Factor Analysis (FA) is an exploratory data analysis method used to search influential underlying factors or latent variables from a set of observed variables. It helps in data interpretations by reducing the number of variables. It extracts maximum common variance from all variables and puts them into a common score.

因子分析(FA)是一种探索性数据分析方法，用于从一组观察到的变量中搜索有影响力的潜在因子或潜在变量。 通过减少变量数量，它有助于数据解释。 它从所有变量中提取最大共同方差，并将它们放入一个共同得分。

Factor analysis is widely utilised in market research, advertising, psychology, finance, and operation research. Market researchers use factor analysis to identify price-sensitive customers, identify brand features that influence consumer choice, and helps in understanding channel selection criteria for the distribution channel.

因子分析广泛应用于市场研究，广告，心理学，金融和运营研究。 市场研究人员使用因素分析来识别价格敏感的客户，识别影响消费者选择的品牌特征，并帮助理解分销渠道的渠道选择标准。

In this tutorial, you are going to cover the following topics:

在本教程中，您将涵盖以下主题：

• Factor Analysis因子分析
• Types of Factor Analysis因子分析的类型
• Determine Number of Factors确定因素数
• Factor Analysis Vs. Principle Component Analysis因子分析与 主成分分析
• Factor Analysis in PythonPython中的因素分析
• Adequacy Test充足性测试
• Interpreting the results解释结果
• Pros and Cons of Factor Analysis因素分析的利弊
• Conclusion结论

For more such tutorials, projects, and courses visit DataCamp:

有关更多此类教程，项目和课程，请访问DataCamp ：

因子分析 (Factor Analysis)

Factor analysis is a linear statistical model. It is used to explain the variance among the observed variable and condense a set of the observed variable into the unobserved variable called factors. Observed variables are modeled as a linear combination of factors and error terms (Source). Factor or latent variable is associated with multiple observed variables, who have common patterns of responses. Each factor explains a particular amount of variance in the observed variables. It helps in data interpretations by reducing the number of variables.

因子分析是线性统计模型。 它用于解释观察变量之间的方差，并将一组观察变量浓缩为称为因子的未观察变量。 观察变量被建模为因子和误差项的线性组合( Source )。 因子或潜在变量与具有共同响应模式的多个观察变量相关。 每个因素都说明了观察变量中的特定方差量。 通过减少变量数量，它有助于数据解释。

Factor analysis is a method for investigating whether a number of variables of interest X1, X2,……., Xl, are linearly related to a smaller number of unobservable factors F1, F2,..……, Fk.

因子分析是一种研究感兴趣的变量X1，X2，……，X1是否与较少数量的不可观察因子F1，F2，……，Fk线性相关的方法。

Source: This image is recreated from an image that I found in factor analysis notes. The image gives a full view of factor analysis.

来源：此图像是根据我在因子分析说明中找到的图像重新创建的。 该图提供了因素分析的完整视图。

Assumptions:

假设：

1. There are no outliers in data.数据中没有异常值。
2. The sample size should be greater than the factor.样本数量应大于因子。
3. There should not be perfect multicollinearity.不应有完美的多重共线性。
4. There should not be homoscedasticity between the variables.变量之间不应有同调性。

因子分析的类型 (Types of Factor Analysis)

• Exploratory Factor Analysis: It is the most popular factor analysis approach among social and management researchers. Its basic assumption is that any observed variable is directly associated with any factor.探索性因素分析：它是社会和管理研究人员中最流行的因素分析方法。 它的基本假设是，任何观察到的变量都与任何因素直接相关。
• Confirmatory Factor Analysis (CFA): Its basic assumption is that each factor is associated with a particular set of observed variables. CFA confirms what is expected on the basis.验证性因素分析(CFA)：其基本假设是每个因素都与一组特定的观察变量相关联。 CFA确认在此基础上的期望。

因子分析如何工作？ (How does factor analysis work?)

The primary objective of factor analysis is to reduce the number of observed variables and find unobservable variables. These unobserved variables help the market researcher to conclude the survey. This conversion of the observed variables to unobserved variables can be achieved in two steps:

因子分析的主要目的是减少观察变量的数量并发现不可观察的变量。 这些未观察到的变量有助于市场研究人员完成调查。 观察变量到未观察变量的这种转换可以通过两个步骤实现：

• Factor Extraction: In this step, the number of factors and approach for extraction selected using variance partitioning methods such as principal components analysis and common factor analysis.

  因子提取：在此步骤中，使用方差划分方法(例如主成分分析和公共因子分析)选择因子的数量和提取方法。

• Factor Rotation: In this step, rotation tries to convert factors into uncorrelated factors — the main goal of this step to improve the overall interpretability. There are lots of rotation methods that are available such as the Varimax rotation method, Quartimax rotation method, and Promax rotation method.

  因子轮换：在此步骤中，轮换尝试将因子转换为不相关的因子-此步骤的主要目标是提高整体的可解释性。 有许多可用的旋转方法，例如Varimax旋转方法，Quartimax旋转方法和Promax旋转方法。

术语 (Terminology)

What is a factor?

是什么因素？

A factor is a latent variable that describes the association among the number of observed variables. The maximum number of factors is equal to a number of observed variables. Every factor explains a certain variance in observed variables. The factors with the lowest amount of variance were dropped. Factors are also known as latent variables or hidden variables or unobserved variables or Hypothetical variables.

因子是一个潜在变量，它描述了观察到的变量数量之间的关联。 因素的最大数量等于观察到的变量的数量。 每个因素都说明观测变量存在一定差异。 方差量最低的因素被删除。 因子也称为潜在变量或隐藏变量或未观察到的变量或假设变量。

What are the factor loadings?

什么是因子负载？

The factor loading is a matrix that shows the relationship of each variable to the underlying factor. It shows the correlation coefficient for observed variables and factors. It shows the variance explained by the observed variables.

因子加载是一个矩阵，显示每个变量与基础因子之间的关系。 它显示了观测变量和因子的相关系数。 它显示了观察到的变量解释的方差。

What is Eigenvalues?

什么是特征值？

Eigenvalues represent variance explained each factor from the total variance. It is also known as characteristic roots.

特征值代表方差，由总方差解释每个因素。 它也被称为特征根。

What are Communalities?

什么是社区？

Commonalities are the sum of the squared loadings for each variable. It represents the common variance. It ranges from 0–1 and value close to 1 represents more variance.

共同点是每个变量的平方加载总和。 它代表共同方差。 它的范围是0到1，接近1的值表示更多的方差。

What is Factor Rotation?

什么是因子旋转？

Rotation is a tool for better interpretation of factor analysis. Rotation can be orthogonal or oblique. It re-distributed the commonalities with a clear pattern of loadings.

轮换是一种更好地解释因子分析的工具。 旋转可以是正交或倾斜的。 它以清晰的负载模式重新分配了共性。

选择因素数 (Choosing the Number of Factors)

Kaiser criterion is an analytical approach, which is based on the more significant proportion of variance explained by a factor that will be selected. The eigenvalue is a good criterion for determining the number of factors. Generally, an eigenvalue greater than 1 will be considered as the selection criteria for the feature.

凯撒(Kaiser)准则是一种分析方法，它基于差异的较大比例(由将要选择的因素解释)。 特征值是确定因子数量的良好标准。 通常，将大于1的特征值视为特征的选择标准。

The graphical approach is based on the visual representation of factors’ eigenvalues also called scree plots. This scree plot helps us to determine the number of factors where the curve makes an elbow.

图形化方法基于因子特征值的可视表示，也称为碎石图。 此卵石图有助于我们确定曲线成为弯头的因素的数量。

Source

资源

因子分析与 主成分分析 (Factor Analysis Vs. Principle Component Analysis)

• PCA components explain the maximum amount of variance while factor analysis explains the covariance in data.PCA组件说明最大方差，而因子分析说明数据中的协方差。
• PCA components are fully orthogonal to each other whereas factor analysis does not require factors to be orthogonal.PCA组件彼此完全正交，而因子分析不需要因子正交。
• PCA component is a linear combination of the observed variable while in FA, the observed variables are linear combinations of the unobserved variable or factor.PCA分量是观察变量的线性组合，而在FA中，观察变量是未观察变量或因子的线性组合。
• PCA components are uninterpretable. In FA, underlying factors are labelable and interpretable.PCA组件无法解释。 在FA中，潜在因素是可标记和可解释的。
• PCA is a kind of dimensionality reduction method whereas factor analysis is the latent variable method.PCA是一种降维方法，而因子分析是潜在变量方法。
• PCA is a type of factor analysis. PCA is observational whereas FA is a modeling technique.PCA是一种因素分析。 PCA是观察性的，而FA是一种建模技术。

Source

资源

使用factor_analyzer软件包在Python中进行因子分析 (Factor Analysis in Python using factor_analyzer package)

导入所需的库 (Import Required Libraries)

# Import required librariesimport pandas as pdfrom sklearn.datasets import load_irisfrom factor_analyzer import FactorAnalyzerimport matplotlib.pyplot as plt

加载数据中 (Loading Data)

Let’s perform factor analysis on BFI (dataset based on personality assessment project), which were collected using a 6 point response scale: 1 Very Inaccurate, 2 Moderately Inaccurate, 3 Slightly Inaccurate 4 Slightly Accurate, 5 Moderately Accurate, and 6 Very Accurate. You can also download this dataset from the following the link: https://vincentarelbundock.github.io/Rdatasets/datasets.html

让我们对BFI(基于人格评估项目的数据集)进行因素分析，这些因素是使用6点回应量表收集的：1个非常不准确，2个中等不准确，3个轻微不准确4个稍微准确，5个中等准确和6个非常准确。 您也可以从以下链接下载此数据集： https : //vincentarelbundock.github.io/Rdatasets/datasets.html

df= pd.read_csv("bfi.csv")

预处理数据 (Preprocess Data)

df.columnsOutput:Index(['A1', 'A2', 'A3', 'A4', 'A5', 'C1', 'C2', 'C3', 'C4', 'C5', 'E1', 'E2','E3', 'E4', 'E5', 'N1', 'N2', 'N3', 'N4', 'N5', 'O1', 'O2', 'O3', 'O4','O5', 'gender', 'education', 'age'],dtype='object')# Dropping unnecessary columnsdf.drop(['gender', 'education', 'age'],axis=1,inplace=True)# Dropping missing values rowsdf.dropna(inplace=True)df.info()Output:<class 'pandas.core.frame.DataFrame'>Int64Index: 2436 entries, 0 to 2799Data columns (total 25 columns):A1    2436 non-null float64A2    2436 non-null float64A3    2436 non-null float64A4    2436 non-null float64A5    2436 non-null float64C1    2436 non-null float64C2    2436 non-null float64C3    2436 non-null float64C4    2436 non-null float64C5    2436 non-null float64E1    2436 non-null float64E2    2436 non-null float64E3    2436 non-null float64E4    2436 non-null float64E5    2436 non-null float64N1    2436 non-null float64N2    2436 non-null float64N3    2436 non-null float64N4    2436 non-null float64N5    2436 non-null float64O1    2436 non-null float64O2    2436 non-null int64O3    2436 non-null float64O4    2436 non-null float64O5    2436 non-null float64dtypes: float64(24), int64(1)memory usage: 494.8 KBdf.head()Output:

充足性测试 (Adequacy Test)

Before you perform factor analysis, you need to evaluate the “factorability” of our dataset. Factorability means “can we found the factors in the dataset?”. There are two methods to check the factorability or sampling adequacy:

在执行因子分析之前，您需要评估我们数据集的“可分解性”。 可分解性意味着“我们可以在数据集中找到这些因素吗？”。 有两种方法可以检查可分解性或抽样是否足够：

• Bartlett’s Test巴特利特测试
• Kaiser-Meyer-Olkin TestKaiser-Meyer-Olkin检验

Bartlett’s test of sphericity checks whether or not the observed variables intercorrelate at all using the observed correlation matrix against the identity matrix. If the test found statistically insignificant, you should not employ a factor analysis.

Bartlett的球形度检验使用观察到的相关矩阵和恒等矩阵来检验观察到的变量是否相互关联。 如果测试发现统计上不重要，则不应使用因子分析。

from factor_analyzer.factor_analyzer import calculate_bartlett_sphericitychi_square_value,p_value=calculate_bartlett_sphericity(df)chi_square_value, p_valueOutput:(18146.065577234807, 0.0)

In Bartlett’s test, the p-value is 0. The test was statistically significant, indicating that the observed correlation matrix is not an identity matrix.

在Bartlett检验中，p值为0。该检验具有统计学意义，表明观察到的相关矩阵不是恒等矩阵。

Kaiser-Meyer-Olkin (KMO) Test measures the suitability of data for factor analysis. It determines the adequacy for each observed variable and for the complete model. KMO estimates the proportion of variance among all the observed variables. Lower proportion id more suitable for factor analysis. KMO values range between 0 and 1. The value of KMO less than 0.6 is considered inadequate.

Kaiser-Meyer-Olkin(KMO)测试测量数据是否适合进行因子分析。 它确定每个观察变量和完整模型的充分性。 KMO估计所有观测变量之间的方差比例。 较低的比例ID更适合因子分析。 KMO值在0到1之间。小于0.6的KMO值被认为是不合适的。

from factor_analyzer.factor_analyzer import calculate_kmokmo_all,kmo_model=calculate_kmo(df)kmo_modelOutput:0.8486452309468382

The overall KMO for our data is 0.84, which is excellent. This value indicates that you can proceed with your planned factor analysis.

我们的数据的总体KMO为0.84，非常好。 该值表示您可以继续进行计划的因素分析。

选择因素数 (Choosing the Number of Factors)

For choosing the number of factors, you can use the Kaiser criterion and scree plot. Both are based on eigenvalues.

要选择因子数量，可以使用Kaiser准则和卵石图。 两者均基于特征值。

# Create factor analysis object and perform factor analysisfa = FactorAnalyzer()fa.analyze(df, 25, rotation=None)# Check Eigenvaluesev, v = fa.get_eigenvalues()ev

Here, you can see only for 6-factors eigenvalues are greater than one. It means we need to choose only 6 factors (or unobserved variables).

在这里，您只能看到6因子特征值大于1。 这意味着我们只需要选择6个因素(或未观察到的变量)。

# Create scree plot using matplotlibplt.scatter(range(1,df.shape[1]+1),ev)plt.plot(range(1,df.shape[1]+1),ev)plt.title('Scree Plot')plt.xlabel('Factors')plt.ylabel('Eigenvalue')plt.grid()plt.show()

The scree plot method draws a straight line for each factor and its eigenvalues. Number eigenvalues greater than one considered as the number of factors.

卵石图方法为每个因子及其特征值绘制一条直线。 大于1的特征值数被视为因子数。

Here, you can see only for 6-factors eigenvalues are greater than one. It means we need to choose only 6 factors (or unobserved variables).

在这里，您只能看到6因子特征值大于1。 这意味着我们只需要选择6个因素(或未观察到的变量)。

执行因素分析 (Performing Factor Analysis)

# Create factor analysis object and perform factor analysisfa = FactorAnalyzer()fa.analyze(df, 6, rotation="varimax")fa.loadings

• Factor 1 has high factor loadings for E1,E2,E3,E4, and E5 (Extraversion)因子1具有E1，E2，E3，E4和E5(外推)的高因子负载
• Factor 2 has high factor loadings for N1, N2, N3, N4, and N5 (Neuroticism)因子2对N1，N2，N3，N4和N5具有较高的因子负荷(神经病)
• Factor 3 has high factor loadings for C1, C2, C3, C4, and C5 (Conscientiousness)因子3对C1，C2，C3，C4和C5具有较高的因子负荷(尽责程度)
• Factor 4 has a high factor loadings for O1, O2, O3, O4, and O5 (Openness)因子4对O1，O2，O3，O4和O5(开放度)的因子负载较高
• Factor 5 has high factor loadings for A1, A2, A3, A4, and A5 (Agreeableness)因子5对A1，A2，A3，A4和A5具有较高的因子负载(令人满意)
• Factor 6 has none of the high loadings for any variable and is not easily interpretable. It's good if we take only five factors.因子6没有任何变量的高负荷，也不容易解释。 如果我们仅考虑五个因素，那就太好了。

Let’s perform a factor analysis for 5 factors.

让我们对5个因素进行因素分析。

# Create factor analysis object and perform factor analysis using 5 factorsfa = FactorAnalyzer()fa.analyze(df, 5, rotation="varimax")fa.loadings

# Get variance of each factorsfa.get_factor_variance()

Total 42% cumulative Variance explained by the 5 factors.

5个因素解释了总计42％的累积方差。

因素分析的利弊 (Pros and Cons of Factor Analysis)

Factor analysis explores large datasets and finds interlinked associations. It reduces the observed variables into a few unobserved variables or identifies the groups of inter-related variables, which help the market researchers to compress the market situations and find the hidden relationship among consumer taste, preference, and cultural influence. Also, It helps in improving the questionnaire for future surveys. Factors make for more natural data interpretation.

因子分析探索大型数据集并找到相互关联的关联。 它可以将观察到的变量简化为几个未观察到的变量，或者识别相互关联的变量组，这有助于市场研究人员压缩市场状况，并找到消费者品味，偏好和文化影响力之间的隐藏关系。 而且，它有助于改进问卷以供将来进行调查。 因素使数据解释更加自然。

The results of the factor analysis are controversial. Its interpretations can be debatable because more than one interpretation can be made of the same data factors. After factor identification and naming of factors requires domain knowledge.

因子分析的结果是有争议的。 它的解释可能是有争议的，因为可以对相同的数据因素进行多种解释。 之后，因素识别和因素命名需要领域知识。

结论 (Conclusion)

Congratulations, you have made it to the end of this tutorial!

恭喜，您已完成本教程的结尾！

In this tutorial, you have learned what factor analysis is. The different types of factor analysis, how does factor analysis work, basic factor analysis terminology, choosing the number of factors, comparison of principal component analysis and factor analysis, implementation in Python using Python FactorAnalyzer package, and pros and cons of factor analysis.

在本教程中，您学习了什么是因子分析。 不同类型的因素分析，因素分析如何工作，基本因素分析术语，选择因素的数量，主成分分析和因素分析的比较，使用Python FactorAnalyzer软件包在Python中的实现以及因素分析的利弊。

I look forward to hearing any feedback or questions. you can ask the question by leaving a comment and I will try my best to answer it.

我期待听到任何反馈或问题。 您可以通过发表评论来提问，我会尽力回答。

Originally published at https://www.datacamp.com/community/tutorials/introduction-factor-analysis

最初发布在https://www.datacamp.com/community/tutorials/introduction-factor-analysis

Reach out to me on Linkedin: https://www.linkedin.com/in/avinash-navlani/

在Linkedin上与我联系： https ： //www.linkedin.com/in/avinash-navlani/