What a Response Association Can Establish
When a variable is incomplete, its response indicator records which values were observed. Comparing that indicator across groups defined by an observed variable is a useful diagnostic. If response rates differ across those groups in the population, the response pattern is related to observed data, which is inconsistent with MCAR. The conclusion is deliberately limited: the association does not tell us whether response also depends on values that are missing.
Suppose \(Y\) is a partially observed outcome, \(X\) is a fully observed covariate, and \(R_Y=1\) when \(Y\) is observed and \(R_Y=0\) otherwise. The comparison of interest is between \(P(R_Y=1\mid X=x)\) across values of \(x\). A response association is a pattern in which these probabilities vary with \(x\). It can be described with response rates, a contingency table, a regression model, or another appropriate summary.
For a fully observed \(X\), MCAR implies that \(R_Y\) and \(X\) are independent: response probability is the same at every value of \(X\). Consequently, a genuine population association between \(R_Y\) and \(X\) contradicts MCAR. MAR allows response probability to vary with \(X\), because \(X\) is observed. MNAR also allows that variation; response may depend on \(X\) and on unobserved values of \(Y\).
Why MCAR Implies Equal Response Rates Across Observed Groups
The implication follows by averaging over the possible values of \(Y\). Under MCAR, response probability is constant for every combination of \(X\) and \(Y\). Therefore, for any group \(X=x\) with positive probability,
The same argument applies to continuous \(Y\), with an integral in place of a sum. Thus, under MCAR, response probability cannot vary across values of a fully observed covariate. In a sample, however, observed response rates will rarely be exactly equal. Random sampling variation can create differences even when the population mechanism is MCAR.
A test of independence can assess whether the sample provides evidence against equal response probabilities. Its null hypothesis is that response is independent of the observed variable under study. Rejection is evidence against MCAR, provided the variable is fully observed and the analysis assumptions are reasonable. It is not a test that selects MAR over MNAR.
Worked Example: Response Rates Differ Across Two Observed Groups
In a hypothetical sample of 200 patients, a fully observed baseline group indicator \(X\) has two categories. The outcome \(Y\) is observed for 72 of 100 patients in group A and 48 of 100 patients in group B. Assume patients are independent observations, group membership is recorded for everyone, and response status is correctly coded.
The response rates are
The observed difference is \(0.72-0.48=0.24\), or 24 percentage points. The response odds are \(72/28\) in group A and \(48/52\) in group B, so their ratio is
In this sample, the odds that \(Y\) is observed are about 2.79 times as high in group A as in group B. This is a descriptive association; it does not mean group membership causes response.
For a Pearson chi-square test of independence, the null hypothesis is equal response probability across groups. The pooled response rate is \(120/200=0.60\), so the expected counts under the null are 60 observed and 40 missing in each group. All expected counts are at least 5, and the independent-observation assumption was specified, so the usual chi-square approximation is reasonable. The statistic is
The corresponding p-value is approximately 0.000532. If response were independent of group in the population, a chi-square statistic at least this large would be unusual. The result is evidence against equal response probabilities and therefore against MCAR for this response indicator and observed group, subject to the stated assumptions. It does not distinguish MAR from MNAR: response could depend on group alone, on the unobserved outcome as well, or on both.
Why an Association Does Not Separate MAR From MNAR
An observed association answers a limited question: does response vary with information we can see? It does not reveal whether the response process also uses information we cannot see. The following two hypothetical mechanisms illustrate the distinction.
Worked Example: The Same Kind of Association Under MNAR
Suppose \(X\) and the binary outcome \(Y\) are observed or missing as described below, and \(X\) is fully observed. In a hypothetical population, \(P(Y=1\mid X=0)=0.20\) and \(P(Y=1\mid X=1)=0.80\). Response probabilities are specified for each combination:
| Group \(X\) | \(P(R_Y=1\mid X,Y=0)\) | \(P(R_Y=1\mid X,Y=1)\) |
|---|---|---|
| 0 | 0.90 | 0.50 |
| 1 | 0.70 | 0.30 |
Within group 0, the probability of response differs by \(Y\), and within group 1 it also differs by \(Y\). Since \(Y\) is unobserved when \(R_Y=0\), response depends on a missing value even after conditioning on observed \(X\): this mechanism is MNAR.
Nevertheless, response is also associated with the observed group. Averaging over the outcome distribution gives
For example, in hypothetical groups of 1,000 people each, these probabilities correspond to expected response counts of 820 and 380. The observed-group association is strong, but the mechanism remains MNAR. A MAR mechanism that depended on \(X\) alone could also produce different response probabilities across these groups. Thus, the association cannot tell those explanations apart.
The converse is equally important: no detected association does not establish MCAR. The response process could depend on a missing value without producing a marginal association with the observed variable being examined.
Worked Example: No Group Association Despite MNAR
Suppose \(X\) is binary and fully observed, and \(Y\) is binary with \(P(Y=1\mid X)=0.50\) in both groups. Let response depend on the unobserved outcome: \(P(R_Y=1\mid Y=0)=0.40\) and \(P(R_Y=1\mid Y=1)=0.80\), regardless of \(X\). This is MNAR because response probability depends on \(Y\).
For either group, the overall response probability is
So the response rate is 0.60 in both groups: this comparison shows no population association between \(R_Y\) and \(X\). Yet among responders,
Among nonresponders, the corresponding probability is
The outcome distribution differs substantially between responders and nonresponders, even though response rates do not vary across \(X\). A comparison with \(X\) alone therefore cannot rule out an MNAR process.
Diagnostics, Comparisons, and Limitations
Response associations are most informative when the compared predictors are genuinely observed for everyone in the analysis and when the response indicator and comparison are clearly defined. If a predictor is itself incomplete, comparing another variable’s response indicator across the predictor’s observed values uses only a selected subset. The resulting association may not represent the full sample, so it should not be treated as a straightforward MCAR diagnostic without considering that selection.
A single simple test can also miss patterns. Response may vary nonlinearly with a continuous covariate, differ only within a subgroup, or relate to a combination of observed variables even when no one-variable comparison is striking. Conversely, testing many associations can produce small p-values by chance. Examine the response rates and their uncertainty, use a model suited to the predictor types and plausible patterns, and interpret the results in light of the study design. A model-based association is still an observed-data diagnostic, not a direct observation of the missing values.
| Approach | What it can say | What it cannot say |
|---|---|---|
| Compare response rates or test independence | Whether response appears to vary with a fully observed variable | Whether the mechanism is MAR rather than MNAR |
| Model response using observed predictors | How response probability varies with the included observed predictors, under the model | Whether unobserved values also influence response |
| Use design knowledge and substantive reasoning | Whether plausible causes of nonresponse are recorded in the observed data | Whether an untestable assumption is guaranteed true |
The practical alternatives complement rather than replace one another. A rate comparison is transparent but may be narrow. A response model can represent several observed predictors and their relationships, but its conclusions depend on how that model is specified. Knowledge of how data were collected can identify likely response drivers that were not measured. None of these observed-data approaches can establish that missing values have no further role in response.
A non-significant result means that the data did not provide sufficient evidence against the tested independence hypothesis at the chosen significance level; it does not show that response probabilities are identical. Small samples, sparse categories, and weak associations can leave a test with low power. A significant result, in turn, is evidence against MCAR only for the response indicator and observed variables actually examined, and only if the analysis is valid. It does not automatically establish that every variable’s missingness follows the same mechanism.
Common Mistake
Wrong reasoning: “Response is associated with an observed predictor, so the data are MAR.” The association is consistent with MAR, but MNAR can produce it too, as the example above showed. Correction: conclude only that the association is evidence against MCAR for the response and predictor examined. Separating MAR from MNAR requires more than observing that response varies with measured information.
Another common error is to interpret a large p-value as proof of MCAR. A test evaluates a particular observed association and has finite power; it does not verify the independence of response from every observed and unobserved data value. Report what was examined and what the result supports, rather than assigning a mechanism solely from a non-significant test.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In a hypothetical study, response rates are 0.65 and 0.50 in two fully observed groups. What does this difference imply if it reflects a real population association?
- A response indicator is associated with a fully observed predictor. Can this result distinguish MAR from MNAR? Explain.
- In a population, response rates are equal across a fully observed group variable. Does that establish MCAR? Give a reason.
- A test of association has p-value 0.18 at significance level \(\alpha=0.05\). What is the appropriate conclusion about MCAR?
- The predictor used in a response-rate comparison is itself observed only for some people. Why does this complicate interpretation?
Solutions. (1) A genuine population association contradicts MCAR for that response indicator, because MCAR implies equal response probabilities across values of a fully observed variable. The difference is 0.15, or 15 percentage points. (2) No. Response may depend on the observed predictor alone, as under a possible MAR mechanism, or on that predictor and missing values, as under MNAR. (3) No. Response can depend on a missing value while remaining independent of this particular observed group variable. Equal rates for one comparison do not rule out other dependencies. (4) At the 0.05 level, do not reject the tested independence hypothesis. The result is not proof of MCAR; the test may have limited power and addresses only the association examined. (5) The comparison is then restricted to people whose predictor values are observed. If predictor observation is selective, that subset may not represent the full sample, so the observed association may not directly describe response patterns in everyone.