating a Confidence Interval
Study Questions 4
Discussion
t-Tests For Significance
Comparing Two Treatment Means
Paired t-Test
Independent Samples t-Test
Calculation Results
Try This!: t-Test Exercises
Ex. 2: t-Test Assuming Equal Variances
Ex. 3: t-Test Assuming Unequal Variances
Ex. 4: Paired t-test
Study Questions 5 and 6
Two-Sample Hypothesis Testing
Test Conclusion
Study Questions 7
Direction of the Mean Difference
Calculate t Results
Linear Additive Model
Assumptions for Linear Models
Confidence Limits
Least Significant Difference
Comparing Several Means
Ex. 5: Determining the LSD
Summary
Reflection
Continuous Data

Data Transformation

Data Transformation
Introduction
Assumptions of ANOVA
3 Main Assumptions
Normality, Independence and Random Distribution of Errors
Study Question 1
Homogeneity of Variances
Linear Additive Model
Additive Treatments
Testing Heterogeneity
Bartlett's Test
Example
Chi-Square Value
Interpretation
Study Question 2
Ex. 1: Evaluating the Variances
Ex. 1: Creating a Pivot Table
Ex. 1: Examining Homogeneity Assumption
Study Question 3
Ex. 1: Plotting the Variance against Means
Ex. 1: Creating a Scatterplot
Prerequisite Knowledge
Ex. 2: Test for Homogeneity of Variances - Using R
Ex. 2: Review of 3 ANOVA's Main Assumptions
Ex. 2: Exercise Introduction
Ex. 2: Read the Data
Ex. 2: Visualize the Data
Ex. 2: Combine into a Single Table
Ex. 2: Graph Means and Variance
Ex. 2: Reshape Data
Ex. 2: Reshape GUI
Ex. 2: Preview Result
Ex. 2: Scatter Plot to Visualize Data
Ex. 2: Bartlett's Test
Ex. 2: Conclusions
Data Transformation
3 Main Transformations
Natural Log: When to Use
Data before Transformation
Transform using Natural Log
Square Root: When to Use
Poisson Distribution
Data before Transformation
Transform using Square Root
Arcsine: When to Use
Data before Transformation
Transform using Arcsine
Ex. 3: Data Transformation using Angular
Ex. 3: Natural Log Transformation
Ex. 3: Square Root Transformation
Ex. 3: Arc Sin (Angular) Transformation
Ex. 3: Data Transformation using R
Ex. 3: Combine All Transformations in One Table
Ex. 3: Bartlett's Test and ANOVA
Ex. 3: LSD and Conclusions
Summary
Reflection
Data Transformation

Distributions and Probability

Distributions and Probability
Introduction
Overview
Objectives
Samples & Populations
How to Sample
Sample Represents a Population
Randomization
Try This: Assess a Population by Sampling
Try This: Assess a Population
Study Question 1
Discussion
Accurate Samples
Histograms & Percentiles
Purpose of Histograms
Creating a Histogram
Ex. 1: Using Histograms (1)
Ex. 1: Using Histograms (2)
Ex. 1: Using Histograms (3)
Ex. 1: Using Histograms (4)
Probability
Taking a Random Sample
Sample Space
Events
Defining Sample Spaces
What Is Probability?
Mathematical Symbols
Mutually Exclusive
Calculating Probability
Non-Mutually Exclusive Events
Joint Probability
Marginal Probability
Conditional Probability
Probability Distributions
Discrete Distribution
Continuous Distribution
Normal Distribution
Most Common, Best-Studied
What it Does
Basic Parameters
Reasons for Widespread Use
Properties of Normal Distribution
Study Question 2
Z-Scores
Definition
Calculation
Interpretation
Study Question 3
Second Example
Normal vs. Non-Normal Distribution
Study Question 4
Other Distributions
Non-Normal Continuous Distributions
Non-Normal, Non-Continuous Distribution
Poisson Distribution Example
Calculations
Binomial Distribution
Summary
Reflection
Distributions and Probability

Linear Correlation, Regression and Prediction

Linear Correlation, Regression and Prediction
Introduction
Correlation
Correlation Coefficient
Scatter Plots
Try This: Correlation
Study Question 1
Correlation: Calculating r
Correlation Example
Correlation Example Calculations
Ex. 1: Correlation Exercise (1)
Ex. 1: Bivariate Set of Data
Ex. 1: Bivariate Set of Data (2)
Ex. 1: Bivariate Set of Data (3)
Ex. 1: Bivariate Set of Data (4)
Ex. 1: Bivariate Set of Data (5)
Ex. 1: Bivariate Set of Data (6)
Discussion: Correlation
Linear Regression
Definition
Regression Lines
Sources of Variation
Estimating Regression Line
Point-Slope Formula
Y-Intercept Formula
Example Calculation: Slope
Example Calculation: Interpretation
Ex. 2: Estimate Regression (1)
Ex. 2: Plotting Data
Ex. 2: Plotting Data (2)
Ex. 2: Plotting Data (3)
Ex. 2: Plotting Data (4)
Ex. 2: Plotting Data (5)
Estimation Formula
Errors
Sum of Squares
Regression and Total SS
Partitioning Variation
Statistical Significance
F-tests
Formula for F
ANOVA Table
Example: ANOVA
Ex. 3: Calculating a Regression Line and Testing the Slope (1)
Ex. 3: Calculating a Regression Line and Testing the Slope (2)
Ex. 3: Calculating a Regression Line and Testing the Slope (2)
Study Question 2
Confidence Limits
Purpose
Equation
Using t-Test
Ex. 4: Confidence Limits
Replicated Regression
Purpose
Example
Error Calculation
Example: ANOVA Table
Ex. 5: ANOVA with Replicated Data
Ex. 5: ANOVA with Replicated Data (2)
Ex. 5: ANOVA with Replicated Data (3)
Ex. 5: ANOVA with Replicated Data (4)
Summary
Reflection
Linear Correlation, Regression and Prediction

Mean Comparisons

Mean Comparisons
Introduction
Comparing Means
Many Approaches
Multiple Comparison Procedures
Planned t-tests or F-tests
Trend Analysis
Least Significant Difference
Stated Level of Significance
Definition
Study Question 1
Formulas
CRD and RCRD
LSD Example
Study Question 2
Study Question 3
Conclusions
Calculations
Steps and Results
Exercise 1
Ex. 1: Calculating LSD and Tukey's HSD
Ex. 1: What are LSDs and HSDs
Ex. 1: Getting Ready
Ex. 1: ANOVA Output
Ex. 1: LSD Test
Ex. 1: HSD Test
Ex. 1: LSD Output
Ex. 1: HSD Output
Ex. 1: Review
Ex. 1: Supplement - Calculate LSD
Ex. 1: Supplement - LSD Calculation Steps
Ex. 1: Supplement - LSD Results
Ex. 1: Supplement - HSD Test
Ex. 1: Supplement - LSD and HSD Resources
Study Question 4
Study Question 5
Multiple Range Tests
Calculating Differences
Definition
How to do HSD
HSD
Study Question 6
Study Question 7
Contrasts
Contrasts - Introduction
Test Equations
Estimating Variance
Linear Combination and Variance of Linear Contrast
Study Question 8
Study Question 9
Study Question 10
Testing
Planned F-Tests
Planned F-Test: Introduction
Corn Example
Assigning Contrast Coefficients 1
Assigning Contrast Coefficients 2
Assigning Contrast Coefficients - Sums
Assigning Contrast Coefficients - Weighting
Assigning Contrast Coefficients - Comparison
Independence of Comparisons
Non-orthogonal Contrasts
Study Question 11
Contrast Sums of Squares
Calculating Contrast SS
Corn Population Example
Mean Square
F-Tests
F-Test Critical Value
Study Question 12
Exercise 2
Ex.2: Calculating Contrasts
Ex. 2: Getting Ready
Ex. 2: Read Data
Ex. 2: Contrast Coefficients
Ex. 2: Contrast Coefficients - Output
Ex. 2: Contrasts for Hybrid Effect
Ex. 2: Interaction
Ex. 2: Compare Yield - 7.5 vs 10
Ex. 2: Compare Yield - 10 vs 12.5
Ex. 2: Review
Ex. 2: Supplement - Decide Comparison(s)
Ex. 2: Supplement - Assign Coefficients
Trend Comparisons
Trend Comparisons - Description
Linear Trend
Quadratic Trend
Contrast Weights
Data Analysis
Summary
Reflection
Mean Comparisons

Multiple Regression

Multiple Regression
Introduction
Observing Variables
Exploring Multiple Variables
Multiple Correlation and Regression
Simple Correlation
Partial Correlation
Correlation Matrix
Total Correlation
Calculating the Correlation
Ex. 1: Correlation-Multiple Regression Analysis
Ex. 1, Step 1
Ex. 1, Step 2
Ex. 1, Step 3
Ex. 1, Step 4
Ex. 1, Step 5
Ex. 1, Step 6
Multiple Regression
Relationships Among Multiple Variables
Example of Multiple Correlation and Regression
Review the Data
Study Questions 1
Partial Coefficients of Determination
Total Coefficients of Determination
Partial Regression Coefficients
Ex. 2: Multiple Regression and Anova Using R (1)
Ex. 2: Multiple Regression and Anova Using R (2)
Ex. 2: Multiple Regression and Anova Using R (3)
Ex. 2: Multiple Regression and Anova (4)
Ex. 2: Multiple Regression and Anova Using R (5)
Ex. 2: Multiple Regression and Anova Using R (6)
Ex. 2: Multiple Regression and Anova Using R (7)
Ex. 2: Multiple Regression and Anova Using R (8)
Ex. 2: Multiple Regression and Anova Using R (9)
Ex. 2: Multiple Regression and Anova Using R (10)
Ex. 2: Multiple Regression and Anova Using R (11)
Ex. 2: Multiple Regression and Anova Using R (12)
Ex. 2: Multiple Regression and Anova Using R (13)
Ex. 2: Multiple Regression and Anova Using R (14)
Ex. 2: Multiple Regression and Anova Using R (15)
Ex. 2: Multiple Regression and Anova Using R (16)
Ex. 2: Multiple Regression and Anova Using R (17)
Ex. 2: Multiple Regression and Anova Using R (18)
Ex. 2: Multiple Regression and Anova Using R (19)
Ex. 2: Multiple Regression and Anova Using R (20)
Ex. 2: Multiple Regression and Anova Using R (21)
Ex. 2: Multiple Regression and Anova Using R (22)
Ex. 2: Multiple Regression and Anova Using R (23)
Ex. 2: Multiple Regression and Anova Using R (24)
Ex. 2: Multiple Regression and Anova Using R (25)
Ex. 2: Multiple Regression and Anova Using R (26)
Ex. 3: Correlation, Multiple Regression and Anova (1)
Ex. 3: Correlation, Multiple Regression and Anova (2)
Ex. 3: Correlation, Multiple Regression and Anova (3)
Ex. 3: Correlation, Multiple Regression and Anova (4)
Ex. 3: Correlation, Multiple Regression and Anova (5)
Ex. 3: Correlation, Multiple Regression and Anova (6)
Ex. 3: Correlation, Multiple Regression and Anova (7)
Ex. 3: Correlation, Multiple Regression and Anova (8)
Ex. 3: Correlation, Multiple Regression and Anova (9)
Testing Multiple Regression
Regression Model Significance
The Whole Regression Relationship
Regression Coefficient Signficance
Ex. 4: Non-Linear Regression and Model Comparison (1)
Ex. 4: Non-Linear Regression and Model Comparison (2)
Ex. 4, Non-Linear Regression and Model Comparison (3)
Ex. 4, Non-Linear Regression and Model Comparison (4)
Ex. 4, Non-Linear Regression and Model Comparison (5)
Ex. 4, Non-Linear Regression and Model Comparison (6)
Ex. 4, Non-Linear Regression and Model Comparison (7)
Ex. 4, Non-Linear Regression and Model Comparison (8)
Ex. 4, Non-Linear Regression and Model Comparison (9)
Ex. 4, Non-Linear Regression and Model Comparison (10)
Ex. 4, Non-Linear Regression and Model Comparison (11)
Ex. 4, Non-Linear Regression and Model Comparison (12)
Ex. 4, Non-Linear Regression and Model Comparison (13)
Ex. 4, Non-Linear Regression and Model Comparison (14)
Ex. 4, Non-Linear Regression and Model Comparison (15)
Ex. 4, Non-Linear Regression and Model Comparison (16)
Problems in Multiple Regression
Examining Problems
Multicollinearity
Polynomial Functions
Polynomial Functions
Polynomial Relationships
Polynomial Regression
Polynomial Example
Variance in the Data
Calculating Polynomial Equations
Ex. 5: Non-Linear Multiple Regression Analysis (1)
Ex. 5: Non-Linear Multiple Regression Analysis (2)
Ex. 5: Non-Linear Multiple Regression Analysis (3)
Ex. 5: Non-Linear Multiple Regression Analysis (4)
Ex. 5: Non-Linear Multiple Regression Analysis (5)
Summary
Reflection
Multiple Regression

Multivariate Analysis

Introduction
Multivariate Analysis
Multivariate Analysis
Measures that Describe Similarities/Dissimilarities Between Units or Variables
Initial Example
Data Sheet for Initial Example
R Output for Initial Example
Calculating Similarities/Dissimilarities for Different Data Types
Calculating Similarities and Dissimilarities in Binary Data
Different Coefficients
Second Example
Marker Data for Second Example
Dissimilarity Matrices
Calculating Similarities and Dissimilarities in Categorical Data
Creating Placeholder Variables
Binary Placeholder Variables
Calculating Similarities or Dissimilarities in Quantitative Data
Euclidean Distance
Manhattan Distance
Euclidean and Manhattan Distance Results
Correlation
Calculating the Correlation
Preparing Data for Statistical Analysis
Preparing Data for Statistical Analysis
Looking for Obvious Inconsistencies
Typical Data Clean-up - Example
Missing Values
Cluster Analysis
Explanation of Cluster Analysis
Agglomerative Hierarchical Clustering
Hierarchical Clustering Example
Different Agglomeration Methods
Cluster Analysis Results
Deciding a Cut-off Height
K-means Clustering
K-means Clustering Example
K-means Cluster Analysis
Distribution of Types
Principal Components Analysis
Principal Components Analysis
PCA Step by Step....
Distribution of Data
Eigenvectors Output Matrix
Display of Principal Components
Percentage of Overall Variance
Calculate the PCA Scores
Perform a Principal Component Analysis
Generate a Scatterplot Matrix
Calculate the Principal Components
Loadings of the Principal Components
Scree Plot
Create a Biplot
Reflection
Mulivariate Analysis

Nonlinear Regression

Nonlinear Regression
Introduction
Approximation of Non-Linear Data
Relationships Among Variables
In Detail - Linear Growth
Interpolating Data
Difference Comparisons
Study Questions 1
Comparing Equations
Functional Relationships
Nonlinear Relationships
Exponential Graph
Study Questions 2
Exponential Relationships
Ex. 1: Calculating the Regression Equation for an Exponential
Ex. 1: Examining the Fit of Data
Ex. 1: Calculating Residuals
Ex. 1: Calculating ANOVA
Ex. 1: Transforming the Data and Calculating Residuals
Ex. 2: Estimating Nonlinear Regression
Ex. 2: Summary of the Model
Ex. 3: Plotting the Exponential Curve
Ex. 3: ANOVA
Study Questions 3
Monomolecular Function
In Detail - Maximum Possible Response
Total Growth Functions - Logistic
Total Growth Functions - Gompertz
Nonlinear Model Calculation
Ex. 4: Estimating Regression Equations
Ex. 4: Plot Monomolecular and Gompertz
Ex. 4: Computation
Selecting the Best Function
Summary of Nonlinear Functions
Summary
Reflection
Nonlinear Regression

Randomized Complete Block Design

Randomized Complete Block Design
Introduction
Blocking
The Rationale for Blocking
Study Questions 1
Heterogeneity
Study Questions 2
Variance of the Error
How to Block
Minimize Field Differences
Study Questions 3
Treatments
Design Control
Randomization
Ex. 1: Randomizing Treatments For a RCBD
Ex. 1: Create a Random Assignment
Ex. 1: Finished Random Assignment
Ex. 1: Plot Plan
Ex. 1: RCBD vs. CRD Randomization
Ex. 1: R Code Functions
Ex. 1: Maize Yield Test
Ex. 1: Creating a Field
Ex. 1: Creating a Vector
Ex. 1: Vector with 3 Entries
Ex. 1: New Matrix with Block
Ex. 1: Ordering the Population in Block
Ex. 1: Filling the Block
Ex. 1: Review RCBD Method 1
Ex. 1: RCBD Method 2
Ex. 1: Creating a Field Matrix
Ex. 1: Finished Field Matrix
Ex. 1: Review RCBD Method 2
Linear Additive Model
RCBD - Linear Additive Model
Differences in Models
Treatment Differences
Estimate Effects Using ANOVA
Ex. 2: Analyze an RCBD Experiment
Ex. 2: Beginning Analysis
Ex. 2: Running ANOVA
Ex. 2: Interpreting Results
Ex. 2: Conclusions
Analysis of Variance
Analysis of Variance for RCBD
Example Using RCBD
Degrees of Freedom
Sum of Squares
Sum of Squares Example
Difference in RCBD and CRD
Study Questions 4
Sum of Squares Table
Mean Squares
F-Values and F-Test
RCBD Analysis Exercises using R
Ex. 3: Two-Factor ANOVA
Ex. 3: Interpreting Results
Ex. 3: Standard Error of the Mean (SEM)
Ex. 3: RCBD - Red Clover Variety Trial
Ex. 4: Mean Comparisons with RCBD
Ex. 4: Calculating LSD
Ex. 4: LSD Calculation Exercise
Ex. 4: Second LSD Calculation
Ex. 4: Interpretation of LSD
Ex. 4: R Output
Ex. 4: Interpret the Results/Make a Decision
Study Questions 5 and 6
Blocking Efficiency
Blocking Vs. CRD Efficiency
Calculating Blocking Efficiency
Calculating Error Mean Square for CRD
Summary
Reflection
Randomized Complete Block Design

The Analysis of Variance (ANOVA)

The Analysis of Variance (ANOVA)
Introduction
One-Factor ANOVA
Purpose of ANOVA
Variance
Example
Discussion
ANOVA Table
ANOVA Table
Sources of Variation
Degrees of Freedom
Study: Degrees of Freedom
Sum of Squares
Sum of Squares Calculations
Sum of Squares - Total SS
Sum of Squares - Residual SS
Study Question 1
One-Way ANOVA (1)
One-Way ANOVA (2)
Discussion: One-Way ANOVA
Mean Squares
Observed F-Ratio
Study Question 2
Testing Hypotheses
Testing Hypotheses - Purpose
Comparing Values - The Critical F-value
Study Question 3
Explanation
Ex. 1: One-Factor ANOVA of a CRD
Ex. 1: Read the Data Set into R
Ex. 1: Exploratory Data Analysis
Ex. 1: Create a Boxplot
Ex. 1: Calculate Coefficient of Variance
Ex. 1: Carry Out ANOVA
Ex. 1: Interpreting the Results
Ex. 2: Wheat Yield Example
Ex. 2: Enter the Data into R
Ex. 2: Interpret the ANOVA
The Linear Additive Model
Purpose
The Linear Model Equation
Application
Visual Guide
Summary
Reflection
For Your Information
Critical F-values
The Analysis of Variance (ANOVA)

Two Factor ANOVAs

Two-Factor ANOVAs
Introduction
Factorial Experiments
Multiple Treatment Factors
Combining Factors
Degrees of Freedom
Interaction
No Interaction
Positive Interaction
Negative Interaction
Linear Additive Model for Two-Factor ANOVA
The Linear Model
True Sources of Variation
Ex. 1: Running an ANOVA for a Two-Factor CRD
Ex. 1: Data Set
Ex. 1: Run the ANOVA
Ex. 1: Make Adjustments
Ex. 1: Two-Way ANOVA
Ex. 1: Run Individual ANOVAs
Ex. 1: Simple Main Effects
Ex. 1: Plot the Interaction
Ex. 1: Interaction Plot
Ex. 1: ANOVA for 10 Hybrids
Ex. 1: Interaction Plot of 10 Hybrids
Ex. 1: Review Questions
Study Questions
ANOVA and Experimental Design
Experimental Design and Analysis
Ex. 2: Randomized Complete Design using R
Ex. 2: Activity Objectives
Ex. 2: Randomize as Pairs
Ex. 2: Randomize the Order
Ex. 2: Matrix Form
Ex. 2: Visualizing Results in Excel
Ex. 2: R Code Glossary
Error Structure
Summary
Reflection
For Your Information
Experimental Design
Two Factor ANOVAS