Analysis of Variance (ANOVA)

• Analysis of Variance is commonly referred to as ANOVA

Variance

• Variance is the average squared deviation of a random variable from its mean
• Functions as the measure of variability within a data set
• Distributions sharing the same mean can exhibit varying degrees of variability
• Applying variance helps quantify the dispersion of data points relative to their mean
• Variance is applicable in descriptive statistics and hypothesis testing

Analysis of Variance

• An experiment is conducted to assess how one or more independent variables influence a single dependent variable
• For example, examining the impact of a practice schedule (independent variable) on a psychological test (dependent variable)
• ANOVA is employed to determine if exposing a sample to the independent variable significantly changes the variance of the dependent variable, compared to variance due to random chance

Experimental Design

• Involves scientifically planning an experiment to explore the impact of independent variables on a dependent variable
• Includes procedures like opting for uncontrolled or controlled trials
• An uncontrolled experiment aims to study the effects of adolescent tobacco smoking habits on adult pulmonary carcinoma
• Controlled investigations involve "fixed" treatment variables, where levels, methods, modes, and timing are investigator-controlled to minimize errors and manage variables affecting the dependent variable

Methods

• Retrospective methods look backward in time to investigate causes of existing events, and involves uncontrolled experiments, using classification variables
• Prospective methods utilize controlled experiments, using "fixed" experimental treatments applied as independent variables
• Pilot experiments using small groups should precede full-scale investigations

Levels and Replications

• Levels comprise chosen amounts or categories applied to the sample
• The number of groups corresponds to varying levels of to a factor
• Members of a random sample are allocated to groups to study the impact on a dependent variable, with each group exposed to a unique level of the factor
• Replications are the application of a factor to more than one individual, thus minimizing experimental errors.

Experiments

• Single-factor experiments explore dependent variable changes due to a single independent variable
• Factorial experiments study the effects of different independent variable combinations on a dependent variable, which can involve two-way, three-way, or four-way designs

Sample Size

• The sample size needs statistical consideration before the experiment
• It should accurately reflect the population, including different types or categories in appropriate proportions
• A smaller has an increased likelihood of excluding rare cases, thus becoming less representative

Randomization

• Randomization is crucial during sampling and while treating with the independent variable

Variables

• Relevant variables not intended to be applied may influence the dependent variable or skew results
• Interferences from these variables need to be minimized through experimental design

Treatments

• Data from experiments need to be subject to statistical tests for analysis, interpretation, inference, and prediction

Analysis of Variance (ANOVA)

• Acts as an extension and generalization of Student's t-test
• ANOVA is more powerful and can simultaneously analyze two or more groups.
• It estimates the correlation the dependent variable to the independent variable and minimizes experimental errors through more rigorous experiment design

ANOVA Purpose

• It assesses variance differences among two or more groups
• One-way ANOVA dissects a sample's total variance to gauge the relative contributions of within-groups variance (caused by random factors) and between-groups variance (influenced by the independent variable)
• The goal is to check if between-groups variance can be explained by the null hypothesis or to differ significantly with the within-group variance
• The fundamental statistic in ANOVA is the variance ratio, known as the F ratio

Classifying ANOVA

• The method of ANOVA hinges on the quantity of independent variables used
• One-way ANOVA assesses how one independent variable affects a dependent variable
• The levels of the independent variable determine the number of groups, where group size equals the number of replications
• One-way ANOVA can be applied to ventilation values measured across three insect groups exposed to different pesticide doses, determining whether or not tracheal ventilation changes

Higher Orders of ANOVA

• Two-way and three-way ANOVAs are used when simultaneously studying multiple independent variables and their effects during an experiment
• The number of groups relates to the number of combinations that can be created by using different levels of the independent variables

Models of ANOVA

• Models are determined by the nature of the independent variables in the experiment
• Model I ANOVA explores "fixed" treatment effects by analyzing dependent variable variances in experiments using controlled treatments as independent variables
• This model is ideal for studying the impact of variables like drugs, hormones, and temperature on physical and chemical aspects

Model II ANOVA:

• ANOVA explores random factors' effects on the dependent variable
• It examines variance in experiments where groups are exposed to varying and randomly assigned variables, such as sex, genotypes, and habitats

Model III ANOVA:

• Also known as Mixed Model of ANOVA; a two-way analysis where some independent variables are "fixed" experimental treatments and others are uncontrolled variables

Assumptions of ANOVA

Assignment

• Randomized assignment experimental designs need population sampling so each individual has an equal chance of being chosen for a group

Distribution

• Dependent variable should have a normal distribution in the population so individual scores will deviate normally

Homogeneity

• Error terms, or deviations of individual scores from the group mean, need independence. This is an alternative form of the assumption that the individual scores occur at random

Homoscedasticity

• The groups must have homogenous variances, and be similarly drawn so error terms for individuals should have dispersion

Additivity

• Factors should produce separate variations of the dependent variable, enabling the analysis of total variance

One-Way ANOVA

• Investigates the effects of an independent variable on the dependent variable
• Determine if the exposure of subjects to varying levels have statistically significant changes in the variance between groups
• Model I and Model II are the two models possible with One-Way ANOVA, regarding the independent variable; it cannot be Model III ANOVA and requires more than one independent variable
• One-way focuses on the computation and interpretation of the ANOVA F statistic, which is equal to the variance ratio between groups and within groups

Explore Analysis of Variance (ANOVA), a statistical method used to examine how independent variables affect a single dependent variable. Learn how ANOVA determines if exposure to an independent variable significantly alters the variance of the dependent variable. Understand the role of variance in measuring data set variability.