The Effect Flipped Sign When I Added a Covariate

Symptoms

• An exposure has a positive unadjusted coefficient and a negative adjusted coefficient.
• A coauthor suggests dropping the covariate or labeling the result multicollinearity.

What this usually means

The unadjusted association and the association conditional on a covariate answer different questions. A reversal may reflect confounding or suppression, but the regression output alone cannot decide which causal interpretation is justified. Adjustment can also introduce bias when a variable is a mediator or collider.

Common causes

• The exposure and covariate are associated, and the covariate is associated with the outcome.
• Adding a covariate silently removes rows with missing values.
• Coding, interactions, influential observations, or a misspecified relationship change the comparison.

Run these checks

1. Compare both models on the same rows and verify reference levels and units.
2. State whether you want a marginal association, a conditional association, or a causal effect.
3. Use substantive knowledge and a causal diagram to decide whether adjustment is appropriate.
4. Inspect scatterplots, uncertainty, predictor correlations, and influence. Assess functional form and interactions.

What not to do

Do not select covariates to restore a preferred direction. A tolerable VIF does not establish correct adjustment, and a large VIF does not by itself explain a reversal. Do not call the adjusted coefficient causal without the necessary design and assumptions.

Treatment options

Report unadjusted and prespecified adjusted estimates with their distinct interpretations. Separate the effects of changing the sample and changing the model. If an adjustment decision is uncertain, present justified sensitivity models rather than optimizing the sign.

Worked example

Eight deliberately constructed observations satisfy z = x + w and y = −x + 2z + e, where x, w, and e are orthogonal. Both models use all eight rows.

The adjusted z coefficient is +2.000; correlation between x and z is .707 and the x VIF is 2.000. Thus a reversal is possible with modest VIF and no sample change. Both focal intervals include zero. This algebraic illustration supplies no real-world causal evidence.

What to tell the reviewers

We compared the unadjusted and adjusted models on identical observations. The sign change therefore reflects adjustment rather than sample exclusion. We describe the different questions the models address, justify the covariate using the study design, and report confidence intervals for both estimates.

See it in R and Python

Both languages construct the same deterministic observations and reproduce the sign reversal, t-based confidence intervals, and VIF.

Python dependencies: NumPy and SciPy. Install with python -m pip install numpy scipy.

# Exact synthetic illustration; base R only. Orthogonal residuals.
x <- rep(c(-1,1),each=4)
w <- rep(c(-1,-1,1,1),2)
e <- rep(c(-1,1),4)
z <- x+w
y <- -x+2*z+e
d <- data.frame(x,z,y)
a <- lm(y~x,d); b <- lm(y~x+z,d)
cat(sprintf('Same rows: unadjusted n=%d; adjusted n=%d\n',nobs(a),nobs(b)))
cat(sprintf('Unadjusted x=%.3f; adjusted x=%.3f; adjusted z=%.3f\n',coef(a)['x'],coef(b)['x'],coef(b)['z']))
vif <- 1/(1-summary(lm(x~z,d))$r.squared)
cat(sprintf('Correlation x,z=%.3f; x VIF=%.3f\n',cor(x,z),vif))
cat(sprintf('Unadjusted x CI=[%.3f,%.3f]; adjusted x CI=[%.3f,%.3f]\n',confint(a)['x',1],confint(a)['x',2],confint(b)['x',1],confint(b)['x',2]))
stopifnot(abs(coef(a)['x']-1)<1e-8,abs(coef(b)['x']+1)<1e-8)

# DASS Analysis Clinic Case 007; Python dependencies: numpy, scipy.
from pathlib import Path
import numpy as np
from scipy import stats, optimize
HERE = Path(__file__).resolve().parent

def ols(X, y):
    b = np.linalg.lstsq(X, y, rcond=None)[0]
    residual = y - X @ b
    df = len(y) - X.shape[1]
    cov = (residual @ residual / df) * np.linalg.inv(X.T @ X)
    se = np.sqrt(np.diag(cov))
    ci = np.column_stack((b-stats.t.ppf(.975, df)*se,b+stats.t.ppf(.975,df)*se))
    return b, cov, ci

def power(n, d):
    # Matches R power.t.test(strict=FALSE): rejection tail in effect direction.
    return stats.nct.sf(stats.t.ppf(.975,2*n-2),2*n-2,d*np.sqrt(n/2))

x=np.repeat([-1.,1.],4); w=np.tile([-1.,-1.,1.,1.],2); e=np.tile([-1.,1.],4)
z=x+w; y=-x+2*z+e
naive=ols(np.column_stack((np.ones(8),x)),y)
adjusted=ols(np.column_stack((np.ones(8),x,z)),y)
vif=1/(1-np.corrcoef(x,z)[0,1]**2)
print('Unadjusted x / CI:',naive[0][1],naive[2][1]); print('Adjusted x / CI:',adjusted[0][1],adjusted[2][1]);print('Adjusted z:',adjusted[0][2],'VIF:',vif)
assert np.isclose(naive[0][1],1) and np.isclose(adjusted[0][1],-1) and np.isclose(vif,2)

Download Python script

Sources

• Tu, Gunnell, & Gilthorpe (2008). Simpson’s paradox, Lord’s paradox, and suppression effects: the reversal paradox. Emerging Themes in Epidemiology, 5, 2.
• R Core Team. Fitting linear models: lm documentation.