ANOVA - Analysis of Variance

ANOVA - Analysis of Variance

• ANOVA is a statistical method used to compare means of more than two groups.
• It assesses the variation between groups compared to the variation within groups.
• ANOVA uses variation (variance, s²) to determine if the means of the groups are significantly different.
• Previously, hypothesis testing for two means was used.
• Now, with multiple means, it's essential to test all possible pairwise comparisons. In essence, comparing every possible pair of the means individually to check if they are equal would result in numerous individual tests needing to be done. ANOVA streamlines this by assessing differences simultaneously and using a single hypothesis test for all groups.

Why Analyze Variance?

• Researchers are often interested in studying the impact of more than two means.
• ANOVA helps achieve this objective and determine the key significant differences between various group means.
• Variation (variance) in the dataset is a crucial tool for understanding the differences between the means.

One-Way ANOVA

• In one-way ANOVA, there's just one independent variable and one dependent variable.
• It's used to observe how a single factor affects the dependent variable across different groups.

ANOVA Technique

• ANOVA is used to test the null hypothesis that all group means are equal, against the hypothesis that at least one mean value differs significantly.
• The data is divided into categories measuring variation between groups (groups are based on the tested variable) and the variation within each group.

Example

• A study examining the impact of temperature on production rate in a plant.
• There are three temperature levels.
• Samples were taken at each level, recording the number of units produced per hour.
• This specific study example uses different temperatures (68°F, 72°F, 76°F) to see if the temperature variable meaningfully impacts the production rate.

Sum of Squares (SS)

• Total Sum of Squares (SSTotal): Overall variation in the data.
• Between-Group Sum of Squares (SSB): Variation between the groups due to the independent variable (temperature).
• Within-Group Sum of Squares (SSW): Variation within each temperature group (in this case, the error or variance that's outside of your tested variable of temperature).
• Important Note: The overall data variation is the sum of the variation between groups and the variation within groups; SSB + SSW = SSTotal.

ANOVA Table

• This table summarizes the data from the analysis with columns for factors like:
  • Source (e.g., Between Groups, Within Groups)
  • Sum of Squares
  • Degrees of Freedom (df)
  • Mean Square (MS)
  • F-Statistic

Degrees of Freedom (df)

• df for Between Groups: is dependent on the number of groups being compared.
• df for Within Groups: is dependent on the sample sizes of each group.
• df for the total sum of squares in the ANOVA procedure is dependent on the total sample size.
• df in the ANOVA table must all add up together to equal your overall total sample size.

Mean Squares (MS)

• MS is the quantity found by dividing the sum of squares by the degrees of freedom.
• MS values for different parts of your experiment.

F-statistic

• Calculated by dividing the Mean Square Between Groups by the Mean Square Within Groups.
• It shows the ratio of the variability between groups to the variability within groups.

Testing the Hypothesis

• The calculated F value is compared to a critical F-value obtained from a statistical table.
• If the calculated F exceeds the critical F-value at the given significance level (typically 0.05), the null hypothesis is rejected. In the case where your F value is not greater than the critical F value, then the null hypothesis cannot be rejected.

Multiple Comparison Procedure

• In cases where the overall ANOVA rejects the null hypothesis, use of a Multiple Comparison Procedure is needed to determine the specific groups between which differences occur.
• Often a Tukey's HSD (Honestly Significant Difference) method is employed. It precisely identifies specific group comparisons where there are meaningful differences.