Why Compare Marginal and Adjusted Associations?
In an observational survey, people with more education may also differ in work experience and occupation. Those features are associated with wages too, so a wage comparison by education alone can combine several patterns. Multiple regression offers a way to describe the education–wage association while holding measured covariates fixed. The resulting coefficient answers a conditional question; it does not, by itself, tell us what would happen if people were assigned different amounts of education.
This tutorial uses a clearly hypothetical survey to distinguish the marginal association, which relates wages to education without adjusting for other variables, from the adjusted association, which compares wages at the same modeled values of experience and occupation. The distinction matters both for interpreting coefficients and for communicating the limits of observational evidence.
The Model and Its Interpretation
Suppose the outcome \(Y_i\) is person \(i\)'s hourly wage in dollars, education \(Edu_i\) is years of completed schooling, and experience \(Exp_i\) is years of work experience. Occupation has three categories: production, professional, and service. Production is the reference group. Define \(I(Professional_i)\) to equal 1 for a professional occupation and 0 otherwise, and define \(I(Service_i)\) in the same way.
The word “adjusted” describes what is included in the regression comparison, not whether the comparison is causal. The coefficient is conditional on the included variables and the chosen model form. For example, it does not adjust for unmeasured family resources, ability, location, or other factors unless they are represented in the model.
Before using conventional regression standard errors and t-based inference, state the assumptions. The conditional mean of wages must be represented adequately by the specified additive linear form; errors must have mean zero conditional on the predictors; observations must be independent; and there must be no perfect linear dependence among the predictors. Conventional standard errors also require constant conditional error variance. For exact finite-sample conventional t inference, assume in addition that, conditional on the predictors, the errors are independent normal variables with the same variance. Large-sample approximate inference can sometimes be used without conditional normality, but it requires suitable regularity conditions; it is not automatically guaranteed by a large sample alone.
Worked Example: The Marginal Education–Wage Slope
Worked Example: The Marginal Education–Wage Slope
Imagine a hypothetical simple random sample of 240 workers. The sample mean education is 14.2 years and the mean hourly wage is $27.60. Suppose the sample sums of squares and cross-products are \(S_{xx}=960\) years-squared and \(S_{xy}=1459.2\) year-dollar units, where \(x\) is education and \(y\) is hourly wage. Calculate the slope of the marginal least-squares line.
For a simple regression of wage on education alone, the slope is \(b_1=S_{xy}/S_{xx}\). Substituting the hypothetical summaries gives \(b_1=1459.2/960=1.52\).
The marginal line’s intercept is \(b_0=\bar{y}-b_1\bar{x}\). Thus, \(b_0=27.60-(1.52)(14.2)=27.60-21.584=6.016\). The fitted marginal line is therefore \(\widehat{Wage}=6.016+1.52(Education)\), with wage measured in dollars per hour and education in years.
In this sample, the marginal line associates one additional year of education with $1.52 higher hourly wage on average. This is a description of the unadjusted linear relationship, not a claim that an additional year of education causes wages to rise by $1.52.
What Adjustment Does
The multiple regression coefficient for education is not generally the same calculation as the marginal slope. It describes the fitted education–wage relationship after accounting for the model’s experience and occupation terms. One way to understand this is to compare workers who have the same modeled experience and occupation, while allowing their education to differ. The coefficient summarizes the model’s common per-year education slope for those conditional comparisons.
A short algebraic justification makes the role of adjustment precise. Let \(Z\) denote the columns for the intercept, experience, and occupation indicators, and let \(x\) be the education column. Regress education on \(Z\) and call the residual vector \(r\). By construction, \(r\) is orthogonal to every column of \(Z\). In the full regression, the least-squares coefficient on education is
To see why, write the full model’s fitted values as \(Z\hat{\gamma}+x\hat{\beta}_1\). Since \(x\) can be decomposed into its fitted part from \(Z\) and residual \(r\), the fitted part from \(Z\) can absorb the component of \(x\) explained by \(Z\). The remaining direction for estimating the education coefficient is \(r\), which is orthogonal to \(Z\). Minimizing the residual sum of squares in that direction gives \(r^\mathsf{T}(Y-Z\hat{\gamma}-x\hat{\beta}_1)=0\). Orthogonality implies \(r^\mathsf{T}Z\hat{\gamma}=0\), and \(r^\mathsf{T}x=r^\mathsf{T}r\); rearranging yields the displayed formula. Thus the adjusted slope relates the part of education not linearly explained by the covariates to the corresponding wage variation. It is not simply a comparison of the raw education and wage columns.
Worked Example: Interpreting an Adjusted Estimate
Suppose a hypothetical regression on the same 240 workers estimates the education coefficient as \(\hat{\beta}_1=1.18\), with conventional standard error \(SE(\hat{\beta}_1)=0.22\) dollars per hour per education year. The model includes experience and the two occupation indicators described above.
Assume the survey observations can reasonably be treated as independent, the conditional mean form is adequate, residual plots show no substantial curvature or changing spread, and there is no severe predictor redundancy or perfect collinearity. For this example, also assume the conditional normal-error model is appropriate for exact t inference. These checks can support the model’s use, but plots and study design cannot prove the assumptions.
There are five estimated regression parameters: an intercept, education, experience, and two occupation coefficients. The residual degrees of freedom are \(df=n-5=240-5=235\). For a 95% confidence interval, use the \(t\) critical value \(t^*_{0.975,235}\approx1.970\). The interval is \(1.18\pm(1.970)(0.22)=1.18\pm0.4334\), or approximately \((0.747,1.613)\) dollars per hour per education year.
Under the model, the interval procedure would cover the conditional education coefficient in 95% of repeated samples. In context, the data are compatible with an adjusted association of about $0.75 to $1.61 higher hourly wage per additional education year, at fixed modeled experience and occupation. This is not a 95% probability statement about the fixed coefficient.
For the two-sided test \(H_0:\beta_1=0\) against \(H_a:\beta_1\ne0\), the test statistic is \(t=(1.18-0)/0.22=5.364\), with 235 degrees of freedom. The two-sided p-value is less than 0.001. Under the null model and the stated inference assumptions, a test statistic at least this far from zero would be very unusual. The result is evidence of an adjusted association, not evidence that education has been isolated as a cause.
Worked Example: Comparing the Two Estimates
Worked Example: Comparing the Two Estimates
The hypothetical marginal slope was $1.52 per education year, while the adjusted estimate was $1.18. Their difference is \(1.52-1.18=0.34\) dollars per hour per year. Relative to the marginal estimate, the adjusted estimate is lower by \((0.34/1.52)\times100\%=22.37\%\), or about 22.4%.
A careful report might say: “The estimated marginal association was $1.52 higher hourly wage per additional education year. After adjustment for experience and occupation, the estimated association was $1.18 per year (95% CI $0.75 to $1.61; \(p<0.001\)).” The comparison shows that the fitted association changed after including the covariates. It does not show that exactly 22.4% of an education effect was confounded or that adjustment removed all bias.
The estimates answer different statistical questions. A shift can reflect the way education, experience, and occupation vary together in the sample, the conditional relationship represented by the model, or model misspecification. The difference alone cannot identify which explanation is responsible.
Why Adjustment Does Not Establish Causation
In an observational survey, education was not randomly assigned. People with different education levels may differ in characteristics that also influence wages. Adjustment addresses only the measured covariates included in an appropriate model; it does not automatically control unmeasured differences, measurement error, selection into the sample, or a misspecified relationship. Nor does a small p-value resolve these design issues. It quantifies evidence about a coefficient under a model and its assumptions.
Occupation deserves particular care. It may be useful to adjust for occupation when the question is the education–wage association among people in the same occupation. But occupation could also lie on a pathway through which education relates to wages. If so, adjusting for it changes the question and may exclude part of the relationship of interest. The right covariates depend on the substantive question and the roles of those variables, not merely on which variables are available.
Common Mistakes and Reporting Practice
- Calling an adjusted coefficient an effect. The model estimates a conditional association. Use causal language only when the study design and causal assumptions justify it.
- Omitting the reference group. Occupation coefficients compare professional and service categories with production, the stated reference category. Their interpretation depends on that coding.
- Ignoring units. State that the education coefficient is in dollars per hour per additional year of education, conditional on experience and occupation.
- Claiming exact t inference from insufficient assumptions. The conventional finite-sample t result relies on the stated error model, including conditional normality, not only on a linear mean and constant variance. If assumptions are not credible, the inference method and its limitations need to be reported.
- Treating the change in slope as proof of confounding. The marginal and adjusted slopes describe different fitted relationships. Their difference alone does not establish why they differ or establish causality.
Check Your Understanding
Answer each question using the model and hypothetical results above.
- What does the adjusted education coefficient of 1.18 mean in wage and education units?
- Why is the marginal slope of 1.52 not interchangeable with the adjusted coefficient?
- Using the reported estimate and standard error, what test statistic tests \(H_0:\beta_1=0\)?
- What additional assumption supports exact finite-sample conventional t inference in the example?
- Give one reason the adjusted coefficient does not by itself establish a causal effect of education on wages.