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**: SS_{T}is divided into SS_{M}and SS_{R}, and there is an interaction term SS_{AB}.**ANOVA without Replication**: SS_{T}is equal to SS_{M}, and there is no interaction term SS_{AB}.

## Reasons for No Interaction Term **SS**_{AB}

_{AB}

There are two perspective to explain why there is no interaction term **SS _{AB}** 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 **SS _{R}** in two-factor ANOVA with replication. Instead, the formula for

**SS**for

_{R}**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

**SS**. For instance, if you have 4 cells,

_{T}**df**=3. Then, there is 1 df for

_{T}**SS**, and 1 df for

_{A}**SS**. Finally, there is only 1 df is left for

_{B}**SS**.

_{R}## Further Reading

The following is the tutorial showing formulas for ANOVA Two-Factor without Replication and how to calculate it by hand and in Excel.

Further, there are related tutorials on this site on Two-factor ANOVA with replication.

Finally, there are two tutorials outside of this site as well.