Partial R-Squared for Added Predictors

Why Compare Predictors Against a Reduced Model?

A regression model may already explain much of the variation in an outcome before a new predictor, or a group of predictors, is added. The useful question is then not simply how much variation the larger model explains overall. It is how much of the variation that remained unexplained by the smaller model is removed by the added predictors. Partial R-squared summarizes that conditional improvement in fit.

For example, suppose a model predicts weekly energy use from outdoor temperature. A larger model also includes building age. The larger model’s overall R-squared describes the proportion of total outcome variation explained by all its predictors together. Partial R-squared instead describes the proportion of the temperature model’s remaining residual variation removed by adding building age. Neither quantity, by itself, establishes that a predictor causes a change in the outcome.

Definition: For two nested least-squares regression models fitted to the same outcome observations, let $RSS_R$ be the residual sum of squares from the reduced model and $RSS_F$ the residual sum of squares from the full model. If $RSS_R>0$, the partial R-squared for the predictors added in the full model is $$ R^2_{\mathrm{partial}}=\frac{RSS_R-RSS_F}{RSS_R} =1-\frac{RSS_F}{RSS_R}. $$ It is the fraction of the reduced model’s residual sum of squares removed by fitting the added predictors.

What the Fraction Measures

The reduced model is the smaller model; the full model contains every predictor in the reduced model and one or more additional predictors. The comparison is conditional on the predictors already in the reduced model. Thus, partial R-squared is not an isolated measure of an added predictor’s relationship with the outcome. It describes what the added predictor or block contributes after accounting for the predictors already included.

The numerator, $RSS_R-RSS_F$, is the reduction in residual sum of squares. The denominator, $RSS_R$, is the residual sum of squares available to be reduced before the added predictors are included. The ratio has no units, even though each RSS has units equal to the square of the outcome’s units. It is often reported as a proportion or percentage.

For ordinary least-squares fitting, adding predictors to a model while keeping all its existing predictors cannot increase RSS: the full model can always reproduce the reduced model’s fit by setting the added coefficients to zero. Therefore, for nested models fitted to the same observations, $RSS_F\leq RSS_R$. If $RSS_R>0$, partial R-squared lies between 0 and 1. A value of 0 means the fitted full model has not reduced RSS; a value of 1 means its RSS is zero. A value near 1 does not necessarily mean that the full model explains nearly all variation in the outcome: it means that little or none of the variation left over from the reduced model remains.

Conditions for a Fair Comparison

The formula is straightforward, but its meaning depends on comparing the right quantities. Use the same outcome, the same observations, and the same response scale for both fits. The models must be nested, and their RSS values must come from least-squares fits using the same fitting convention, including the same treatment of the intercept. If one fit omits observations that the other uses, their RSS values do not represent a clean comparison of added predictors.

For the calculation itself, no normality assumption or hypothesis test is required. Partial R-squared is a description of the in-sample fit comparison. Regression-model assumptions become important for other purposes, such as inference or generalizing beyond the observed data, but they are not needed simply to compute this ratio. In each example below, the stated RSS values are from hypothetical least-squares fits on the same outcome observations; that information establishes the comparability needed for the calculation.

Worked Example: One Added Predictor

In hypothetical data on 50 houses, a reduced model predicts sale price from floor area. A full model also includes distance to the city centre. Both models are fitted to the same 50 houses, and their residual sums of squares, measured in squared thousands of dollars, are $RSS_R=240$ and $RSS_F=180$. Calculate and interpret partial R-squared.

Check the comparison. The outcome and the 50 observations are the same in both fits, and the full model contains the reduced model’s floor-area predictor plus distance. The models are nested and comparable.

Substitute into the definition.

R^2_{\mathrm{partial}} =\frac{RSS_R-RSS_F}{RSS_R} =\frac{240-180}{240} =\frac{60}{240} =0.25.

The added distance predictor removes 25% of the residual sum of squares left by the floor-area model in these 50 houses. This is a conditional, in-sample statement. It does not say that distance explains 25% of the total variation in sale prices, nor does it show that changing a house’s distance would cause its sale price to change.

Connection to Overall R-Squared

For models with an intercept fitted to the same outcome observations, overall R-squared is $1-RSS/TSS$, where $TSS$ is the total sum of squares around the sample mean. The partial R-squared comparison can be expressed in terms of the two overall R-squared values because the $TSS$ is the same for both fits:

R^2_{\mathrm{partial}} =\frac{RSS_R-RSS_F}{RSS_R} =\frac{(1-R^2_R)TSS-(1-R^2_F)TSS}{(1-R^2_R)TSS} =\frac{R^2_F-R^2_R}{1-R^2_R}.

Here $R^2_R$ and $R^2_F$ are the overall R-squared values for the reduced and full models. This relation highlights the difference in denominators: the change in overall R-squared is measured against total outcome variation, while partial R-squared is measured against the variation left unexplained by the reduced model. The relation assumes the usual centered R-squared definition and a positive reduced-model RSS; it should not be applied automatically to a no-intercept R-squared definition.

Worked Example: Same Improvement, Different Denominator

In another hypothetical analysis, models predicting examination scores use the same students and outcome. The reduced model has $TSS=1{,}000$ score-points squared and $RSS_R=400$; after adding a block of study-behaviour predictors, the full model has $RSS_F=300$. Calculate both partial R-squared and the change in overall R-squared.

Partial R-squared. The added block removes $400-300=100$ score-points squared from the reduced model’s residual sum of squares. The fraction removed is

R^2_{\mathrm{partial}} =\frac{400-300}{400} =\frac{100}{400} =0.25.

So the study-behaviour block removes 25% of the variation that remained unexplained by the reduced model.

Overall R-squared values. The reduced model has $R^2_R=1-400/1{,}000=0.60$, and the full model has $R^2_F=1-300/1{,}000=0.70$. Overall R-squared increases by $0.70-0.60=0.10$, or 10 percentage points. Equivalently,

\frac{R^2_F-R^2_R}{1-R^2_R} =\frac{0.70-0.60}{1-0.60} =\frac{0.10}{0.40} =0.25.

The 10-percentage-point increase is a share of total score variation; the 25% partial R-squared is a share of the reduced model’s residual variation. Both describe the same fit change with different baselines.

Why the Starting Model Matters

Partial R-squared depends on which predictors are already in the reduced model. The same added predictor can remove a different fraction of residual variation when added to different reduced models. This is not a flaw in the calculation: it reflects the fact that the question is conditional on a specified starting model. Always identify the reduced model when reporting a partial R-squared.

Worked Example: Interpreting a Block Added to an Existing Model

In hypothetical data on 80 patients, a reduced model predicts a blood-pressure outcome from age and baseline blood pressure. A full model adds three medication-category indicators. Both fits use the same 80 patients and the same outcome scale. The reduced-model RSS is 500 and the full-model RSS is 425, in squared millimetres of mercury.

The full model contains the reduced model’s predictors plus the three indicators, so the models are nested. Since $425\leq500$, the reported fit change is consistent with least-squares fitting. The calculation is

R^2_{\mathrm{partial}} =\frac{500-425}{500} =\frac{75}{500} =0.15.

In this hypothetical sample, the medication-category block removes 15% of the residual sum of squares remaining after age and baseline blood pressure are included. This statement concerns the block as a whole, relative to that reduced model. It does not mean each indicator accounts for 15%, and it does not establish that medication category has a statistically significant association or a causal effect. Those are different questions.

A Short Justification of the Formula

The definition follows directly from asking what fraction of the reduced model’s residual variation has been removed. Before adding predictors, the amount available for removal is $RSS_R$. The amount removed is the decrease in residual sum of squares, $RSS_R-RSS_F$. A fraction is the amount removed divided by the starting amount:

\text{fraction removed} =\frac{\text{starting residual sum of squares}-\text{ending residual sum of squares}} {\text{starting residual sum of squares}} =\frac{RSS_R-RSS_F}{RSS_R}.

Rearranging gives $1-RSS_F/RSS_R$. The subtraction is important: $RSS_F/RSS_R$ is the fraction of the reduced model’s residual sum of squares that remains, not the fraction removed. If the models have identical fitted values, then $RSS_F=RSS_R$ and partial R-squared is 0, as the definition should imply.

Common Mistakes and Practical Meaning

Using total variation as the denominator. Dividing $RSS_R-RSS_F$ by $TSS$ gives the increase in overall R-squared for standard models with an intercept, not partial R-squared. Partial R-squared divides by $RSS_R$.
Calling partial R-squared the share of all outcome variation explained. It is the share of the reduced model’s residual variation removed by the added predictors. Use overall R-squared for the proportion of total variation explained by a model under the standard intercept-based definition.
Comparing fits from different observations. A difference in RSS may reflect a changed sample as well as added predictors. For the intended comparison, fit both models to the same observations and outcome.
Interpreting a fit measure as evidence of causation or significance. Partial R-squared summarizes in-sample fit improvement. It is not, by itself, a p-value, a test decision, or a causal estimate.

If $RSS_R=0$, the formula is undefined because there is no residual sum of squares in the reduced model to use as a baseline. If $RSS_R>0$, the least-squares nesting condition $RSS_F\leq RSS_R$ ensures the result is between 0 and 1. A large value can be practically informative when a substantial fraction of the reduced model’s remaining error is removed, but its importance still depends on the outcome, the starting model, and the purpose of the analysis.

Key takeaway: Partial R-squared is $(RSS_R-RSS_F)/RSS_R$. It answers a specific question: what fraction of the reduced model’s residual sum of squares is removed by the added predictors? Keep that conditional denominator distinct from the total-variation denominator used by overall R-squared.

Check Your Understanding

Use the definition and identify the comparison baseline in each answer.

A reduced model has $RSS_R=360$, and a nested full model has $RSS_F=270$, using the same observations. Calculate partial R-squared and interpret it in one sentence.
Why is a partial R-squared of 0.20 not generally the same as a 20-percentage-point increase in overall R-squared?
What conditions about the outcome, observations, and models should be checked before comparing their RSS values?
If a reduced model has $RSS_R=100$ and the full model has $RSS_F=100$, what is partial R-squared, and what does it indicate about the fitted models?
Explain why a positive partial R-squared alone does not show that the added predictors cause a change in the outcome.