**ANOVA Two-Factor without Replication** is used for a design of two factors (e.g., **Factor A** and **Factor B**) and only 1 observation in each cell.

For instance, both **Factor A** and **Factor B** have two levels, leading to 4 cells in total. Each cell only has 1 observations (see below).

## Variance Partitioning

Two-Factor ANOVA without Replication (**SS _{T}**) can be broken down into 3 part,

**SS**,

_{A}**SS**, and

_{B}**SS**. The following is the figure showing the variance partitioning.

_{R}## Formulas

The following shows the formulas for each components, including

\( SS_T=SS_A+SS_B+SS_R \)

For ** SS_{T}**, you can use the following formula.

\( SS_T=\sum \sum (x_{ij}-\bar{x})^2 \)

where,

- \( i \) represents \( i^{th} \) row, \( j \) represents \( j^{th} \) column.
- \( x_{ij} \) represents observation at \( i^{th} \) row and \( j^{th} \) column.
- \( \bar{x}\) is the grand mean of all the cells.

For **SS _{A}**, the following is the formula.

\( SS_A= \sum n_i (\bar{x_i} -\bar{x})^2 \)

where,

- \( n_i \) represents the number of observations for \( i^{th} \) row.
- \( \bar{x_i}\) is the mean of observations for \( i^{th} \) row.
- \( \bar{x}\) is the grand mean of all the cells.

For ** SS_{B}**, the following is the formula.

\( SS_B= \sum n_j (\bar{x_j} -\bar{x})^2 \)

where,

- \( n_j \) represents the number of observations for \( j^{th} \) column.
- \( \bar{x_j}\) is the mean of observations for \( j^{th} \) column.
- \( \bar{x}\) is the grand mean of all the cells.

The formula for **SS _{R}** 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.

Note that, since \( x_{ij} \) is exactly the same as \( \bar{x_{ij}}\), you can not use the the same formula for **SS _{R}** in two-factor ANOVA with replication. Instead,

**SS**here is the similar to interaction effect

_{R }**SS**in Two-Factor ANOVA with replication.

_{AB}## Calculate by Hand

The following is the data example used here.

We can calculate the row and column means and grand mean as follow.

We can calculate **SS _{A}**,

**SS**, and

_{B}**SS**as follows.

_{R}SS_{T}= (10-9.75)^{2} + (20-9.75)^{2} +(5-9.75)^{2} +(4-9.75)^{2} =160.75

SS_{A}= 2(15-9.75)^{2} + 2(14.5-9.75)^{2} =110.25

SS_{B}= 2(7.5-9.75)^{2} + 2(12-9.75)^{2} =20.25

SS_{R}= (10-7.5-15+9.75)^{2} + (20-12-15+9.75)^{2} +(5-7.5-14.5+9.75)^{2} +(4-12-14.5+9.75)^{2} =30.25

Thus, we can see that SS_{A}+SS_{B}+SS_{R} =160.75 = SS_{R}.

We can calculate **MS _{A}**,

**MS**, and

_{B}**MS**as follows. Since all the degree of freedom is 1 for all of them. We can get the following.

_{R}MS_{A}= SS_{A} / df_{A}=110.25/1=110.25

MS_{A} =SS_{B} /df_{B} =20.25/1=20.25

MS_{R }= SS_{R} /df_{R}= 30.25/1=30.25

Finally, we can calculate the * F-ratios* as follows.

*F*_{A}=MS_{A} / MS_{R} =110.25/30.25=3.64

*F*_{B}=MS_{B} / MS_{R} =20.25/30.25=0.67

You can compare the output here with the output shown below using Excel. You can see that the results are the same.

## Calculate using Excel

This section is to explain how to do ANOVA Two-Factor without Replication in Excel.

**Step 1**: Select menu options in Excel

You can select **ANOVA: Two-Factor without Replication** in Excel and click **OK**.

**Step 2**: Set data range in Excel

Then, you select **A1 to C3** to the Input Range, click “**Lables**” and then “**OK**.”

**Step 3:** Interpretation of the Excel output from two-factor ANOVA without Replication

The following are the null and alternative hypotheses for the effect of **City**.

**Null Hypothesis**: City 1 and City 2 do not differ in sales.**Alternative Hypothesis**: City 1 and City 2 do differ in sales.

The following are the null and alternative hypotheses for the effect of **Chain Brand**.

**Null Hypothesis**: Chain Brand A and Chain Brand B do not differ in sales.**Alternative Hypothesis**: Chain Brand A and Chain Brand B do differ in sales.

The following is the output from Excel. For the effect of **City** on **sales**, the p-value is greater than 0.05, and thus we fail to reject the null hypothesis. In other words, City 1 and City 2 do differ in sales.

For the effect of **Chain Brand** on **sales**, the p-value is greater than 0.05 and thus we fail to reject the null hypothesis. In other words, Chain Brand A and Chain Brand B do differ in sales.

## Further Reading

There are tutorials on this site on Two-factor ANOVA with replication. You can check them out to have a better understanding on this topic.

- Two-Way ANOVA: Formula and Example (with replication)
- Two-Way ANOVA in Excel (with replication)

Finally, the following is a reference outside of this site.