# 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