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Basics of ANOVA

An ANOVA test is a way to find out if survey or experiment results are significant. In other words, they help you to figure out if you need to reject the null hypothesis or accept the alternative hypothesis.

Basically, you are testing groups to see if there is a difference between them. Example of when you might want to test different groups:

  • A group of psychiatric patients are trying three different therapies: counseling, meditation and biofeedback. You want to see if one therapy is better than the others.

  • A manufacturer has two different processes to make light bulbs. They want to know if one process is better than other.

What does One-way or Two-way mean?

One-way or two-way refers to the number of independent variables in your Analysis of Variance test.

1. One-way has one independent variable (with 2 levels). For example: brand of cereal,

2. Two-way has two independent variable (it can have multiple levels). For example: brand of cereal, calories.

What is group or levels?

Groups or levels are different groups within the same independent variable. In the above example, your levels for “brand of cereal” might be Lucky Charms, Raisin Bran and Cornflakes – a total of three levels. Your levels for “Calories” might be sweetened, unsweetened- a total of two levels.

Let’s say you are studying if an alcoholic support group and individual counseling combined is the most effective treatment for lowering alcohol consumption. You might split the study participants into three groups or levels:

· Medication only,

· Medication and counseling,

· Counseling only

Your dependent variable would be the number of alcoholic beverages consumed per day.

If your groups or levels have a hierarchical structure (each levels has unique subgroups), then use a nested ANOVA for the analysis.

What does Replication means?

It’s whether you are replicating (i.e. duplicating) your test(s) with multiple group. With a two way ANOVA with replication, you have two groups and individuals within that group are doing more than one thing (i.e. two groups of students from two colleges taking two test). If you only have one group taking two test, you would use without replication.

Types of Tests:

There are two main types: one-way and two-way. Two-way tests can be with or without replication.

· One-way ANOVA between groups:Used when you want to test two groups to see if there’s a difference between them.

· Two-way ANOVA without replication: Used when you have one group and you’re double testing that same group. For example, you are testing one set of individuals before and after they take a medication to see if it works or not.

· Two- way ANOVA with replication: Two groups, and the members of those groups are doing more than one thing. For example, two groups of patients from different hospitals trying two different therapies.


Use a one-way ANOVA when you have collected data about one categorical independent variable and one quantitative dependent variable. The independent variable should have at least three levels (i.e. at least three different groups or categories). It is compare two means from two independent (unrelated) groups using the F-distribution.

The null hypothesis for the test is that the two means are equal. Therefore, a significant result means that the two means are unequal. ANOVA tells you if the dependent variable changes according to the level of the independent variable.

For example:

  • Your independent variable is social media use,and you assign groups to low, medium, and high levels of social media use to find out if there is a difference in hours of sleep per night.

  • Your independent variable is brand of soda,and you collect data on Coke, Pepsi, Sprite, and Fanta to find out if there is a difference in the price per 100ml.

  • You independent variable is type of fertilizer,and you treat crop fields with mixtures 1, 2 and 3 to find out if there is a difference in crop yield.

The null hypothesis (H0) of ANOVA is that there is no difference among group means. The alternate hypothesis (Ha) is that at least one group differs significantly from the overall mean of the dependent variable. If you only want to compare two groups, use a t-test instead.

How does an ANOVA test work?

ANOVA determines whether the groups created by the levels of the independent variable are statistically different by calculating whether the means of the treatment levels are different from the overall mean of the dependent variable.

If any of the group means is significantly different from the overall mean, then the null hypothesis is rejected.

ANOVA uses the F-test for statistical significance. This allows for comparison of multiple means at once, because the error is calculated for the whole set of comparisons rather than for each individual two-way comparison (which would happen with a t-test).

The F-test compares the variance in each group mean from the overall group variance. If the variance within groups is smaller than the variance between groups, the F-test will find a higher F-value, and therefore a higher likelihood that the difference observed is real and not due to chance.

Assumptions of ANOVA

The assumptions of the ANOVA test are the same as the general assumptions for any parametric test:

1. Independence of observations: the data were collected using statistically-valid methods, and there are no hidden relationships among observations. If your data fail to meet this assumption because you have a confounding variable that you need to control for statistically, use an ANOVA with blocking variables.

2. Normally-distributed response variable: The values of the dependent variable follow a normal distribution.

3. Homogeneity of variance: The variation within each group being compared is similar for every group. If the variances are different among the groups, then ANOVA probably isn’t the right fit for the data.

Two Ways ANOVA

A Two Way ANOVA is an extension of the One Way ANOVA. With a One Way, you have one independent variable affecting a dependent variable. With Two Way ANOVA, there are two independents. Use a two way ANOVA when you have one measurement variable (i.e. a quantitative variable) and two nominal variable.

In other words, if your experiment has a quantitative outcome and you have two categorical explanatory variables, a two way ANOVA is appropriate.

For example, you might want to find out if there is an interaction between income and gender for anxiety level at job interviews. The anxiety level is the outcome, or the variable that can be measured. Gender and Income are the two categorical variables. These categorical variables are also the independent variables, which are called factors in a Two Way ANOVA.

The factors can be split into levels. In the above example, income level could be split into three levels: low, middle and high income. Gender could be split into three levels: male, female, and transgender. Treatment groups are all possible combinations of the factors. In this example there would be 3 x 3 = 9 treatment groups.


Above text I mentioned some example why we used ANOVA test. Basically, ANOVA test gives an estimate of how much variation in the dependent variable that can be explained by the independent variable.

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