# Three-Way ANOVA

Three-Way ANOVA | |
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**Three-way ANOVA** (also **three-factor ANOVA**) is a statistic device to calculate the relationship between variables and stands for variances analysis ). It regulates and helps to understand the interplay of three factors on a result in a special way which allows us to determine the cause of change in the analysis - a chance or factors influence (Bingham, 1992, s. 52).

Three-way ANOVA can be applicated not only in economy, science, medicine but also in many other disciplines (Klar, 2013, s. 46).

Three-way ANOVA includes three main factors: A, B, C, and variable x. All of the factors and interactions require hypothesis tests (Gravetter, 2009, s. 506).

## Hypothesis

The first group of hypothesis refers to the equality of the mean reply for groups of factor A. It is presented as (Macho, 2006, s.43):

**\(H_0:\)** \(α_1=α_2⋯=α_I\)

**\(H_1:\)** at least one \(α_i\) is different,

**\(i=1, 2,..., I\)**.

In case of the equality of the mean reply for groups of factor B, the formula for the hypothesis is as follows:

**\(H_0:\)** \(β_1=β_2=⋯=β_J\)

**\(H_1:\)** at least one \(β_j\) is different,

\(j=1, 2,..., J\).

In case of the equality of the mean reply for groups of factor C, the hypothesis is:

**\(H_0:\)** \(γ_1=γ_2=⋯=γ_K\)

**\(H_1:\)** at least one \(γ_k\) is different,

\(k=1, 2,..., K\).

The second group of hypothesis refers to the interaction of the factors:

**\(H_0:\)** \((αβ)_{ij}=0\)

**\(H_1:\)** at least one \((αβ)_{ij}≠0\),

**\(H_0:\)** \((αγ)_{ik}=0\)

**\(H_1:\)** at least one \((αγ)_{ik}≠0\),

**\(H_0:\)** \((βγ)_{jk}=0\)

**\(H_1:\)** at least one \((βγ)_{jk}≠0\),

**\(H_0:\)** \((αβγ)_{ijk}=0\)

**\(H_1:\)** at least one \((αβγ)_{ijk}≠0\).

*i*describes the group i of the factor of A:

**\(i= 1, 2, 3, 4,..., I\)**,

*j*describes the group j of factor B:

**\(j= 1, 2, 3, 4,..., J\)**,

*k*describes group k of factor C:

**\(k=1, 2, 3, 4,..., K\)**,

**\(α_i\)**describes the deviations of groups of factor A from the overall mean (μ) due to factor A:

**\(α_i=0\)**,

**\(β_j\)**describe the deviations of groups in factor B from the overall mean μ due to factor B:

**\(β_j=0\)**,

**\((αβ)_{ij}\)**describes the interaction between factors A and B:

**\((αβ)_{ij}=0\)**,

**\((αγ)_{ik}\)**describes the interaction between factors A and C:

**\((αγ)_{ik}=0\)**,

**\((βγ)_{jk}\)**describes the interaction between factors B and C:

**\((βγ)_{jk}=0\)**,

**\((αβγ)_{ijk}\)**describes the three-way interaction between factors A, B, and C:

**\((αβγ)_{ijk}=0\)**.

As we can perceive, parameters with two subscripts calculate the interaction between two factors (for example \((αβ)_{ij}\)) (Fox, 1984, s. 126). Three-way interactions are expressed by the three subscripts, for example \((αβγ)_{ijk}\) (Devore, 2015, S.460). To calculate three-factor ANOVA, first of all, we have to calculate two-factors interactions for the variables (O’Mahony, 1986, s. 211).

## Other information

In search of interaction finding its conditions is an important step. When there is a connection, the analysis may be required to expand apart from ANOVA (Strickland, 2014 s. 273).

The interpretation of data is not so easy when there are interactions. The basic mistake is to make calculations of significance or estimated treatment effects as a nominal value (Scheffe, 1959, s. 354).

To avoid the masking principal effect by interactions and to augment understanding is used graphical method (Houser, 2016, s. 290).

Data transformation may cause the removal of interactions in some cases (Strickland, 2016, s. 329).

## References

- Bingham A.H., Ek C.W. (1992).
*Emission Testing of Aerial Devices and Associated Equipment Used in the Utility Industries*, ASTM International - Devore J.L. (2015).
*Probability and Statistics for Engineering and the Sciences*, Cengage Learning - Fox J. (1984).
*Linear Statistical Models and Related Methods: With Applications to Social Research*, Wiley - Gravetter J.F., Wallnau L.B. (2009).
*Statistics for the Behavioral Sciences*, Cengage Learning - Hauser J. (2016).
*Nursing Research: Reading, Using and Creating Evidence*, Jones & Bartlett Learning - Klar R., Opitz O. (2013).
*Classification and Knowledge Organization: Proceedings of the 20th Annual Conference of the Gesellschaft für Klassifikation e.V., University of Freiburg, March 6–8, 1996*, Springer Science & Business Media - Macho S. (2006).
*The Impact of Home Internet Access on Test Scores*, Cambria Press - O’Mahony M. (1986).
*Sensory Evaluation of Food: Statistical Methods and Procedures*, CRC Press - Scheffe H. (1959).
*Analysis of Variance*, John Wiley & Sons - Strickland J. (2014).
*Predictive Modeling and Analytics*, Lulu.com - Strickland J. (2016).
*Predictive Analytics using R*, Lulu.com

**Author:** Dominika Paś