Why Income Response Can Vary Across People
Income is often not reported by everyone asked. A survey may find that response rates differ by age or education: for example, younger respondents may be more likely to answer, or people with more education may be more likely to complete the income item. Such differences do not, by themselves, mean that the missingness mechanism depends on the unreported income. The key question is whether age and education account for the response pattern, leaving no further dependence on income after conditioning on them.
Let \(Y\) denote income, \(A\) age, \(E\) education, and \(R\) the income-response indicator, where \(R=1\) means income is observed and \(R=0\) means it is missing. In this tutorial, age and education are assumed to be observed for everyone in the relevant survey records. The focus is the mechanism for this one potentially missing variable, income.
This definition allows response probabilities to differ across age-education groups. For example, the probability of reporting income could be 0.90 for one group and 0.60 for another. What matters is that, within each such group, the probability is the same for different income values. The condition is about the response mechanism, not about whether income itself is associated with age or education.
A Hypothetical Response-Rate Table
Suppose a hypothetical survey records age group and education for all 1,000 sampled people. For illustration, imagine the response probabilities are determined by the age-education group alone. The following counts are consistent with that pattern.
| Age group | Education group | Sampled | Income observed | Observed response rate |
|---|---|---|---|---|
| Younger | Non-college | 300 | 240 | 0.800 |
| Younger | College | 200 | 180 | 0.900 |
| Older | Non-college | 250 | 175 | 0.700 |
| Older | College | 250 | 200 | 0.800 |
The rates are calculated within groups: for younger, non-college participants, \(240/300=0.800\); for older, college participants, \(200/250=0.800\). Overall, \(795\) of \(1{,}000\) people report income, so the overall response rate is \(795/1{,}000=0.795\). The overall rate summarizes the sample, but the group-specific rates show how response varies with observed covariates.
Worked Example: Classifying a Group-Based Response Pattern
Use the table above. Assume, as a hypothetical mechanism, that each person's probability of reporting income is exactly the probability assigned to that person's age-education group, with no additional dependence on income. Is this MAR given age and education?
Step 1: State the assumed probabilities. For the four groups, the response probabilities are 0.800, 0.900, 0.700, and 0.800, respectively. For example, in the younger, non-college group, the assumption is \(P(R=1\mid Y=y,A=\text{younger},E=\text{non-college})=0.800\) for every supported \(y\).
Step 2: Compare with the definition. Within each fixed age-education group, the assumed response probability does not change with income. Thus \(P(R=1\mid Y=y,A=a,E=e)=P(R=1\mid A=a,E=e)\) for each group.
Conclusion. Under the stated hypothetical mechanism, income nonresponse is MAR given age and education. The table's observed rates alone do not prove this assumption: they do not reveal whether response also varies with the incomes that are missing.
Justifying the Conditional-Distribution Implication
The MAR condition has a useful implication. If income response is MAR given age and education, and people in a group have a positive chance of responding, then among people with the same age and education, the income distribution among respondents matches the income distribution in that group. This follows directly from the conditional probability rule.
The positive-response condition matters: if no one in a particular age-education group reports income, the group’s income distribution cannot be learned from its respondents because there are none. In practice, very small response probabilities can also make conclusions or estimates unstable, even when they are not exactly zero.
This result is conditional on age and education. It does not say that respondents and nonrespondents have the same income distribution when all age and education groups are pooled together. Respondents and nonrespondents can have different mixes of age and education groups, and income can be associated with those groups.
A Logistic Model for Response
A response model can express the same idea using observed predictors. For a binary response indicator, a logistic model specifies the probability of response as a function of predictors through the log-odds. The relevant MAR formulation includes age and education but no further dependence on income. This is a description of the assumed response mechanism; fitting such a model to observed response indicators does not establish that unreported income has no additional role.
Worked Example: Interpreting a Hypothetical Response Model
Suppose a hypothetical survey uses the following response model, where \(A\) is age in years and \(C=1\) indicates college education while \(C=0\) indicates non-college education: $$ \operatorname{logit}\{P(R=1\mid Y,A,C)\} =-0.40-0.30\left(\frac{A-40}{10}\right)+0.25C. $$ Assume the model is the full response mechanism: it has no additional income term.
Step 1: Calculate a probability. For a 40-year-old non-college person, \((A-40)/10=0\) and \(C=0\), so the log-odds are \(-0.40\). The response probability is $$ \frac{\exp(-0.40)}{1+\exp(-0.40)} =\frac{0.6703}{1+0.6703} =0.4013. $$
Step 2: Compare another group. For a 60-year-old college-educated person, \((A-40)/10=2\) and \(C=1\). The log-odds are \(-0.40-0.30(2)+0.25=-0.75\), giving a response probability of $$ \frac{\exp(-0.75)}{1+\exp(-0.75)} =\frac{0.4724}{1+0.4724} =0.3208. $$
Step 3: Interpret the model. Holding education fixed, a ten-year increase in age multiplies the odds of response by \(\exp(-0.30)=0.7408\) under this model. The response probabilities differ with observed age and education, but the assumed model contains no additional dependence on income. Under that assumption, it describes MAR given those observed covariates. The coefficients concern income response, not an effect of age or education on income.
Why Marginal Patterns Can Mislead
A crude comparison of response rates across income categories, if income categories could be known for everyone, can differ from the conditional comparison within age-education groups. That difference can arise because income and the observed covariates are associated, while response varies across those covariates. The next hypothetical table makes the arithmetic explicit.
Worked Example: A Marginal Difference With No Within-Group Difference
Imagine complete follow-up made income category known for everyone in a hypothetical sample. Let \(S_1\) and \(S_2\) denote two age-education groups. The response counts are:
| Group | Income category | People | Income observed | Response rate |
|---|---|---|---|---|
| \(S_1\) | Low | 20 | 18 | 0.900 |
| \(S_1\) | High | 80 | 72 | 0.900 |
| \(S_2\) | Low | 80 | 40 | 0.500 |
| \(S_2\) | High | 20 | 10 | 0.500 |
Within \(S_1\), the response rate is \(18/20=0.900\) for low income and \(72/80=0.900\) for high income. Within \(S_2\), the rates are \(40/80=0.500\) and \(10/20=0.500\). So response does not vary by income within either group. But the marginal rates differ: among low-income people, \(58/100=0.580\) respond; among high-income people, \(82/100=0.820\) respond.
The marginal difference occurs because the low-income group contains more people from \(S_2\), where response is lower, while the high-income group contains more from \(S_1\), where response is higher. In an actual income-nonresponse setting, income categories for nonrespondents are unavailable, so this full table cannot be produced from ordinary observed data. The example shows why a marginal association, even if it could be assessed, would not by itself settle the conditional MAR question.
What Diagnostics Can and Cannot Show
Because age and education are observed for respondents and nonrespondents, response patterns across those variables can be examined. For example, one can tabulate response proportions by age and education categories or fit a response-indicator model with those observed predictors. These checks can reveal whether response is associated with the recorded covariates, whether some groups have very low response, and whether a simple model misses obvious patterns.
Such diagnostics address only the part of the response pattern related to observed information. When income is missing, its values for nonrespondents are not available to compare with those for respondents within age-education groups. Consequently, observed data alone generally cannot verify that the response probability is independent of unreported income after conditioning on age and education. MAR is a substantive assumption about that unobserved comparison, informed by survey design and subject-matter knowledge as well as observed-data checks.
The conditioning set must also be appropriate. If relevant recorded predictors of response are omitted, the claim that response depends only on age and education may be implausible. Conversely, adding a variable to the conditioning set can only support this particular formulation if that variable is observed for the people whose income is missing. The later material on auxiliary variables treats their role more fully.
Common Mistakes and Limitations
- Wrong reasoning: “Response rates differ by age, so the data are not MAR.” Correction: MAR permits response to depend on observed age and education. The defining question is whether income adds dependence after those variables are fixed.
- Wrong reasoning: “Response rates look similar across observed age and education groups, so MAR is proven.” Correction: this can be evidence about observed patterns, but it cannot show whether missing income itself affects response within those groups.
- Wrong reasoning: “If respondents and nonrespondents have different overall income distributions, the mechanism cannot be MAR.” Correction: MAR is conditional. Different age-education group mixes can produce different marginal distributions even when the within-group income distributions agree.
- Limitation: sparse groups. Very few responses in some age-education combinations make response patterns difficult to assess and can weaken practical support for analyses relying on those groups.
- Limitation: an incomplete conditioning set. The stated MAR condition concerns age and education. If other observed variables are relevant to response, this particular condition may not adequately describe the process.
The earlier material on Missing At Random establishes the general mechanism framework; here the central point is its concrete interpretation for income: response may vary with observed age and education, but not additionally with income once those variables are held fixed. The distinction is a conditional one, and observed response rates cannot settle the dependence on values that were not reported.
Check Your Understanding
Answer each question using the definition and examples above. Brief solutions follow the questions.
- In a hypothetical age-education group, the assumed response probability is 0.75 for every income value. What does this imply for the MAR condition within that group?
- A survey has response rates of 0.85 for younger adults and 0.60 for older adults, with education held fixed. Does this pattern rule out MAR given age and education? Explain.
- In one group, 45 of 60 people report income. Calculate the response rate and state what additional condition is needed for the group’s respondents to represent its income distribution under the derivation above.
- Observed response rates are similar across age and education categories. Can this establish that response is independent of income within those categories? Why or why not?
Solutions. (1) It satisfies the within-group MAR equality if the probability remains 0.75 for every supported income value; response does not depend on income after conditioning on the group. (2) No. MAR permits response rates to vary with observed age, provided that within each age-education group response does not additionally depend on income. (3) The rate is \(45/60=0.750\). The group must have a positive response probability, and the MAR condition must hold; then the respondent income distribution equals the group's income distribution. (4) No. Similar observed rates only describe patterns by recorded variables. Missing income values prevent directly establishing independence between response and income within those categories.