Why the Selection Mechanism Matters
A regression fitted to complete cases uses only records for which the outcome and all predictors in the specified model are observed. Those cases may not represent the full data set in every respect. The key question is therefore not simply how many records remain, but whether the relationship between the outcome and predictors among the remaining records is still the relationship the analysis aims to estimate.
There are mechanisms under which complete-case regression can retain that relationship. If complete-case status is unrelated to the analysis data, the selected records have the same joint distribution as the full population. Even when selection depends on predictors, the conditional outcome relationship can remain unchanged. In contrast, selection that depends on the outcome, even after accounting for the predictors, can alter the relationship being fitted.
The Regression Target and Complete-Case Selection
The linear model here describes the conditional mean of \(Y\) given the predictors. In an error formulation, \(Y=\beta_0+\beta_1X+\varepsilon\), the essential condition for the stated mean relationship is \(E(\varepsilon\mid X)=0\). The model must also be appropriately specified for the intended target: for example, a straight-line conditional mean is not an adequate description if the intended relationship has important curvature that the model omits.
For complete-case regression to estimate the same conditional relationship, a useful sufficient condition is that selection is independent of the outcome once the predictors are fixed: \(R\perp Y\mid X\). This condition allows the probability of being complete to vary with \(X\). It rules out additional selection on the outcome, or on outcome-related information, after conditioning on the predictors in the model. The predictors must also have adequate representation among complete cases: in particular, the probability of being complete should be positive across the predictor values needed to estimate the target, and the selected design must contain enough variation to fit the regression.
Two Selection Settings That Can Preserve the Relationship
Missing completely at random (MCAR) means that complete-case status is independent of the analysis variables. In the notation above, \(R\perp (X,Y)\). Since selection does not favor any values of the predictors or outcome, the joint distribution among complete cases is the same as in the full population. The population regression relationship is therefore retained.
Selection depending only on predictors is less restrictive. If \(R\perp Y\mid X\), then the chance a record is complete may differ across predictor values, but at a fixed value of \(X\) the outcome distribution is unchanged. This condition includes MCAR as a special case, but also allows, for example, people with higher recorded predictor values to be more likely to have complete records.
When the conditional mean model is correct, preserving \(E(Y\mid X)\) preserves its regression parameters, provided the complete cases give enough information to estimate them. The distribution of \(X\) among complete cases can nevertheless differ from its population distribution. That change may affect precision, and it can leave parts of the target predictor range poorly represented. A regression line estimated from the remaining cases is not guaranteed to equal the line estimated from the full data in any one finite sample.
Worked Example: MCAR and a Different Sample Estimate
Suppose, purely hypothetically, four records have predictor values \(x=0,1,2,3\) and outcomes \(y=1,2,5,6\). Assume every record has the same probability of being complete, independently of its \(x\) and \(y\) values. In one possible realization, the records with \(x=0,1,3\) are complete.
For all four records, \(\bar{x}=1.5\) and \(\bar{y}=3.5\). The ordinary least-squares slope is \(b_1=S_{xy}/S_{xx}\), where \(S_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})\) and \(S_{xx}=\sum_i(x_i-\bar{x})^2\). Here,
For the three complete records, \(\bar{x}=4/3\) and \(\bar{y}=3\). Their sums of squares and cross-products are
Thus the complete-case slope is \(8/(14/3)=12/7\approx1.714\), and its intercept is \(3-(12/7)(4/3)=5/7\approx0.714\). These estimates differ from the full-data estimates. That difference does not show that MCAR changed the population regression target: random selection can produce a different finite sample and therefore a different fitted line. MCAR preserves the target in the selection mechanism, not the exact numerical estimate from every realized subset.
Worked Example: Selection Depending on a Predictor
Consider a hypothetical model \(Y=5+2X+\varepsilon\), with \(E(\varepsilon\mid X)=0\). Suppose records are more likely to be complete at \(X=0\) or \(1\) than at \(X=2\) or \(3\), but selection depends on \(X\) alone. At each fixed predictor value, the expected outcome among complete records is still the model mean:
For a concrete illustration, imagine these eight complete records: \((0,4),(0,6),(1,6),(1,8),(2,8),(2,10),(3,10),(3,12)\). Within each \(x\), the outcomes lie one unit below and above \(5+2x\), so their average equals the stated conditional mean. Across these records, \(\bar{x}=1.5\), \(\bar{y}=8\), \(S_{xx}=10\), and \(S_{xy}=20\). Therefore,
This fitted line matches the conditional mean in this illustration. The arithmetic is not a guarantee that every selected sample will reproduce the line exactly; rather, it shows why predictor-dependent selection need not change the conditional relationship. Unequal selection across \(X\) can still change how much information is available at different predictor values.
Why Conditional Independence Preserves the Mean
The central justification is that selection does not change the outcome distribution at a fixed predictor value when \(R\perp Y\mid X\). For values of \(X\) where complete cases are possible, the conditional density obeys
Under \(R\perp Y\mid X\), the selection probability in the numerator is \(P(R=1\mid x)\), so it cancels with the denominator. Hence \(f(y\mid x,R=1)=f(y\mid x)\). Taking the mean of both sides gives
If the latter conditional mean is the correctly specified regression function, the selected cases follow that same conditional mean function. For a linear regression, its intercept and slopes therefore remain the same target. This argument concerns the conditional relationship, not the distribution of \(X\), and not the size of the standard errors.
When Selection Involves the Outcome
If completeness depends on \(Y\) even after fixing \(X\), the cancellation in the derivation no longer applies. The distribution of outcomes among complete cases at a given predictor value can be shifted or otherwise reshaped. In the error formulation, this often means \(E(\varepsilon\mid X,R=1)\ne0\). A regression through the selected cases may then estimate a different conditional relationship.
The direction and size of the distortion depend on the selection rule and on the outcome distribution. Outcome-related selection does not logically guarantee that every regression coefficient changes: special mechanisms can preserve a slope, for example, while shifting an intercept. But preservation is no longer justified by the conditional-independence argument, and coefficients can change substantially.
Worked Example: A High-Outcome Inclusion Rule Changes the Slope
Suppose a hypothetical full data set has two predictor values, \(X=0\) and \(X=1\). At each value, three outcomes are generated by \(Y=10+2X+\varepsilon\), with errors \(-2,0,2\). The full-data mean outcome at \(X=0\) is \(10\), and at \(X=1\) it is \(12\). With a binary predictor, the least-squares slope is the difference between the two group means, so the full-data slope is \(12-10=2\), and the intercept is \(10\).
Now suppose a record is complete only when \(Y\ge12\). At \(X=0\), only the outcome \(12\) remains. At \(X=1\), outcomes \(12\) and \(14\) remain, with mean \((12+14)/2=13\). The complete-case slope is therefore
The selected-data slope is \(1\), rather than the full-data slope \(2\), because the inclusion rule selects outcomes differently at the two predictor values. In this example, selection depends directly on the outcome and the conditional mean among complete cases is not the original line.
Conditions to Check and a Common Pitfall
Before treating a complete-case regression as an estimate of the full-population relationship, check the following conditions. They are assumptions about how records became complete and about the regression target; the observed complete cases alone generally cannot establish them.
- State the target model. Identify the outcome, predictors, and conditional mean relationship the regression is intended to describe.
- Assess the selection assumption. MCAR is sufficient. More generally, selection may depend on \(X\), provided \(R\perp Y\mid X\) for the predictors in the target model.
- Check support and estimability. Complete cases must occur across the predictor values needed for the target, and the selected data must have enough independent predictor variation to estimate the coefficients.
- Keep model specification in view. Preserving a conditional relationship does not make an incorrectly specified linear form correct.
- Separate the target from precision. Even when the coefficient target is preserved, having fewer cases or a different predictor distribution can reduce precision and affect uncertainty calculations.
A common mistake is to reason, “The complete cases have a different distribution of predictor values, so their regression must be biased.” A different predictor distribution by itself does not establish that the conditional regression relationship changed. If selection depends only on the modeled predictors and the conditional outcome relationship is preserved, the same regression parameters remain the target, subject to adequate support and a correctly specified model.
The opposite mistake is to assume that complete-case regression is safe because the selected sample is large. A large selected sample does not remove selection bias when inclusion depends on the outcome in a way that changes \(E(Y\mid X,R=1)\). Sample size can improve precision around the selected-data relationship without making that relationship equal the population target. Regression describes an association under the stated model; preservation of its coefficients does not by itself establish a causal effect.
Check Your Understanding
Use the definitions and conditions above to answer these questions.
- In the notation here, what does \(R=1\) mean, and how does MCAR relate \(R\) to \(X\) and \(Y\)?
- How can selection probabilities vary with \(X\) while leaving the conditional outcome relationship unchanged?
- Why does a complete-case slope that differs from a full-data slope in one finite sample not, by itself, disprove the MCAR condition?
- What role does positive probability of completeness across the relevant predictor range play in estimating the target regression?
- In the threshold example, why does selecting records with \(Y\ge12\) change the fitted slope?
- Can outcome-related selection always be said to change every regression coefficient? Explain why or why not.