Analysis of variance (ANOVA) is a statistical technique that is used to check if the means of two or more groups are significantly different from each other.
Basically, we’re testing groups to see if there’s a difference between them.
For eg: A group of psychiatric patients are trying three different therapies: counseling, medication and biofeedback. You want to see if one therapy is better than the others, then you will use ANOVA test. It checks the impact of one or more factors by comparing the means of different samples and gives you the result.
All the sample means are equal, or the factor does not have any significant effect on the groups
At least one of the sample means is different from another, thus factor has a significant effect on one of the groups.
The statistic which measures if the means of different samples are significantly different or not is called the F-Ratio.
F = Between group variability / Within group variability
This F-statistic calculated here is compared with the F-critical value for making a conclusion. If the value of the calculated F-statistic is more than the F-critical value (for a specific α/significance level), then we reject the null hypothesis and can say that the treatment has a significant effect.
One way / Two-way ANOVA
One-way or two-way refers to the number of independent variables (IVs) in your Analysis of Variance test.
One-way has one independent variable/factor .
For example: brand of cereal.
A one way ANOVA will tell you that at least two groups were different from each other. But it won’t tell you which groups were different. If your test returns a significant f-statistic, you may need to run an ad hoc test to tell you exactly which groups had a difference in means.
Two-way has two independent variables/factors.
For example: brand of cereal, calories. It is most appropriate when experiment has a quantitative outcome and you have two categorical explanatory variables.
MANOVA (Multi-variate ANOVA) is just an ANOVA with several dependent variables. It’s purpose is to find out if the response variable (i.e. your dependent variable) is changed by manipulating the independent variable.
For eg: Suppose you want to find out if a difference in textbooks affects students’ scores in math and science. So here, improvements in math and science means that there are two dependent variables, with textbook as one independent variable. Hence, here a MANOVA is appropriate.