10.2
One-Way ANOVA
A one-way ANOVA test compares the means of three or more samples defined by one factor.
Consider the average fuel consumption of cars from three companies. Here, the samples are defined by one factor—the company.
For cars from different companies driven in summer and winter, a one-way ANOVA cannot simultaneously test for two factors—company and season.
In general, begin by stating the null hypothesis that the sample means are equal, and the alternative hypothesis that the sample means are unequal.
Next, compute the variance between the samples and the variance within the samples, and calculate the F statistic.
F statistic values far from 1 lead to smaller P-values. This occurs when the variance within samples is small or the variance between samples is high. Thereby, we infer inequality of sample means, rejecting the null hypothesis.
Alternatively, F statistic values closer to 1 lead to larger P-values. This occurs when the variance between samples is close to the variance within samples. Thereby, we infer equality of sample means, failing to reject the null hypothesis.
One-way ANOVA analyzes more than three samples categorized by one factor. For example, it can compare the average mileage of sports bikes. Here, the data is categorized by one factor - the company. However, one-way ANOVA cannot be used to simultaneously compare the sample mean of three or more samples categorized by two factors. An example of two factors would be sports bikes from different companies driven in different terrains, such as a desert or snowy landscape. Here, two-way ANOVA is used since two factors are involved, namely company and terrain.
Two hypotheses, namely the null and the alternative hypothesis, are stated before analyzing samples using one-way ANOVA. The null hypothesis states that the means of the samples used during analysis are equal, while the alternative hypothesis states that the sample means are unequal. After stating the two hypotheses, the variances between samples and within samples are calculated. The variance between samples is computed as the variance of the sample means multiplied by the sample size, n. The variance within samples is computed as the average of the sample variances.
Next, the F statistic is calculated as the ratio of the variance between samples to the variance within samples. If the value of the F statistic is greater than 1, smaller P-values are obtained. This occurs when the variance between samples is high or the variance within samples. From this, it is inferred that the sample means are unequal, and the null hypothesis is rejected. If the value of the F statistic is closer to or equal to 1, larger P-values are obtained. This happens when the variance between the samples is close to or equal to the variance within the samples. In such a case, it is inferred that the sample means are equal, so one fails to reject the null hypothesis.
This text is adapted from Openstax, Introductory Statistics, Section 13.1 One way ANOVA
A one-way ANOVA test compares the means of three or more samples defined by one factor.
Consider the average fuel consumption of cars from three companies. Here, the samples are defined by one factor—the company.
For cars from different companies driven in summer and winter, a one-way ANOVA cannot simultaneously test for two factors—company and season.
In general, begin by stating the null hypothesis that the sample means are equal, and the alternative hypothesis that the sample means are unequal.
Next, compute the variance between the samples and the variance within the samples, and calculate the F statistic.
F statistic values far from 1 lead to smaller P-values. This occurs when the variance within samples is small or the variance between samples is high. Thereby, we infer inequality of sample means, rejecting the null hypothesis.
Alternatively, F statistic values closer to 1 lead to larger P-values. This occurs when the variance between samples is close to the variance within samples. Thereby, we infer equality of sample means, failing to reject the null hypothesis.
From Chapter 10:
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10.2 : One-Way ANOVA
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10.1 : What is an ANOVA?
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10.3 : One-Way ANOVA: Equal Sample Sizes
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10.4 : One-Way ANOVA: Unequal Sample Sizes
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10.5 : Multiple Comparison Tests
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10.6 : Bonferroni Test
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10.7 : Two-Way ANOVA
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