# One-way ANOVA Test

An One-way ANOVA Test is an ANOVA test that compares the means of three or more samples using the F distribution.

**AKA:**One-way Analysis of Variance, One-Factor ANOVA, Between Subjects ANOVA.**Context:**- It can be expressed as:
- Null Hypothesis:
*all k population means are equal*i.e. [math]\displaystyle{ H_0:\forall_{k\in S}\; :\mu_1 = \mu_2 = \mu_3 = \cdots = \mu_k \quad (k=1,\cdots,N) }[/math];

- Alternative hypothesys:
*at least one of the k population means is not equal to the others*i.e. [math]\displaystyle{ H_1:\forall_{k\in S}\;\exists_{i \in S}: \;\mu_i\ne \mu_k\; \quad (k=1,\cdots,m;\; i=1,\cdots,p;\; p\lt m\lt N) }[/math]

where S the set of N samples and [math]\displaystyle{ \mu_i }[/math] is the population mean i-th sample, [math]\displaystyle{ \mu_k }[/math] is the population mean k-th sample.

- Null Hypothesis:

- It can be expressed as:
**Example(s):**- STATA Support One-Way ANCOVA example(s): http://campusguides.lib.utah.edu/c.php?g=160853&p=1054192

**Counter-Example(s):****See:**ANOVA Algorithm, Mean Value, F-Distribution, Statistical Inference, MANOVA, ANCOVA, Hypothesis Testing.

## References

### 2019

- (KSU, 2019) ⇒ Kent State University Libraries. (2019, Mar 18). "SPSS Tutorials: One-Way ANOVA" . Retrieved:2019-04-12.
- QUOTE: The One-Way ANOVA ("analysis of variance") compares the means of two or more independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. One-Way ANOVA is a parametric test.
This test is also known as:

- QUOTE: The One-Way ANOVA ("analysis of variance") compares the means of two or more independent groups in order to determine whether there is statistical evidence that the associated population means are significantly different. One-Way ANOVA is a parametric test.

- The variables used in this test are known as:
- Dependent variable
- Independent variable (also known as the grouping variable, or factor)

- This variable divides cases into two or more mutually exclusive levels, or groups

- The variables used in this test are known as:

### 2016

- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/One-way_analysis_of_variance Retrieved 2016-07-03
- In statistics,
**one-way analysis of variance**(abbreviated one-way ANOVA**) is a technique used to compare means of three or more samples (using the F distribution). This technique can be used only for numerical data.**The ANOVA tests the null hypothesis that samples in two or more groups are drawn from populations with the same mean values. To do this, two estimates are made of the population variance. These estimates rely on various assumptions (see below). The ANOVA produces an F-statistic, the ratio of the variance calculated among the means to the variance within the samples. If the group means are drawn from populations with the same mean values, the variance between the group means should be lower than the variance of the samples, following the central limit theorem. A higher ratio therefore implies that the samples were drawn from populations with different mean values.

Typically, however, the one-way ANOVA is used to test for differences among at least three groups, since the two-group case can be covered by a t-test (Gosset, 1908). When there are only two means to compare, the t-test and the F-test are equivalent; the relation between ANOVA and

*t*is given by*F*=*t*^{2}. An extension of one-way ANOVA is two-way analysis of variance that examines the influence of two different categorical independent variables on one dependent variable.**::**Assumptions

- In statistics,

- The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:
- Response variable residuals are normally distributed (or approximately normally distribution).
- Variances of populations are equal.
- Responses for a given group are independent and identically distributed normal random variables (not a simple random sample (SRS)).

- The results of a one-way ANOVA can be considered reliable as long as the following assumptions are met:

- (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Brown%E2%80%93Forsythe_test Retrieved 2016-07-30
- In statistics, when a usual one-way ANOVA is performed, it is assumed that the group variances are statistically equal. If this assumption is not valid, then the resulting
*F*-test is invalid.

- In statistics, when a usual one-way ANOVA is performed, it is assumed that the group variances are statistically equal. If this assumption is not valid, then the resulting

### 2009

- (Heiberger, R. M., & Neuwirth, E.) ⇒ Richard M. Heiberger and Erich Neuwirth (2009). “One-way anova". In R through excel (pp. 165-191). Springer New York.[1]
- One-way ANOVA (analysis of variance) is a technique that generalizes the two-sample t-test to three or more samples. We test the hypotheses (specified here for k=6 samples) about population means [math]\displaystyle{ μ j }[/math]:

[math]\displaystyle{ H_0:\quad \mu_1=\mu_2=\mu_3=\mu_4=\mu_5=\mu_6 }[/math]

- [math]\displaystyle{ H_1:\quad \textrm{Not all} \; μ_j\;\textrm{are equal}\; (j=1:6) }[/math]
- The test is based on the observed sample means [math]\displaystyle{ \overline{x}_j }[/math].

### 1996

- (Christensen, 1996) ⇒ Ronald Christensen (1996). “One-way ANOVA". In Plane Answers to Complex Questions (pp. 79-93). Springer New York. [2]
- A one-way ANOVA model can be written

- [math]\displaystyle{ y_{ij}=\mu+a_i+e_{ij} \quad i=1,...,t, \quad j=1,...,Ni, }[/math]
- where [math]\displaystyle{ E(e_{ij}) = 0 }[/math], [math]\displaystyle{ Var(e_{ij} ) = \sigma^2 }[/math], and [math]\displaystyle{ Cov(e_{ij} , e_{i'j'}) = 0 }[/math] when [math]\displaystyle{ (i, j) \neq(i', j') }[/math]. For finding tests and confidence intervals, the e
*ij*'s are assumed to have a multivariate normal distribution.

- where [math]\displaystyle{ E(e_{ij}) = 0 }[/math], [math]\displaystyle{ Var(e_{ij} ) = \sigma^2 }[/math], and [math]\displaystyle{ Cov(e_{ij} , e_{i'j'}) = 0 }[/math] when [math]\displaystyle{ (i, j) \neq(i', j') }[/math]. For finding tests and confidence intervals, the e