When you have a research study with 2 factors (regardless of the number of levels for each factor), you will conduct a two factor ANOVA. The two factor ANOVA will determine the presence or absence of a main effect for each factor and the presence or absence of an interaction.

As with the other ANOVAs presented earlier, the two way ANOVA uses an F-ratio. The difference is that you need to calculate an F-ratio for each factor and the F-ratio for the interaction. The F-ratio is always made up of the same basic principle:

MS_{between_treatments} is made up of the variance between the treatment means due to an effect of the treatment. This variance encompasses variance from each of the two factors alone and variance from the interaction of the factors. MS_{within_treatments} is made up of the variance you would expect from chance.

A source table can be used to present a two factor ANOVA. Here is an example of fictional data from a study examining the effects of room color (IV1 levels: red, blue) and room temperature (IV2 levels: 70 degrees and 80 degrees) on ratings of emotion. This study had 80 total participants.

	SS	df	MS	F
Btwn Treatments	51	3
Main Effect IV1	27	1	27	9
Main Effect IV2	6	1	6	2
Interaction	18	1	18	6
Within Treatments	162	76	3
Total	213	79

Using the information in the source table, you can conduct a hypothesis test. You will be conducting an F test for each factor (IV1 and IV2) and the interaction, so you will start by setting up hypotheses for each of these, find separate critical values, and make separate decisions on the statistical significance of each.

STEP ONE: SET UP HYPOTHESES

For IV1, room color

H₀: µ_red = µ_blue

H₁: µ_red ≠ µ_blue

For IV2, room temperature

H₀: µ₇₀ = µ₈₀

H₁: µ₇₀ ≠ µ₈₀

For the interaction of room color and room temperature

H₀: There is no interaction between room color and room temperature.

H₁: There is an interaction between room color and room temperature.

STEP TWO: SET THE CRITICAL REGION

For IV1 (room color) you need the df_IV1 and df_{within_treatments}. You can get these from the source table.

	SS	df	MS	F
Btwn Treatments	51	3
Main Effect IV1	27	1	27	9
Main Effect IV2	6	1	6	2
Interaction	18	1	18	6
Within Treatments	162	76	3
Total	213	79

In the F-Table, the df_IV1 is the df_numerator and the df_{within_treatment} is the df_denominator. The critical value for these values and an alpha value of 0.05 is equal to 3.97. An F-table is also available on page 541 in the Morling book.

For IV2 (room temperature), you need the df_IV2 and df_{within_treatments}. You can get these from the source table.

	SS	df	MS	F
Btwn Treatments	51	3
Main Effect IV1	27	1	27	9
Main Effect IV2	6	1	6	2
Interaction	18	1	18	6
Within Treatments	162	76	3
Total	213	79

In the F-Table the critical value for these values and an alpha value of 0.05 is equal to 3.97. An F-table is also available on page 541 in the Morling book.

For the interaction, you need the df_Interaction and df_{within_treatments}. You can get these from the source table.

	SS	df	MS	F
Btwn Treatments	51	3
Main Effect IV1	27	1	27	9
Main Effect IV2	6	1	6	2
Interaction	18	1	18	6
Within Treatments	162	76	3
Total	213	79

In the F-Table the critical value for these values and an alpha value of 0.05 is equal to 3.97. An F-table is also available on page 541 in the Morling book.

STEP THREE: CALCULATE YOUR STATISTICS

This has already been conducted and presented in the source table.

STEP FOUR: MAKE A DECISION

Compare each critical value to the calculated value of F in the source table. Based on this example, all three critical values ended up being the same 3.97. This value should be compared to the value of F for IV1, IV2, and the interaction.

You can see that the F value for IV1 and the interaction are greater than the critical value for F, and the F value for IV2 is less than the critical value for F. Thus, there is a main effect for room color (IV1) but not room temperature(IV2), and there is a significant interaction between room color and room temperature.

This analysis should be presented in APA style. It could look like this:

We were testing the effects of room color and room temperature on ratings of emotion. The two factor ANOVA showed a main effect of room color, F (1, 76) = 9, p < 0.05 and a significant interaction between room color and room temperature, F (1, 76) = 6, p < 0.05. A main effect of room temperature was not found, F (1, 76) = 2, p > 0.05.

*NOTE: In most cases, means and standard deviations of the data would also be presented in the results section in a table or in the text.