FMPH 102: Biostatistics in Public Health Study Notes

FMPH 102: Biostatistics in Public Health - One-Way ANOVA for Three or More Independent Means

Continuous Variables Inference for Means

• Inference methods for various types of samples:

  • One Sample

  • Two Paired Samples

  • Two Independent Samples

  • Three or More Independent Samples

Overview of Three or More Independent Samples

• When the goal is to compare the means of a continuous variable across 3 or more independent samples.

Example Study: Pets and Owner’s Stress

• Study Details:

  • Reference: Allen et al. (1991)

  • Investigative focus on the effect of pet dogs on owner's stress levels.

  • Data Source: PetStress.sav

  • Participants: 45 women dog owners subjected to stress through mental arithmetic.

  • Response Variable: Heart rate (in bpm).

  • Conditions for Participants:

  • Pet dog present (n=15)

  • Human friend present (n=15)

  • Neither present (n=15)

  • Research Question: Does the presence of a pet dog help in coping with stress?

Heart Rate Results from the Study

• Mean Heart Rates Indicated for Groups:

  • Pets: 73.5 bpm

  • Friends: 91.3 bpm

  • Neither: 82.5 bpm

• Conclusion: Pets seem helpful, while friends seem to induce more stress.

Two-Step Analysis for k=3 Groups

1. Step 1: Test overall hypothesis for equal means.

   • Null Hypothesis (H0): μ1 = μ2 = μ3

   • Alternative Hypothesis (Ha): At least one mean is different (i.e., μ1≠μ2, μ1≠μ3, or μ2≠μ3).

   • If H0 is not rejected, conclude no evidence of group differences.

   • If H0 is rejected, proceed to Step 2.

2. Step 2: Implement post-hoc analysis for pairwise comparisons.

   • Null Hypotheses for Pairwise Comparisons:

     • H01: μ1 = μ2 (Pet vs. Friend)

     • H02: μ1 = μ3 (Pet vs. Neither)

     • H03: μ2 = μ3 (Friend vs. Neither)

   • Apply Bonferroni Correction to adjust significance levels for multiple comparisons.

   • Adjusted significance level: α = 0.05 divided by the number of comparisons.

One-Way ANOVA

• The two-step procedure outlined above exemplifies the One-way ANOVA technique.

• It is a statistical method used to test if there are significant differences between the means of three or more independent groups.

One-Way ANOVA Model Assumptions

1. Assumption #1: For k=3 independent samples, data should be normally distributed for each group.

   • Notation:

     • X1 ~ N(μ1, σ1)

     • X2 ~ N(μ2, σ2)

     • X3 ~ N(μ3, σ3)

   • Alternatively, if sample sizes are large: n1, n2, n3 ≥ 30.

2. Assumption #2: The groups should have equal variances.

   • Notation: σ1 = σ2 = σ3 = σ

   • Practical check: Ratios of standard deviations should satisfy smax/smin < 2 for all group pairs.

Analysis of Variance

• ANOVA evaluates whether the means of different groups differ significantly.

  • Variability within the dataset is categorized into:

  • Variance Between Groups (Mean Square Between): Represents variance due to the interaction between groups.

  • Variance Within Groups (Mean Square Within): Represents variance within each individual group.

• The F-test statistic is computed as the ratio of these two variances.

Detailed Computation of Mean Square Components

• Mean Square Between (MSB): Measures variation among sample means.

• Formula for MSB:
  $MSB = rac{SSB}{dfB}$

• Where:

  • SSB is the Sum of Squares Between Groups,

  • dfB (degrees of freedom Between groups) = k - 1.

• Mean Square Within (MSW): Measures how much variability exists from one observation to another within the same group.

• Formula for MSW:
  $MSW = rac{SSW}{dfW}$

• Where:

  • SSW is the Sum of Squares Within Groups,

  • dfW (degrees of freedom Within groups) = n - k.

F-Test Statistic Calculation

• F-test statistic is calculated as:
  $F_{stat} = rac{MSB}{MSW}$

Example Computation: Pets & Stress Study

• SSB Calculation:

• $SSB = 15 imes (73.48 - 82.44)^{2} + 15 imes (91.33 - 82.44)^{2} + 15 imes (82.53 - 82.44)^{2} = 2390$

• Degrees of Freedom Between (dfB): $dfB = k - 1 = 2$

• MSB Calculation: $MSB = rac{SSB}{dfB} = rac{2390}{2} = 1195$

• SSW Calculation:

• $SSW = 14 imes 9.9702^{2} + 14 imes 8.3402^{2} + 14 imes 9.2412^{2} = 3561$

• Degrees of Freedom Within (dfW): $dfW = n - k = 42$

• MSW Calculation: $MSW = rac{SSW}{dfW} = rac{3561}{42} = 84.8$

• Final F-statistic Calculation:
  $F_{stat} = rac{MSB}{MSW} = rac{1195}{84.8} = 14.1$

How to Interpret the F-Test Results

• To read the F-table and determine significance:

  • df1 = number of groups - 1,

  • df2 = total number of samples - number of groups.

  • For the study, we have: F(2, 42) = critical value against which F-stat will be compared.

Post-Hoc Comparisons Insight

• After establishing significant results from the F-test, utilize post-hoc tests to ascertain which groups differ. This includes:

  • Employing Two Independent Samples t-Tests.

  • Applying Bonferroni correction for elevated stringencies.

Alternate Methods for Multiple Comparisons Correction

• Other than Bonferroni, various methods exist such as the Benjamini-Hochberg False Discovery Rate (FDR) correction, which balances robustness with power in multiple testing scenarios.

Summary of One-Way ANOVA

1. Compare means across k ≥ 3 independent groups.

2. Test H0: μ1 = μ2 = … = μk.

   • F-statistic: $F_{stat} = rac{MSB}{MSW}$

3. If H0 is not rejected, no significant differences exist.

4. Post-hoc analysis conducts pairwise comparisons with necessary corrections to p-values.

5. Ensure assumptions regarding normality and homogeneity of variance are met before performing the ANOVA.

Final Note on ANOVA

• Recognize the practical implications of the One-Way ANOVA, as conducting this test necessitates larger sample sizes and greater caution in interpretation due to correction for multiple comparisons.