the guide to understanding the analysis of variance

the guide to understanding the analysis of variance


ANOVA is the abbreviation for analysis of variance, analysis of variance in English. Developed by the British statistician Sir Ronald Aylmer Fisher at the beginning of the 20th century, ANOVA brings together statistical models, including the F test, or Fisher’s test. The ANOVA method is used to study the dependence ratio of a quantitative variable to one or two qualitative variables.

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As part of a marketing study, for example, the analysis of variance can be applied to the opening rate of the newsletter. The company obtains an open rate of 50%; this data, to be used for improvement purposes, must be refined. It is for this purpose that the marketing team uses the ANOVA test. The aim is to determine whether factors such as age, sex or geographical sector influence the opening rate of the newsletter. The ANOVA table highlights whether or not a relationship of dependence of the opening rate on the age, gender or geographical sector factor: the company that observes a relationship of dependence of the rate on the gender factor deduces from this that the content of the newsletter does not does not interest women as much as men. The marketing team can work on this aspect to better satisfy its audience.

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What is the ANOVA test?

The ANOVA test is a succession of statistical formulas that tests two hypotheses. The null hypothesis highlights the equality of the means: the qualitative variable has no influence on the quantitative variable. The alternative hypothesis makes it possible to observe that an average deviates significantly from the others.

When to use the ANOVA test?

ANOVA tests the homogeneity of the mean of the quantitative variable studied on the different values ​​of the qualitative variable. The analysis of variance, if it leads to a result far from zero, makes it possible to reject the null hypothesis: the qualitative variable effectively influences the quantitative variable.

The ANOVA method is used in quantitative studies in many fields, to test or verify hypotheses. Illustrations:

  • In the example of the newsletter, the marketing team collects a lot of data related to the open rate. In particular, the company knows the geographical area and gender of each subscriber to the newsletter. The ANOVA test on the geographic sector factor does not highlight any dependency: in Bordeaux, Paris and Le Havre, the average is similar. The ANOVA test on the gender factor, on the other hand, reveals a significant difference: the analysis of variance enables it to identify this influencing factor.
  • A national survey conducted by a polling institute reveals that 60% of French people are equipped with a connected television. ANOCA makes it possible to know whether the geographical factor and the level of income are decisive. From one city to another, no significant average difference is observed. The averages, on the other hand, differ to a large extent according to income. The analysis of variance thus concludes that there is a correlation between the fact of owning a connected television and the level of income.
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  • A sports education establishment studies the performance of its students to improve it. The establishment endeavors to compare the sports results of several groups of candidates, each group following a specific diet. The F test leads to a ratio of variances close to 1: the diet factor does not impact sports results. The establishment is testing another hypothesis: performance varies according to the frequency of training. The analysis of variance verifies this second hypothesis.

These examples are schematic illustrations of the ANOVA method, in its univariate version. In fact, Fisher’s test compares variances not only between samples, but also within samples. Advanced tests, moreover, allow performing multivariate analyses, which test the relationship of dependence of a quantitative variable on several qualitative factors. Automatic calculation tools make it possible to obtain the results of the variance analysis in the form of an easily usable table, the user contenting himself with entering his experience data.

How to interpret an ANOVA table?

The ANOVA table is the end result of a succession of complex calculation formulas. It presents three types of usable digital data:

  • The degrees of freedom or dof.
  • The result noted F.
  • The significance noted p: this value, obtained thanks to the data ddl and F, constitutes the ratio of variance which confirms or invalidates the hypothesis tested. If the p-value is less than 0.05, the null hypothesis that the means are equal is likely to be rejected. That is, the qualitative variables have a significant effect on the quantitative variable: at least one mean stands out to a large extent within the sample.
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