4 · Data Cleaning & Missing Data · Missing Data: Concepts & Diagnosis

MCAR: missing completely at random

MCAR means the probability that a value is missing does not depend on observed or unobserved values. Under true MCAR, complete-case analysis can remain unbiased for many estimands, although it wastes information and reduces precision.

Reference lessonPython exampleVisual explanation

Intuition first

What this concept means in practice

The practical value of MCAR: missing completely at random comes from understanding both the transformation and the boundary around it: what information is allowed to enter, what assumption is being made, and how you know the result is still valid after the transformation.

A beginner-friendly way to reason about it is to start with a tiny case where the correct result can be checked independently. Once the mechanism is clear, scale the exact same reasoning to larger tables, pipelines or models.

PurposeMCAR is a useful benchmark assumption, but it is often stronger than real data justify.

MechanismAsk whether missingness appears unrelated to measured variables and the missing value itself; compare complete and incomplete rows and investigate collection processes.

EvidenceInspect intermediate and final output; compare with an independent expectation.

Main cautionDo not label data MCAR merely because a statistical test is non-significant.

Mechanism

Trace the operation from input to decision

Ask whether missingness appears unrelated to measured variables and the missing value itself; compare complete and incomplete rows and investigate collection processes.

1Input→

2Apply rule→

3Inspect state→

4Validate→

5Use result

Key rule

P(R=1 | X,Y)=P(R=1), where R indicates observability.

Visual explanation

Make the structure visible

The interactive view uses a concept-specific plot when the topic maps naturally to one; otherwise it uses a workflow view instead of leaving a broken placeholder.

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Practical example

Where you would use it

A small random hardware packet loss affects sensor readings independently of participant condition and measured signal level.

Use when

MCAR is a useful benchmark assumption, but it is often stronger than real data justify.

Pitfall

What can make the result misleading

Watch out

Do not label data MCAR merely because a statistical test is non-significant.

A useful diagnostic question is: Could the same code still run successfully if the analytical assumption were wrong? If yes, add an explicit validation check rather than relying on execution success.

Implementation

Miniature Python example

Keep the example small enough that you can inspect each stage manually.

Python

# Purpose: demonstrate MCAR: missing completely at random with a small, inspectable example.
# Follow the comments and printed stages to connect each operation with its result.
# Import the library or helper used in this example.
# Step 1 — Import the module so its functions/classes are available to the rest of this example.
import numpy as np
# Import the library or helper used in this example.
# Step 2 — Import the module so its functions/classes are available to the rest of this example.
import pandas as pd
# Import the library or helper used in this example.
# Step 3 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.compose import ColumnTransformer
# Import the library or helper used in this example.
# Step 4 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.impute import SimpleImputer
# Import the library or helper used in this example.
# Step 5 — Import only the named objects needed by the following steps, keeping dependencies explicit.
from sklearn.preprocessing import OneHotEncoder

# Create a small labelled dataset that is easy to inspect by eye.
# Step 6 — Construct `X` as a tabular object with named columns for inspectable analysis.
X = pd.DataFrame({
    "age": [22,25,np.nan,31,35,38,41,44,48,np.nan,56,60],
    "income": [42,45,48,np.nan,55,59,62,66,70,74,np.nan,82],
    "city": ["A","B","A","B",None,"C","A","C","B","A","C",None]
})
# Print this intermediate result so you can verify the workflow step by step.
# Step 7 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 1 · Shape:", X.shape)
# Print this intermediate result so you can verify the workflow step by step.
# Step 8 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 1 · Missing counts:", X.isna().sum().to_dict())
# Store this intermediate value with a descriptive name for the next step.
# Step 9 — Compute the right-hand expression and store its result in `pre` for the next step.
pre = ColumnTransformer([
    ("num", SimpleImputer(strategy="median"), ["age","income"]),
    ("cat", SimpleImputer(strategy="most_frequent"), ["city"])
], verbose_feature_names_out=False)
# Learn the transformation from this data and apply it in one step.
# Step 10 — Fit the transformation on the training input and immediately transform that same input.
Xt = pre.fit_transform(X)
# Print this intermediate result so you can verify the workflow step by step.
# Step 11 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 2 · Missing after imputation:", int(pd.isna(Xt).sum()))
# Print this intermediate result so you can verify the workflow step by step.
# Step 12 — Display the current value explicitly so the result/state can be inspected during execution.
print("STEP 3 · First four transformed rows:")
# Print this intermediate result so you can verify the workflow step by step.
# Step 13 — Display the current value explicitly so the result/state can be inspected during execution.
print(Xt[:4])

Expected / illustrative output

STEP 1 · Shape: (12, 3)
STEP 1 · Missing counts: {'age': 2, 'income': 2, 'city': 2}
STEP 2 · Missing after imputation: 2
STEP 3 · First four transformed rows:
[[22.0 42.0 'A']
 [25.0 45.0 'B']
 [39.5 48.0 'A']
 [31.0 60.5 'B']]

Implementation checklist

Before you move on

Can you state what data or object enters the operation?
Can you explain what changes and what must remain invariant?
Have you checked the result on a tiny case you can verify independently?
Have you considered the main failure mode described above?
Can the operation be reproduced from code/formulas and documented assumptions?