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One-Way ANOVA

One-way analysis of variance (ANOVA) is used to compare the means of more than two groups. The model that one-way ANOVA assumes is that if \(x_{ij}\) is observation from the \(j^{th}\) subject from the \(i^{th}\) group, where \(i = 1, \dots, k\) and \(j = 1, \dots n\),

\[ x_{ij} = \mu_i + \epsilon_{ij} \]

where \(\epsilon_{ij}\) is the random error. We assume that the random errors are i.i.d normal random variables whose mean is 0 and variance is \(\sigma^2\).

When we compare the means of each treatment group using one-way ANOVA, we can compare them simultaneously, which answers the question 'is one of the means different from the rest', or compare them pairwise. When pairwise comparisons are performed, the type I error rate is increased. This calculator takes that into account by adjusting the type I error of each pairwise test so that the overall type I error is less than the desired alpha.

Power Calculation Parameters

To input multiple values, seperate them by a comma.

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Power Calculation Explanation

Solve For
The unknown you are interested in solving for.
Comparison
If you desire simultaneous or pariwise comparisons of the means.
N
The sample size used to test the hypothesis.
Alpha
The \(\alpha\) (Type I error rate) level of the hypothesis test.
Power
The power (1 - Type II error rate) of the hypothesis test.
Means
The means of each group. Seperate each mean a comma.
Standard Deviation
The standard deviation of the sample (\(\sigma^2\) in the above model).
Known Standard Deviation
If the standard deviation is known or estimated (usually it is estimated).

Calculation Results

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Comparison: {{ comparison }}

Means: {{ mu }}

Known Standard Deviation

Unknown Standard Deviation

N Alpha Power Standard Deviation
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Power Graph

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References

  • Chow, S., Shao, J., & Wang, H. (2003), Sample size calculations in clinical research, New York: Marcel Dekker.