ANOVA (Analysis of Variance)

Module Seven: ANOVA (Analysis of Variance)

• ANOVA is a statistical method used to analyze variance.
• The module covers Single-Factor ANOVA and Two-Way ANOVA.

Objectives

• Understand terms used in Hypothesis Testing specifically in ANOVA.
• Perform testing the difference among three means for Single-Factor ANOVA.
• Perform testing two independent variables to its dependent variables.

New Statistical Notation

• ANOVA is analysis of variance.
• A factor is an independent variable.
• A level is condition of the independent variable.
• A treatment is condition of the independent variable.
• A treatment effect are the independent variable differences.
• The number of levels in a factor is denoted by k.

Overview of ANOVA

• Analysis of variance is a parametric procedure to find significant differences. It can be used with experiments containing two or more sample means.
• In experiments only having two independent variable conditions, either a t-test or an ANOVA serve as solutions.

One-Way ANOVA

• One-way ANOVA is for experiments testing only one independent variable.

Types of Factors

• Between-subjects factor is a factor studied using independent samples in all conditions.
• A between-subjects factor uses formulas for a between-subjects ANOVA.
• Within-subjects factor is when a factor is studied using related (dependent) samples in all levels, uses the within-subjects ANOVA formulas.

Experiment-Wise Error

• The overall probability of making a Type I error in an experiment is the experiment-wise error rate.
• The experiment-wise error rate equals a when making a t-test to compare two means in an experiment.

Comparing Means

• Applying multiple t-tests in an experiment with more than two means results in an experiment-wise error rate much larger than the selected value.
• ANOVA makes comparing means from all factor levels and allows keeping the experiment-wise error rate equal to a.

Assumptions of the One-Way Between-Subjects ANOVA

• Only one independent variable is in the experiment and all conditions have independent samples.
• Ratio scores are used to measure dependent variables that are normally distributed.
• The variances of all populations represented are homogeneous.

Statistical Hypotheses

• H₀: μ₁ = μ₂ =…= μk
• Hₐ: not all μs are equal

The F-Test

• F is the statistic for the ANOVA.
• The significance of Fobt indicates among the means and at least two of them differ significantly.
• Specific means differences isn't indicated and may require post hoc comparisons if the F-test is significant.

Post Hoc Comparisons

• Post hoc comparisons are like t-tests.
• All possible pairs of means from a factor are compared to determine which means differ significantly, one pair at a time.

Components of ANOVA

• There are two potential sources of variance.
• Scores differ from each other despite participants being in the same condition because of variance within groups.
• Scores differ from each other because coming from different conditions that is called the variance between groups.
• Mean square within groups describes the variability of scores within the conditions of an experiment.
• Mean square between groups describes the variability between the means of levels.

Performing the ANOVA (Sum of Squares)

• The computations require several sums of squared deviations which is called the sum of squares, symbolized by SS.

Performing the ANOVA, computing Fobt

• Compute the total sum of squares:
  • SS tot = ΣΧ² tot -((ΣΧ tot)²/N)
• Compute the sum of squares between groups:
  • SSbn = Σ((sum of scores in the column)²/n of scores in the column) - ((ΣΧ tot)²/N)
• Compute the sum of squares within groups:
  • SSwn = SStot - SSbn
• Compute the degrees of freedom (df):
  • The degrees of freedom between groups equals k - 1
  • The degrees of freedom within groups equals N - k
  • The degrees of freedom total equals N - 1
• Compute the mean squares:
  • MSbn = SSbn/df bn
  • MS wn = SS wn df wn
• Compute Fobt:
  • Fobt = MSbn MSwn

The F-Distribution

• The F-distribution is the sampling distribution showing the various values of F when H₀ is true. All conditions represent one population

Critical F Value

• The critical value of F (Fcrit) depends on degrees of freedom.
• dfbn = k - 1, is the Numerator
• dfwn= N - k, is the Denominator
• The a selected (e.g. a 0.05 or a 0.01).
• The F-test is always a one-tailed test.

Performing Post Hoc Comparisons

• When the ns in the levels of the factor are not equal, use Fisher's protected t-test.
• To obtain HSD, use the k and (df of MSwn);
• For the HSD, when the ns in all levels of the factor are equal, use the Tukey HSD (Honestly Significant Difference) multiple comparisons test (samples with 30<).

Additional Procedures in the One-Way ANOVA

• For a single u use The computational formula for the confidence interval.
• Graphs of means from three conditions of an independent variable.
• To find the proportion of variance in the dependent variable that is accounted for by a factor compute:
  • n² = SSbn/SS tot

Key Terms

• analysis of variance
• ANOVA
• between-subjects ANOVA
• between-subjects factor
• error variance
• eta squared
• experiment-wise error rate
• factor
• F-distribution
• Fisher's protected t-test
• F ratio
• level
• mean square between groups
• mean square within groups
• multivariate statistics
• one-way ANOVA
• post hoc comparisons
• sum of squares
• treatment
• treatment variance
• Tukey’s HSD multiple comparisons test
• univariate statistics
• within-subjects ANOVA

Two-Way Analysis of Variance (ANOVA)

• ANOVA provides a very flexible hypothesis testing procedure evaluated to the mean differences produced in a research study with two or more independent variables. Tests is significant with each independent variable/s and the interactions between.
• A factorial design occurs when two or more factors.
• The main effect of factor A (row) is known as the A-effect and defines the row.
• The main effect of factor B (column) is known as the B-effect and defines the column.
• The interaction (the A x B interaction) are the interactions of row & column.

Components of the two-way ANOVA table

• Treatment groups and their size. -size: columns * rows cell counts corresponding to a treatment group.
• Each cell corresponds to a treatment group with a uniquely treated sample.
• Marginal means is the mean of each cell and factor main effects defines differences between marginal means.
• The interaction Effect is the effect between the two independent variable.

Repeated Measure Analysis of Variance

• A repeated measures ANOVA (also known as a two-factor repeated measures ANOVA, two-factor or two-way ANOVA with repeated measures, or within-within-subjects ANOVA) compares the mean differences between groups that have been split on two within-subjects factors (also known as independent variables).
• The Repeated measures ANOVA procedure analyzes groups of related dependent variables that represent different measurements of the same attribute. The order in which you specify within-subjects factors is important and each factor constitutes a level within the previous factor.
• A repeated measures ANOVA is used to determine whether or not there is a statistically significant difference between the means of three or more groups in which the same subjects show up in each group.
• Used is two specific situations:
  • Measuring mean scores of subjects during three or more time points.
  • Measuring mean scores of subjects under three different conditions