SSBA II Unit 3: Analysis of Variance

• SSBA II – Major 08 – UNIT THREE introduces Analysis of Variance.

Variance

• Variance is the average squared deviation of a random variable from its mean.
• Variability is measured by variance.
• Distributions sharing a mean can exhibit different levels of variability.
• Data point spread from the average value is measured using variance.
• Variance concepts apply to descriptive statistics, hypothesis testing, and more.

Analysis of Variance (ANOVA)

• Experiment designs study the effects of independent variables on a single dependent variable.
• E.g., Practice schedule effects (independent) on psychological test results (dependent).
• ANOVA determines if independent variable exposure significantly enhances dependent variable variance above random factor variance.

Experimental Design

• The scientific planning of an experiment explores the effect of independent variables on a specific dependent variable.

Opting for uncontrolled or controlled experiments

• Studying the effects of adolescent tobacco smoking habits on adult pulmonary carcinoma is one of the uncontrolled experiment example.
• Controlled investigations use "fixed" treatment variables with levels/methods/timings rigorously controlled to minimize errors.
• Controlled application contains or eliminates relevant variables affecting the dependent variable.

Choosing Retrospective or Prospective Methods

• Retrospective methods are frequently used for uncontrolled experiments with classification variables beyond investigator control with backward tracking exploring past events already in existence.
• Prospective methods are mainly in controlled experiments, where "fixed" experimental treatments are applied under investigator control.

Pilot Experiment

• Using small groups initially must be carried out before wider investigation.

Levels and Replications

• Levels are amounts/intensities/categories of the independent variable applied, the number (k) of individual groups determined by the number of factor levels applie
• Random allocation assigns individuals from a population sample to k groups.
• To minimize experimental errors apply each factor level to multiple individuals i.e. replication of each factor level.
• The group size representing replications is determined by the desired replication number.
• In an experiment with three levels of a factor each with ten replications should consist of three groups of 10 individuals randomly chosen.
• A sample of 310 or 30 cases (nk) is drawn from a population.

Single-Factor and Factorial Experiments

• A single-factor experiment studies dependent variable changes from exposure of sample groups to single independent variable levels
• In contrast, factorial experiments study the effects of combinations of independent variable levels on a dependent variable.
• Factorial experiments are two-way, three-way, or four-way classifications, based on the number of factors.

Sample Size

• Working out sample size statistically before drawing a sample for the experiment is important.
• Ensuring that the sample being large enough is representative of the overall population makes sure that the sample includes different categories/types in the same proportions as they exist inherently.
• A smaller sample increases the probability of excluding rare types, making it subsequently less representative.

Randomization

• Randomization should be ensured when sampling and treating with the independent variable.

Minimizing Effects of Relevant Variables

• Effects of relevant variables that interfere with the aimed investigation need countering/minimizing by proper experiment design and proceedings.

Statistical Treatments

• The data obtained in investigations must undergo statistical tests for analysis, interpretation, inference, and prediction.

Analysis of Variance (ANOVA)

• ANOVA is an extension of Students's t-test.

• ANOVA is more powerful than the t-test.

• ANOVA can be applied to two or more groups simultaneously.

• ANOVA estimates the strength of association between dependent and independent variables.

• ANOVA reduces experimental errors by designing experiments to meet assumptions rigorously.

• ANOVA tests the variance differences between two or more groups.

• The one-way ANOVA analyzes total sample variance (s²t) components.

• This estimates within-groups variance (s²w), also known as uncontrolled random factors.

• It also considers between-groups variance (s²b); influenced by the independent variable. The aim is to determine if s²b differs significantly from s²w based on the null hypothesis testing from the F ratio i.e. the basic ANOVA statistic.

Classification of ANOVA

• ANOVA methods vary with the number of independent variables in the experiment.

• One-way ANOVA investigates the effects of a single one independent variable on the dependent variable.

• Levels of the independent variable determine the number of groups in an experiment.

• Size of each group equals the given level of the independent variable's replications.

• It can be used to measure tracheal ventilation values (dependent variable) on three insect groups (20 each) exposed to three doses of pesticide (independent variable) -to check if it changes ventilation significantly.

• Two-way and three-way ANOVAs are higher-order and are used in factorial experiments to check effects of the simultaneous effects of more than one independent variable.

• The number of groups used is akin to the number of combinations of different levels of the independent variables, each such combination being applied to one of the groups.

• The group size is akin to the desired number of replications of each factor combination.

Models of ANOVA

• ANOVA models vary by nature of the independent variables.

Fixed Model or Model I ANOVA

• Explores "fixed" or controlled treatment effects.
• It analyzes the variance when using "fixed" experimental treatments as independent variables.
• Model I studies the effects of chosen and controlled levels of drugs/hormones/temperature etc on aspects of organisms such as activity.
• It can check effects of practices, learning methods, etc., on performance.

Random Model or Model II ANOVA

• Explores chosen random factors' effects on the dependent variable of the experiment.
• It analyzes variance where groups are exposed to sex, race, age etc i.e. uncontrolled classification variables beyond investigator control.

Mixed Model or Model III ANOVA

• Model III is generally a two-way or higher order ANOVA, where some independent variables are "fixed" treatment(s) while others are uncontrolled variables unlike Model I, II that can be one-way or higher.

Assumptions of ANOVA

• Random assignment should give population members an equal chance of group selection.
• Random assignment should be independent of others' choices.
• Randomization of treatment should be ensured for different levels of the independent variable.

Normal Distribution

• The dependent variable should follow a normal distribution in the population.
• Deviations of individual scores from respective means i.e. error terms should be normally distributed.

Independent of Errors

• Error terms (deviations from group means) should be independent.
• Individual scores occur at random and independent of each other.

Homoscedasticity

• The assumption of homoscedasticity implies that experimental groups have homogeneous variances initially.
• This assumes sampling from same/similar populations, so initial variances estimate population variance, differing only in sampling errors and the error terms of individuals of different groups must posses homogeneous dispersion.

Additivity

• Separate variations from different factors, alongside independent variables should sum up to total variation of the dependent variable.
• This enables analysis of the total variance into components.

One-Way ANOVA

• One-way ANOVA investigates a single independent variable’s effects on the dependent variable and also finds whether there's a significance on significant differences in variance between groups.
• Dependent on how independent variables are classified (i.e "fixed" or uncontrolled ) the one-way ANOVA can be either a model I or model II, with model III not being permissible with one independent variables.

Computation of Variance Ratio

• One-way ANOVA computes and interprets the F statistic; i.e. the variance ratio betweens and within groups.

This section introduces Analysis of Variance (ANOVA). Variance measures the spread of data points from the average value. ANOVA determines if independent variable exposure significantly enhances dependent variable variance above random factor variance.