Analysis of Results in a One-Way ANOVA Experiment
To conduct a One-Way ANOVA, several steps are followed to decompose the total variability into explanatory and error components.
Step 1: Model Formulation
The statistical model for One-Way ANOVA is:
yij = μ + αi + εij
- yij is the observed response for the j-th subject in the i-th group.
- μ is the overall mean.
- αi is the effect of the i-th treatment.
- εij is the random error
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Step 2: Hypotheses Formulation
Null hypotheses (H0)
Alternative Hypothesis (Ha)
At least one of the means is different.
(μ1=μ2=…=μk)
Step 3: Variability Decomposition
- Total Sum of Squares (SST): measures the total variability in the data relative to the overall mean.
- Sum of Squares Between Groups (SSB): measures the variability between the means of different groups.
- Sum of Squares Within Groups (SSW): measures the variability within each group.
SST=SSB+SSW
Step 4: Calculation of Degrees of Freedom (df)
- For SSB: dfSSB = k−1
- For SSW: dfSSW = N−k
- For SST: dfSST = N−1
where k is the number of groups and N is the total number of observations.
Step 5: Calculation of Mean Squares (MS)
- Mean Square Between Groups (MSB): MSB = SSB / dfSSB
- Mean Square Within Groups (MSW): MSW = SSW / dfSSW
Step 6: Calculation of the F Statistic
F = MSB / MSW
Step 7: Decision Making
If the value of F is greater than the critical value of F obtained from the F distribution tables for a given significance level (e.g., 0.05), the null hypothesis that the means of the groups are equal is rejected. Otherwise, the null hypothesis cannot be rejected.