What Is Analysis of Variance?

Analysis of Variance (ANOVA for short), also known as "variation analysis", was invented by RAFisher and used to test the significance of the difference between two or more samples. Due to the influence of various factors, the data obtained from the study showed fluctuations. The causes of fluctuations can be divided into two categories, one is the uncontrollable random factors, and the other is the controllable factors that influence the results in the research.

The basic idea of analysis of variance is: by analyzing the contribution of variation from different sources to the total variation, so as to determine the influence of controllable factors on the research results.

Example analysis:

Below we use a simple example to illustrate the basic idea of analysis of variance:

Such as

1. The assumptions of the analysis of variance are:

(1) The samples under each processing condition are random.

(2) The samples under each processing condition are

The main uses of analysis of variance are: significance test for differences in means, separation of related factors and estimation of their effect on total variation, analysis of interactions between factors, homogeneity test of variance.

In scientific experiments, the effects of different experimental conditions or processing methods on experimental results are often discussed. It is usually to compare the differences between sample means under different experimental conditions. For example, the medical community studies the effects of several drugs on a certain disease; agricultural studies on the effects of factors such as soil, fertilizer, and sunshine time on the yield of a certain crop; and the insecticidal effects of different chemicals on crop pests, etc. To solve.

Covariance analysis

A complex thing, many factors often restrict each other and depend on each other. The purpose of the analysis of variance is to find out the factors that have a significant impact on the thing, the interaction between the factors, and the optimal level of the significant factors through data analysis. Variance analysis is a technique that decomposes the total "variation" between data according to each specified source of variation in a comparable array. For the measure of variation, the sum of squared deviations is used. The analysis of variance method is to decompose the sum of squares of the partial separations that can be traced back to the specified source from the sum of the squares of the total dispersion. This is an important idea.

If the test hypothesis is rejected after analysis of variance, it can only indicate that the population mean of multiple samples is not equal or all equal. To get more detailed information between the means of each group, a pair-wise comparison of multiple sample means should be performed on the basis of analysis of variance.

Pairwise comparison of multiple sample means

The method of q-test is commonly used in pairs between multiple sample means, which is the Newman-keuls method. The basic steps are: establish test hypothesis-> sample mean sorting-> calculate q value-> check q-bound table judgment result . Because it mimics the Student's distribution, it is also called the SNK q test.

Pairwise comparisons between multiple experimental groups and a control group

Compare the mean of multiple experimental groups with a control group. If the purpose is to reduce the type II error, it is better to use the least significant difference method (LSD method); if the purpose is to reduce the type I error, it is better to use the new For the complex range method, the former looks at the t-boundary table and the latter looks at the q'-boundary table ^[1] .