Assignment: CFA with Missing Data

Software Integration

Overview

In this assignment, you will simulate psychological scale data, introduce missing data for one group, and then fit a confirmatory factor analysis (CFA) model. You will compare your estimated parameters to the true values from the simulation to evaluate how well your model recovers the population parameters.

Learning Objectives

• Practice simulating psychological data with known parameters
• Understand the impact of missing data on parameter estimation
• Gain experience running CFA models in Mplus or lavaan
• Critically evaluate model results against known true values

Instructions

Part 1: Data Simulation (20 points)

Create a simulated dataset with the following characteristics:

1. Sample size: N = 400
2. Groups: Two groups (e.g., “Treatment” and “Control”), with 200 participants each
3. Latent construct: One latent factor (e.g., “Wellbeing”)
4. Items: 5 Likert-scale items (1-5 scale)
5. True factor loadings: Choose your own values between 0.5 and 0.9 for each item

Starter code:

library(tidyverse)

set.seed(YOUR_STUDENT_ID)  # Use your student ID for reproducibility
n <- 400

# Create your simulation here
data_sim <- tibble(
    id = 1:n,
    group = rep(c("Treatment", "Control"), each = n/2),
    # Add latent trait
    theta = rnorm(n, mean = 0, sd = 1)
) %>%
    mutate(
        # Generate 5 Likert items with YOUR chosen factor loadings
        # item1 = round(pmin(5, pmax(1, 3 + λ1 * theta + rnorm(n, 0, error1)))),
        # ... continue for all 5 items
    )

Deliverable: Report your chosen factor loadings (λ₁ through λ₅) in a table.

Part 2: Introduce Missing Data (20 points)

After creating your complete dataset, introduce missing data only for the Treatment group.

Requirements:

• Make approximately 20% of the item responses missing for the Treatment group
• The Control group should have no missing data
• Missing data should be Missing Completely at Random (MCAR)

Starter code:

# Function to introduce MCAR missing data
introduce_missing <- function(x, prop_missing = 0.20) {
    n <- length(x)
    missing_idx <- sample(1:n, size = round(n * prop_missing))
    x[missing_idx] <- NA
    return(x)
}

# Apply only to Treatment group
data_missing <- data_sim %>%
    mutate(
        across(
            starts_with("item"),
            ~ if_else(group == "Treatment", introduce_missing(.), .)
        )
    )

# Check missing data pattern
data_missing %>%
    group_by(group) %>%
    summarise(across(starts_with("item"), ~ mean(is.na(.))))

Deliverable: Report the percentage of missing data for each item by group.

Part 3: CFA Model (30 points)

Fit a one-factor CFA model to your data with missing values. You may use either Mplus or lavaan (or both for extra credit).

Option A: lavaan

library(lavaan)

# Select only item variables
data_cfa <- data_missing %>%
    select(starts_with("item"))

# Define model
cfa_model <- '
    Wellbeing =~ item1 + item2 + item3 + item4 + item5
'

# Fit model with FIML for missing data
fit <- cfa(cfa_model, 
           data = data_cfa, 
           std.lv = TRUE,
           missing = "fiml")  # Full Information Maximum Likelihood

summary(fit, standardized = TRUE, fit.measures = TRUE)

Option B: Mplus

TITLE: CFA with Missing Data

DATA: FILE IS cfa_data.dat;

VARIABLE: 
    NAMES ARE item1 item2 item3 item4 item5;
    USEVARIABLES ARE item1-item5;
    MISSING ARE ALL (-999);  ! Or your missing value code

ANALYSIS:
    ESTIMATOR = ML;

MODEL:
    Wellbeing BY item1* item2 item3 item4 item5;
    Wellbeing@1;

OUTPUT: STANDARDIZED;

Deliverable:

1. Report your model syntax
2. Report fit indices (χ², df, p-value, CFI, TLI, RMSEA)
3. Report unstandardized and standardized factor loadings

Part 4: Compare to True Values (30 points)

Create a comparison table and answer the reflection questions.

Comparison Table:

Item	True λ	Estimated λ	Difference	% Bias
item1	?	?	?	?
item2	?	?	?	?
item3	?	?	?	?
item4	?	?	?	?
item5	?	?	?	?

Note: % Bias = ((Estimated - True) / True) × 100

Reflection Questions:

1. How close were your estimated factor loadings to the true values?
2. Did the model under- or over-estimate the loadings? Is there a pattern?
3. How might the missing data in the Treatment group have affected your estimates?
4. What would you expect to happen if you increased the missing data rate to 40%?
5. Why is simulation useful for understanding statistical methods?

Extra Credit (10 points)

Choose one of the following:

1. Run the CFA in both Mplus and lavaan and compare the results
2. Run the analysis on the complete data (before introducing missing values) and compare to your results with missing data
3. Try a different missing data mechanism (e.g., MAR - missing at random, where missingness depends on age or another variable) and discuss how it affects estimates

Submission

Submit the following:

1. Your R script or Quarto document with all code
2. Your Mplus input and output files (if using Mplus)
3. A brief report (1-2 pages) with:
   • Your true parameter values
   • Missing data summary
   • Model results
   • Comparison table
   • Answers to reflection questions

Grading Rubric

Component	Points
Data simulation with correct structure	20
Missing data correctly introduced	20
CFA model correctly specified and run	30
Comparison table and reflection	30
Total	100
Extra credit	+10