When Income Itself Affects Whether It Is Reported
Suppose a survey asks for annual income and records respondents’ age and education. If, among people of the same age and education, the chance of reporting income still varies with the income amount itself, income nonresponse is missing not at random (MNAR). For example, people with higher incomes might be less likely to report their income, even after accounting for age and education.
The distinction is about the variables that remain relevant to response after conditioning on what is observed. A response rate that differs by age or education does not by itself imply MNAR: those are observed characteristics and may explain the difference under a missing-at-random (MAR) mechanism. The defining issue here is residual dependence on the unreported income.
The response probability may also vary with age, education, or other observed characteristics. MNAR does not mean that income is the only predictor of response; it means that conditioning on the observed characteristics does not remove the dependence on income. The earlier material on MAR establishes the contrasting case: when \(P(R=1\mid Y,X)=P(R=1\mid X)\), income is MAR relative to the observed \(X\).
How Income-Dependent Response Changes the Respondent Distribution
Let \(f(y\mid x)\) describe the income distribution among people with observed characteristics \(X=x\). The distribution among respondents with those same characteristics is obtained by weighting each income value by its chance of response. This selection effect follows directly from conditional probability.
The denominator is the response probability for a person with characteristics \(x\); it averages the income-specific response probability over that group’s income distribution. For discrete income values, the integral is replaced by a sum. If high incomes have lower response probabilities, the respondent distribution gives those incomes less weight than the full group distribution does.
The corresponding respondent mean is a conditional mean for a randomly selected person, not necessarily the expectation of a finite sample’s realized respondent mean:
When response probability decreases with income, this respondent mean is no greater than the full-group mean, provided the mean exists. To see why, fix \(x\), write \(\pi(Y)=\pi(Y,x)\), and let \(Y'\) be an independent draw from the same income distribution as \(Y\mid X=x\). If \(\pi\) is nonincreasing, then \((Y-Y')(\pi(Y)-\pi(Y'))\leq 0\) for every pair of values. Taking expectations gives \(2\operatorname{Cov}(Y,\pi(Y)\mid X=x)\leq 0\). Since \(E(Y\pi(Y)\mid x)=E(Y\mid x)E(\pi(Y)\mid x)+\operatorname{Cov}(Y,\pi(Y)\mid x)\), dividing by the positive response probability shows that the respondent mean is no greater than \(E(Y\mid X=x)\). The inequality is strict when the covariance is negative.
Worked Example: Lower Response at Higher Income
Hypothetical data. Consider one fixed age-and-education group. A randomly selected person from this group has a 50% chance of income $40,000 and a 50% chance of income $100,000. The response probability is 0.80 at $40,000 and 0.40 at $100,000. These probabilities differ at fixed observed characteristics, so this mechanism is MNAR.
The group’s full-population mean income is \(0.50(\$40{,}000)+0.50(\$100{,}000)=\$70{,}000\). The probability that a randomly selected person responds is \(0.50(0.80)+0.50(0.40)=0.60\).
The respondent mean, defined for a randomly selected person conditional on responding, is
Correction: the numerator’s second term is \(0.50(0.40)(\$100{,}000)=\$20{,}000\), and the first term is \(0.50(0.80)(\$40{,}000)=\$16{,}000\). Thus the correct calculation is
The respondent mean is $10,000 below the full-group mean because the higher-income value is less likely to be observed. This is the conditional mean for a randomly selected person given response. In a finite sample with independent random response, the realized respondent mean has a random numerator and denominator; the ratio above is not generally the expected value of that finite-sample ratio.
The example also illustrates why the response-weighted distribution matters: the income-specific probabilities act on the full income distribution before it is normalized among respondents. It would be incorrect to treat the reported incomes as a random sample from the group’s full income distribution without accounting for that selection.
A Response Model Can State the Dependence Clearly
A mathematical response model can make the MNAR feature explicit. For illustration, suppose response follows a logistic model. Let \(y\) measure income in units of $10,000, and consider people with the same age and education:
This is a hypothetical model, not a model estimated from observed survey data. Its negative income coefficient says that response odds decrease as income increases within this fixed observed-characteristic group.
Worked Example: Interpreting a Logistic Response Model
At income $40,000, \(y=4\). The model’s linear predictor is \(2.58-0.30(4)=1.38\), so the response probability is
At income $100,000, \(y=10\). The linear predictor is \(2.58-0.30(10)=-0.42\), giving
Both probabilities are between 0 and 1, but they are not equal at the same age and education. The model therefore describes MNAR income response. For a $10,000 increase in income, the response odds are multiplied by \(\exp(-0.30)\approx0.741\), a decrease of about 25.9% in the odds, conditional on the fixed observed characteristics. This is a statement about odds, not a 25.9 percentage-point decrease in response probability.
The logistic form is one way to express an assumed response mechanism; MNAR is not defined by choosing a particular regression family. Any response function that varies with unreported income at fixed observed characteristics has the relevant MNAR feature. A functional form, such as a linear term on the log-odds scale, is an additional modeling assumption, not a conclusion guaranteed by the missingness label.
Why Observed Data Cannot Usually Verify MNAR
When income is missing, its value is unavailable precisely for the people whose response behavior is most informative about income-dependent nonresponse. The observed data reveal the response indicator, reported incomes, and observed characteristics. They do not directly reveal how response probability varies with the missing incomes. Consequently, the same observed data can be consistent with different full-data income distributions and different response mechanisms.
Worked Example: Two Mechanisms With the Same Observed Data
Hypothetical observed pattern. Within one fixed age-and-education group, suppose the observed-data probabilities are \(P(R=1,Y=\$40{,}000)=0.25\), \(P(R=1,Y=\$80{,}000)=0.25\), and \(P(R=0)=0.50\). The response rate is 0.50, and half of reported incomes are $40,000 while half are $80,000.
Possible MAR explanation. Suppose the full group has probability 0.50 at each income and each person responds with probability 0.50, regardless of income. Then the two observed respondent probabilities are \(0.50(0.50)=0.25\) each, and the nonresponse probability is 0.50. The full-group mean is \(0.50(\$40{,}000)+0.50(\$80{,}000)=\$60{,}000\).
Possible MNAR explanation. Instead, suppose the full group has probability 0.40 at $40,000 and 0.60 at $80,000. Let response probabilities be 0.625 and \(5/12\approx0.417\), respectively. Then \(0.40(0.625)=0.25\) and \(0.60(5/12)=0.25\), so the observed respondent probabilities are unchanged. Nonresponse has probability \(0.40(0.375)+0.60(7/12)=0.15+0.35=0.50\), also unchanged. Yet this response probability varies with income, and the full-group mean is \(0.40(\$40{,}000)+0.60(\$80{,}000)=\$64{,}000\).
Both explanations produce the same observed pattern, but one is MAR and the other is MNAR, with different full-group means. Therefore, the observed response rate and respondent income distribution alone cannot determine which mechanism generated the data or identify the full income distribution without additional assumptions or information.
This non-identifiability is the key practical limitation. A response model involving missing income cannot generally be estimated from the observed income-response pairs alone, because the income values for \(R=0\) are absent. Evidence from administrative records, a follow-up that obtains income from some initial nonresponders, or other external information can help assess the mechanism. Without such information, conclusions that depend on the unobserved part of the response process require explicit assumptions and sensitivity analysis.
What Diagnostics Can and Cannot Show
Observed characteristics can still help describe the response process. For example, response rates that differ by age or education are evidence against a mechanism in which response is wholly unrelated to those observed characteristics. They do not, by themselves, establish MNAR. The differences might be explained by those characteristics, leaving income response MAR conditional on them; or response might also depend on unreported income.
- Check response patterns by observed characteristics. Large differences can reveal that an analysis assuming response is unrelated to those characteristics is implausible.
- Do not treat a lack of observed associations as proof of MAR. Dependence on an unreported income value may remain invisible in the observed data.
- Look for information beyond the usual observed records. Validation data or targeted follow-up can provide information about income among at least some initial nonresponders.
- Check support. If some income range has zero chance of response, respondent data contain no observed incomes from that range. Recovering its distribution then requires strong additional assumptions.
These checks are diagnostic, not a test that conclusively classifies the mechanism. In particular, no comparison of respondent incomes with missing incomes is available unless information about at least some missing incomes has been obtained from another source.
Implications for Imputation and Analysis
Standard MICE analyses commonly rely on a MAR assumption conditional on variables included in the imputation models. If income response depends on income itself after conditioning on age and education, merely including age and education does not remove that dependence. An imputation model using only observed predictors can therefore fail to represent this MNAR selection process.
An MNAR analysis needs additional structure: for example, an explicit assumption about how response changes with unreported income, informed where possible by external data. Such an assumption cannot usually be learned from the observed data alone. Different plausible assumptions can lead to different results, so the sensitivity of conclusions to them should be made clear. MNAR does not mean that imputation is impossible; it means the missingness mechanism requires assumptions beyond the observed-data information used under MAR.
The main alternatives differ in what they assume about response. MCAR requires response not to depend on the data values; MAR allows dependence on observed values but not on missing values after conditioning on observed information; MNAR permits dependence on the unreported income even after that conditioning. These are assumptions about the data-generating process, not labels determined by the percentage of income values that are missing.
| Mechanism for income response | Condition at fixed observed characteristics \(X\) | What it implies here |
|---|---|---|
| MCAR | Response does not depend on income or \(X\) | No systematic response selection by income or observed characteristics |
| MAR | \(P(R=1\mid Y,X)=P(R=1\mid X)\) | Income may be associated with response before conditioning, but not after conditioning on \(X\) |
| MNAR | \(P(R=1\mid Y,X)\) varies with \(Y\) for some fixed \(X\) | Response still depends on unreported income after conditioning on observed characteristics |
Common Mistake and Key Takeaway
Common mistake: “Response rates differ by age, so income nonresponse must be MNAR.” This does not follow. Age is observed, so a response difference by age may be consistent with MAR once age is conditioned on. The relevant MNAR question is whether response probability still varies with income among people with the same observed characteristics.
A related calculation pitfall is to call the ratio of expected respondent income totals to expected respondent counts the expected finite-sample respondent mean. With a random number of respondents, the realized mean is a random ratio, and its expectation need not equal the ratio of expectations. The selection-weighted formula above is the conditional mean income for a randomly selected person given response; it can also describe a large-sample limiting respondent mean under suitable sampling conditions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Within a fixed age-and-education group, the response probability is 0.70 for incomes below $60,000 and 0.40 for incomes at or above $60,000. Is this MAR or MNAR relative to age and education? Explain.
- In a hypothetical group, half the people have income $30,000 and half have income $90,000. Their response probabilities are 0.80 and 0.20, respectively. Find the response probability and the conditional mean income for a randomly selected respondent.
- Why does observing that response rates differ by education not, on its own, establish MNAR?
- Two mechanisms produce the same response rate and the same observed respondent incomes, but one mechanism is MAR and one is MNAR. What does this imply about classifying MNAR from observed data alone?
Solutions. (1) It is MNAR: at fixed age and education, response probability changes with income. (2) The response probability is \(0.50(0.80)+0.50(0.20)=0.50\). The respondent conditional mean is \(\{0.50(0.80)(\$30{,}000)+0.50(0.20)(\$90{,}000)\}/0.50 =(\$12{,}000+\$9{,}000)/0.50=\$42{,}000\). This is a conditional mean for a randomly selected person given response, not necessarily the expected realized mean in a finite sample with a random number of respondents. (3) Education is observed, so a response difference by education can be explained by conditioning on education under MAR; MNAR requires remaining dependence on the unreported income at fixed observed characteristics. (4) The missingness mechanism is generally not identifiable from observed data alone; distinguishing the mechanisms requires additional assumptions or information about missing incomes.