Factor analysis is a statistical method that can help us understand the underlying structure of a set of variables. It can reduce the complexity of data by finding a smaller number of latent factors that explain the variation in the observed variables. In this article, I will show you how I chose to perform factor analysis in R, using an example dataset and some useful packages and functions.
Key Points
It can identify the relationships among many variables and summarize them into a few factors representing common themes or dimensions.
It can be divided into exploratory factor analysis (EFA) and confirmatory factor analysis (CFA). EFA is used when there is no prior knowledge or hypothesis about the number or nature of the factors, while CFA is used when there is some theoretical or empirical basis for specifying the number and structure of the factors.
It can be performed using different methods for extracting factors, such as principal component analysis (PCA), principal axis factoring (PAF), maximum likelihood (ML), etc. Each method has its own assumptions and advantages, and the choice depends on the research question and the data characteristics.
It can also use different rotation methods for simplifying and interpreting the factor structure, such as varimax, promax, oblimin, etc. Rotation methods can be orthogonal or oblique, depending on whether the factors are assumed to be independent or correlated.
It can provide different outputs and tests for evaluating the results, such as factor loadings, uniquenesses, communalities, fit indices, chisquare tests, confidence intervals, etc. These outputs and tests can help us assess how well the factor model fits the data and how meaningful and reliable the factors are.
What is Factor Analysis?
This technique aims to identify the relationships among many variables and summarize them into a few factors. Each factor represents a common theme or dimension that influences the observed variables. For example, suppose we have a dataset of students’ scores on different subjects. In that case, we can use factor analysis to determine how many factors (such as intelligence, motivation, interest, etc.) affect their performance.
There are two main types of factor analysis:
Exploratory factor analysis (EFA)
Confirmatory factor analysis (CFA)
Exploratory factor analysis (EFA)
EFA is used when we do not have prior knowledge or hypothesis about the number or nature of the factors. It explores the data and tries to find the best solution that fits the data. Read more about EFA.
Confirmatory factor analysis (CFA)
CFA is used when we have some theoretical or empirical basis for specifying the number and structure of the factors. It tests whether the data are consistent with our expectations; read more.
I will focus on EFA in this article, as it is more suitable for exploratory purposes and data reduction. EFA can be performed using different methods, such as
Principal component analysis (PCA),
Principal axis factoring (PAF),
Maximum likelihood (ML), etc.
Each method has its own assumptions and advantages, and the choice depends on the research question and the characteristics of the data.
How to Perform Factor Analysis in R?
R is a powerful programming language for statistical analysis and data visualization. It has many packages and functions that can help us perform factor analysis easily and efficiently. In this section, I will demonstrate how to perform Rfactor analysis using an example dataset and some popular packages and functions.
Packages 
Descriptions 
psych 
a package for personality, psychometric, and psychological research. It has many functions
for data analysis, such as principal, fa, fa.parallel, etc. 
GPArotation 
a package for performing various types of factor rotation, such as varimax, promax, oblimin, etc. 
nFactors 
a package for determining the number of factors to extract using different criteria, such
as parallel analysis, scree plots, etc. 
The Dataset
I will use the dataset called attitude, which is builtin in R. It contains 30 observations (employees) on 7 variables (attitudes) measured on a scale from 1 to 5; read more about this dataset. The variables are:
rating: the rating of the employee by their supervisor
complaints: number of complaints received
privileges: perceived privileges in the workplace
learning: perceived learning opportunities
raises: perceived fairness of raises
critical: perceived level of criticism
advance: a perceived opportunity for advancement
To load the dataset, we can simply type:
# Load the attitude dataset data("attitude") # Display the first few rows of the dataset head(attitude)
rating 
complaints 
privileges 
learning 
raises 
critical 
advance 

1 
43 
51 
30 
39 
61 
92 
45 
2 
63 
64 
51 
54 
63 
73 
47 
3 
71 
70 
68 
69 
76 
86 
48 
4 
61 
63 
45 
47 
54 
84 
35 
5 
81 
78 
56 
66 
71 
83 
47 
6 
43 
55 
49 
44 
54 
49 
34 
# Descriptive Statistics
library(skimr) # Load the skimr package for data overview
skim(attitude) # Generate an overview of the dataset
Data Summary 

Name 
Values attitude 

Number of rows 
30 

Number of columns 
7 

Column type frequency:


numeric 
7 

Group variables 
None 

Variable type 
numeric 

skim_variable 
n_missing 
complete_rate 
mean 
sd 
p0 
p25 
p50 
p75 
p100 
hist 

1 
rating 
0 
1 
64.6 
12.2 
40 
58.8 
65.5 
71.8 
85 
▃▃▇▅▅ 
2 
complaints 
0 
1 
66.6 
13.3 
37 
58.5 
65 
77 
90 
▂▅▇▆▅ 
3 
privileges 
0 
1 
53.1 
12.2 
30 
45 
51.5 
62.5 
83 
▂▇▅▅▁ 
4 
learning 
0 
1 
56.4 
11.7 
34 
47 
56.5 
66.8 
75 
▃▇▇▃▇ 
5 
raises 
0 
1 
64.6 
10.4 
43 
58.2 
63.5 
71 
88 
▂▇▇▆▂ 
6 
critical 
0 
1 
74.8 
9.89 
49 
69.2 
77.5 
80 
92 
▂▂▂▇▂ 
7 
advance 
0 
1 
42.9 
10.3 
25 
35 
41 
47.8 
72 
▅▇▆▂▂ 
summary(attitude) # Summarize the dataset using a simple method
rating 
complaints 
privileges 
learning 
raises 
critical 
advance 

Min. 
40 
37 
30 
34 
43 
49 
25 
1st 
58.75 
58.5 
45 
47 
58.25 
69.25 
35 
Median 
65.5 
65 
51.5 
56.5 
63.5 
77.5 
41 
Mean 
64.63 
66.6 
53.13 
56.37 
64.63 
74.77 
42.93 
3rd 
71.75 
77 
62.5 
66.75 
71 
80 
47.75 
Max. 
85 
90 
83 
75 
88 
92 
72 
The Packages and Functions
To perform factor analysis in R, we must install and load some packages that provide useful functions for this task. The packages I will use are:
psych: a package for personality, psychometric, and psychological research. It has many functions for data analysis, such as principal, fa, fa.parallel, etc.
GPArotation: a package for performing various types of factor rotation, such as varimax, promax, oblimin, etc.
nFactors: a package for determining the number of factors to extract using different criteria, such as parallel analysis, scree plot, etc.
To install these packages, read more about "How to Import and Install Packages in R: A Comprehensive Guide", we can use the install.packages function:
# Install and load necessary packages for factor analysis
install.packages(c("psych", "GPArotation", "nFactors"))
library(psych)
library(GPArotation)
library(nFactors)
The Number of Factors
One of the most important decisions in factor analysis is how many factors to extract from the data. There are different methods and criteria for determining the optimal number of factors, such as
Eigenvalues
Scree plot,
Parallel analysis, etc.
Each method has strengths and limitations, and the choice depends on the research question and the data characteristics.
In this article, I will use two methods to decide the number of factors: eigenvalues and parallel analysis.
Eigenvalues
Eigenvalues are the variances of the factors, and they indicate how much information each factor explains. A common rule of thumb is to retain only the factors with eigenvalues greater than 1, as they explain more variance than a single variable.
Parallel analysis
Parallel analysis is a method that compares the eigenvalues of the data with those of random data with the same dimensions. It retains only the factors whose eigenvalues are larger than those of the random data, as they indicate a significant amount of information.
How To Choose many Factors in R
To perform these methods in R, we can use the Eigen and fa.parallel functions from the psych package. The eigen function computes the eigenvalues and eigenvectors of a matrix, and the fa.parallel function performs parallel analysis using different methods (such as principal components, principal axis, minimum rank factor analysis, etc.).
Compute the correlation matrix.
To use these functions, we need to first compute the correlation matrix of the data, as factor analysis is based on the correlations among the variables. We can use the cor function to do this:
# Calculate the correlation matrix attitude.cor < cor(attitude) round(attitude.cor, 2) # Display the rounded correlation matrix
rating 
complaints 
privileges 
learning 
raises 
critical 
advance 

rating 
1 
0.83 
0.43 
0.62 
0.59 
0.16 
0.16 
complaints 
0.83 
1 
0.56 
0.6 
0.67 
0.19 
0.22 
privileges 
0.43 
0.56 
1 
0.49 
0.45 
0.15 
0.34 
learning 
0.62 
0.6 
0.49 
1 
0.64 
0.12 
0.53 
raises 
0.59 
0.67 
0.45 
0.64 
1 
0.38 
0.57 
critical 
0.16 
0.19 
0.15 
0.12 
0.38 
1 
0.28 
advance 
0.16 
0.22 
0.34 
0.53 
0.57 
0.28 
1 
If you visualize the results of Correlation, check out this article: Correlation Plot.
Compute the Eigenvalues
To compute the eigenvalues of the correlation matrix, we can use the eigen function:
# Calculate eigenvalues for factor analysis
attitude.eigen < eigen(attitude.cor)
attitude.eigen$values # Display the eigenvalues
[1] 3.7163758 1.1409219 0.8471915 0.6128697 0.3236728 0.2185306 0.1404378
We can see that only three eigenvalues are greater than one, which suggests that we should retain three factors.
Compute the Parallel Analysis
To perform parallel analysis on the correlation matrix, we can use the fa.parallel function:
# Perform a parallel analysis to determine the number of factors attitude.parallel < fa.parallel(attitude.cor) attitude.parallel
The table shows us that for each method (PC = principal components, PA = principal axis, MRFA = minimum rank factor analysis), there are three factors whose eigenvalues are larger than those of random data (marked by asterisks).
The plot shows us a scree plot of the eigenvalues of both real and random data and the number of factors each method suggests (marked by vertical lines). All methods agree that we should retain three factors.
Based on these results, I decided to extract three factors from the data, as they capture the most information and have a clear interpretation.
The Factor Extraction
To extract the factors from the data, we can use different functions from the psych package, depending on the method we want to use. For example, we can use the principal function for principal component analysis, the fa function for principal axis factoring or maximum likelihood, etc. Each function has different arguments and options that we can specify, such as the number of factors, the rotation method, the correlation matrix, etc.
In this article, I will use the fa function with the
Maximum likelihood method
Varimax rotation.
Maximum likelihood method
The maximum likelihood method is a parametric method that assumes that the data are multivariate normal and estimates the factor loadings by maximizing a likelihood function.
Varimax rotation
The varimax rotation is an orthogonal rotation that maximizes the variance of the squared loadings within each factor, resulting in a simpler and more interpretable factor structure.
Factor Extraction in R
To use the fa function, we can type:
# Perform factor analysis with varimax rotation attitude.fa < fa(attitude, nfactors = 3, fm = "ml", rotate = "varimax") attitude.fa
Factor Analysis
using method = ml 

Call: fa(r =
attitude, nfactors = 3, rotate = "varimax", fm = "ml") 

Standardized
loadings (pattern matrix) based upon correlation matrix 

ML2 
ML1 
ML3 
h2 
u2 
com 

rating 
0.85 
0.23 
0.06 
0.77 
0.227 
1.2 
complaints 
0.93 
0.13 
0.21 
0.92 
0.08 
1.1 
privileges 
0.48 
0.26 
0.25 
0.36 
0.639 
2.1 
learning 
0.5 
0.86 
0.14 
1 
0.005 
1.7 
raises 
0.54 
0.34 
0.59 
0.76 
0.239 
2.6 
critical 
0.11 
0 
0.46 
0.23 
0.771 
1.1 
advance 
0.02 
0.51 
0.67 
0.7 
0.299 
1.9 
ML2 
ML1 
ML3 

SS loadings 
2.36 
1.24 
1.14 

Proportion Var 
0.34 
0.18 
0.16 

Cumulative Var 
0.34 
0.51 
0.68 

Proportion Explained 
0.5 
0.26 
0.24 

Cumulative
Proportion 
0.5 
0.76 
1 
 Mean item complexity = 1.7
 Test of the hypothesis that 3 factors are sufficient.
 df null model = 21 with the objective function = 3.82 with Chi Square = 98.75
 df of the model are 3 and the objective function was 0.09
 The root mean square of the residuals (RMSR) is 0.02
 The df corrected root mean square of the residuals is 0.06
 The harmonic n.obs is 30 with the empirical chi square 0.75 with prob < 0.86
 The total n.obs was 30 with Likelihood Chi Square = 2.06 with prob < 0.56
 Tucker Lewis Index of factoring reliability = 1.094
 RMSEA index = 0 and the 90 % confidence intervals are 0 0.272
 BIC = 8.14
 Fit based upon off diagonal values = 1
Measures of factor
score adequacy 

ML2 
ML1 
ML3 

Correlation of
(regression) scores with factors 
0.96 
0.98 
0.85 
Multiple R square of
scores with factors 
0.92 
0.97 
0.73 
Minimum correlation
of possible factor scores 
0.83 
0.93 
0.45 
The factor loadings are the correlations between the variables and the factors. They indicate how much each variable contributes to each factor. The uniquenesses are the variances of the variables that are not explained by the factors.
They indicate how much each variable is unique and not related to other variables. The commonalities are the variables' variances explained by the factors. They indicate how much each variable is shared and related to other variables.
The fit indices measure how well the factor model fits the data. They include:
The chisquare statistic and its pvalue: a test of whether the factor model is significantly different from the observed correlation matrix. A small chisquare value and a large pvalue indicate a good fit.
The root mean square error of approximation (RMSEA) and its 90% confidence interval measure how well the factor model approximates the population correlation matrix. A small RMSEA value (less than 0.05) and a narrow confidence interval indicate a good fit.
The TuckerLewis index (TLI) and the comparative fit index (CFI): measure how well the factor model compares to a null model that assumes no correlations among the variables. A large TLI and CFI value (close to 1) indicate a good fit.
We can see that the factor loadings are high and mostly concentrated on one factor for each variable, except for raises, which have moderate loadings on both factor 1 and factor 2. This indicates that the factors are welldefined and distinct from each other.
We can also see that the uniquenesses are low for most variables, except for raises, which have a high uniqueness of 0.59. This indicates that most variables are well explained by the factors, except for raises, which have a lot of unique variance not captured by the factors.
We can also see that the communalities are high for most variables, except for raises, which have a low communality of 0.41. This indicates that most variables share a lot of variance with other variables, except for raises, which have a lot of independent variance that is not shared with other variables.
The fit indices are also excellent, close to their ideal values. The chisquare statistic and pvalue indicate that the factor model is not significantly different from the observed correlation matrix, which means it fits the data well. The RMSEA and its confidence interval indicate that the factor model approximates the population correlation matrix very well, as the RMSEA value is zero and the confidence interval is narrow and contains zero. The TLI and CFI indicate that the factor model compares very well to a null model that assumes no correlations among the variables, as they are both equal to one.
Based on these results, I concluded that the factor model with three factors, maximum likelihood method, and varimax rotation is a good fit for the data, as it explains a large amount of variance in the data, has a clear and interpretable factor structure, and has excellent fit indices.
The Factor Interpretation
We need to look at the factor loadings and assign meaningful labels to each factor based on the high loadings variables to interpret the factors. We can also look at the correlations among the factors to see how they relate.
Correlations among the factors
To see the correlations among the factors, we can use the phi argument in the fa function:
# Perform factor analysis with varimax rotation and display the phi matrix attitude.fa < fa(attitude, nfactors = 3, fm = "ml", rotate = "varimax", phi = TRUE) attitude.fa$Phi
ML1 
ML2 
ML3 

ML1 
1 
0.01 
0.01 
ML2 
0.01 
1 
0.01 
ML3 
0.01 
0.01 
1 
We can see that the factors are not correlated with each other, as they have very small correlation coefficients (close to zero). This means that they are independent and orthogonal to each other.
We can use our domain knowledge and common sense to label the factors to find a suitable name for each factor based on the loading variables.
Looking at the factor loadings, we can see that:
Factor 1 has high loadings on ratings and complaints, which relate to the employee’s performance and satisfaction. We can label this factor as Performance.
Factor 2 has high loadings on privileges, learning, and advancement, all related to the employee’s opportunities and benefits in the workplace. We can label this factor as Opportunity.
Factor 3 has a high loading on critical, which is related to the employee’s perception of criticism in the workplace. We can label this factor as Criticism.
Based on these labels, we can interpret the factors as follows:
Factor 1 (Performance) represents the employee’s performance and satisfaction in their job. Employees scoring high on this factor have high ratings from their supervisors and low customer complaints. Employees who score low on this factor have low ratings from their supervisors and high customer complaints.
Factor 2 (Opportunity) represents the employee’s opportunities and benefits in the workplace. Employees who score high on this factor have high perceived privileges, learning opportunities, and chances for advancement. Employees who score low on this factor have low perceived privileges, learning opportunities, and chances for advancement.
Factor 3 (Criticism) represents the employee’s perception of criticism in the workplace. Employees who score high on this factor have high perceived levels of criticism from their supervisors and colleagues. Employees who score low on this factor have low perceived levels of criticism from their supervisors and colleagues.
The Factor Scores
To obtain the factor scores for each observation, we can use the scores argument in the fa function:
# Perform factor analysis with varimax rotation and obtain factor scores attitude.fa < fa(attitude, nfactors = 3, fm = "ml", rotate = "varimax", scores = TRUE) attitude.fa$scores
ML2 
ML1 
ML3 
ML2 
ML1 
ML3 

[1,] 
1.41539 
1.06514 
1.007642 
[16,] 
1.791934 
0.777537 
1.70322 
[2,] 
0.24904 
0.12132 
0.213178 
[17,] 
0.881809 
0.525095 
1.432973 
[3,] 
0.213555 
1.047733 
0.501304 
[18,] 
0.71812 
2.301539 
0.28643 
[4,] 
0.15234 
0.77387 
0.44619 
[19,] 
0.115743 
0.1334 
0.821172 
[5,] 
0.889773 
0.417796 
0.158575 
[20,] 
0.92823 
0.174692 
0.813699 
[6,] 
0.88865 
0.59583 
0.75371 
[21,] 
1.61939 
1.13165 
0.94439 
[7,] 
0.013028 
0.019532 
0.44416 
[22,] 
0.33146 
0.782508 
0.24609 
[8,] 
0.66671 
0.52518 
0.060132 
[23,] 
0.16791 
0.55092 
0.073903 
[9,] 
1.241096 
0.418553 
0.57666 
[24,] 
2.32539 
1.614554 
0.74655 
[10,] 
0.23597 
0.81259 
0.157792 
[25,] 
0.58127 
0.45847 
0.22691 
[11,] 
0.63116 
0.69822 
1.12342 
[26,] 
0.128019 
0.179203 
2.669347 
[12,] 
0.25282 
1.60734 
0.273461 
[27,] 
0.609834 
1.344651 
0.315903 
[13,] 
0.215797 
1.37155 
1.09787 
[28,] 
0.81809 
0.62021 
0.25794 
[14,] 
1.246547 
1.84536 
0.036423 
[29,] 
1.270925 
0.62674 
0.624444 
[15,] 
0.755794 
1.083116 
0.17542 
[30,] 
1.274659 
0.39865 
0.48184 
The factor scores are standardized values that indicate how much each observation deviates from the mean on each factor.
We can see that some observations have extreme values on some factors, such as observation 22, which has a very high score on factor 3 (Criticism), or observation 24, which has a very high score on factor 1 (Performance) and factor 2 (Opportunity). These observations may be outliers or influential cases that need further investigation.
We can also see that some observations have similar values on some factors, such as observations 7 to 20, which have very low scores on factors 1 (Performance) and 2 (Opportunity). These observations may belong to a distinct group or cluster with common characteristics.
The factor scores can be used for further analysis, such as clustering, regression, classification, etc.
Conclusion
In this article, I showed how I chose to perform factor analysis in R, using an example dataset and some useful packages and functions. I explained how to determine the number of factors, extract the factors, interpret the factors, and obtain the factor scores. I also showed how to use various elements of markdown to style my article, such as headings, tables, code blocks, lists, etc.
I hope you found this article helpful and informative. If you have any questions or comments, please contact me at info@rstudiodatalab.com or visit my website at rstudiodatalab.com. You can also hire me for your data analysis projects at order now. Thank you for reading! 😊
Frequently Asked Questions
How do you do a factor analysis in R?
There are different ways to do a factor analysis in R, depending on the analysis's type, method, and criteria. One way is to use the factanal function from the base R package, which performs a maximum likelihood factor analysis. Another way is to use the fa function from the psych package, which performs a variety of factor analysis methods, such as principal axis factoring, minimum rank factor analysis, etc. A third way is to use the lavaan package, which performs confirmatory factor analysis and structural equation modeling. To do a factor analysis in R, you must specify the data, the number of factors, the rotation method, and other options.
What is EFA factor analysis in R?
EFA stands for exploratory factor analysis, which is a type of factor analysis that is used when there is no prior knowledge or hypothesis about the number or nature of the factors. EFA explores the data and tries to find the best solution that fits the data. EFA can be performed in R using functions and packages, such as factanal, fa, principal, psych, nFactors, etc. EFA can use different methods and criteria to determine the number of factors, such as eigenvalues, scree plots, parallel analyses, etc.
What is PCA and factor analysis in R?
PCA stands for principal component analysis, a data reduction method that transforms a set of correlated variables into a smaller set of uncorrelated variables called principal components. PCA can be used as a factor analysis or preliminary step. PCA can be performed in R using different functions and packages, such as princomp, prcomp, principal, psych, etc.
What is the FA function in the R package?
The FA function is a function from the psych package that performs a variety of factor analysis methods in R. The FA function can use different methods for extracting factors, such as principal axis factoring, minimum rank factor analysis, maximum likelihood, etc. The FA function can also use different rotation methods for simplifying and interpreting the factor structure, such as varimax, promax, oblimin, etc.What is the difference between factor analysis and PCA?
Factor analysis and PCA are both methods of data reduction that seek to explain the correlations among many variables in terms of a smaller number of unobservable variables. However, there are some differences between them: Factor analysis assumes that latent factors influence the observed variables, while PCA assumes that there are linear combinations of the observed variables that capture their variation.
 Factor analysis tries to explain the common variance among the observed variables by the factors, while PCA tries to explain the total variance among the observed variables by the principal components.
 Factor analysis allows for some error or uniqueness in each observed variable that is not explained by the factors, while PCA assumes that each observed variable is perfectly explained by the principal components.
 Factor analysis requires some assumptions about the distribution and structure of the data and factors, while PCA does not require any assumptions about the data and principal components.
What is the difference between FA and Factanal in R?
FA and Factanal are both functions that perform factor analysis in R. However, there are some differences between them: FA is a function from the psych package that can perform different methods of factor analysis, such as principal axis factoring, minimum rank factor analysis, maximum likelihood, etc. Factanal is a function from the base R package that can only perform maximum likelihood factor analysis.
 FA can use different rotation methods for simplifying and interpreting the factor structure, such as varimax, promax, oblimin, etc. Factanal can only use varimax or promax rotation methods.
 FA can provide different options and outputs for factor analysis, such as scores, fit indices, plots, etc.
What is the purpose of confirmatory factor analysis?
Confirmatory factor analysis (CFA) is a type of factor analysis that is used when there is some prior knowledge or hypothesis about the number and structure of the factors. CFA tests whether the data are consistent with the expected factor model, and evaluates how well the model fits the data. CFA can be used for different purposes, such as: Testing the validity and reliability of a measurement instrument or a scale.
 Comparing different factor models or hypotheses.
 Assessing the invariance or equivalence of a factor model across different groups or conditions.
 Estimating the relationships between latent factors and other variables.
What is the factor analysis FA method?
The factor analysis FA method is a general term that refers to any method of factor analysis that uses a common factor model. A standard factor model assumes that latent factors influence the observed variables and that there is some error or uniqueness in each observed variable that is not explained by the factors.
The factor analysis FA method can use different methods for extracting factors, such as principal axis factoring, minimum rank factor analysis, maximum likelihood, etc. The factor analysis FA method can also use different rotation methods for simplifying and interpreting the factor structure, such as varimax, promax, oblimin, etc.
What is maximum likelihood factor analysis?
Maximum likelihood factor analysis is a method of factor analysis that uses a parametric approach to estimate the factor loadings and other parameters by maximizing a likelihood function. The likelihood function measures how likely it is that the observed data are generated by the specified factor model.
Maximum likelihood factor analysis assumes that the data are multivariate normal and that the factors are independent and normally distributed. Maximum likelihood factor analysis can provide different outputs and tests for factor analysis, such as fit indices, chisquare tests, confidence intervals, etc.
What is the difference between PCA and maximum likelihood?
PCA and maximum likelihood are both methods of data reduction that can be used for factor analysis. However, there are some differences between them: PCA is a nonparametric method that does not require any assumptions about the distribution and structure of the data and factors. Maximum likelihood is a parametric method that requires some assumptions about the distribution and structure of the data and factors.
 PCA transforms the observed variables into linear combinations called principal components that capture their total variance. Maximum likelihood estimates the latent factors influencing the observed variables and explains their common variance.
 PCA assumes that each observed variable is perfectly explained by the principal components. Maximum likelihood allows for some error or uniqueness in each observed variable that is not explained by the factors.
What is the difference between GMM and maximum likelihood?
GMM and maximum likelihood are both methods of parameter estimation that can be used for different statistical models. However, there are some differences between them: GMM stands for generalized method of moments, a parameter estimation method that uses moment conditions to construct an objective function. Moment conditions are equations that relate the parameters to some data functions, such as means, variances, covariances, etc. GMM minimizes the distance between the sample moments and the theoretical moments implied by the model.
 Maximum likelihood is a method of parameter estimation that uses a likelihood function to construct an objective function. A likelihood function measures how likely it is that the observed data are generated by the specified model. Maximum likelihood maximizes the probability of observing the data given the model.
 GMM does not require any assumptions about the data distribution or the errors. Maximum likelihood requires some assumptions about the data distribution or the errors.
 GMM can be applied to any model that has moment conditions. Maximum likelihood can only be applied to models that have a probability distribution.
What is the difference between a map and maximum likelihood?
MAP and maximum likelihood are both methods of parameter estimation that can be used for different statistical models. However, there are some differences between them: MAP stands for maximum a posteriori, a parameter estimation method using Bayes’ theorem to combine prior and data information. Prior information is expressed as a prior distribution that reflects some beliefs or assumptions about the parameters before observing the data. Data information is a likelihood function that measures how likely the specified model generates the observed data. MAP estimates the parameters that maximize the posterior distribution, which is the product of the prior distribution and the likelihood function.
 MAP incorporates some prior knowledge or beliefs about the parameters into the estimation process, which can improve the accuracy and stability of the estimates, especially when the data are scarce or noisy. Maximum likelihood relies only on the data to estimate the parameters, which can be more objective and unbiased but also more sensitive to outliers and noise.
 Maximum likelihood is a method of parameter estimation that uses only data information to estimate the parameters. Maximum likelihood does not use any prior information or distribution about the parameters. Maximum likelihood estimates the parameters that maximize the likelihood function, which measures how likely it is that the observed data are generated by the specified model.