Yishuo School District (23) | SPSS Statistical Analysis (33) Covariance Analysis
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在上一期，我们简要介绍了什么是多因素方差分析，并且举了一个实际案例来说明多因素方差分析的SPSS步骤。
但是，我们在学习单因素与多因素的方差分析时也发现，无论是单因素方差分析还是多因素方差分析，都有一些人为控制的变量。但在实际问题中，某些随机因素是很难人为控制的，但是不管它们，它们又会对结果产生影响，从而可能导致错误的结论。
In the last issue, we briefly introduced what multifactor ANOVA is, and gave a practical case to illustrate the SPSS steps of multifactor ANOVA.
However, when we study single factor and multi factor ANOVA, we also find that there are some artificially controlled variables in both single factor and multi factor ANOVA. However, in practical problems, some random factors are difficult to control, but regardless of them, they will affect the results, which may lead to wrong conclusions.
这就要求我们排除这些随机变量对结果造成的影响，也就是协方差分析。协方差分析是将很难控制的因素作为协变量，在排除协变量影响的条件下，分析控制变量对观测变量的影响，从而更加准确地对控制变量进行分析和评价。
This requires us to exclude the impact of these random variables on the results, that is, covariance analysis. Covariance analysis takes factors that are difficult to control as covariates, and analyzes the influence of control variables on observation variables without the influence of covariates, so as to analyze and evaluate control variables more accurately.
协方差分析仍然沿袭方差分析的思想，并在分析观测变量离差时，考虑了协变量的影响，认为观测变量的变动受4个方面的影响，即控制变量的独立作用、控制变量的交互作用、协变量的作用和随机因素的作用，并在剔除协变量的作用后再分析控制变量对观测变量的影响。
Covariance analysis still follows the idea of analysis of variance. When analyzing the variance of observation variables, it considers the influence of covariates, and believes that the change of observation variables is affected by four aspects, namely, the independent role of control variables, the interaction of control variables, the role of covariates, and the role of random factors. After removing the role of covariates, it analyzes the influence of control variables on observation variables.
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协方差分析问题也属于推断统计中的假设检验问题，其基本步骤与假设检验一致。这里就不再过多赘述。我们来看一个实际案例。
Covariance analysis is also a hypothesis testing problem in inferential statistics, and its basic steps are consistent with hypothesis testing. I will not repeat it here. Let's look at a practical case.
如图所示，已知一个班级三组学生的入学成绩和分别接受了三种不同教学方法后的数学成绩，试研究这三组学生再接受了不同的教学方法后数学成绩是否有显著性差异。
As shown in the figure, given the entrance scores of three groups of students in a class and the math scores after receiving three different teaching methods, try to study whether the math scores of these three groups of students after receiving different teaching methods have significant differences.
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第一步，分析，入学成绩在实际生活中肯定会对最后成绩有一定程度上的影响，这里着重分析的是不同教学方法的影响，应将入学成绩（数学基础）的影响剔除，考虑使用协方差分析。
The first step is to analyze that the entrance score will certainly have a certain impact on the final score in real life. Here, we focus on the impact of different teaching methods. We should exclude the impact of the entrance score (mathematical basis) and consider using covariance analysis.
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第二步，数据组织，将姓名、数学、入学成绩和组别分别定义为“name”、“math”、“entrance”、“group”，并设置其标签为中文名称，将数据输入并保存为文件。
The second step is data organization. Define the name, math, enrollment score and group as "name", "math", "entry" and "group" respectively, set their labels as Chinese names, and input and save the data as a file.
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第三步，进行协方差分析的设置
首先我们需要先进行检验协方差分析的前提条件，即各组方差具有齐性。我们按照下图所示步骤进行方差齐性检验，检验结果其显著性概率P值为0.131>0.05，因此认为各组的方差具有齐性。
Step 3: Set the covariance analysis
First of all, we need to test the precondition of covariance analysis, that is, the variance of each group is homogeneous. We carried out the homogeneity test of variance according to the steps shown in the following figure, and the test result showed that the P value of the significance probability was 0.131>0.05, so it was considered that the variance of each group was homogeneous.
然后从下图我们可以看出，group与entrance交互作用的显著性概率P=0.784>0.05，因此认为它们之间没有交互作用，即协变量与控制变量之间没有交互作用（另一前提条件）。因此模型不再选择group*entrance。
Then we can see from the figure below that the significant probability of interaction between group and entry P=0.784>0.05, so we think there is no interaction between them, that is, there is no interaction between covariates and control variables (another precondition). Therefore, group * entry is no longer selected for the model.
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第四步，主要结果及分析
从下图中可以看出，group所对应的显著性概率值p=0.000<0.05，说明分组情况（不同的教学方法）对数学成绩有显著性影响，而entrance所对应的显著性概率值P=0.782>0.05，说明入学成绩对最后的成绩无显著性影响。
Step 4: Main results and analysis
It can be seen from the figure that the significance probability value corresponding to the group p=0.000<0.05, indicating that the grouping (different teaching methods) has a significant impact on the math scores, while the significance probability value corresponding to the entrance P=0.782>0.05, indicating that the enrollment score has no significant impact on the final scores.
下期预告：本期，我们学习了
协方差分析。
下一期，我们将会学习
多元方差分析的理论和实例。
Preview of the next issue: In this issue, we learned about covariance analysis. In the next issue, we will learn the theory and examples of multivariate analysis of variance.
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参考资料：百度百科，《SPSS 23 统计分析实用教程》
翻译：百度翻译
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