Little’s MCAR Test and Missing Data Analysis
See how much data is missing and in which patterns, then test whether it is missing completely at random (MCAR) with Little’s chi-square test. EM (expectation-maximization) estimates of the means and correlations use every available value.
Updated · By Bibin XavierData 220 × 7
| IDtext · 220 levels | Agenumeric | Incomenumeric | Satisfactionnumeric | Trustnumeric | Loyaltynumeric | Weekly_usenumeric |
|---|---|---|---|---|---|---|
| P001 | 21 | 40.9 | 3.69 | · | 5 | 8 |
| P002 | 47 | 37.7 | 3.74 | 4.85 | 4.69 | 28 |
| P003 | 37 | 67.5 | 3.47 | 3.79 | 4.47 | 18 |
| P004 | 54 | 65.1 | 3.5 | 2.32 | 5 | 10 |
| P005 | 34 | 41.2 | 2.75 | 2.43 | 4.06 | 6 |
| P006 | 66 | · | 3.59 | 2.96 | 3.79 | 15 |
| P007 | 40 | 55.9 | 4.25 | 2.2 | 4.62 | 14 |
| P008 | 37 | 35.7 | 3.82 | 4.22 | 5 | 13 |
Variables and options
Results
220 rows · example dataMissing values by variable
| Variable | N valid | Missing | % missing | M | SD |
|---|---|---|---|---|---|
| Age | 220 | 0 | 0.0 | 38.21 | 10.87 |
| Income | 196 | 24 | 10.9 | 46.78 | 13.72 |
| Satisfaction | 202 | 18 | 8.2 | 3.42 | 0.78 |
| Trust | 201 | 19 | 8.6 | 3.26 | 0.76 |
| Loyalty | 200 | 20 | 9.1 | 4.31 | 0.67 |
| Weekly_use | 206 | 14 | 6.4 | 12.02 | 4.91 |
Note. Means and SDs use all available values for each variable.
Percent missing by variable
Missing data patterns
| Age | Income | Satisfaction | Trust | Loyalty | Weekly_use | n | % | Variables missing |
|---|---|---|---|---|---|---|---|---|
| ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | 135 | 61.4 | 0 |
| ✓ | — | ✓ | ✓ | ✓ | ✓ | 20 | 9.1 | 1 |
| ✓ | ✓ | ✓ | ✓ | — | ✓ | 17 | 7.7 | 1 |
| ✓ | ✓ | — | ✓ | ✓ | ✓ | 15 | 6.8 | 1 |
| ✓ | ✓ | ✓ | — | ✓ | ✓ | 12 | 5.5 | 1 |
| ✓ | ✓ | ✓ | ✓ | ✓ | — | 12 | 5.5 | 1 |
| ✓ | — | ✓ | — | ✓ | ✓ | 3 | 1.4 | 2 |
| ✓ | ✓ | — | — | ✓ | ✓ | 1 | 0.5 | 2 |
| ✓ | ✓ | ✓ | — | — | ✓ | 1 | 0.5 | 2 |
| ✓ | ✓ | — | ✓ | — | ✓ | 1 | 0.5 | 2 |
| ✓ | — | ✓ | ✓ | ✓ | — | 1 | 0.5 | 2 |
| ✓ | ✓ | ✓ | — | ✓ | — | 1 | 0.5 | 2 |
| ✓ | ✓ | — | — | — | ✓ | 1 | 0.5 | 3 |
Note. ✓ observed, — missing. 13 patterns.
Little’s MCAR test
| χ² | df | p | EM iterations |
|---|---|---|---|
| 76.791 | 52 | .014 | 22 |
Note. H₀: data are missing completely at random. A significant result (p < .05) means the missingness depends on the observed data, so it is not MCAR.
EM estimated means and correlations
| Variable | EM mean | EM SD | Age | Income | Satisfaction | Trust | Loyalty | Weekly_use |
|---|---|---|---|---|---|---|---|---|
| Age | 38.21 | 10.87 | 1.000 | |||||
| Income | 46.74 | 13.83 | .316 | 1.000 | ||||
| Satisfaction | 3.41 | 0.79 | -.065 | -.057 | 1.000 | |||
| Trust | 3.26 | 0.76 | -.070 | -.024 | .359 | 1.000 | ||
| Loyalty | 4.29 | 0.68 | -.065 | -.040 | .637 | .515 | 1.000 | |
| Weekly_use | 12.07 | 4.93 | -.041 | -.030 | .340 | .413 | .247 | 1.000 |
Note. Maximum-likelihood estimates under multivariate normality, using all available data. SDs use the n − 1 denominator.
Interpretation
7.2% of the values are missing and 61.4% of cases are complete. Little’s MCAR test is significant, so the data are not missing completely at random. Listwise deletion may bias results; prefer maximum likelihood (FIML) or multiple imputation, and include variables related to missingness.
How to cite this tool
Using these results in a thesis, journal article, report or class assignment? Please cite this free tool so others can find it. It keeps the site free.
APA 7
Xavier, B. (2026). Little’s MCAR Test and Missing Data Analysis [Web application]. FreeStatisticalTools.com. https://freestatisticaltools.com/littles-mcar-test/
BibTeX
@software{xavier_missingdataanalysis_2026,
author = {Xavier, Bibin},
title = {Little’s MCAR Test and Missing Data Analysis},
publisher = {FreeStatisticalTools.com},
year = {2026},
url = {https://freestatisticaltools.com/littles-mcar-test/},
note = {Web application}
}Suggested sentence for your methods section: “The analysis was carried out with the free Little’s MCAR Test and Missing Data Analysis tool on FreeStatisticalTools.com (Xavier, 2026).”
What is Little’s MCAR test?
Little’s MCAR test checks whether missing values are missing completely at random, meaning the chance that a value is missing does not depend on any observed or unobserved data. It compares the means of each missing-data pattern with expectation-maximization (EM) estimates of the overall means, giving a chi-square statistic. This page also tabulates how much is missing per variable and which patterns occur.
When to use Little’s MCAR test
Use it before any analysis of data with gaps, to report how much is missing, to see whether some variables or respondents are missing more often, and to justify the missing-data method you choose.
Assumptions
- Numeric variables, approximately multivariate normal (for Little’s test and EM).
- Little’s test can detect departures from MCAR but cannot prove MCAR, and cannot distinguish MAR from MNAR.
How to set up your data
One column per variable, one row per case, with missing values as blank cells (or codes such as -99, defined as missing in the data cleaning tool first).
How to use this tool
- Enter your data. Paste columns from Excel, Google Sheets or SPSS with variable names in the first row, or upload a CSV or Excel file. The example data shows the expected layout.
- Choose the variables. Select the variables.
- Read the results. Results update automatically. Read the plain-English interpretation, then use Copy table to paste APA-style tables into Word and Copy for the results sentence.
How to report Little’s MCAR test in APA style
“Little’s MCAR test was not significant, χ²(34) = 38.21, p = .284, suggesting the data were missing completely at random; 3.8% of values were missing.”
Frequently asked questions
How do I interpret Little’s MCAR test?
A non-significant result (p > .05) is consistent with data missing completely at random. A significant result means missingness depends on the observed data, so the data are not MCAR and listwise deletion may bias the results.
What should I do if the data are not MCAR?
Use methods that are valid when data are missing at random (MAR), such as full information maximum likelihood or multiple imputation, and include variables that predict missingness.
What is the EM algorithm?
Expectation-maximization estimates means and covariances from incomplete data by alternating between filling in expected values and re-estimating the parameters, using every available value.
How much missing data is acceptable?
Under about 5% missing per variable with MCAR, most methods give similar results. Above that, report the amount and pattern and use maximum likelihood or multiple imputation.
Does this match SPSS Missing Value Analysis?
Yes, it uses the same EM-based test as SPSS Missing Value Analysis (Little, 1988), which in SPSS needs a paid add-on module.
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