Yishuo School District (24) | SPSS Statistical Analysis (34) Multivariate Analysis of Variance
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SPSS统计分析
上一期，我们一起学习了协方差分析的相关知识，这一期，我们一起来了解多元方差分析的有关内容。
In the last issue, we learned about covariance analysis together. In this issue, we learned about multivariate analysis of variance.
多元方差分析是研究多个控制因素（自变量）与多个因变量相互关系的一种统计分析方法，又称为多变量分析。多元分析实质上是单变量统计方法的发展和推广，适用于研究控制因素同时对两个或者两个以上的因变量产生影响的情况，用来分析控制因素取不同水平时这些因变量的均值是否存在显著性差异。
Multivariate analysis of variance is a statistical analysis method to study the relationship between multiple control factors (independent variables) and multiple dependent variables, also known as multivariate analysis. In essence, multivariate analysis is the development and promotion of univariate statistical methods. It is applicable to the study of the influence of control factors on two or more dependent variables at the same time, and is used to analyze whether there is significant difference in the mean values of these dependent variables when the control factors are at different levels.
多元方差分析的基本原理同一元方差分析相似，是将总变异按照其来源（或实验设计）分为多个部分，从而检验各个因素对因变量的影响以及各因素间的交互作用。在这个过程中可以分析每个因素的作用，也可以分析因素之间的交互作用、协方差，以及各控制因素与协变量之间的交互作用。
The basic principle of multivariate analysis of variance is similar to that of univariate analysis of variance, which is to divide the total variation into multiple parts according to its source (or experimental design), so as to test the influence of each factor on the dependent variable and the interaction between each factor. In this process, we can analyze the role of each factor, the interaction between factors, covariance, and the interaction between control factors and covariates.
多元方差分析的优点是可以在一次研究中同时检验具有多个水平的多个因素各自对因变量的影响以及各因素之间的交互作用。
The advantage of multivariate analysis of variance is that it can simultaneously test the influence of multiple factors with multiple levels on the dependent variables and the interaction between the factors in a single study.
在方差分析中，要求样本必须满足独立、正态、等方差的总体，而对于多元方差分析而言，由于涉及多个因变量，除了要求每个因变量满足以上条件之外，还必须满足以下条件：
In variance analysis, the sample must meet the requirements of independent, normal, and equal variance population. For multiple variance analysis, because multiple dependent variables are involved, in addition to the requirements that each dependent variable meet the above conditions, the following conditions must also be met:
● 各变量之间具有相关性
Correlation between variables
●每一组都有相同的方差——协方差矩阵
Each group has the same variance covariance matrix
●各因变量为多元正态分布
All dependent variables are multivariate normal distribution
多元方差分析的步骤与单因素方差分析和协方差分析比较相近，下面通过具体实例来说明。
The steps of multivariate analysis of variance are similar to those of single factor analysis of variance and analysis of covariance, which are explained by specific examples below.
某科研所研究某树种在不同海拔、不同施肥量情况下的苗高增加量和地径增加量的差别，将海拔设置为3个水平，并将施肥量也设在了3个水平，将两个因素组合成9个组合，每个组合重复三次。试分析海拔和施肥量对苗高增加量和地径增加量的影响，并分析海拔与施肥量是否存在交互作用。
A scientific research institute studied the difference between the increase of seedling height and the increase of ground diameter of a tree species at different altitudes and different fertilization amounts, set the altitude at three levels, and set the fertilization amount at three levels. The two factors were combined into nine combinations, and each combination was repeated three times. The effects of altitude and fertilization amount on the increase of seedling height and ground diameter were analyzed, and whether there was interaction between altitude and fertilization amount was analyzed.
第一步，分析，这是一个两个控制因素对两个因变量影响的分析，是一个多元方差分析问题，我们按照题目组织4列数据，并保存数据文件。
The first step is analysis. This is an analysis of the influence of two control factors on two dependent variables. It is a multiple analysis of variance problem. We organize four columns of data according to the topic and save the data file.
第二步，选择菜单“分析->一般线性模型->多变量”，将“苗高增加量”和“地径增加量”移入因变量框，将“海拔”和“施肥量”移入固定因子框。打开“事后比较”对话框，将“海拔”和“施肥量”移入到“下列各项的事后检验”列表框，并勾选“假定等方差”选项组中的“LSD”复选框。打开“选项”对话框，在“显示”选项组中，勾选“齐性检验”复选框，完成设置并运行。
Step 2: select "Analysis ->General Linear Model ->Multivariable" from the menu, move "Increase in Seedling Height" and "Increase in Ground Diameter" into the dependent variable box, and move "Elevation" and "Fertilization Amount" into the fixed factor box. Open the "Post comparison" dialog box, move "Altitude" and "Fertilization amount" into the "Post inspection of the following items" list box, and check the "LSD" check box in the "Assumed equal variance" option group. Open the "Options" dialog box, and in the "Display" option group, check the "Homogeneity inspection" check box to complete the setting and run.
第三步，主要结果及分析，从误差方差的莱文等同性检验可以看出，苗高增加量和地径增加量的显著性概率P值分别为0.344与0.166，均大于显著性水平0.05，说明两者在各组总体方差具有齐性，满足方差分析的前提条件。
The third step is the main results and analysis. It can be seen from the Levin's equality test of error variance that the P values of the significance probabilities of the increase in seedling height and the increase in ground diameter are 0.344 and 0.166 respectively, which are greater than the significance level of 0.05, indicating that the overall variances of the two groups are homogeneous, meeting the preconditions of ANOVA.
根据多变量检验结果，可看出海拔与施肥量两个主效应的4种检验显著性概率均小于0.05，说明海拔与施肥量对苗高增加量和地径增加量有显著性影响。而“海拔*施肥量”的4种检验的显著性概率均大于0.05，说明两者对苗高增加量和地径增加量的影响不存在交互作用。
According to the results of multivariate test, it can be seen that the four test significance probabilities of the two main effects of altitude and fertilizer amount are all less than 0.05, indicating that altitude and fertilizer amount have significant effects on the increase of seedling height and ground diameter. The significance probability of the four tests of "altitude * fertilization amount" is greater than 0.05, indicating that there is no interaction between them on the increase of seedling height and the increase of ground diameter.
根据主体间效应的检验结果，苗高增加量在海拔和施肥量上的显著性概率分别为0.002和0.000，说明苗高增加量在海拔和施肥量上均存在显著性差异；地径增加量在海拔和施肥量上的显著性概率分别为0.018和0.000，说明地径增加量在海拔和施肥量上均存在显著性差异；而苗高增加量与地径增加量在“海拔*施肥量”上的显著性概率为0.237和0.058均大于0.05，说明海拔与施肥量的交互作用在苗高增加量与地径增加量上无显著性差异。
According to the test results of inter subject effect, the significant probabilities of seedling height increase in altitude and fertilization amount are 0.002 and 0.000 respectively, which indicates that there are significant differences in both altitude and fertilization amount; The significant probabilities of the increment of ground diameter in altitude and fertilization amount were 0.018 and 0.000, respectively, indicating that there were significant differences in the increment of ground diameter in altitude and fertilization amount; However, the significant probabilities of the increase of seedling height and ground diameter in the "altitude * fertilizer amount" were 0.237 and 0.058, both greater than 0.05, indicating that the interaction between altitude and fertilizer amount had no significant difference in the increase of seedling height and ground diameter.
从海拔的多重比较结果分析可以看出，可看出苗高增加量在海拔1与2、1与3、2与3上的显著性概率分别为0.927、0.002和0.002，说明苗高增加量在海拔1与3、2与3上存在显著性差异，在1与2上没有显著性差异；同时，可以看出地径增加量在海拔1与3、2与3之间存在显著性差异，而在1与2上没有显著性差异。
From the analysis of multiple comparison results of altitude, it can be seen that the significant probabilities of seedling height increase at altitudes 1 and 2, 1 and 3, 2 and 3 are 0.927, 0.002 and 0.002 respectively, indicating that there is a significant difference in seedling height increase at altitudes 1 and 3, 2 and 3, but there is no significant difference at altitudes 1 and 2; At the same time, it can be seen that there is a significant difference in the increment of ground diameter between elevations 1 and 3, 2 and 3, but there is no significant difference between elevations 1 and 2.
同理，我们在施肥量的多重比较分析结果种可以看出，苗高施肥量在施肥量1与2、1与3和2与3上均存在显著性差异；地径增加量在施肥量1与2、1与3上存在显著性差异，而在2与3上没有显著性差异。
In the same way, we can see from the results of multiple comparative analysis of fertilizer amount that the fertilizer amount for seedling height has significant differences in fertilizer amount 1 and 2, 1 and 3, and 2 and 3; There were significant differences in the increment of ground diameter between fertilization amount 1 and 2, 1 and 3, but there was no significant difference between fertilization amount 2 and 3.
下期预告：本期，我们学习了
多元方差分析。
下一期，我们将会学习新的一章
关于相关分析的理论和实例。
Preview of the next issue: In this issue, we learned the multiple variance analysis. In the next issue, we will learn a new chapter about the theory and examples of correlation analysis.
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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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