One-Way Between-Subjects ANOVA

One-Way Between-Subjects ANOVA

• Analysis of Variance (ANOVA) tests if the variance of scores between two or more groups is greater than the variance within groups

  • It is also an F test

• Formula for F:

  • F = variance between groups / variance within groups

• Variance within groups is "error variance" or "unexplained variance"

  • It's variance that the group/treatment/IV cannot explain
  • It's not literally an error or mistake

• One-way ANOVAs might have multiple groups, but only examine one independent variable (IV)

  • A two-way ANOVA = 2 IVs
  • A three-way ANOVA = 3 IVs, and so on

• One-way between-subjects ANOVAs with only 2 groups are similar to independent-samples t tests

  • With only 2 groups, the F test stat equals the t test stat squared, i.e., F = t^2

• Three assumptions must be met for one-way between-subjects ANOVAs to be valid:

  • The dependent variable (DV) is measured with a ratio or interval scale
  • The samples are drawn from normally distributed populations
  • The samples are drawn from populations with equal (homogenous) variances

Calculating F

• An F test is one-tailed and cannot be negative
• Steps for hypothesis testing and calculating F
  • Step 1: State the hypothesis
    • Null Hypothesis: There is NO significant difference among exercise conditions in the population means of mood
    • Alternative hypothesis: There IS a significant difference between at least two exercise conditions in the population means of mood
  • Step 2: State what your N, n, k, IV, and DV are. Example:
    • N = total number of participants in the study = 12
    • n = number of participants per exercise group = 4
    • k = number of exercise groups = 3 (per exercise condition)
    • IV = exercise condition
    • DV = mood
  • Step 3: Indicate your means for each group and the grand mean
  • Step 4: Calculate SS(Total)
    • Calculate the sum of squares (SS), which is the sum of the squared deviations around some point
    • Formula: SSTotal = Σχ² - (Σx)² / N
  • Step 5: Calculate SS(IV)
    • Formula: SSIV = ηΣ(Μιν - MGM)2
  • Step 6: Calculate SS(Error)
    • SSError = SSTotal - SSIV
  • Step 7: Create a summary table and indicate whether or not to reject the null hypothesis
  • Step 8: Write Results
    • Result in APA format: There was significant difference among the exercise conditions' mood, F(2,9) = 5.23, p<0.05
• Omnibus test: A statistical test that examines multiple groups at once
  • The study of exercise conditions for a difference in mood is an omnibus test because it examines 3 exercise groups at once
  • An omnibus test reveals whether a significant difference exists, but not which groups are significantly different from each other
  • A significant F indicates the largest group mean is significantly larger than the smallest group mean.
  • Post hoc tests are required to confirm and find other different pairs of means

Post Hoc Test

• Post hoc test: a statistical test performed after an initial test
  • Post hoc is Latin for "after this"
  • Post hoc tests usually control for experimentwise alpha (α)
• Experimentwise alpha: The aggregated alpha or probability of making a Type 1 error.
• Fisher's least significant difference (LSD): A post hoc test involving a t-test, also known as Protected t
  • Conducted ONLY if F is significant; this requirement provides Inflated Type 1 error protection
  • Fisher's LSD doesn't control for experimentwise alpha like other post hoc tests, and too many Fisher's LSDs shouldn't be conducted

Effect Size

• Omega-squared (ω)²: A measure of effect size describing the variance in one variable that is associated with another variable.

• Formula:

  • ω² = (SSIV - dfIV (MSError)) / (SSTotal + MSError)

• Example: Approximately 41% of the variation in mood is associated with exercise group, (w)^2 = 0.41

One-Way Within-Subjects ANOVA

• Similar to Between-Subjects ANOVA, is a statistical test of whether the variance of scores between two or more groups is greater than the variance within groups
  • Like between, referred to as an F test
  • F = Variance Between groups/Variance Within groups.
• The "one" in one-way ANOVA indicates 1 IV is being tested
  • A two-way ANOVA = 2 IVS
  • A three-way ANOVA = 3 IVS, etc
• One-way ANOVAs have 2, 3, 4, or any number of groups.
• One-way within-subjects ANOVAS with only 2 groups are like dependent-samples t tests
  • When there are only 2 groups, the F test stat equals the t test stat squared, i.e., F = t^2
• Homogeneity of covariance assumption only applies to cases with more that two scores for each subject.

Calculating a Within-Subjects F

• Steps for calculating a Within-Subjects F
  • Step 1: State Hypothesis
    • There is NO significant difference among prices in the population means of wine taste ratings.
    • There IS a significant difference between at least two prices in the population means of wine taste ratings.
  • Step 2: State your N, n, k, IV, and DV
  • Step 3: Indicate your means for each condition and the grand mean
  • Step 4: Calculate SS(Total)
    • Formula: SSTotal = 2x2 - (Σx)2 / N
  • Step 5: Calculate SS(Subjects)
    • Formula: SSSubjects = kΣ(MSubjects - MGM)2
  • Step 6: Calculate SS(IV)
    • Formula: SSIV = ηΣ(Μιν – MGM)2
  • Step 7: Calculate SS(error)
    • Formula: SSError = SSTotal - SSSubjects - SSIV
  • Step 8: Create a summary table and indicate whether or not to reject null hypothesis
  • Result in APA format: "There was a significant difference in wine taste ratings among the price conditions, F(2,6) = 22.08, p<0.01
• Omnibus test: A statistical test that examines multiple groups at once
  • Reveals whether a significant difference exists, but not which groups are significantly different.

Post Hoc Tests

• Required to confirm if other pairs of means are significantly different from each other
• After use of Fisher's LSD to determine if the expensive and moderately priced wines were rated significantly better than the inexpensive wine.

Partial omega-squared (w^2p)

• calculating and interpreting

Two-Way Between-Subjects ANOVA

• A statistical test of whether the variance of scores between two or more groups is greater than the variance within groups
  • It's also known as an F test
• Factorial ANOVA:An ANOVA that examines at least two independent variables
  • A factor is another word for independent variables
• Examples
  • What is the effect of drug (drug A, drug B), and dose (low, medium, high) on symptoms? This is a 2x3 ANOVA
    • One IV (drug) has 2 levels (A,B), and the other IV (dose) has 3 levels (low, medium, high). Thus, this is a 2x3 ANOVA
• Assumptions that must be met for two-way between-subjects ANOVAs to be valid:
  • (1) The DV is measured with a ratio or interval scale
  • (2) Samples are drawn from normally distributed populations
  • (3) Samples are drawn from populations with equal (homogenous) variances

Calculating

• State each level of the IV with their respective hypothesis:
  • There is no significant difference between sunscreen use groups in the population means of skin cancer symptoms
• For each H₁, replace “no” with “a” in the null hypotheses above
• State your N, n, k, IV, and the DV your testing. (number of participants, groupings, variables)
• State your means for each condition and the grand mean.
• Calculate SS(Total): -SSTotal = Σ𝑥²−(Σ𝑥)²/𝑁
• Calculate SS(IV₁) 𝑆𝑆𝐼𝑉₁ =𝑛∗𝑘𝐼𝑉₂Σ(𝑀𝐼𝑉₁−𝐺𝑀)² -𝑆𝑆Sunscreen =𝑛∗𝑘𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒Σ(𝑀𝑆𝑢𝑛𝑠𝑐𝑟𝑒𝑒𝑛−𝐺𝑀)²
• Calculate SS(IV₂) 𝑆𝑆𝐼𝑉₂ =𝑛∗𝑘𝐼𝑉₁Σ(𝑀 𝐼𝑉₂−𝐺𝑀)² -𝑆𝑆𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 =𝑛∗𝑘𝑆𝑢𝑛𝑠𝑐𝑟𝑒𝑒𝑛Σ(𝑀𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒−𝐺𝑀)²
• Calculate SS(Cells)
  • 𝑆𝑆𝐶𝑒𝑙𝑙𝑠=𝑛Σ(𝑀𝐶𝑒𝑙𝑙−𝐺𝑀)²
• Calculate SS(IV₁xIV₂ AKA interaction) 𝑆𝑆𝐼𝑉₁𝑥𝐼𝑉₂ =𝑆𝑆𝐶𝑒𝑙𝑙𝑠 −𝑆𝑆𝐼𝑉₁−𝑆𝑆 𝐼𝑉₂
  • 𝑆𝑆𝑆𝑢𝑛𝑠𝑐𝑟𝑒𝑒𝑛𝑥𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 =𝑆𝑆𝐶𝑒𝑙𝑙𝑠 -𝑆𝑆𝑆𝑢𝑛𝑠𝑐𝑟𝑒𝑒𝑛−𝑆𝑆𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒
• Calculate SS(Error) 𝑆𝑆𝐸𝑟𝑟𝑜𝑟 =𝑆𝑆𝑇𝑜𝑡𝑎𝑙−𝑆𝑆𝐶𝑒𝑙𝑙𝑠
• Create a summary table and indicate whether or not to reject null hypothesis(s)
• (MS = Mean Square = SS/df)
• (F = variance among groups/variance within groups)
  • 𝐹=𝑀𝑆𝐺𝑟𝑜𝑢𝑝/𝑀𝑆𝐸𝑟𝑟𝑜𝑟) -Result in APA format

Main effect

• effect of one IV averaged across the levels of the other IV(s)
• number of main effects equals the number of IVs (but determining whether they're significant requires F tests)
• A two-way ANOVA has 2 IVS and therefore 2 main effects, a three-way ANOVA has 3 IVS and therefore 3 main effects, and so forth

Simple effect

• effect of one IV at one level of another IV
• If the F test for the interaction is significant, test simple effects. If the interaction is not significant don't test simple effects.
• Calculate Simple effects 𝑆𝑆𝐿𝑎𝑡𝑖 𝑡𝑢𝑑𝑒 𝑎𝑡 𝑁𝑜𝑆𝑢𝑛𝑠𝑐𝑟𝑒𝑒𝑛 =2[(6−4)²+(4−4)²+(2−4)²]=16
• Conducting Fisher's LSD tests