Confirmatory Factor Analysis (CFA)

Created:December 4, 2025

Last Updated:January 6, 2026

This calculator performs comprehensive Confirmatory Factor Analysis (CFA), a statistical method used to test and validate hypothesized factor structures in measurement models. Unlike exploratory factor analysis, CFA requires you to specify which variables load on which factors based on theory or prior research.

What You'll Get:

Model Fit Indices: Chi-square, CFI, TLI, RMSEA, and SRMR to evaluate model fit
Standardized Factor Loadings: Strength of relationships between observed variables and latent factors
Significance Tests: Statistical significance of all parameter estimates
Factor Correlations: Relationships between latent factors
Modification Indices: Suggestions for improving model fit
APA-Formatted Report: Professional statistical reporting ready for publication

Pro Tip: CFA requires a theoretical basis for your factor structure. If you're exploring factor structures without prior hypotheses, use Exploratory Factor Analysis instead. Good model fit is indicated by CFI/TLI e 0.90, RMSEA d 0.08, and SRMR d 0.08.

Ready to test your measurement model? (2-factor cognitive ability model) to see CFA in action, or upload your own data to validate your theoretical factor structure.

Software Implementation Differences

CFA models can be identified using different parameterization strategies, which produce equivalent models but different parameter estimates:

Marker Variable Approach (default): First indicator of each factor fixed to 1.0, latent variances freely estimated. Used by lavaan (R) and semopy (Python) by default
Standardized Latent Variable Approach: Latent variances fixed to 1.0, all loadings freely estimated. Use std.lv=TRUE in R lavaan
Important: Both approaches produce identical fit indices (CFI, TLI, RMSEA, SRMR, χ²) and represent the same underlying model
This calculator uses the marker variable approach. To match results in R, use cfa(model, data) without std.lv=TRUE
SPSS Amos, Mplus, EQS: May use different default identification strategies
SRMR Calculation: Minor differences (typically < 0.005) may occur across software due to different formulas for converting model-implied covariances to correlations. This does not affect model interpretation.

Calculator

1. Load Your Data

2. Select Variables

Select Variables for Analysis:

Selected: 0 of 0 variables

3. Define Factor Structure

Important: Assign each variable to exactly one factor based on your theoretical model. Each factor should have at least 2-3 indicators for model identification.

Note: Please avoid using spaces in factor names. Use underscores instead (e.g., Cognitive_Ability).

Factor Name:

Assigned Variables (0):

4. Analysis Options

Standardize Data (Recommended)

Related Calculators

Exploratory Factor Analysis Calculator

Principal Component Analysis Calculator

Cronbach's Alpha Calculator

Correlation Coefficient Calculator

Learn More

Definition

Confirmatory Factor Analysis (CFA) is a multivariate statistical technique used to test whether a hypothesized factor structure fits the observed data. Unlike EFA, CFA requires researchers to specify in advance which variables load on which factors based on theory, making it a hypothesis-testing approach ideal for scale validation and measurement model assessment.

When to Use CFA

Use Confirmatory Factor Analysis when you want to:

Test theoretical models: Verify that your data supports a specific factor structure derived from theory
Validate measurement instruments: Confirm that a questionnaire or scale measures the intended constructs
Compare competing models: Evaluate which of several theoretical models best fits your data
Assess construct validity: Demonstrate that variables cluster as expected based on theory
Prepare for SEM: CFA is often the first step before structural equation modeling

EFA vs CFA vs PCA: Key Differences

Aspect	EFA	CFA	PCA
Purpose	Seeks to explain correlations among variables using underlying latent factors. Separates shared variance from unique variance.	Tests a pre-specified factor structure based on theory. Confirms hypotheses about relationships between observed variables and latent factors.	Focuses on explaining total variance and creating orthogonal components. Does not distinguish shared from unique variance.
Approach	Exploratory - discovers underlying structure without prior hypotheses	Confirmatory - tests specific hypothesized factor structures	Descriptive - reduces dimensionality for data simplification
When to Use	Theory building, scale development, understanding construct validity	Theory testing, validating measurement models, assessing model fit to data	Data reduction, feature extraction, eliminating multicollinearity

Interpreting Model Fit Indices

Comparative Fit Index (CFI) & Tucker-Lewis Index (TLI)

e 0.95: Excellent fit
0.90 - 0.95: Acceptable fit
< 0.90: Poor fit

Root Mean Square Error of Approximation (RMSEA)

d 0.05: Excellent fit
0.05 - 0.08: Acceptable fit
0.08 - 0.10: Mediocre fit
> 0.10: Poor fit

Standardized Root Mean Square Residual (SRMR)

d 0.08: Good fit
> 0.08: Poor fit

Factor Loadings & Model Modification

Factor Loadings:

Standardized loadings e 0.7 indicate strong relationships
Loadings 0.5-0.7 are moderate and generally acceptable
Loadings < 0.5 may indicate weak indicators
All loadings should be statistically significant (p < .05)

Modification Indices:

MI values > 3.84 suggest parameters that could improve model fit if freed. However, modifications should only be made if theoretically justified. Data-driven modifications without theoretical rationale can lead to overfitting and non-generalizable results.

Example Code

R Code (lavaan)

library(lavaan)
library(tidyverse)
library(semPlot)

# Psychological test scores data
data <- tibble(
  verbal = c(65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73),
  numerical = c(62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70),
  logical = c(68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77),
  spatial = c(58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69),
  memory = c(70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81)
)

# Define CFA model (2-factor structure)
# Factor 1: Cognitive reasoning (verbal, numerical, logical)
# Factor 2: Spatial-memory abilities (spatial, memory)
model <- '
  # Latent factors
  Cognitive =~ verbal + numerical + logical
  Spatial_Memory =~ spatial + memory
'

# Fit the CFA model
fit <- cfa(model, data = data, std.lv = TRUE)

# View results
summary(fit, fit.measures = TRUE, standardized = TRUE)

# Get fit indices
fitMeasures(fit, c("chisq", "df", "pvalue", "cfi", "tli", "rmsea", "srmr"))

# Standardized loadings
standardizedSolution(fit)

# Modification indices
modindices(fit, sort = TRUE, minimum.value = 3.84)

# Path diagram
semPaths(fit, what = "std", edge.label.cex = 0.8,
         curvePivot = TRUE, layout = "tree")

Python Code (semopy)

Python

import pandas as pd
import numpy as np
from semopy import Model, semplot
import matplotlib.pyplot as plt
import semopy

# Psychological test scores data
data = pd.DataFrame({
    'verbal': [65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73],
    'numerical': [62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70],
    'logical': [68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77],
    'spatial': [58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69],
    'memory': [70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81]
})

# Define CFA model (2-factor structure)
# Factor 1: Cognitive reasoning (verbal, numerical, logical)
# Factor 2: Spatial-memory abilities (spatial, memory)
model_spec = """
# Latent factors
Cognitive =~ verbal + numerical + logical
Spatial_Memory =~ spatial + memory
"""

# Fit the CFA model
model = Model(model_spec)
model.fit(data)

# Model fit statistics
stats = semopy.calc_stats(model)
print("Model Fit Statistics:")
print(stats)

# Parameter estimates
estimates = model.inspect()
print("Parameter Estimates:")
print(estimates)

# Factor loadings
print("Factor Loadings:")
loadings = estimates[estimates['op'] == '~'][['lval', 'rval', 'Estimate', 'Std. Err', 'z-value', 'p-value']]
print(loadings)

Verification

Confirmatory Factor Analysis (CFA)

Created:December 4, 2025

Last Updated:January 6, 2026

What You'll Get:

Model Fit Indices: Chi-square, CFI, TLI, RMSEA, and SRMR to evaluate model fit
Standardized Factor Loadings: Strength of relationships between observed variables and latent factors
Significance Tests: Statistical significance of all parameter estimates
Factor Correlations: Relationships between latent factors
Modification Indices: Suggestions for improving model fit
APA-Formatted Report: Professional statistical reporting ready for publication

Ready to test your measurement model? (2-factor cognitive ability model) to see CFA in action, or upload your own data to validate your theoretical factor structure.

Software Implementation Differences

CFA models can be identified using different parameterization strategies, which produce equivalent models but different parameter estimates:

Marker Variable Approach (default): First indicator of each factor fixed to 1.0, latent variances freely estimated. Used by lavaan (R) and semopy (Python) by default
Standardized Latent Variable Approach: Latent variances fixed to 1.0, all loadings freely estimated. Use std.lv=TRUE in R lavaan
Important: Both approaches produce identical fit indices (CFI, TLI, RMSEA, SRMR, χ²) and represent the same underlying model
This calculator uses the marker variable approach. To match results in R, use cfa(model, data) without std.lv=TRUE
SPSS Amos, Mplus, EQS: May use different default identification strategies
SRMR Calculation: Minor differences (typically < 0.005) may occur across software due to different formulas for converting model-implied covariances to correlations. This does not affect model interpretation.

Calculator

1. Load Your Data

2. Select Variables

Select Variables for Analysis:

Selected: 0 of 0 variables

3. Define Factor Structure

Important: Assign each variable to exactly one factor based on your theoretical model. Each factor should have at least 2-3 indicators for model identification.

Note: Please avoid using spaces in factor names. Use underscores instead (e.g., Cognitive_Ability).

Factor Name:

Assigned Variables (0):

4. Analysis Options

Standardize Data (Recommended)

Related Calculators

Exploratory Factor Analysis Calculator

Principal Component Analysis Calculator

Cronbach's Alpha Calculator

Correlation Coefficient Calculator

Learn More

Definition

When to Use CFA

Use Confirmatory Factor Analysis when you want to:

Test theoretical models: Verify that your data supports a specific factor structure derived from theory
Validate measurement instruments: Confirm that a questionnaire or scale measures the intended constructs
Compare competing models: Evaluate which of several theoretical models best fits your data
Assess construct validity: Demonstrate that variables cluster as expected based on theory
Prepare for SEM: CFA is often the first step before structural equation modeling

EFA vs CFA vs PCA: Key Differences

Aspect	EFA	CFA	PCA
Purpose	Seeks to explain correlations among variables using underlying latent factors. Separates shared variance from unique variance.	Tests a pre-specified factor structure based on theory. Confirms hypotheses about relationships between observed variables and latent factors.	Focuses on explaining total variance and creating orthogonal components. Does not distinguish shared from unique variance.
Approach	Exploratory - discovers underlying structure without prior hypotheses	Confirmatory - tests specific hypothesized factor structures	Descriptive - reduces dimensionality for data simplification
When to Use	Theory building, scale development, understanding construct validity	Theory testing, validating measurement models, assessing model fit to data	Data reduction, feature extraction, eliminating multicollinearity

Interpreting Model Fit Indices

Comparative Fit Index (CFI) & Tucker-Lewis Index (TLI)

e 0.95: Excellent fit
0.90 - 0.95: Acceptable fit
< 0.90: Poor fit

Root Mean Square Error of Approximation (RMSEA)

d 0.05: Excellent fit
0.05 - 0.08: Acceptable fit
0.08 - 0.10: Mediocre fit
> 0.10: Poor fit

Standardized Root Mean Square Residual (SRMR)

d 0.08: Good fit
> 0.08: Poor fit

Factor Loadings & Model Modification

Factor Loadings:

Standardized loadings e 0.7 indicate strong relationships
Loadings 0.5-0.7 are moderate and generally acceptable
Loadings < 0.5 may indicate weak indicators
All loadings should be statistically significant (p < .05)

Modification Indices:

Example Code

R Code (lavaan)

library(lavaan)
library(tidyverse)
library(semPlot)

# Psychological test scores data
data <- tibble(
  verbal = c(65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73),
  numerical = c(62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70),
  logical = c(68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77),
  spatial = c(58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69),
  memory = c(70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81)
)

# Define CFA model (2-factor structure)
# Factor 1: Cognitive reasoning (verbal, numerical, logical)
# Factor 2: Spatial-memory abilities (spatial, memory)
model <- '
  # Latent factors
  Cognitive =~ verbal + numerical + logical
  Spatial_Memory =~ spatial + memory
'

# Fit the CFA model
fit <- cfa(model, data = data, std.lv = TRUE)

# View results
summary(fit, fit.measures = TRUE, standardized = TRUE)

# Get fit indices
fitMeasures(fit, c("chisq", "df", "pvalue", "cfi", "tli", "rmsea", "srmr"))

# Standardized loadings
standardizedSolution(fit)

# Modification indices
modindices(fit, sort = TRUE, minimum.value = 3.84)

# Path diagram
semPaths(fit, what = "std", edge.label.cex = 0.8,
         curvePivot = TRUE, layout = "tree")

Python Code (semopy)

Python

import pandas as pd
import numpy as np
from semopy import Model, semplot
import matplotlib.pyplot as plt
import semopy

# Psychological test scores data
data = pd.DataFrame({
    'verbal': [65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73],
    'numerical': [62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70],
    'logical': [68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77],
    'spatial': [58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69],
    'memory': [70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81]
})

# Define CFA model (2-factor structure)
# Factor 1: Cognitive reasoning (verbal, numerical, logical)
# Factor 2: Spatial-memory abilities (spatial, memory)
model_spec = """
# Latent factors
Cognitive =~ verbal + numerical + logical
Spatial_Memory =~ spatial + memory
"""

# Fit the CFA model
model = Model(model_spec)
model.fit(data)

# Model fit statistics
stats = semopy.calc_stats(model)
print("Model Fit Statistics:")
print(stats)

# Parameter estimates
estimates = model.inspect()
print("Parameter Estimates:")
print(estimates)

# Factor loadings
print("Factor Loadings:")
loadings = estimates[estimates['op'] == '~'][['lval', 'rval', 'Estimate', 'Std. Err', 'z-value', 'p-value']]
print(loadings)

Verification

Aspect

EFA

CFA

PCA

Purpose

Seeks to explain correlations among variables using underlying latent factors. Separates shared variance from unique variance.

Tests a pre-specified factor structure based on theory. Confirms hypotheses about relationships between observed variables and latent factors.

Focuses on explaining total variance and creating orthogonal components. Does not distinguish shared from unique variance.

Approach

Exploratory - discovers underlying structure without prior hypotheses

Confirmatory - tests specific hypothesized factor structures

Descriptive - reduces dimensionality for data simplification

When to Use

Theory building, scale development, understanding construct validity

Theory testing, validating measurement models, assessing model fit to data

Data reduction, feature extraction, eliminating multicollinearity

library(lavaan) library(tidyverse) library(semPlot) # Psychological test scores data data <- tibble( verbal = c(65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73), numerical = c(62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70), logical = c(68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77), spatial = c(58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69), memory = c(70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81) ) # Define CFA model (2-factor structure) # Factor 1: Cognitive reasoning (verbal, numerical, logical) # Factor 2: Spatial-memory abilities (spatial, memory) model <- ' # Latent factors Cognitive =~ verbal + numerical + logical Spatial_Memory =~ spatial + memory ' # Fit the CFA model fit <- cfa(model, data = data, std.lv = TRUE) # View results summary(fit, fit.measures = TRUE, standardized = TRUE) # Get fit indices fitMeasures(fit, c("chisq", "df", "pvalue", "cfi", "tli", "rmsea", "srmr")) # Standardized loadings standardizedSolution(fit) # Modification indices modindices(fit, sort = TRUE, minimum.value = 3.84) # Path diagram semPaths(fit, what = "std", edge.label.cex = 0.8, curvePivot = TRUE, layout = "tree")

import pandas as pd import numpy as np from semopy import Model, semplot import matplotlib.pyplot as plt import semopy # Psychological test scores data data = pd.DataFrame({ 'verbal': [65, 72, 58, 68, 75, 62, 70, 66, 71, 69, 63, 74, 60, 67, 73], 'numerical': [62, 68, 55, 65, 71, 60, 68, 63, 70, 66, 61, 72, 58, 64, 70], 'logical': [68, 74, 60, 70, 78, 65, 73, 69, 75, 71, 66, 76, 62, 69, 77], 'spatial': [58, 65, 52, 62, 68, 56, 64, 60, 66, 63, 58, 67, 54, 61, 69], 'memory': [70, 76, 63, 72, 80, 68, 75, 71, 77, 73, 69, 79, 65, 72, 81] }) # Define CFA model (2-factor structure) # Factor 1: Cognitive reasoning (verbal, numerical, logical) # Factor 2: Spatial-memory abilities (spatial, memory) model_spec = """ # Latent factors Cognitive =~ verbal + numerical + logical Spatial_Memory =~ spatial + memory """ # Fit the CFA model model = Model(model_spec) model.fit(data) # Model fit statistics stats = semopy.calc_stats(model) print("Model Fit Statistics:") print(stats) # Parameter estimates estimates = model.inspect() print("Parameter Estimates:") print(estimates) # Factor loadings print("Factor Loadings:") loadings = estimates[estimates['op'] == '~'][['lval', 'rval', 'Estimate', 'Std. Err', 'z-value', 'p-value']] print(loadings)

Aspect

EFA

CFA

PCA

Purpose

Seeks to explain correlations among variables using underlying latent factors. Separates shared variance from unique variance.

Tests a pre-specified factor structure based on theory. Confirms hypotheses about relationships between observed variables and latent factors.

Focuses on explaining total variance and creating orthogonal components. Does not distinguish shared from unique variance.

Approach

Exploratory - discovers underlying structure without prior hypotheses

Confirmatory - tests specific hypothesized factor structures

Descriptive - reduces dimensionality for data simplification

When to Use

Theory building, scale development, understanding construct validity

Theory testing, validating measurement models, assessing model fit to data

Data reduction, feature extraction, eliminating multicollinearity

Confirmatory Factor Analysis (CFA)

What You'll Get:

Software Implementation Differences

Calculator

1. Load Your Data

2. Select Variables

3. Define Factor Structure

4. Analysis Options

Related Calculators

Exploratory Factor Analysis Calculator

Principal Component Analysis Calculator

Cronbach's Alpha Calculator

Correlation Coefficient Calculator

Learn More

Definition

When to Use CFA

EFA vs CFA vs PCA: Key Differences

Interpreting Model Fit Indices

Comparative Fit Index (CFI) & Tucker-Lewis Index (TLI)

Root Mean Square Error of Approximation (RMSEA)

Standardized Root Mean Square Residual (SRMR)

Factor Loadings & Model Modification

Example Code

R Code (lavaan)

Python Code (semopy)

Verification

View Verification Details

Confirmatory Factor Analysis (CFA)

What You'll Get:

Software Implementation Differences

Calculator

1. Load Your Data

2. Select Variables

3. Define Factor Structure

4. Analysis Options

Related Calculators

Exploratory Factor Analysis Calculator

Principal Component Analysis Calculator

Cronbach's Alpha Calculator

Correlation Coefficient Calculator

Learn More

Definition

When to Use CFA

EFA vs CFA vs PCA: Key Differences

Interpreting Model Fit Indices

Comparative Fit Index (CFI) & Tucker-Lewis Index (TLI)

Root Mean Square Error of Approximation (RMSEA)

Standardized Root Mean Square Residual (SRMR)

Factor Loadings & Model Modification

Example Code

R Code (lavaan)

Python Code (semopy)

Verification

View Verification Details

Confirmatory Factor Analysis (CFA)

What You'll Get:

Software Implementation Differences

Calculator

1. Load Your Data

2. Select Variables

3. Define Factor Structure

4. Analysis Options

Related Calculators

Exploratory Factor Analysis Calculator

Principal Component Analysis Calculator

Cronbach's Alpha Calculator

Correlation Coefficient Calculator

Learn More

Definition

When to Use CFA

EFA vs CFA vs PCA: Key Differences

Interpreting Model Fit Indices

Comparative Fit Index (CFI) & Tucker-Lewis Index (TLI)

Root Mean Square Error of Approximation (RMSEA)

Standardized Root Mean Square Residual (SRMR)

Factor Loadings & Model Modification

Example Code

R Code (lavaan)

Python Code (semopy)

Verification