The two-way ANOVA is an extension of the one-way ANOVA. It is a statistical test performed on three or more samples categorized by two factors - a row factor and a column factor. Ronald Fischer mentioned it in 1925 in his book 'Statistical Methods for Researchers.'

The two-way ANOVA analysis initially begins by stating the null hypothesis that there is an interaction effect between the two factors of a dataset. This effect can be visualized using line segments formed by joining the means for each factor. If the line segments are not parallel, an interaction between the two factors exists. In other words, the two factors simultaneously affect the values in a given dataset. If the two lines are parallel, then no interaction effect is observed. Calculating the F statistic for interaction effect can confirm this graphical representation. If the calculated P-value of the F statistic is greater than a specific significance level (for example, P-value = 0.05), one can fail to reject the null hypothesis.

Next, the effect of each factor on the data values is determined. In other words, it is checked if either the row factor or the column factor affects the data in the dataset. This is done by separately stating the null hypothesis and calculating the F statistic for each factor. If the P-value computed from the F statistic of a specific factor is lower than a chosen significance level (for example, P-value = 0.05), then that factor is said to affect the data values in a given dataset significantly.