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.