Chapter 11-B

ANOVA: Analysis of Variance

Introduction to ANOVA

Analysis of Variance (ANOVA) is a statistical method used to test whether there are significant differences between the means of three or more independent groups. It is commonly applied in experimental and observational studies to compare group means and determine if at least one group mean is different from the others.

• Purpose: To test the null hypothesis that all group means are equal.

• Applications: Comparing means across different treatments, locations, or time periods.

Key Terms and Definitions

• Factor: The categorical independent variable (e.g., plant, engine type).

• Level: The different groups or categories within a factor.

• Response Variable: The quantitative outcome measured (e.g., AQI, dBA, number of accidents).

• Between-group Variation: Variation due to differences between group means.

• Within-group Variation: Variation within each group (random error).

Constructing the ANOVA Table

ANOVA Table Structure and Formulas

The ANOVA table summarizes the sources of variation, their degrees of freedom, sum of squares, mean squares, F-statistic, and p-value.

Source	df	SS	MS	F	p-Value
Between	$k-1$	SSB	MSB	$F = \frac{MSB}{MSW}$	Computed
Within	$N-k$	SSW	MSW
Total	$N-1$	SST

• SSB (Sum of Squares Between): $SSB = SST - SSW$

• SSW (Sum of Squares Within): $SSW = \sum_{i=1}^k \sum_{j=1}^{n_i} (x_{ij} - \bar{x}_i)^2$

• SST (Total Sum of Squares): $SST = \sum_{i=1}^k \sum_{j=1}^{n_i} (x_{ij} - \bar{x})^2$

• MSB (Mean Square Between): $MSB = \frac{SSB}{k-1}$

• MSW (Mean Square Within): $MSW = \frac{SSW}{N-k}$

• F-statistic: $F = \frac{MSB}{MSW}$

• p-value: Probability of observing an F-statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

Computing the F-statistic

• Calculate the mean square between groups (MSB) and within groups (MSW).

• Compute the F-statistic: $F = \frac{MSB}{MSW}$

• Compare the F-statistic to the critical value from the F-distribution or use the p-value for decision-making.

p-value Calculation: $\text{p-value} = Fcdf(F, 1000, df_B, df_W)$

Steps in ANOVA Hypothesis Testing

Step 1: Hypothesize – Formulation

• Null Hypothesis ($H_0$): All group means are equal. $H_0: \mu_1 = \mu_2 = \cdots = \mu_k$

• Alternative Hypothesis ($H_A$): At least one group mean is different. $H_A:$ Not all $\mu_i$ are equal

Step 2: Prepare – Check Conditions

• Set the significance level $\alpha$ (e.g., 0.05, 0.02, 0.15).

• Random samples are required.

• Independence between and within groups (random assignment).

• Normality of the response variable within each group.

Step 3: Compute (Statistics) to Compare

• Calculate the ANOVA table values (SS, df, MS, F, p-value).

• Use the formulas above to compute each component.

Step 4: Interpret and Conclude

• Compare the p-value with $\alpha$:

  • If p-value $< \alpha$, reject the null hypothesis ($H_0$).

  • If p-value $\geq \alpha$, do not reject the null hypothesis ($H_0$).

• State the conclusion in the context of the problem.

Worked Examples

Example 1: Air Quality Index (AQI) in Plants

A researcher collects AQI readings from three plants (A, B, C) to test if the mean AQI varies by plant.

AQI Readings	Plant A	Plant B	Plant C
	75	43	6
	39	80	11
	49	33	19
	28	52	22
	27	47	12

• Step 1: $H_0: \mu_A = \mu_B = \mu_C$; $H_A:$ Not all means are equal.

• Step 2: $\alpha = 0.02$; independence and normality assumed.

• Step 3: ANOVA table constructed as follows:

Source	df	SS	MS	F	p-Value
Between	2	2984.3323	1492.1661	4.871	0.028
Within	12	3565.1533	297.0961
Total	14	6549.4866

• Step 4: Since p-value (0.028) > $\alpha$ (0.02), do not reject $H_0$. Not enough evidence that the means are different at 2% significance.

• At $\alpha = 0.05$, since 0.028 < 0.05, reject $H_0$ at 5% significance.

Example 2: Sound Levels in Engines

Testing if mean decibel readings differ among three engine types (P, Q, R).

Decibel Readings (dBA)	Engine P	Engine Q	Engine R
	79	90	35
	65	80	20
	30	40	40
	55	60	25

• Step 1: $H_0: \mu_P = \mu_Q = \mu_R$; $H_A:$ Not all means are equal.

• Step 2: $\alpha = 0.02$; independence and normality assumed.

• Step 3: ANOVA table constructed as follows:

Source	df	SS	MS	F	p-Value
Between	2	2767.545	1383.773	2.466	0.129
Within	9	374.111	41.568
Total	11	3141.656

• Step 4: Since p-value (0.129) > $\alpha$ (0.02), do not reject $H_0$. Not enough evidence that the means are different at 2% significance.

• At $\alpha = 0.15$, since 0.129 < 0.15, reject $H_0$ at 15% significance.

Example 3: Annual Accidents in Plants

Testing if the mean number of annual accidents differs among four plants (A, B, C, D).

# of Annual Accidents	Plant A	Plant B	Plant C	Plant D
	2	2	5	6
	3	1	3	9
	5	2	6	8
	4	1	7	8

• Step 1: $H_0: \mu_A = \mu_B = \mu_C = \mu_D$; $H_A:$ Not all means are equal.

• Step 2: $\alpha = 0.05$; independence and normality assumed.

• Step 3: ANOVA table constructed as follows:

Source	df	SS	MS	F	p-Value
Between	3	54.1500	18.0500	3.8201	0.0307
Within	16	75.6000	4.7250
Total	19	129.7500

• Step 4: Since p-value (0.0307) < $\alpha$ (0.05), reject $H_0$. There is evidence that not all means are equal at 5% significance.

• At $\alpha = 0.02$, since 0.0307 > 0.02, do not reject $H_0$.

Summary Table: Decision Rules in ANOVA

p-value	Significance Level ($\alpha$)	Decision
p-value < $\alpha$	e.g., 0.05	Reject $H_0$
p-value > $\alpha$	e.g., 0.05	Do not reject $H_0$

Conclusion

ANOVA is a powerful tool for comparing means across multiple groups. By following the structured steps—formulating hypotheses, checking assumptions, computing the ANOVA table, and interpreting results—researchers can make informed decisions about group differences. Always ensure assumptions are met for valid results.

Additional info: The examples provided illustrate the application of ANOVA in real-world scenarios, including environmental and industrial studies. The formulas and tables are standard for one-way ANOVA.