Analysis of Variance

What is Analysis of Variance?

ANOVA (Analysis of Variance) is a statistical technique used to compare means between two or more groups. It assesses whether the means of these groups are significantly different from each other by examining the variation within each group and comparing it to the variation between groups.

ANOVA is commonly used to compare means across multiple groups, such as in experimental studies with different treatment conditions or observational studies with categorical predictors. It provides a robust method for assessing group differences while controlling for Type I error rates.

Null and Alternative Hypotheses

In ANOVA, the null hypothesis (H0) states no significant difference in means between the groups, while the alternative hypothesis (H1) suggests that at least one group’s mean is different from the others.

F-Statistic

The F-statistic is calculated by comparing the variability between group means to the variability within the groups, essentially comparing the model’s explained variance to the unexplained variance.

Variation Between Group Means (Explained Variance): This reflects how much the group’s means deviate from the overall mean. A larger variance between groups suggests that the group means are not all the same, implying that the group factor significantly affects the dependent variable.
Variation Within Groups (Unexplained Variance): This represents the random variation, noise, or error within each of the groups. The variation in observations within the same group can’t be explained by the group effect alone.

Degrees of Freedom

Degrees of freedom (DF) are parameters that help quantify the variability in the data and are used in calculating the F-statistic and determining statistical significance. The degrees of freedom in ANOVA are split into two main types:

Degrees of Freedom Between Groups (DFbetween):
- This refers to the number of independent comparisons that can be made between the group means.
- It is calculated as the number of groups minus one ( $k-1$ ), where $k$ is the total number of groups.
- DFbetween accounts for the variability in the data due to the differences among the group means.
Degrees of Freedom Within Groups (DFwithin) or Degrees of Freedom for Error:
- This represents the total number of observations minus the number of groups ( $N-k$ ), where $N$ is the total number of observations across all groups.
- DFwithin reflects the variability within the groups themselves, which is not explained by the group membership.

Sum of Squares

ANOVA (Analysis of Variance) is a statistical method to analyze the differences among group means and their associated procedures. It helps to determine whether the means of several groups are equal, and it does this by breaking down the total variability among the data into two components: variability between groups and variability within groups.

Total Sum of Squares (SST): This is the overall variability in the data without regard to group divisions. It’s the sum of the squared differences between each observation and the overall mean of the data.
Sum of Squares Between Groups (SSbetween): This represents the variability due to the interaction between the different groups. It’s calculated by summing the squared differences between each group’s mean and the overall mean, weighted by each group’s size. This component reflects the extent to which the group means differ from each other and the overall mean.
Sum of Squares Within Groups (SSwithin) or Sum of Squares Error (SSE): This measures the variability within each group, that is, the sum of the squared differences between each observation in a group and the mean of that group. It represents the random variation, error, or noise within each group.

Mean Squares

Mean squares (MS) are statistical measures used in the context of an analysis of variance (ANOVA), which is a method used to analyze the differences among group means in a sample. The mean square is calculated by dividing the sum of squares (SS) by the corresponding degrees of freedom (df). In ANOVA, there are typically two types of mean squares that are calculated:

Mean Square Between Groups (MSB): This value is obtained by dividing the sum of squares between groups (SSB) by its degrees of freedom, which is usually the number of groups minus one. It reflects the variance between the group means and is used to assess the variation due to the interaction between the groups.
Mean Square Within Groups (MSW) or Mean Square Error (MSE): This is calculated by dividing the sum of squares within groups (SSW) by its degrees of freedom, often the total number of observations minus the number of groups. This measures the variance within the individual groups and represents the error or noise level in the data.

Assumptions

ANOVA assumes that the populations from which the groups are sampled are normally distributed, have equal variances (homogeneity of variances), and that observations within each group are independent.

Post-hoc Tests

If ANOVA indicates a significant difference between groups, post-hoc tests such as Tukey’s HSD (Honestly Significant Difference) or Bonferroni tests can be conducted to determine which specific group means differ significantly.

Example of Analysis of Variance

Let’s say we use three different fertilizers (A, B, and C) to grow plants. We want to determine if there is a significant difference in the average height of plants when using these fertilizers. We conduct an experiment in which we randomly assign 30 plants to three groups, with each group receiving one of the three fertilizers.

After a certain period of time, we measure the height of each plant in centimeters. The data collected is as follows:

Group A (Fertilizer A): 25 cm, 28 cm, 26 cm, 27 cm, 29 cm
Group B (Fertilizer B): 30 cm, 32 cm, 31 cm, 33 cm, 29 cm
Group C (Fertilizer C): 27 cm, 28 cm, 26 cm, 29 cm, 30 cm

To analyze this data using ANOVA, we first state the null hypothesis (H0) and alternative hypothesis (Ha):

H0: There is no significant difference in the average height of plants among the three fertilizers.
Ha: There is a significant difference in the average height of plants among the three fertilizers.

Next, we calculate the sum of squares within groups (SSW), sum of squares between groups (SSB), degrees of freedom (df), mean square within groups (MSW), mean square between groups (MSB), and the F-statistic.

After performing the ANOVA calculation, we obtain an F-statistic value and compare it to the critical F-value from the F-distribution table at a chosen significance level (e.g., 0.05). If the calculated F-value is greater than the critical F-value, we reject the null hypothesis and conclude that there is a significant difference in the average height of plants among the fertilizers. Otherwise, we fail to reject the null hypothesis if the calculated F-value is less than the critical F-value.

In this example, if the ANOVA analysis indicates a significant difference, we can further investigate using post-hoc tests (e.g., Tukey’s HSD test) to determine which specific pairs of fertilizers have significantly different effects on plant height.