Contents :( taken from Baidu Encyclopedia)
First, the basic concept
Second, type:
1, ANOVA
2, two-way analysis of variance
3, analysis of covariance
First, the basic concept
Analysis of variance also known as "analysis of variance" or "F test" for the significance test by two or more sample mean differences.
Fundamental analysis of variance is considered the primary source of the difference between the mean of two different treatment groups:
(1) test conditions, i.e., the difference caused by the different treatments, referred to differences between groups. And the sum of the squared deviations of the mean of each group and the average of the total represented by a variable, denoted by SSb, the degree of freedom between groups dfb.
(2) random error, as measured differences between individual differences or errors caused by differences in the group referred to, and the sum of the squared deviations of the mean of each group within the set of values represented by a variable variable, SSw denoted, group internal degrees of freedom dfw.
The total sum of squared deviations SSt = SSb + SSw.
SSw-group, divided between the respective degrees of freedom SSb group (the group dfw = nm, between groups dfb = m-1, where n is the total number of samples, m is the number of groups), which give the mean square MSw and the MSb, one kind of It is a process where no effect, i.e. each set of samples are from the same population, MSb / MSw≈1. Another case is the processing does have a role in the results between groups is the mean square error due to the different treatments lead to a common, i.e., each of the samples from different populations. So, MSb >> MSw (far greater than).
MSb / MSw F ratio distribution configuration. F value and its comparison with the threshold value, to infer whether the sample from each of the same overall.
Second, type:
1, ANOVA
Is used to study whether a control variable for different levels of observed variables had a significant impact. Effects of single factors on the observed variables, so called one-way ANOVA analysis.
For example, analysis of different fertilizer whether to bring a significant impact on crop yields, study regional differences affect women's fertility study the effect of education on wages and other income. These issues can be the single factor analysis of variance to get an answer.
The first step in the analysis of variance (ANOVA) was clear observation and control variables. For example, the above problems observed variables are crop production, fertility rate, wage income; control variables were fertilizer, region, education.
The second step is the analysis of variance (ANOVA) variance analysis of observed variables. Analysis of variance that: the observed changes in variable values can be affected by both the control variables and random variables. Accordingly, analysis of variance of the observed variables from the total sum of squares from the decomposition of the group and the sum of squares and square sum of deviations between groups of two parts, expressed as mathematical form: SST = SSA + SSE.
The third step is the analysis of variance (ANOVA) and the proportion of the total sum of squares occupied by each part by comparing the observed variables, control variables to infer whether to bring a significant impact to the observed variables.
principle:
The total sum of squares of the observed variables, sum of squares and if large proportion between groups, then the changes observed variables is mainly caused by the control variable may be interpreted by the main control variable, the control variable to the observed variables bring a significant effect; the other hand, if the between groups sum of squares and the proportion of small, then the observed changes in the control variable is not primarily due to variable, not primarily be explained by the control variable, the control variable does not give different levels observed variables bring significant impact on the observed changes in the value of the variable is caused by a random variable factors.
The basic analysis steps
:
1, the null hypothesis is proposed: H0-- no difference; significant difference H1 -
2. Select the test statistic: analysis of variance test statistic used is F statistic, namely F test.
3, test statistic is calculated and observed values of probability P values: The purpose of this step is to calculate the test statistic of the observed values and the corresponding value of the probability P.
4, a given level of significance, and make decisions
2, two-way analysis of variance
Multivariate analysis of variance to study whether two and two or more control variables have a significant influence on the observed variables. Influence of several factors on the observed variables, so called multi-factor analysis of variance. Multivariate analysis of variance not only able to analyze a number of factors independent of observed variables, better able to analyze the interaction of multiple control factors can have a significant impact on the distribution of observed variables, and then eventually find favor optimal combination of observed variables.
For example: analysis of different varieties, when the effect of different amounts of fertilizer crop yield, crop yields may be observed as a variable, and a variety of fertilizer as the control variable. The use of multi-factor analysis of variance, of different varieties, how different amount of fertilizer is affecting crop yields and fertilizer which species and what further research is to improve the level of optimal combination of crop yields.
further analysis:
1, the establishment of non-saturation model
2, comparative analysis of the mean
3, the control variable graphical analysis of the interaction
3, analysis of covariance
Analysis of covariance those artificially controlling factors difficult to control as the covariate, and under exclusion of the influence of covariates on the observed variables, analyze the role of the control variable (controlled) on the observed variables, thus more accurately control factors Evaluation.
Covariance analysis is still along for the basic idea of the analysis of variance, and in the analysis of observed variables worse, considering the effect of covariates, man-made changes in the observed variables affected four areas: the role of independent control variables, control variables interactions, the role of the role of covariates and random factors, and, after deducting the impact of covariates, and then control the impact analysis variables.
Analysis of variance of the null hypothesis are: linear influence of covariates on the observed variable is not significant; in covariates discounting the effect of the condition, the overall mean of observed variables under the control of each horizontal variables no significant difference between the control variable of each level of observation effect of variables simultaneously zero. The test statistic still use the F statistic, which is the mean square due to various random factors mean square ratio