Why the Missing Variable Matters
A regression record can be incomplete in different ways. A response value may be missing while all predictors are recorded, or a predictor may be missing while the response is known. Both situations can prevent a record from entering an ordinary regression fit, but they do not remove the same information or raise the same concerns about selection.
The first task is therefore not to choose a method for filling gaps. It is to identify which variables are incomplete, which records remain usable for the intended model, and what could explain the missingness. A missing response directly prevents calculation of that record’s regression residual. A missing predictor prevents evaluation of the model’s fitted value for that record. In either case, simply retaining or discarding records without examining the pattern can be misleading.
What Ordinary Regression Can Use
For context, consider the usual linear regression model
Here, \(\beta_0,\ldots,\beta_k\) describe the conditional mean of the response given the predictors, and \(\varepsilon_i\) is the part of the response not explained by that mean. Ordinary least squares calculates residuals from records for which both \(Y_i\) and all the predictors are available. A record with an observed response but a missing predictor cannot be given the usual fitted value for this model; a record with all predictors but a missing response has no residual to calculate.
This mechanical limitation is not the same as saying that every incomplete record is useless. For example, a record with missing \(Y\) may still contribute to summaries of its observed predictors. A record with missing \(X_j\) may still have a recorded response and other recorded predictors. Those values can matter for understanding the missingness and for analyses designed to handle it. But neither kind of record can simply be treated as a complete observation in the ordinary fit.
Missing Responses: A Conditional Relationship May Be Preserved
Suppose all predictors are observed but some responses are not. Define \(R_Y=1\) if the response is observed and \(R_Y=0\) otherwise. If, given the predictors, response observation does not depend on the value of \(Y\), then the observed responses can still represent the conditional response distribution at each predictor value. In that situation, fitting the regression to observed responses loses information and usually precision, but need not change the conditional regression relationship.
This claim is conditional on the predictors, not a claim that response missingness is harmless in every setting. If, among records with the same predictor values, responses are more or less likely to be observed depending on the response itself, the observed responses can have a different distribution from all responses. The fitted regression may then be biased. This distinction connects to the missingness concepts discussed in the earlier material on MCAR, MAR, and MNAR in Regression.
Even when the conditional mean is preserved, fewer observed responses generally mean less information and less precise estimates. Also, a valid conditional relationship in the observed data does not guarantee that every predictor region is represented: if some values or combinations of predictors have no observed responses, the regression cannot learn the relationship there from observed outcomes alone.
Worked Example: Fitting a Simple Regression With Missing Responses
Suppose a hypothetical data set records study hours \(x\) and quiz score \(y\), in points, for six students. Two quiz scores are missing; the four observed pairs are \((1,2)\), \((2,4)\), \((4,5)\), and \((5,7)\). Calculate the ordinary least-squares line using the observed responses.
The fitted line is \(\hat{y}=b_0+b_1x\), where \(b_1=S_{xy}/S_{xx}\) and \(b_0=\bar{y}-b_1\bar{x}\). For the four observed pairs, \(\bar{x}=(1+2+4+5)/4=3\) hours and \(\bar{y}=(2+4+5+7)/4=4.5\) points.
The sums of squared and cross-deviations are
Therefore, \(b_1=11/10=1.1\) points per study hour, and \(b_0=4.5-(1.1)(3)=1.2\) points. The fitted line is \(\hat{y}=1.2+1.1x\). This calculation describes the observed-response cases. Whether it estimates the relationship for all six students depends on how the two scores came to be missing; the arithmetic alone cannot establish that.
Missing Predictors: Observed Responses Still May Not Enter the Fit
Now suppose a response is observed but at least one predictor is missing. The response remains useful information, but ordinary regression with the specified predictors still cannot use that row as a complete observation. This is especially important when several predictors are required: a record can be complete for one model and incomplete for another. A regression of \(Y\) on \(X_1\) may have more usable records than a regression of \(Y\) on both \(X_1\) and \(X_2\).
Dropping incomplete rows can also change which predictor values and responses are represented. Whether this changes the regression estimates depends on the relationship between inclusion in the complete-case sample and the outcome variation around the regression mean, as well as on whether there is adequate representation across predictor values. For example, if inclusion depends on the response in a way related to the regression error, the selected cases need not preserve the original conditional relationship. The earlier material on Conditions for Valid Complete-Case Regression treats the conditions for that analysis in more detail.
A useful way to state the key concern is in terms of the regression error. Let \(S=1\) indicate that a record is complete for the model. If complete-case selection leaves the conditional mean of the error equal to zero given the predictors, then the regression mean relationship is preserved among selected cases. If instead selection makes the errors systematically positive or negative at particular predictor values, the fitted relationship can be distorted. Because a missing predictor is itself unavailable, determining whether this condition is plausible requires careful examination of the observed variables and missingness patterns; it cannot be decided merely by noting that some responses are recorded.
Worked Example: A Recorded Response With a Missing Predictor
In a hypothetical data set, a researcher models quiz score \(y\) using study hours \(x\). The four complete pairs are \((1,3)\), \((2,4)\), \((3,6)\), and \((4,7)\). A fifth student has a recorded score of 5 points but no recorded study-hours value. Find the ordinary least-squares line based on complete pairs, and explain the role of the fifth record.
For the four complete pairs, \(\bar{x}=(1+2+3+4)/4=2.5\) hours and \(\bar{y}=(3+4+6+7)/4=5\) points. The deviation sums are
Thus, \(b_1=7/5=1.4\) points per study hour and \(b_0=5-(1.4)(2.5)=1.5\) points. The fitted line is \(\hat{y}=1.5+1.4x\). The fifth student’s score cannot be paired with a predictor value to calculate a residual or add a point to this ordinary fit. It should not be discarded from the data record: the observed score may help describe missingness or be useful in an analysis that handles missing predictors. It does not, by itself, reveal the missing study-hours value.
Start With the Missingness Pattern, Not a Single Count
A single statement such as “10% of records are incomplete” can hide very different situations. The missing percentage for the response answers a different question from the missing percentage for each predictor, and neither necessarily equals the percentage excluded from a particular regression. The model’s predictor list determines which missing values prevent a record from being used.
An initial audit should therefore identify missingness variable by variable and show combinations of incomplete variables. It should also distinguish records with observed responses but incomplete predictors from records with missing responses. Then compare observed characteristics across the relevant groups, using the available measurements and clear denominators. These comparisons can reveal patterns worth investigating, but they do not prove that missingness is harmless or establish its mechanism. This is consistent with the limits of missingness diagnostics covered earlier.
Worked Example: Different Models, Different Complete-Case Counts
A hypothetical data set contains 12 records and three variables: response \(Y\), predictor \(X_1\), and predictor \(X_2\). A missingness-pattern summary is:
| Pattern | Number of records |
|---|---|
| All three variables observed | 6 |
| \(Y\) missing only | 1 |
| \(X_1\) missing only | 2 |
| \(X_2\) missing only | 1 |
| \(Y\) and \(X_1\) missing | 1 |
| \(X_1\) and \(X_2\) missing | 1 |
The counts sum to \(6+1+2+1+1+1=12\). There are two records with missing \(Y\): one with \(Y\) missing only and one with \(Y\) and \(X_1\) missing. There are four with missing \(X_1\): \(2+1+1=4\). For a model of \(Y\) on \(X_1\), seven records are complete: the six with all variables observed plus the one with \(X_2\) missing. For a model of \(Y\) on both predictors, only six are complete.
This audit immediately shows why “the complete-case sample size” must be stated for a specified model. It also suggests distinct initial summaries: compare observed \(Y\) for records with and without observed \(X_1\), and examine the availability of each predictor separately. Such comparisons describe the observed data and help identify questions about selection; they do not establish that either regression is unbiased.
Common Mistakes and Useful First Checks
- Assuming missing outcomes and predictors are interchangeable. Both can exclude a record from an ordinary fit, but missing outcomes leave no response residual, while missing predictors prevent calculation of the specified fitted value.
- Calling every excluded record useless. A partial record may contribute to summaries or to an analysis designed for incomplete data, even when it cannot enter the ordinary complete-case fit.
- Assuming observed-response regression is automatically unbiased. It is not enough to know that predictors are observed. If response observation depends on the response after conditioning on predictors, the observed conditional relationship may differ.
- Using one overall missingness percentage. Report which variable is missing and the complete-case count for the actual predictor set in the model.
- Filling a missing predictor with zero without justification. Zero is a real value, not a generic symbol for “not recorded.” Replacing a missing value this way changes the data and does not solve the missing-data problem.
A disciplined first pass is to specify the response and predictors, tabulate missingness for each variable and each joint pattern, calculate the usable count for the intended model, and compare observed characteristics across relevant missingness groups. Keep the interpretation modest: these steps reveal where information is absent and what patterns are visible, but they do not alone determine the correct remedy.
Check Your Understanding
Use the distinctions in this tutorial to answer the following questions.
- A record has all predictors observed but no response. Why can it not contribute a residual to ordinary least squares, and what information might it still provide?
- A response is observed, but one predictor in a multiple regression is missing. Why can’t the row enter the ordinary complete-case fit for that model?
- What condition on response observation given the predictors preserves the conditional mean relationship among observed responses?
- A model uses \(Y\), \(X_1\), and \(X_2\). Why might its complete-case sample be smaller than the sample for a model using only \(Y\) and \(X_1\)?
- Why does comparing records with and without observed outcomes or predictors not, by itself, prove that complete-case regression is unbiased?