Why a One-Predictor Screen Can Mislead
A simple screening rule might say: keep a candidate predictor only if it has a sufficiently strong association with the outcome when examined by itself. The rule is appealing because it is easy to apply. But it answers a different question from whether that predictor contributes useful information alongside other predictors. When candidate predictors are related to one another, their separate associations with the outcome can conceal the pattern that appears in a multivariable model.
This is a reason not to use univariate screening as a specification rule, not a claim that every candidate term belongs in every model. Candidate terms should be considered in light of the research question, the intended use, and the meaning of the variables. The issue here is narrower: a weak marginal association alone does not establish that a predictor is irrelevant after adjustment.
How Predictor Relationships Can Mask an Association
Consider a linear model with two predictors:
Here \(Y_i\) is the outcome for observation \(i\), \(X_{1i}\) and \(X_{2i}\) are predictors, and \(\varepsilon_i\) is the error. For the coefficient calculations below, assume the conditional mean is described by this linear form and the observed predictors are not perfectly linearly related. The latter condition ensures that the two slopes can be separately estimated. Inference, such as a test or confidence interval, requires additional assumptions about the errors; the examples focus on the fitted slopes and do not rely on such inference.
For clarity, use centered variables and define \(S_{11}=\sum_i(X_{1i}-\bar{X}_1)^2\), \(S_{22}=\sum_i(X_{2i}-\bar{X}_2)^2\), \(S_{12}=\sum_i(X_{1i}-\bar{X}_1)(X_{2i}-\bar{X}_2)\), and \(S_{1Y}\) and \(S_{2Y}\) as the corresponding predictor-outcome cross-products. The two-predictor least-squares normal equations are:
Solving these equations gives the fitted coefficient of \(X_2\):
By contrast, the slope from regressing \(Y\) on \(X_2\) alone is \(S_{2Y}/S_{22}\). The two expressions are generally different. In particular, a predictor can share outcome-related variation with another predictor, so its marginal association can be small even when its conditional coefficient is not.
Worked Examples
Worked Example: A Zero Marginal Correlation and a Nonzero Conditional Slope
The following are hypothetical, constructed observations. Let \(X_1\) and \(X_2\) be two measured predictors and \(Y\) an outcome. The table gives eight centered observations.
| Observation | \(X_1\) | \(X_2\) | \(Y\) |
|---|---|---|---|
| 1 | -1 | -2 | -0.5 |
| 2 | -1 | -2 | 0.5 |
| 3 | -1 | 0 | 1.5 |
| 4 | -1 | 0 | 2.5 |
| 5 | 1 | 0 | -2.5 |
| 6 | 1 | 0 | -1.5 |
| 7 | 1 | 2 | -0.5 |
| 8 | 1 | 2 | 0.5 |
Marginal result for \(X_2\). The cross-product of \(X_2\) with \(Y\) is \(S_{2Y}=(-2)(-0.5)+(-2)(0.5)+0(1.5)+0(2.5)+0(-2.5)+0(-1.5)+2(-0.5)+2(0.5)=0\). Also, \(S_{22}=(-2)^2+(-2)^2+2^2+2^2=16\). Therefore, the slope from regressing \(Y\) on \(X_2\) alone is \(S_{2Y}/S_{22}=0/16=0\).
To calculate the marginal correlation, the outcome sum of squares is \(S_{YY}=\sum_iY_i^2=18\). Thus \(r_{X_2,Y}=S_{2Y}/\sqrt{S_{22}S_{YY}}=0/\sqrt{16(18)}=0\). A rule retaining only predictors with \(|r|\geq 0.30\), for example, would reject \(X_2\).
Conditional result. The other cross-products are \(S_{11}=8\), \(S_{12}=8\), and \(S_{1Y}=-8\). The two-predictor normal equations are therefore \(8\hat{\beta}_1+8\hat{\beta}_2=-8\) and \(8\hat{\beta}_1+16\hat{\beta}_2=0\). Subtracting the first equation from the second gives \(8\hat{\beta}_2=8\), so \(\hat{\beta}_2=1\); substituting into the first gives \(\hat{\beta}_1=-2\). The determinant \(S_{11}S_{22}-S_{12}^2=8(16)-8^2=64>0\), so the predictors are not perfectly linearly related and both slopes are estimable.
In this fitted model, a one-unit increase in \(X_2\) corresponds to a one-unit increase in the fitted mean of \(Y\), holding \(X_1\) fixed. This constructed example shows why a marginal correlation cutoff can discard a predictor with a nonzero conditional coefficient. It does not establish a causal effect.
Worked Example: A Weak Correlation but a Larger Adjusted Slope
Suppose a hypothetical set of 30 observations has centered cross-products \(S_{11}=100\), \(S_{22}=100\), \(S_{12}=80\), \(S_{1Y}=-20\), \(S_{2Y}=20\), and \(S_{YY}=100\). Think of \(X_1\) as a baseline score, \(X_2\) as exposure hours, and \(Y\) as a later score. The summaries are internally consistent with a full-rank two-predictor fit: \(S_{11}S_{22}-S_{12}^2=10{,}000-6{,}400=3{,}600>0\).
The marginal slope for exposure hours is \(S_{2Y}/S_{22}=20/100=0.20\) score points per hour. The marginal correlation is \(r_{X_2,Y}=20/\sqrt{100(100)}=0.20\). Under a marginal-correlation cutoff of \(|r|\geq0.30\), this predictor would not pass.
For the adjusted coefficient, substitute the cross-products into the two-predictor formula: \(\hat{\beta}_2=[100(20)-80(-20)]/[100(100)-80^2]=3{,}600/3{,}600=1.00\) score point per exposure hour, holding the baseline score fixed. The other fitted slope is \(\hat{\beta}_1=[100(-20)-80(20)]/3{,}600=-1.00\).
The marginal slope is small partly because observations with more exposure also tend to have different baseline scores, and baseline score is related to the outcome. Once the baseline score is held fixed in the model, the fitted exposure slope is 1.00 rather than 0.20. The numbers demonstrate a difference in fitted associations; they do not by themselves show that exposure causes a score change.
Worked Example: Group Composition Hides a Within-Group Difference
Suppose a hypothetical outcome is recorded for people in two baseline-score groups and two exposure groups. The table gives the number of observations and the mean outcome score in each group combination.
| Baseline group | Exposure group | Number observed | Mean outcome score |
|---|---|---|---|
| Lower | No | 10 | 50 |
| Higher | No | 90 | 70 |
| Lower | Yes | 60 | 60 |
| Higher | Yes | 40 | 80 |
The marginal mean for the no-exposure group is \([10(50)+90(70)]/100=6{,}800/100=68\). The marginal mean for the exposure group is \([60(60)+40(80)]/100=6{,}800/100=68\). The marginal difference is zero.
Within the lower baseline group, the mean difference is \(60-50=10\) points. Within the higher baseline group, it is \(80-70=10\) points. A regression using indicators for the two groups has a fitted exposure coefficient of 10 points, with baseline group held fixed. The exposure groups have different mixes of baseline categories, so their overall means conceal the within-group difference. This is another way a one-predictor screen can miss a conditional association.
These summaries illustrate an association pattern, not a causal conclusion. The table alone does not justify saying that exposure produced the differences.
Why the Screen Is Not a Specification Rule
A univariate screen treats each candidate predictor as if its value could be judged without reference to the other terms. But the question answered by a marginal slope or correlation is not the same as the question answered by a multivariable coefficient. The difference matters when predictors share information, when their relationships with the outcome point in opposite directions, or when their distributions differ across groups.
The choice of screening measure matters, too. A slope has units and changes when measurement units change; a correlation is unit-free but still describes a marginal relationship. A cutoff on one cannot be substituted for a cutoff on the other. Neither measure alone determines whether a term belongs in a purpose-driven specification.
There is also a practical danger in screening many predictors and keeping only those that look strong in separate analyses: this can discard a term that contributes after adjustment and make the candidate model depend on the observed sample’s particular pattern. Instead, first define candidate terms in relation to the question and the measurement process. Then assess the specified model in the context of its purpose. A weak one-predictor result may be relevant information, but it is not a complete evaluation of a predictor’s role.
Common Mistakes
- Equating marginal and conditional association. A weak association in a one-predictor fit does not imply a weak coefficient when other predictors are held fixed.
- Calling every numerical cutoff a correlation cutoff. A slope cutoff and a correlation cutoff are different rules. State which quantity is being screened and, for a correlation, calculate it using both variables’ variability.
- Interpreting an adjusted coefficient as causal. Holding measured predictors fixed in a regression is not, by itself, enough to establish causation.
- Using a single result as an automatic inclusion or exclusion decision. A marginal calculation does not replace a rationale for the specification based on the research question and intended use.
Check Your Understanding
Use the definitions and calculations in this tutorial to answer the following questions.
- In your own words, distinguish a marginal association from a conditional association.
- A predictor has marginal slope zero in a sample. Does that establish that its coefficient must be zero in a multiple regression? Explain.
- Why does the formula for a multiple-regression coefficient include the cross-product \(S_{12}\)?
- In the group-composition example, what were the marginal mean difference and the within-group differences?
- Explain why a rule based on \(|r|\geq0.30\) is not interchangeable with a rule based on a slope of at least 0.30.