# Difference between ANOVA Two-Factor with Replication and without Replication

You use ANOVA: Two-Factor without Replication when each cell only has 1 observation, but use Two-Factor with Replication when each cell has more than 1 observation.

For instance, there are two factors, City and Chain Brand. Each has 2 levels, leading to 4 cells (see the figure below).

Each cell in the left-hand data table has 5 observations, and thus we use ANOVA: Two-Factor with Replication.

In contrast, each cell in the right-hand data table has only 1 observation, and thus we use ANOVA: Two-Factor without Replication.

## ANOVA Two-Factor with Replication and without Replication: Difference in Variance Partitioning

The following figure shows the difference in variance partitioning. In particular,

• ANOVA with Replication: SST is divided into SSM and SSR, and there is an interaction term SSAB.
• ANOVA without Replication: SST is equal to SSM, and there is no interaction term SSAB.

## Reasons for No Interaction Term SSAB

There are two perspective to explain why there is no interaction term SSAB for ANOVA: Two-Factor Without Replication.

First, since each cell only one observation, $$x_{ij}$$ is exactly the same as $$\bar{x_{ij}}$$. Thus, you can not use the the same formula for SSR in two-factor ANOVA with replication. Instead, the formula for SSR for ANOVA two-Factor without Replication is as follows.

$$SS_R = \sum \sum (x_{ij}-\bar{x_i}-\bar{x_j}+\bar{x})^2$$

where,

• $$x_{ij}$$ represents observation at $$i^{th}$$ row and $$j^{th}$$ column.
• $$\bar{x_i}$$ is the mean of observations for $$i^{th}$$ row.
• $$\bar{x_j}$$ is the mean of observations for $$j^{th}$$ row.
• $$\bar{x}$$ is the grand mean of all the cells.

Second, we can consider it from the degree of freedom perspective. The total degree of freedom is n-1 for SST. For instance, if you have 4 cells, dfT=3. Then, there is 1 df for SSA, and 1 df for SSB. Finally, there is only 1 df is left for SSR.