Why is the homogeneity test of variance a necessary condition for variance analysis?

Theoretically, homogeneity of variance is the premise of variance analysis.

For example, simply check the data before analysis, such as whether it contains outliers. The existence of outliers may affect the analysis results, so it needs to be processed before analysis. In addition, the analysis of variance also needs to meet the requirements that all populations obey the normal distribution, all population variance are the same (satisfying the homogeneity of variance) and the observed values are independent (satisfying the examples). So let's test it next.

Outlier test

Check the abnormal values in the data. As the name implies, outliers are quite different from other values in the same group (such as more than 3 times standard deviation, etc.). ). The existence of outliers will make the overall mean and standard deviation inaccurate, which may lead to inaccurate analysis of the final results, especially in small sample research. In the example, * * * has three sets of data, each with 15 samples, and * * has 45 samples, so check the outliers. There are many methods to check outliers, including descriptive analysis to check the standard deviation, scatter plot and box plot. Among them, the graphic method is more intuitive, so here we use SPSSAU visual block diagram to describe it:

Block diagram results:

As can be seen from the above results, there is an abnormal value in the two groups of "Retail" and "Tourism", and there is no abnormality in the data of "Airlines". See below for specific abnormal values:

It can be seen that the abnormal value of data in "Retail" is 99, and the abnormal value of data in "Tourism" is 68. Next, perform the screening process:

After the outliers are processed, whether the data meet the preconditions of variance analysis is tested respectively. First, the normality test is carried out.

test of normality

There are many methods to test normal distribution, including normality test, graphic p-p diagram, q-q diagram and so on. Generally speaking, the normality test is the most stringent, so SPSSAU is used to test the normality, and the results are as follows:

Because it is a small sample analysis (the sample size is less than 50), it is enough to test the results of S-W (Shapiro-wilk) test. For large sample data, K-S(Kolmogorov-Smimov) or J-B (Jarque–Bera) test can be considered. As can be seen from the results, the P values of the three groups of data are all greater than 0.05, and all three groups of data are normal. If normality is not satisfied, nonparametric test can be used for analysis, and then variance homogeneous test is used as follows.

test for homogeneity of variance

The results of variance homogeneity test using SPSSAU are as follows:

As can be seen from the results, the homogeneity test of variance is used in all three groups, and the final F value is 2.797, and the P value is 0.073 greater than 0.05, which shows that there is little difference in data fluctuation among the three groups, and they are homogeneous in variance. Through the analysis, it is found that the data meet the conditions of variance analysis (nonparametric test or welch variance or Brown-Forsythe variance can be used when the homogeneity of variance is not satisfied) and meet the conditions of one-way variance analysis.