One-way Analysis of Variance 📂Statistical Test

One-way Analysis of Variance

Hypothesis Testing ¹

In the experimental design where there are $k$ treatments, suppose that we obtain $n_{j}$ samples from each treatment for a total of $n = n_{1} + \cdots + n_{k}$ samples. Assume that the samples from the $j = 1 , \cdots , k$-th treatment are independent and randomly follow a normal distribution with $N \left( \mu_{j} , \sigma_{j}^{2} \right)$, and that the population variance of each normal distribution is the same, denoted by $\sigma^{2} = \sigma_{1}^{2} = \cdots = \sigma_{k}^{2}$. In one-way ANOVA for comparing the population means among groups, the hypothesis testing is as follows.

$H_{0}$: $\mu_{1} = \cdots = \mu_{k}$
$H_{1}$: At least one $\mu_{j}$ is different from the other population means.

Test Statistic

Suppose we have an ANOVA table under a completely randomized design.

Source	df	SS	MS	F
Treatments	$k-1$	SST	MST	MST/MSE
Error	$n-k$	SSE	MSE
Total	$n-1$	TSS

The test statistic is as follows. $$ F = {\frac{ \text{MST} }{ \text{MSE} }} = {\frac{ \text{SST} / (k - 1) }{ \text{SSE} / (n - k) }} $$ This test statistic follows an F-distribution $F \left( k - 1 , n - k \right)$ with degrees of freedom $(k-1), (n-k)$ assuming the null hypothesis is true.

Explanation

Let the treatment means be $\bar{x}_{j} := \sum_{i} x_{ij} / n_{j}$ and the overall mean be $\bar{x} := \sum_{ij} x_{ij} / n$. $$ \begin{align*} \text{SST} =& \sum_{j=1}^{k} n_{j} \left( \bar{x}_{j} - \bar{x} \right)^{2} \\ \text{SSE} =& \left( n_{1} - 1 \right) s_{1}^{2} + \cdots + \left( n_{k} - 1 \right) s_{k}^{2} \\ \text{MST} =& {\frac{ \text{SST} }{ k - 1 }} \\ \text{MSE} =& {\frac{ \text{SSE} }{ n - k }} \\ F =& {\frac{ \text{MST} }{ \text{MSE} }} = {\frac{ \text{SST}/ (k - 1) }{ \text{SSE} / (n - k) }} \end{align*} $$ For the derivation of the test statistic, refer to the F-test in ANOVA.

Example

One-way ANOVA is used to determine if there are differences in population means due to treatments. Let’s analyze the heights of three girl groups debuted by Korea’s K-POP agency STARSHIP under a completely randomized design. The null hypothesis is that the mean heights of the groups are the same, while the alternative hypothesis is that at least one group’s mean height is different.

SISTAR: {Bora: 164cm, Hyorin: 163cm, Soyou: 168cm, Dasom: 167cm}
WJSN: {Seola: 165cm, Bona: 163cm, Exy: 166cm, Soobin: 156cm, Luda: 158cm, Dawon: 167cm, Eunseo: 170cm, Yeoreum: 161cm, Dayoung: 161cm, Yeonjung: 166cm}
IVE: {Yujin: 173cm, Gaeul: 164cm, Rei: 169cm, Wonyoung: 173cm, Liz: 171cm, Leeseo: 165cm}
The heights of the group KiiiKiii, debuted in 2025, are undisclosed.

SISTAR	WJSN	IVE
164	165	173
163	163	164
168	166	169
167	156	173
	158	171
	167	165
	170
	161
	161
	166

The overall average height is 165.5cm, and the group averages are SISTAR 165.5cm, WJSN 163.3cm, and IVE 169.2cm.

Although one can easily see that IVE has the highest average height, to determine if this difference is statistically significant, we need to complete the ANOVA table and conduct an F-test. The total number of members is the sample size $n = 4 + 10 + 6 = 20$, and the number of groups is $k = 3$.

Source	df	SS	MS	F
Treatments	$2$	SST	SST/$2$	MST/MSE
Error	$17$	SSE	SSE/$17$
Total	$19$	TSS

$$ \begin{align*} \text{SST} =& 4 \cdot (165.5 - 165.5)^{2} + 10 \cdot (163.3 - 165.5)^{2} + 6 \cdot (169.2 - 165.5)^{2} &= 129.1 \\ \text{SSE} =& 3 \cdot 17 + 9 \cdot 168.1 + 5 \cdot 76.8 &= 261.9 \\ F =& {\frac{ 129.1 / 2 }{ 261.9 / 17 }} = {\frac{ 64.5 }{ 15.4 }} &= 4.19 \end{align*} $$

If the significance level is $\alpha = 5 \%$, the lower bound of the critical region is $F_{2,17} (0.05) = 3.59$, which is $F = 4.19 > 3.59 = F_{2,17} (0.05)$, thus the null hypothesis can be rejected. In other words, at least one group’s average height is different from the others.

Verification

These results can also be reproduced using Excel.

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Experimental Design	Parametric Methods	Non-parametric Methods
Randomized Design	One-way ANOVA	Kruskal–Wallis $H$ Test
Randomized Block Design	Two-way ANOVA	Friedman $F_{r}$ Test