Analysis of Variance (ANOVA) - SoojungHong/StatisticalMind GitHub Wiki

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It may seem odd that the technique is called "Analysis of Variance" rather than "Analysis of Means." As you will see, the name is appropriate because inferences about means are made by analyzing variance.

ANOVA is used to test general rather than specific differences among means. This can be seen best by example.

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 alternate hypothesis. Basically, you’re testing groups to see if there’s a difference between them. Examples of when you might want to test different groups:

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. A manufacturer has two different processes to make light bulbs. They want to know if one process is better than the other. Students from different colleges take the same exam. You want to see if one college outperforms the other.

One way ANOVA

An Analysis of Variance Test or an ANOVA is a generalization of the t-tests to more than 2 groups. Our null hypothesis states that there are equal means in the populations from which the groups of data were sampled. More succinctly: μ1=μ2=...=μn for n groups of data. Our alternative hypothesis would be that any one of the equivalences in the above equation fail to be met.

In this ANOVA test, we are dealing with an F-Statistic and not a p-value. Their connection is integral as they are two ways of expressing the same thing. When we set a significance level at the start of our statistical tests (usually 0.05), we are saying that if our variable in question takes on the 5% ends of our distribution, then we can start to make the case that there is evidence against the null, which states that the data belongs to this particular distribution.

The F value is the point such that the area of the curve past that point to the tail is just the p-value. Therefore: Pr(>F)=p

For more information on the choice of 0.05 for a significance level

Two way ANOVA

In a Two-Way ANOVA, there are two variables to consider. The question is whether our variable in question (tooth length len) is related to the two other variables supp and dose by the equation: len=supp+dose+supp×dose

Python code example

http://benalexkeen.com/comparative-statistics-in-python-using-scipy/

Analysis of Variance (ANOVA)

ANOVA is used to compare the means of three or more samples.

While you could do multiple T-tests, as you increase the numbers of T-tests you do, you are more likely to encounter a Type I error. If you have a p value of 0.05 for each T-test, once you have run three T-tests, your p is effectively 0.143. ANOVA controls for this error such that the Type I error rate remains at 5%.

An ANOVA will provide an F-statistic which can, along with degrees of freedom, be used to calculate a p value.

ANOVAs assume independence of observations, homogeneity of variances and normally distributed observations within groups.

This is implemented in scipy by as f_oneway(). Example

We will use R’s Plant Growth Data Set for our ANOVA.

The null hypothesis is that there is no difference between the means of the weights of dried plants under control and 2 different treatment conditions. In [4]:

ctrl = [4.17, 5.58, 5.18, 6.11, 4.5, 4.61, 5.17, 4.53, 5.33, 5.14] trt1 = [4.81, 4.17, 4.41, 3.59, 5.87, 3.83, 6.03, 4.89, 4.32, 4.69] trt2 = [6.31, 5.12, 5.54, 5.5, 5.37, 5.29, 4.92, 6.15, 5.8, 5.26]

stats.f_oneway(ctrl, trt1, trt2)

Out[4]:

F_onewayResult(statistic=4.846087862380136, pvalue=0.015909958325622899)

We report this using the degrees of freedom (between), which is k-1, and the degrees of freedom within, which is N-k.

So we report the result of our ANOVA as (F(2, 27), p=0.016).

As p < 0.05, we can reject the null hypothesis that there is no difference between the means of the weights of dried plants under control and each of the 2 different treatment conditions.