Two-way analysis of variance is a hypothesis test that is used to compare the means of several groups. It includes two independent variables and one dependent variable. The two independent variables are categorical in nature, whereas the one dependent variable is continuous in nature.
One-way ANOVA: When the test relates to comparing the group means of a dependent variable based on different levels of one categorical variable, it is called one-way analysis.
Two-way ANOVA: When the test relates to comparing the group means of a dependent variable based on different levels of two categorical variables, it is called two-way analysis.
Two-way ANOVA is a type of hypothesis test; it is called the analysis of variance. Though the name includes the term variance, its hypothesis is not related to variance. ANOVA is used to compare the mean differences of the continuous variables based on two different categorical variables. The aim is to understand how each of the categorical variables affects the continuous variable. It helps to understand if the interaction of the two categorical variables has a significant influence on the continuous variable.
Here, the continuous variable is called the dependent variable, and the two categorical variables are considered the independent variables. The aim is to see if there is a significant relation among these variables.
alculations in Two-Way ANOVA:
The test of two-way Anova starts with the following sets of hypotheses
Null hypothesis1: The continuous variable does not depend on the categorical variable 1.
Alternate hypothesis1: The continuous variable depends on the categorical variable 1.
Null hypothesis2: The continuous variable does not depend on the categorical variable 2.
Alternate hypothesis2: The continuous variable depends on the categorical variable 2.
Null hypothesis3: The continuous variable does not depend on the interaction of the two categorical variables.
Alternate hypothesis3: The continuous variable depends on the interaction of the two categorical variables.
Next, the basic calculations are done to separately find the mean values of each category, each row, and each column.
These values are then used to find the sum of squared differences. The sum of squared differences is denoted by SS. SS for the first factor, second factor, interaction, and error are calculated. All these four sum of squared differences add up to give the sum of squared differences in total.
Further, the means of these sums are calculated. The formula used is the sum of squared differences divided by the degree of freedom.
Mean squared error is denoted by MS. MS of factor, interaction, and error are separately calculated.
Once the mean squared error values are obtained, the test statistic value for each pair of hypothesis is obtained.
Test statistic is obtained by dividing the MS of factor by the MS of the error term.
Thus, the test statistic used for two-way ANOVA is F statistic. This value is based on the two degrees of freedom (numerator and denominator).
The F-critical value is calculated based on the two degrees of freedom. The degree of freedom of the numerator is related to the factor and the degree of freedom of the denominator is related to the error term.
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