# Three-Way ANOVA

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, p. 52).

Three-way ANOVA can be applicated not only in economy, science, medicine but also in many other disciplines (Klar, 2013, p. 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, p. 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, p. 43):

${\displaystyle H_{0}:}$ ${\displaystyle \alpha _{1}=\alpha _{2}...=\alpha _{I}}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle \alpha _{i}}$ is different,

${\displaystyle i=1,2,...,I}$.

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

${\displaystyle H_{0}:}$ ${\displaystyle \mathrm {B} _{1}=\mathrm {B} _{2}=...=\mathrm {B} _{J}}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle \mathrm {B} _{j}}$ is different,

${\displaystyle j=1,2,...,J}$.

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

${\displaystyle H_{0}:}$ ${\displaystyle \gamma _{1}=\gamma _{2}=...=\gamma _{K}}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle \gamma _{k}}$ is different,

${\displaystyle k=1,2,...,K}$.

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

${\displaystyle H_{0}:}$ ${\displaystyle (\alpha \mathrm {B} )_{ij}=0}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle (\alpha \mathrm {B} )_{ij}\neq 0}$,

${\displaystyle H_{0}:}$ ${\displaystyle (\alpha \gamma )_{ik}=0}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle (\alpha \gamma )_{ik}\neq 0}$,

${\displaystyle H_{0}:}$ ${\displaystyle (\mathrm {B} \gamma )_{jk}=0}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle (\mathrm {B} \gamma )_{jk}\neq 0}$,

${\displaystyle H_{0}:}$ ${\displaystyle (\alpha \mathrm {B} \gamma )_{ijk}=0}$

${\displaystyle H_{1}:}$ at least one ${\displaystyle (\alpha \mathrm {B} \gamma )_{ijk}\neq 0}$.

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

${\displaystyle i=1,2,3,4,...,I}$,

• j describes the group j of factor B:

${\displaystyle j=1,2,3,4,...,J}$,

• k describes group k of factor C:

${\displaystyle k=1,2,3,4,...,K}$,

• ${\displaystyle \alpha _{i}}$ describes the deviations of groups of factor A from the overall mean (μ) due to factor A:

${\displaystyle \alpha _{i}=0}$,

• ${\displaystyle \mathrm {B} _{j}}$ describe the deviations of groups in factor B from the overall mean μ due to factor B:

${\displaystyle \mathrm {B} _{j}=0}$,

• ${\displaystyle (\alpha \mathrm {B} )_{ij}}$ describes the interaction between factors A and B:

${\displaystyle (\alpha \mathrm {B} )_{ij}=0}$,

• ${\displaystyle (\alpha \gamma )_{ik}}$ describes the interaction between factors A and C:

${\displaystyle (\alpha \gamma )_{ik}=0}$,

• ${\displaystyle (\mathrm {B} \gamma )_{jk}}$ describes the interaction between factors B and C:

${\displaystyle (\mathrm {B} \gamma )_{jk}=0}$,

• ${\displaystyle (\alpha \mathrm {B} \gamma )_{ijk}}$ describes the three-way interaction between factors A, B, and C:

${\displaystyle (\alpha \mathrm {B} \gamma )_{ijk}=0}$.

As we can perceive, parameters with two subscripts calculate the interaction between two factors (for example ${\displaystyle (\alpha \mathrm {B} )_{ij}}$) (Fox, 1984, p. 126). Three-way interactions are expressed by the three subscripts, for example ${\displaystyle (\alpha \mathrm {B} \gamma )_{ijk}}$ (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, p. 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 p. 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, p. 354).

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

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

## Examples of Three-Way ANOVA

• A Three-Way ANOVA can be used to analyze the results of a survey in which respondents were asked to rate a product on three different dimensions. The three dimensions can be represented by the three independent variables: customer satisfaction, ease of use, and value for money. By analyzing the data using a Three-Way ANOVA, the researcher can determine whether there is a statistically significant relationship between the three dimensions and the overall rating of the product.
• Another example of a Three-Way ANOVA is the analysis of a clinical trial in which participants are randomly divided into three treatment groups. The three factors that are analyzed are the type of treatment, the gender of the participant, and the age of the participant. By analyzing the results of the trial using a Three-Way ANOVA, the researcher can determine whether the type of treatment, the gender of the participant, or the age of the participant has a statistically significant impact on the outcome of the trial.
• A third example of a Three-Way ANOVA is the analysis of the effect of three different fertilizers on the growth of a crop. The three independent variables are the type of fertilizer, the amount of fertilizer applied, and the type of soil. By analyzing the data using a Three-Way ANOVA, the researcher can determine whether the type of fertilizer, the amount of fertilizer applied, or the type of soil has a statistically significant effect on the growth of the crop.

A Three-Way ANOVA (also known as three-factor ANOVA) is a statistical tool used to analyze the relationship between multiple variables. It offers a number of advantages, including:

• Ability to look at complex relationships between three or more variables, helping to identify which factors are causing changes in results.
• Allows researchers to determine whether differences in the results are due to chance or factors influencing the data.
• Can be used to investigate interactions between variables and identify relationships that would not be evident in a two-way ANOVA.
• Analyzes variance in results over time, allowing researchers to identify trends in data sets.
• More accurate than two-way ANOVA in cases where more than two variables are involved.
• Easy to use and interpret, saving time and resources.

## Limitations of Three-Way ANOVA

Despite the usefulness of Three-Way ANOVA, there are some limitations associated with it. These limitations include:

• The assumptions of normality, homogeneity of variance, and independence must be met for accurate results. If these assumptions are not met, the results of the analysis may be unreliable.
• Three-Way ANOVA does not provide information about the specific differences between the groups, only whether or not the differences exist. Therefore, post-hoc testing must be used to determine the specific differences between the groups.
• Three-Way ANOVA is not recommended for small sample sizes as the results may not be reliable.
• Three-Way ANOVA is limited in its ability to detect interactions between the factors. Therefore, additional tests may need to be used to identify interactions.

## Other approaches related to Three-Way ANOVA

In addition to Three-Way ANOVA, there are several other approaches that can be used to analyze the relationship between variables. These include:

• Factorial ANOVA: This is a type of ANOVA which examines the effects of two or more factors on a given dependent variable. It allows for more than one independent variable to be tested at the same time, making it useful for analyzing interactions between multiple variables.
• Multivariate Analysis of Variance (MANOVA): This is a statistical technique which is used to compare multiple dependent variables simultaneously. It is used to look at the relationship between multiple independent variables and a single dependent variable.
• Mixed ANOVA: This is a type of ANOVA that combines within-subjects and between-subjects factors into one analysis. It can be used to investigate the effects of different treatments on a dependent variable, or to investigate the effects of different independent variables on a dependent variable.

In conclusion, Three-Way ANOVA is one of several approaches used to analyze the relationship between variables. Other approaches include Factorial ANOVA, Multivariate Analysis of Variance, and Mixed ANOVA. Each of these approaches has its own strengths and limitations, and can be used to gain a better understanding of the complex relationships between variables.

 Three-Way ANOVA — recommended articles Analysis of variance — Parametric analysis — Logistic regression model — Principal component analysis — Exploratory factor analysis — Correlational study — Quantitative variable — Measurement method — Multivariate data analysis

## References

Author: Dominika Paś