Statistics for the Behavioral Sciences Lecture Notes

Analysis of Variance One-Way Within-Subjects (Repeated-Measures) Design

Chapter Outline

  • Observing the Same Participants Across Groups
  • Selecting Related Samples: The Within-Subjects Design
  • Sources of Variation and the Test Statistic
  • Degrees of Freedom
  • The One-Way Within-Subjects ANOVA
  • Post Hoc Comparisons: Bonferroni Procedure
  • Measuring Effect Size
  • The Within-Subjects Design: Consistency and Power
  • APA in Focus: Reporting the F Statistic, Significance, and Effect Size

Observing the Same Participants Across Groups

  • One-Way Within-Subjects ANOVA
    • A statistical procedure used to test hypotheses for one factor with two or more levels regarding the variance among group means.
    • Utilized when the same participants are observed at each level of a factor and the variance in any one population is undefined.
    • The term “one-way” indicates that one factor is being tested.
    • The term “within-subjects” implies that the same participants are observed across each group.

Selecting Related Samples: The Within-Subjects Design

  • Figure 13.1: The Within-Subjects Design
    • In this design, a researcher assigns the same participants to each level of one factor or group.
    • Every participant in the sample is observed at each level of a factor, ensuring related samples.

Sources of Variation and the Test Statistic

  • The test statistic for any one-way ANOVA follows a general form applicable to within-subjects designs.
  • There are three sources of variation in a one-way within-subjects design:
    • Between Groups
    • Within Groups
    • Between Persons

Degrees of Freedom

  • The total degrees of freedom (df) for a one-way within-subjects ANOVA are calculated as: extTotaldf=(kimesn)−1ext{Total df} = (k imes n) - 1
    • Where:
    • k = number of groups
    • n = number of participants per group
  • The total df is divided into three parts corresponding to each source of variation:
    • Degrees of Freedom Between Groups (dfBGdf_{BG})
    • dfBG=k−1df_{BG} = k - 1
    • Degrees of Freedom Between Persons (dfBPdf_{BP})
    • dfBP=n−1df_{BP} = n - 1
    • Degrees of Freedom Error (dfEdf_E)
    • dfE=dfBGimesdfBP=(k−1)(n−1)df_E = df_{BG} imes df_{BP} = (k - 1)(n - 1)

The One-Way Within-Subjects ANOVA

  • The hypotheses are consistent with those for a between-subjects ANOVA:
    • Null hypothesis (H0): Mean ratings for each advertisement do not vary in the population.
    • Alternative hypothesis (H1): Mean ratings for each advertisement do vary in the population.
  • Four assumptions must be made to compute the one-way within-subjects ANOVA:
    1. Normality
    • Data in the population from which samples are taken are normally distributed.
    1. Independence Within Groups
    • Participants are independently observed within groups.
    1. Homogeneity of Variance
    • The variance within each population is equal.
    1. Homogeneity of Covariance
    • Participant scores across groups are related, as the same participants are used in multiple conditions.

Example 13.1: The One-Way Within-Subjects ANOVA

  • Scenario: Assessing the impact of three ads on teenage smoking cessation. Ratings on a scale from 1 (not at all effective) to 7 (very effective) are gathered from participants viewing each ad:
    • No cues condition: Ad that only uses words.
    • Generic cues condition: Ad featuring an abstract picture.
    • Smoking-related cues condition: Ad showing a teenager smoking and coughing.
  • Conduct a one-way within-subjects ANOVA at a significance level of .05.
Step 1: State the Hypotheses
  • H0: Mean ratings for each advertisement do not vary in the population.
  • H1: Mean ratings for each advertisement do vary in the population.
Step 2: Set the Criteria for a Decision
  • Significance level: extα=0.05ext{α} = 0.05
  • Calculate the degrees of freedom:
    • dfBG=k−1=3−1=2df_{BG} = k - 1 = 3 - 1 = 2
    • dfBP=n−1=7−1=6df_{BP} = n - 1 = 7 - 1 = 6
    • dfE=2imes6=12df_E = 2 imes 6 = 12
  • Critical value is 3.89.
Step 3: Compute the Test Statistic
  • The test statistic follows specific stages to compute the mean square.
    1. Preliminary Calculations
    2. Intermediate Calculations
    3. Compute the Sum of Squares
    4. Complete the F Table
Example 13.1: Preliminary Calculations
  • Table 13.3: Preliminary Calculations for the One-Way Within-Subjects ANOVA in Example 13.1:
    • Blank table with participant ratings for ads leading to calculated sums, such as:
    • extΣxextNoCues=21ext{Σx}_{ ext{No Cues}} = 21
    • extΣxextGenericCues=28ext{Σx}_{ ext{Generic Cues}} = 28
    • extΣxextSmoking−RelatedCues=42ext{Σx}_{ ext{Smoking-Related Cues}} = 42
    • extΣxextTotal=91ext{Σx}_{ ext{Total}} = 91
Example 13.1: Intermediate Calculations
  • Calculating sums and means using derived values from preliminary calculations such as:
    1. For k × n:
    • rac{( ext{Σ}x_{+})^2}{k imes n} = rac{(91)^2}{3 imes 7} = 394.33
    1. Further calculations yield intermediate values leading to mean squares.
Example 13.1: Compute the Sum of Squares (SS)
  • Between Groups (SSBG):
    • SSBG=[2]−[1]=427−394.33=32.67SSBG = [2] - [1] = 427 - 394.33 = 32.67
  • Total (SST):
    • SST=[3]−[1]=451−394.33=56.67SST = [3] - [1] = 451 - 394.33 = 56.67
  • Between Persons (SSBP):
    • SSBP=[4]−[1]=407−394.33=12.67SSBP = [4] - [1] = 407 - 394.33 = 12.67
  • Within Groups (Error, SSE):
    • SSE=SST−SSBG−SSBP=56.67−32.67−12.67=11.33SSE = SST - SSBG - SSBP = 56.67 - 32.67 - 12.67 = 11.33
Example 13.1: Complete the F Table
  • Table 13.4: Tracking variations, mean squares, and F obtained values leading to a completed table.
    • Example values for each source of variation recorded with significance indicated by an asterisk (*) in the table.
Example 13.1: Step 4: Make a Decision
  • Compare obtained value to critical value:
    • Obtained value (17.30) exceeds critical value (3.89).
    • Decision: Reject the null hypothesis as it falls in the rejection region.

Post Hoc Comparisons: Bonferroni Procedure

  • Purpose: Post hoc tests are conducted after a significant one-way ANOVA to determine which specific group means differ.
  • The Bonferroni procedure adjusts the alpha level (Type I error rate) for multiple comparisons:
    • Testwise alpha: The adjusted alpha for each comparison made on the same dataset. - Formula: Testwise alpha = racextαmrac{ ext{α}}{m}, where mm is the number of pairwise comparisons.

Using the Bonferroni Procedure

  • Steps to compute the Bonferroni Post Hoc comparisons:
    1. Calculate the testwise alpha and find critical values.
    2. Compute related-samples t-tests for each pairwise comparison.
Example 13.1: Pairwise Comparisons
  • Comparison results detailed for pairs such as:
    • Generic Cue and No Cue: 4.0−3.0=1.04.0 - 3.0 = 1.0
    • Smoking Cue and Generic Cue: 6.0−4.0=2.06.0 - 4.0 = 2.0
    • Smoking Cue and No Cue: 6.0−3.0=3.06.0 - 3.0 = 3.0
  • Testwise alpha computed as:
    • ext{α} = rac{0.05}{3} = 0.0167
    • Critical value from Table B.2 based on df = 6 is ±3.143.
Example 13.1: Related-Samples t-Test Results
  • Calculating t-values for comparisons such as:
    • Generic Cue and No Cue: tobtt_{obt} calculation leading to retain the null hypothesis.
    • Smoking-Related Cue and Generic Cue: tobtt_{obt} calculated as -3.378 leading to rejecting the null hypothesis.
    • Smoking-Related Cue and No Cue: tobtt_{obt} calculated as -9.709, also rejecting the null hypothesis.

Measuring Effect Size

  • An ANOVA assesses if group means vary significantly, but effect size measures how much of this variability can be accounted for by different levels of a factor.
  • Two Key Measures of Effect Size:
    • Partial Eta-Squared (extηp2ext{η}^2_p)
    • Partial Omega-Squared (extω2ext{ω}^2)
Computing Effect Sizes
  • For Partial Eta-Squared:
    • Remove the sum of squares between persons in the total sum of squares for denominator calculations.
  • For Partial Omega-Squared:
    • Similar approach as above with adjustments made post hoc.

The Within-Subjects Design: Consistency and Power

  • The within-subjects design offers increased power to detect effects compared to between-subject designs.
  • Power is enhanced due to reduced error terms in the test statistic denominator when factors lead to consistent responding between groups.
  • Rules governing power in relation to sample size and variations:
    1. Increasing levels of a factor (kk) increases power.
    2. Decreasing error variance (σ2σ^2) increases power.
    3. Increasing the number of repeated measures(nn) increases power.

APA in Focus: Reporting Results

  • Reporting results from a one-way within-subjects ANOVA should follow specific guidelines:
    • Summarize test statistics, df, and p-value.
    • Report effect sizes for significant findings.
    • Present means and standard deviations in a figure or summarized table.
Example of APA Reporting
  • For findings in Example 13.1:
    • A one-way analysis of variance showed significant variations in ratings of effectiveness among advertisements, F(2,12) = 17.38, p < .05 ext{ (} η^2 = 0.69).
    • Using the Bonferroni post hoc test, significant differences were found for ads with smoking-related cues against other ads (p < .05).
    • Means and standard deviations should also be referenced in conclusion tables rendered in APA style.