Analysis of Variance

Understanding analysis of variance One-way ANOVA

The one-way ANOVA (analysis of variance) is a statistical test used to evaluate whether there is a difference between the averages (MEANS) of two groups or variables. For example; one could use a one-way analysis of variance to examine whether the height of students differs based on their ethnicities such as Black, White and Hispanic. This means examining three different height groups, the heights of Black, White and Hispanic students and then comparing their means/ averages. The ANOVA tests for mean differences/ variances usually determine whether at least two groups were different. This test will fail to tell your which of these specific groups were different. To compliment this test, post Hoc analysis such as the Tukey's posts hoc analysis or the Least Significant Difference (LSD) test are used to help determine which specific groups were statistically significantly different from each other. The general formula for analysis of variance is;

F= MST/ MSE

Where we have;

F= the test statistic of ANOVA (Basically the coefficient)

MST= the mean sum of squares due to treatment

MSE= the mean squared error (sum of squares due to error)

When you decide that you are going to conduct a one-way analysis of variance test, the first step to determine whether the data being used for the analysis can actually be used to conduct an ANOVA test. This involves checking for various assumptions that are required to get valid and accurate results. You should note that, in many instances, real life data will often violate a lot of these assumptions. There are ways to overcome these violations as well. Do not be alarmed if most of the times, the assumptions are not met. These assumptions are;

1. The dependent variables being examined should be measured as an interval or ratio variables (this means that they are scale of continuous variables).
2. The independent variable should be a categorical variable with two or more independent groups. For instance, heigh measured from three groups, black, white and Hispanic Students. Each of these groups will be independent from each other (they are not related in any way).
3. As a compliment to the independence of groups, there should be independence of observations. This means that neither the groups nor individual observations should be related to one another.
4. The data being utilized, i.e., both variables should not have significant and extreme outliers. Outliers my end up distorting the results.
5. The variables being tested should be nearly normally distributed. This is tested in a histogram, where it shows almost a bell-shaped curve for approximately normally distributed variables.
6. There needs to exist a homogeneity of variances. This is usually the assumptions that the sample or population variances (the distribution and spread of the observations around their mean/ average estimate) of the two samples/ groups are considered to be equal.

Some of these assumptions can be checked by simply observing the data. And others can be checked by use of different tests in the statistical tools or software being used. The most important assumptions to consider are the assumption 1, 2 and 3 above.

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