listwise deletion

Missing Data Mechanisms: A Primer

May 11th, 2021 by

Missing data are a widespread problem, as most researchers can attest. Whether data are from surveys, experiments, or secondary sources, missing data abounds.

But what’s the impact on the results of statistical analysis? That depends on two things: the mechanism that led the data to be missing and the way in which the data analyst deals with it.

Here are a few common situations:

Subjects in longitudinal studies often start, but drop out before the study is completed. There are many reasons for this:   they have moved out of the area (nothing related to the study), died (hopefully not related to the study), no longer see personal benefit to participating, or do not like the effects of the treatment.

Surveys suffer missing data in many ways. When participants refuse to answer the entire survey or parts of it; do not know the answer to, or accidentally skip an item. Some survey researchers even design the study so that some questions are asked of only a subset of participants.

Experimental studies have missing data when a researcher is simply unable to collect an observation. Bad weather conditions may render observation impossible in field experiments. A researcher becomes sick or equipment fails. Data may be missing in any type of study due to accidental or data entry error. A researcher drops a tray of test tubes. A data file becomes corrupt.

Most researchers are very familiar with one (or more) of these situations.

Why Missing Data Matters

Missing data cause problems because most statistical procedures require a value for each variable. When a data set is incomplete, the data analyst has to decide how to deal with it.

The most common decision is to use complete case analysis (also called listwise deletion). This means analyzing only the cases with complete data. Individuals with data missing on any variables are dropped from the analysis.

It has advantages–it is easy to use, is very simple, and is the default in most statistical packages. But it has limitations.

It can substantially lower the sample size, leading to a severe lack of power. This is especially true if there are many variables involved in the analysis, each with data missing for a few cases.

Possibly worse, it can also lead to biased results, depending on why and in which patterns the data are missing.

Missing Data Mechanisms

The types of missing data fit into three classes, which are based on the relationship between the missing data mechanism and the missing and observed values. These badly-named classes are important to understand because the problems caused by missing data and the solutions to these problems are different for the three classes.

Missing Completely at Random

The first is Missing Completely at Random (MCAR). MCAR means that the missing data mechanism is unrelated to the values of any variables, whether missing or observed.

Data that are missing because a researcher dropped the test tubes or survey participants accidentally skipped questions are likely to be MCAR.

If the observed values are essentially a random sample of the full data set, complete case analysis gives the same results as the full data set would have. Unfortunately, most missing data are not MCAR.

Missing Not at Random

At the opposite end of the spectrum is Missing Not at Random. Although you’ll most often see it called this, I prefer the term Non-Ignorable (NI). NI is a name that is not so easy to confuse with the other types, but it also tells you its primary feature. It means that the missing data mechanism is related to the missing values.

And this is something you, the data analyst, can’t ignore without biasing results.

It occurs sometimes when people do not want to reveal something very personal or unpopular about themselves. For example, if individuals with higher incomes are less likely to reveal them on a survey than are individuals with lower incomes, the missing data mechanism for income is non-ignorable. Whether income is missing or observed is related to its value.

But that’s not the only example. When the sickest patients drop out of a longitudinal study testing a drug that’s supposed to make them healthy, that’s non-ignorable.

Or an instrument can’t detect low readings, so gives you an error, also non-ignorable.

Complete case analysis can give highly biased results for NI missing data. If proportionally more low and moderate income individuals are left in the sample because high income people are missing, an estimate of the mean income will be lower than the actual population mean.

Missing at Random

In between these two extremes is Missing at Random (MAR). MAR requires that the cause of the missing data is unrelated to the missing values but may be related to the observed values of other variables.

MAR means that the missing values are related to observed values on other variables. As an example of CD missing data, missing income data may be unrelated to the actual income values but are related to education. Perhaps people with more education are less likely to reveal their income than those with less education.

A key distinction is whether the mechanism is ignorable (i.e., MCAR or MAR) or non-ignorable. There are excellent techniques for handling ignorable missing data. Non-ignorable missing data are more challenging and require a different approach.


First Published 2/24/2014;
Updated 5/11/21 to give more detail.

Member Training: Missing Data

December 1st, 2020 by

Missing data causes a lot of problems in data analysis. Unfortunately, some of the “solutions” for missing data cause more problems than they solve.


Member Training: Confusing Statistical Terms

February 28th, 2020 by

Learning statistics is difficult enough; throw in some especially confusing terminology and it can feel impossible! There are many ways that statistical language can be confusing.

Some terms mean one thing in the English language, but have another (usually more specific) meaning in statistics.  (more…)

Linear Mixed Models for Missing Data in Pre-Post Studies

August 30th, 2016 by

In the past few months, I’ve gotten the same question from a few clients about using linear mixed models for repeated measures data.  They want to take advantage of its ability to give unbiased results in the presence of missing data.  In each case the study has two groups complete a pre-test and a post-test measure.  Both of these have a lot of missing data.

The research question is whether the groups have different improvements in the dependent variable from pre to post test.

As a typical example, say you have a study with 160 participants.

90 of them completed both the pre and the post test.

Another 48 completed only the pretest and 22 completed only the post-test.

Repeated Measures ANOVA will deal with the missing data through listwise deletion. That means keeping only the 90 people with complete data.  This causes problems with both power and bias, but bias is the bigger issue.

Another alternative is to use a Linear Mixed Model, which will use the full data set.  This is an advantage, but it’s not as big of an advantage in this design as in other studies.

The mixed model will retain the 70 people who have data for only one time point.  It will use the 48 people with pretest-only data along with the 90 people with full data to estimate the pretest mean.

Likewise, it will use the 22 people with posttest-only data along with the 90 people with full data to estimate the post-test mean.

If the data are missing at random, this will give you unbiased estimates of each of these means.

But most of the time in Pre-Post studies, the interest is in the change from pre to post across groups.

The difference in means from pre to post will be calculated based on the estimates at each time point.  But the degrees of freedom for the difference will be based only on the number of subjects who have data at both time points.

So with only two time points, if the people with one time point are no different from those with full data (creating no bias), you’re not gaining anything by keeping those 72 people in the analysis.

Compare this to a study I also saw in consulting with 5 time points.  Nearly all the participants had 4 out of the 5 observations.  The missing data was pretty random–some participants missed time 1, others, time 4, etc.  Only 6 people out of 150 had full data.  Listwise deletion created a nightmare, leaving only 6 people in the data set.

Each person contributed data to 4 means, so each mean had a pretty reasonable sample size.  Since the missingness was random, each mean was unbiased.  Each subject fully contributed data and df to many of the mean comparisons.

With more than 2 time points and data that are missing at random, each subject can contribute to some change measurements.  Keep that in mind the next time you design a study.


When Listwise Deletion works for Missing Data

February 25th, 2014 by

You may have never heard of listwise deletion for missing data, but you’ve probably used it.

Listwise deletion means that any individual in a data set is deleted from an analysis if they’re missing data on any variable in the analysis.

It’s the default in most software packages.

Although the simplicity of it is a major advantage, it causes big problems in many missing data situations.

But not always.  If you happen to have one of the uncommon missing data situations in which (more…)

Do Top Journals Require Reporting on Missing Data Techniques?

June 3rd, 2011 by

Q: Do most high impact journals require authors to state which method has been used on missing data?

I don’t usually get far enough in the publishing process to read journal requirements.

But based on my conversations with researchers who both review articles for journals and who deal with reviewers’ comments, I can offer this response.

I would be shocked if journal editors at top journals didn’t want information about the missing data technique.  If you leave it out, they’ll either assume you didn’t have missing data or are using defaults like listwise deletion. (more…)