Two-Way ANOVA — Fill in the Blanks: 18-Env-B7 Environmental Sampling and Analysis, May 2017, Q3

Question 3: Two-Way ANOVA — Fill in the Blanks (10 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A 3×4 factorial design (Location × Season) with 2 replications per cell, so $N=3\times4\times2=24$ total observations. Printed values: $MS_A=80.17$, $SS_B=12.46$, $MS_{err}=3.79$, $SS_{total}=262.96$.

Find. Every blank cell of the ANOVA table, plus the significance conclusion for Location, Season and their interaction at α=0.05.

Approach. Fill the degrees of freedom from the design structure first ($df_A=a-1$, $df_B=b-1$, $df_{AB}=(a-1)(b-1)$, $df_{err}=ab(n-1)$, $df_{tot}=N-1$), back out each SS from whichever of $SS=MS\times df$ or $SS_{AB}=SS_{tot}-SS_A-SS_B-SS_{err}$ applies, then compute each $F=MS/MS_{err}$ and compare to the supplied F-table.

Part (a) — Step 1: degrees of freedom. $df_A=3-1=2$, $df_B=4-1=3$, $df_{AB}=2\times3=6$, $df_{err}=3\times4\times(2-1)=12$, $df_{tot}=24-1=23$ (check: $2+3+6+12=23$ ✓).
Step 2 — Location (A). $SS_A=MS_A\times df_A=80.17\times2=160.34$. $F_A=MS_A/MS_{err}=80.17/3.79=21.15$.
Step 3 — Season (B). $MS_B=SS_B/df_B=12.46/3=4.153$. $F_B=MS_B/MS_{err}=4.153/3.79=1.096$.
Step 4 — Error. $SS_{err}=MS_{err}\times df_{err}=3.79\times12=45.48$.
Step 5 — Interaction (AB), by subtraction. $$SS_{AB}=SS_{tot}-SS_A-SS_B-SS_{err}=262.96-160.34-12.46-45.48=44.68$$ $$MS_{AB}=SS_{AB}/df_{AB}=44.68/6=7.447, \qquad F_{AB}=MS_{AB}/MS_{err}=7.447/3.79=1.965$$
Step 6 — compare to F-critical (α=0.05, from the supplied table). $$\boxed{F_A=21.15 > F_{0.05}(2,12)=3.885 \Rightarrow \text{Location is significant}}$$ $$F_B=1.10 < F_{0.05}(3,12)=3.490 \Rightarrow \text{Season is NOT significant}, \qquad F_{AB}=1.97 < F_{0.05}(6,12)=2.996 \Rightarrow \text{Interaction is NOT significant}$$

Final Results — completed 2-way ANOVA table
Source	SS	DF	MS	F	Significant at 5%?
Location (A)	160.34	2	80.17	21.15	Yes
Season (B)	12.46	3	4.153	1.10	No
Interaction (AB)	44.68	6	7.447	1.97	No
Error	45.48	12	3.79	—	—
Total	262.96	23	—	—	—

Conclusion. Only Location has a statistically significant effect on the measured response at the 5% level; neither Season nor the Location×Season interaction is significant. This means the response varies meaningfully from one sampling location to another, but not systematically across the four seasons sampled, and the location effect does not itself depend on which season the samples were taken in (no significant interaction).

Topic: Two-way (factorial) ANOVA table reconstruction.

Key relations: $df_A=a-1$, $df_B=b-1$, $df_{AB}=df_A\cdot df_B$, $df_{err}=ab(n-1)$; $SS=MS\times df$; $SS_{tot}=SS_A+SS_B+SS_{AB}+SS_{err}$; $F=MS_{effect}/MS_{err}$.

Why this works: A balanced factorial design partitions total variability additively into main effects, their interaction and residual (error) variability; because the partition is additive, any one missing sum of squares can always be recovered from the total and the rest, which is exactly the "fill in the blanks" structure of this question.

Common pitfall: Forgetting that $df_{AB}=df_A\times df_B$ (not $df_A+df_B$), and comparing $F$ ratios against the wrong pair of degrees of freedom in the F-table — the numerator df is the effect's own df, the denominator df is always $df_{err}$.

Source: Walpole, Myers, Myers & Ye, Probability & Statistics for Engineers and Scientists, ch. 13 (design of experiments, factorial ANOVA).

Question 3 of 7: Two-Way ANOVA — Fill in the Blanks

Question 3: Two-Way ANOVA — Fill in the Blanks (10 marks)