Design of Experiments & One-Way ANOVA – Comprehensive Study Notes

Introduction to Analysis of Variance (ANOVA)

• Purpose
  • Evaluate mean differences among two or more populations/treatments.
  • Draw population-level conclusions from sample data.
• Advantages
  • One omnibus test controls experiment-wise \alpha when comparing multiple means.
• Core idea
  • Partition total variability into between-treatments and within-treatments components and compare the two via an F-ratio.

Observational vs. Experimental Studies

• Observational study
  • Researcher merely records data; no manipulation.
  • Example: Counting number of pets owned by passers-by in a New-York street to decide if more pet-food stores are needed.
• Experimental study
  • Researcher actively manipulates explanatory factor(s) and observes response.
  • Goal: Demonstrate cause-and-effect.
  • Vitamin C & cold-prevention example (1976)
  • 868 children randomly split:
    • Experimental group: 1 000-mg Vitamin C daily.
    • Control group: visually identical placebo.
  • Mean colds per child: 0.38 vs.0.37 → no significant difference.

Terminology & Principles of Experimental Design

• Experimental Unit – object on which measurement is taken (here, each child).
• Factor – controlled explanatory variable (Vitamin C intake).
• Level – particular setting/intensity of a factor (vitamin vs. placebo).
• Treatment – specific combination of factor levels (two in the example).
• Response – outcome variable measured (number of colds Y).

One-Way ANOVA (Completely Randomized Design)

Hypotheses

• H0:\; \mu1=\mu2=\cdots=\muk
• H_A:\; At least two population means differ.

Logic of the F-Ratio

• F=\dfrac{\text{Variance Between Treatments}}{\text{Variance Within Treatments}}
• Interpretations
  • H_0 true ⇒ systematic effect \approx 0 ⇒ F\approx1 (not 0!).
  • H_A true ⇒ numerator inflated ⇒ F\gg1.
• Avoids inflation of type-I error across multiple comparisons.

Partitioning the Sum of Squares

• Total variation: SS_{\text{Total}} – dispersion of all individual scores around grand mean.
• Between-treatments: SS_T – dispersion of treatment means around grand mean.
• Within-treatments (error): SS_E – dispersion of individual scores around their own treatment mean.
• Relationship: SS{\text{Total}} = SST + SS_E.

Notation for Data Layout

• k – number of treatments.
• n_i – sample size in treatment i; equal n for a balanced design.
• Ti – treatment total; \bar yi – treatment mean; s_i^2 – treatment variance.
• G=\sum T_i = \sum Y – grand total.
• N=\sum n_i – total observations.

ANOVA Table Structure

Formulae for Sum of Squares (computational)

• SS_{\text{Total}} = \sum Y^2 - \dfrac{G^2}{N}
• SST = \sum{i=1}^k \dfrac{Ti^2}{ni} - \dfrac{G^2}{N}
• SSE = SS{\text{Total}} - SS_T

Decision Rules

• Critical-value: Reject H0 if F{\text{obs}} > F_{\alpha;(k-1,\,N-k)} (right-tailed).
• p-value: Reject if p < \alpha.

Worked Example – Children’s Attention Span

• 12 children, 3 meal plans (4 per plan): No Breakfast (NB), Light Breakfast (LB), Full Breakfast (FB).
• Raw data (in minutes):
  • NB: 8, 7, 9, 13 ( T_1=37 )
  • LB: 14, 16, 12, 17 ( T_2=59 )
  • FB: 10, 12, 16, 15 ( T_3=53 )
• Calculations
  • G=149, G^2/N=1850.0833
  • SS_{\text{Total}}=122.9167
  • SS_T=64.6667
  • SS_E=58.25
  • MST=32.3333, MSE=6.4722
  • F_{\text{obs}}=5.00
• Critical value F{0.05;(2,9)}=4.26 ⇒ 5.00>4.26 ⇒ Reject H0.
  Conclusion: Mean attention span differs across at least two meal plans.

Assumptions & Conditions for One-Way ANOVA

• Populations are normally distributed (check boxplots/histograms/QQ-plot).
• Homogeneity of variances across groups.
• Independence of observations.
• Data measured on interval/ratio scale.

Post-Hoc (Multiple-Comparison) Tests

• Applied only when ANOVA rejects H_0.
• Identify which specific means differ.

Bonferroni Confidence Intervals

1. Number of pairwise comparisons: J=\binom{k}{2}.
2. Standard error for pair i,j: s{\bar yi-\bar yj}=\sqrt{MSE\left(\dfrac{1}{ni}+\dfrac{1}{nj}\right)}.
3. Bonferroni CI:
   \bar yi-\bar yj \;\pm\; t{\alpha^}\, s{\bar yi-\bar yj}, \qquad \alpha^=\dfrac{\alpha}{2J},\; df=N-k.
4. If CI contains 0 ⇒ difference not significant.

Example continuation (attention span, MSE=6.4722,\;ni=n_j=4):

• Standard error =\sqrt{6.4722\times(1/4+1/4)}=1.80.
• t_{\alpha^},\; \alpha^=0.05/(2\times3)=0.00833 gives t\approx2.93.
• CIs
  • NB–LB: (9.25-14.75)\pm2.93\times1.80 = (-10.77, -0.22) ⇒ significant.
  • NB–FB: (9.25-13.25)\pm2.93\times1.80 = (-9.27, 1.27) ⇒ not significant.
  • LB–FB: (14.75-13.25)\pm2.93\times1.80 = (-3.77, 6.77) ⇒ not significant.

Learning-Check Insights

• ANOVA lets researchers compare several treatments with one test (True).
• Under H_0 the expected F is 1, not 0 (False statement on 0).
• Large F arises from large mean differences & small within-group variance.
• Post-hoc tests are not needed when H_0 is not rejected.
• Given report F(2,27)=5.36 ⇒ total participants N=30 (since df_{\text{total}}=29).

Kruskal-Wallis Test (Non-Parametric Alternative)

• Objective: Test whether k independent samples originate from populations with the same center (median/location).
• Used when ANOVA assumptions (normality/equal variance) are doubtful.

Hypotheses (Two-Tailed)

• H_0: All k populations are identical.
• H_A: At least two populations differ in location.

Test Procedure

1. Rank all N observations from smallest to largest; assign average rank to ties.
2. Compute rank sums T_i for each group.
3. Test statistic
   H = \dfrac{12}{N(N+1)}\sum{i=1}^{k}\dfrac{Ti^2}{n_i} - 3(N+1).
4. Decision rules
   • Critical-value: Reject H0 if H>\chi^2{\alpha;\,k-1} (right-tailed).
   • p-value: Reject if p<\alpha.

Example – Gasoline Mileage

• 3 gasoline types (A,B,C); small samples, normality questionable.
• Computed H=9.555, k-1=2.
• Critical value \chi^2_{0.05;2}=5.991.
• Since 9.555>5.991 (and p<0.01<0.05) ⇒ Reject H_0.
  Conclusion: Mileage differs among gasoline types.

Ethical & Practical Considerations

• Random assignment ensures internal validity.
• Proper control of family-wise \alpha (ANOVA + post-hoc) prevents overstating significance.
• Non-parametric alternatives safeguard inference when data violate assumptions.

Key Numerical & Statistical References

• H0:\;\mu1=\mu2=\cdots=\muk, H_A: not all equal.
• SS{\text{Total}} = SST + SS_E.
• MST=SST/(k-1), MSE=SSE/(N-k).
• F = MST / MSE.
• Bonferroni adjustment: \alpha^*=\alpha/(2J) with J=\binom{k}{2}.
• Kruskal-Wallis statistic: H = \dfrac{12}{N(N+1)}\sum \dfrac{Ti^2}{ni} - 3(N+1).