Quick Answer: Semi Partial Correlation Result

The main analysis tested whether G2 uniquely predicts G3 after controlling for G1, studytime, failures and absences. The dataset contains 649 complete cases. Before controls, the zero-order correlation between G2 and G3 is r = 0.918548, which is a very strong positive relationship.

After removing from G2 the part explained by the controls, the semi-partial correlation between G3 and the residualized part of G2 is sr = 0.397992. The squared semi-partial correlation is sr² = 0.158397. This means that G2 uniquely explains about 15.84% additional variance in G3 after G1, studytime, failures and absences are already included.

What Is Semi Partial Correlation?

Semi partial correlation measures the relationship between an outcome variable and the unique part of a predictor after that predictor has been adjusted for other variables. It is also called part correlation because it correlates the outcome with only the unique part of the focal predictor.

In this guide, the focal predictor is G2. The controls are G1, studytime, failures and absences. First, G2 is predicted from the controls. Then the leftover part of G2, called residualized G2, is correlated with G3. That correlation is the semi-partial correlation.

Semi partial correlation is closely connected to hierarchical regression. When a focal predictor is added to a model after controls, the squared semi-partial correlation equals the increase in model R². In this example, adding G2 after the controls increases R² by 0.158397. Therefore, sr² = 0.158397.

This topic connects with Correlation in Python, Correlation in R, Correlation in SPSS, Correlation in Excel, Correlation Matrix, Correlation vs Regression, Multiple Linear Regression, Regression Residual Analysis, Effect Size, p-value and Confidence Interval.

Why Use Semi Partial Correlation?

Use semi partial correlation when you want to know how much unique variance a predictor adds to an outcome after other variables are already considered. It is especially helpful in hierarchical regression, model comparison and explanatory research.

For students and researchers, semi partial correlation is useful because it tells a different story from ordinary correlation. The zero-order G2–G3 correlation is extremely high, but semi partial correlation asks a sharper question: how much does G2 still add after G1, studytime, failures and absences are already controlled?

Semi Partial vs Partial Correlation

Partial correlation and semi partial correlation are often confused. Both use control variables, but they residualize different parts of the analysis.

The key practical difference is that partial r² and semi partial sr² answer different variance questions. Partial r² is about the remaining residual variance after controls. Semi partial sr² is about the total outcome variance uniquely added by the focal predictor. That makes semi partial sr² especially useful for model-change interpretation.

Semi Partial Correlation Formula

The residual definition of semi partial correlation is:

In this guide:

The unique variance formula is:

Using the verified model values:

Because the coefficient for G2 in the full model is positive, the semi partial correlation is positive:

The F-change test for one added predictor is:

Here, df1 = 1, df2 = 643, F change = 681.556962, and p = 5.437936e-103. In a normal report, this is written as p < .001.

Null and Alternative Hypotheses

The semi partial correlation hypothesis asks whether the focal predictor adds unique explanatory value after controls are already included.

Dataset and Variables Used

The analysis uses the student performance dataset. The outcome is G3 final grade. The focal predictor is G2 second-period grade. The controls are G1 first-period grade, studytime, failures and absences.

This is a strong example for semi partial correlation because G2 is clearly related to G3, but it also overlaps with G1 and other academic variables. Semi partial correlation separates the unique part of G2 from the part it shares with the controls.

Verified Semi Partial Correlation Results

The verified workbook and Python report agree on the main values for the semi partial correlation analysis.

The result is strong because G2 adds substantial explanatory power beyond the controls. A unique variance value of 15.84% is large in many applied educational-data settings, especially after G1 is already included as a control.

Reduced Model vs Full Model

Semi partial correlation is best understood through model comparison. The reduced model includes only the controls. The full model includes the controls plus the focal predictor G2.

The reduction in SSE is also important. The reduced model has SSE = 2081.963098. The full model has SSE = 1010.679276. Adding G2 reduces unexplained error by about 1071.283822. This supports the conclusion that G2 adds a meaningful amount of explanatory value.

Full Model Coefficients

The full regression model includes G1, studytime, failures, absences and G2. The focal predictor G2 has the largest standardized beta in the full model.

The standardized beta for G2 is 0.798796, much larger than the standardized beta for G1, studytime, failures or absences. This confirms that G2 is the dominant unique predictor of G3 in the full model.

Python Chart-by-Chart Interpretation

The Python output contains six charts that explain the complete semi partial correlation workflow: raw zero-order relationship, residualized predictor scatterplot, R² change, correlation comparison, standardized coefficients and residual diagnostics.

Python Chart 1: Zero-Order Scatterplot

This chart shows the raw association between G2 and G3. The relationship is very strong and positive. Students with higher G2 scores tend to have higher G3 scores. This visual pattern matches the zero-order correlation of r = 0.918548.

The chart is the starting point of the analysis. It tells us that G2 and G3 are strongly related before any controls are considered. However, this chart does not tell us how much of that relationship is unique to G2 after G1, studytime, failures and absences are already included.

That is why the semi partial analysis moves beyond this scatterplot. The zero-order relationship is real and strong, but the research question is about unique contribution, not just raw association.

Python Chart 2: Semi Partial Scatterplot with Residualized Predictor

This chart is the core semi partial correlation visual. The x-axis represents the part of G2 that remains after G2 has been predicted from G1, studytime, failures and absences. The y-axis remains the original G3 outcome.

The positive trend shows that the unique part of G2 still relates strongly to G3. The semi partial correlation is sr = 0.397992. This is smaller than the zero-order correlation, but it is still meaningful because it represents unique predictor information after controls.

The chart helps readers understand the difference between ordinary and semi partial correlation. Ordinary correlation uses raw G2. Semi partial correlation uses only the residualized part of G2, then correlates that part with G3.

Python Chart 3: R-Squared Change

This chart shows the unique variance story directly. The reduced controls-only model has R² = 0.692166. The full model with controls plus G2 has R² = 0.850563. The increase is ΔR² = 0.158397.

This chart is one of the most important visuals because ΔR² equals sr². It shows that adding G2 improves the model by about 15.84% of total G3 variance after the controls have already explained their share.

The correct conclusion is that G2 adds substantial unique explanatory value. The model is not merely statistically significant; it also improves prediction in a practically meaningful way.

Python Chart 4: Correlation Comparison

This chart compares three related coefficients: zero-order r, partial r and semi partial sr. The zero-order correlation is 0.918548, the partial correlation is 0.717325, and the semi partial correlation is 0.397992.

The drop from zero-order r to semi partial sr is expected. The semi partial coefficient removes from G2 the part explained by the controls, so it focuses only on the unique part of G2. Even after this removal, the relationship remains positive and meaningful.

This chart is excellent for teaching because it prevents confusion. A reader can see that all three values answer different questions. Zero-order r answers the raw relationship question. Partial r answers the residualized-both-sides question. Semi partial sr answers the unique-predictor-contribution question.

Python Chart 5: Full Model Standardized Coefficients

The standardized coefficient chart shows that G2 is the strongest predictor in the full model. The standardized beta for G2 is approximately 0.798796. G1 has a smaller positive beta of about 0.118505, while failures has a small negative beta of about -0.040084.

This chart supports the semi partial result. If G2 has the largest unique standardized effect, it makes sense that adding G2 produces a large R² change. The chart also shows that studytime is not statistically strong in the full model, even though it is conceptually important.

For reporting, this chart should be used to explain predictor importance within the full regression model. The semi partial coefficient gives unique variance; standardized beta gives relative coefficient strength after standardization.

Python Chart 6: Full Model Residuals vs Fitted

The residuals versus fitted chart checks whether the full regression model behaves reasonably. In a good model, residuals should be scattered around zero without a strong curved pattern. This chart helps evaluate linearity, constant variance and unusual points.

Because semi partial correlation is connected to regression, residual diagnostics are important. A large and significant semi partial correlation can still be affected by nonlinearity, outliers or heteroskedasticity. This chart reminds the reader not to rely only on sr and p-value.

The full model has strong explanatory power, but diagnostics should still be reviewed before final reporting in a research paper or thesis.

R Chart-by-Chart Interpretation

The R assets provide a colorful validation workflow for the same semi partial correlation analysis. The R PDF link returned 404 during verification, but the R chart URLs are included as the assets provided for the post.

R Chart 1: Colorful Zero-Order Scatterplot

The R zero-order scatterplot repeats the raw G2–G3 relationship. The upward pattern is strong and supports the zero-order r value of 0.918548.

This chart confirms that the strong raw association is visible in both Python and R chart styles. It is a good opening figure for readers because it shows why G2 is a promising predictor before controls are introduced.

R Chart 2: Colorful Semi Partial Scatterplot

The R residualized-predictor scatterplot shows the semi partial relationship directly. The trend remains positive, indicating that the unique part of G2 is still associated with G3.

This chart gives a visual explanation of sr = 0.397992. The relationship is weaker than the raw zero-order scatterplot but remains meaningful because it represents the unique portion of G2 after control variables have been removed from it.

R Chart 3: Colorful R-Squared Change

The R R-squared change chart visually confirms the model improvement. The reduced model explains about 69.22% of the variance in G3. The full model explains about 85.06%. The difference is 15.84%.

This is the most direct effect-size interpretation for semi partial correlation. It tells the reader exactly how much additional variance G2 contributes after the controls.

R Chart 4: Colorful Correlation Comparison

The R correlation comparison chart separates the three coefficient types. It helps readers understand that semi partial sr is not supposed to equal the zero-order correlation or the partial correlation.

The main interpretation is that G2 is highly related to G3 at the raw level, remains strongly related after mutual residualization, and still contributes unique variance when only G2 is residualized.

R Chart 5: Colorful Standardized Coefficients

The R standardized coefficient chart confirms that G2 is the dominant predictor. Its standardized beta is far larger than the other predictors in the full model.

This chart should be interpreted with the R² change chart. The coefficient chart shows predictor strength, while the R² change chart shows the unique amount of outcome variance added by G2.

R Chart 6: Colorful Residuals vs Fitted

The R residuals versus fitted plot provides a diagnostic check for the regression model behind the semi partial correlation. It helps detect nonlinearity, unequal residual spread and unusual cases.

The chart belongs in the post because semi partial correlation is not only a correlation idea; it is also a regression-model idea. Good reporting should include both effect size and model diagnostics.

SPSS Output and Workflow Interpretation

The SPSS output link is included in the resources section as provided. During verification, the SPSS PDF link returned a 404 response, so the numerical SPSS interpretation below is based on the verified Excel workbook and Python report values, not on additional unread SPSS-only output.

Open the SPSS Semi Partial Correlation Output PDF

In SPSS, semi partial correlation is usually read from the Part column in the regression coefficients table. The Part value for the focal predictor is the semi partial correlation. Squaring the Part value gives the unique variance explained by that predictor.

In SPSS-style reporting, the result should be written as a hierarchical regression or part-correlation result: adding G2 after G1, studytime, failures and absences significantly increased R² by 0.158, with sr = .398, p < .001.

Excel Worked File Explanation

The Excel workbook is a fully worked semi partial correlation file. It includes the dataset, inputs, working data, regression formulas, semi partial result table, verified results table, chart data and interpretation guidance.

Download the Semi Partial Correlation Fully Worked Excel File

Download the Semi Partial Correlation Keyword Workbook

Excel Formula Logic

The workbook calculates semi partial correlation in two equivalent ways. First, it correlates G3 with residualized G2. Second, it calculates R² change and takes the signed square root.

The Excel interpretation sheet summarizes the result in one sentence: G2 made a significant unique contribution to G3 after controlling G1, studytime, failures and absences, sr = 0.398, sr² = 0.158, p < .001.

Python, R, SPSS and Excel Workflows

The same semi partial correlation result can be reproduced in Python, R, SPSS and Excel. The important requirement is that the same outcome, focal predictor, controls and missing-value rule are used.

Code Blocks and Excel Formulas

Python Code for Semi Partial Correlation

import pandas as pd
import numpy as np
from scipy import stats
from sklearn.linear_model import LinearRegression

df = pd.read_csv("dataset.csv")

outcome = "G3"
focal = "G2"
controls = ["G1", "studytime", "failures", "absences"]

work = df[[outcome, focal] + controls].dropna().copy()
for col in [outcome, focal] + controls:
    work[col] = pd.to_numeric(work[col], errors="coerce")
work = work.dropna()

Y = work[outcome].to_numpy()
X_focal = work[focal].to_numpy()
X_controls = work[controls].to_numpy()

# Zero-order correlation
zero_r, zero_p = stats.pearsonr(X_focal, Y)

# Residualize focal predictor on controls
model_focal = LinearRegression().fit(X_controls, X_focal)
focal_pred = model_focal.predict(X_controls)
focal_resid = X_focal - focal_pred

# Semi-partial correlation: outcome with residualized focal predictor
sr, sr_p_simple = stats.pearsonr(Y, focal_resid)
sr2 = sr ** 2

# Reduced model: controls only
reduced_model = LinearRegression().fit(X_controls, Y)
yhat_reduced = reduced_model.predict(X_controls)
reduced_r2 = reduced_model.score(X_controls, Y)

# Full model: controls + focal
X_full = work[controls + [focal]].to_numpy()
full_model = LinearRegression().fit(X_full, Y)
yhat_full = full_model.predict(X_full)
full_r2 = full_model.score(X_full, Y)

r2_change = full_r2 - reduced_r2

n = len(work)
p_full = len(controls) + 1
df1 = 1
df2 = n - p_full - 1
f_change = (r2_change / df1) / ((1 - full_r2) / df2)
p_change = stats.f.sf(f_change, df1, df2)

print("N:", n)
print("Zero-order r:", zero_r)
print("Semi-partial sr:", sr)
print("sr squared:", sr2)
print("Reduced R2:", reduced_r2)
print("Full R2:", full_r2)
print("R2 change:", r2_change)
print("F change:", f_change)
print("df1:", df1, "df2:", df2)
print("p-value:", p_change)

R Code for Semi Partial Correlation

df <- read.csv("dataset.csv", stringsAsFactors = FALSE)

vars <- c("G3", "G2", "G1", "studytime", "failures", "absences")
work <- na.omit(df[vars])
work[] <- lapply(work, as.numeric)

# Zero-order correlation
zero_test <- cor.test(work$G2, work$G3, method = "pearson")

# Residualize G2 on controls
focal_resid <- resid(lm(G2 ~ G1 + studytime + failures + absences, data = work))

# Semi-partial correlation
sr_test <- cor.test(work$G3, focal_resid, method = "pearson")
sr <- unname(sr_test$estimate)
sr2 <- sr^2

# Reduced and full models
reduced <- lm(G3 ~ G1 + studytime + failures + absences, data = work)
full <- lm(G3 ~ G1 + studytime + failures + absences + G2, data = work)

reduced_r2 <- summary(reduced)$r.squared
full_r2 <- summary(full)$r.squared
r2_change <- full_r2 - reduced_r2

anova_change <- anova(reduced, full)

cat("Zero-order r:", unname(zero_test$estimate), "\n")
cat("Semi-partial sr:", sr, "\n")
cat("sr squared:", sr2, "\n")
cat("Reduced R2:", reduced_r2, "\n")
cat("Full R2:", full_r2, "\n")
cat("R2 change:", r2_change, "\n")
print(anova_change)
summary(full)

SPSS Syntax for Semi Partial Correlation

* Semi Partial Correlation / Part Correlation in SPSS.
* Outcome: G3.
* Focal predictor: G2.
* Controls: G1 studytime failures absences.

OUTPUT CLOSE ALL.
OUTPUT NEW NAME=Semi_Partial_Correlation_Output.

DESCRIPTIVES VARIABLES=G3 G2 G1 studytime failures absences
  /STATISTICS=MEAN STDDEV MIN MAX.

CORRELATIONS
  /VARIABLES=G3 G2 G1 studytime failures absences
  /PRINT=TWOTAIL
  /MISSING=PAIRWISE.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 studytime failures absences
  /METHOD=ENTER G2
  /STATISTICS COEFF OUTS R ANOVA CHANGE ZPP CI(95)
  /DEPENDENT G3
  /SAVE PRED RESID.

* In the Coefficients table, read the Part column for G2.
* Part for G2 is the semi-partial correlation.
* Part squared is the unique variance added by G2.

OUTPUT EXPORT
  /CONTENTS EXPORT=VISIBLE
  /PDF DOCUMENTFILE='Semi-Partial-Correlation-SPSS-Output.pdf'.

Excel Formula Patterns

Outcome:
G3_range

Focal predictor:
G2_range

Controls:
G1_range, studytime_range, failures_range, absences_range

Semi-partial sr:
=CORREL(G3_range, residualized_G2_range)

Residualized G2:
=G2_actual - G2_predicted_from_controls

Reduced R squared:
=RSQ(predicted_G3_controls_only, G3_range)

Full R squared:
=RSQ(predicted_G3_full_model, G3_range)

R squared change:
=Full_R2 - Reduced_R2

sr squared:
=sr^2

Check:
sr^2 should equal R_squared_change

df2:
=N - number_of_full_model_predictors - 1

F change:
=(R2_change/1)/((1-Full_R2)/df2)

p-value:
=F.DIST.RT(F_change,1,df2)

Decision:
=IF(p_value<0.05,"Significant unique contribution","Not significant")

Assumptions and Diagnostics

Semi partial correlation is based on regression, so its assumptions overlap with multiple linear regression assumptions. The coefficient should be interpreted after reviewing model fit, residual behavior and predictor overlap.

Because G1 and G2 are both grade variables, multicollinearity and shared academic-performance information should be discussed carefully. The result is still strong, but the report should make clear that G2 is being evaluated after G1 and other controls are already in the model.

How to Report Semi Partial Correlation

A complete report should include the outcome, focal predictor, controls, sr, sr² or ΔR², model R² values, F change, degrees of freedom, p-value and interpretation.

Common Mistakes in Semi Partial Correlation Interpretation

Downloads and Resources

External References

For additional learning, review regression textbooks and software documentation on part correlation, partial correlation, hierarchical regression, R-squared change, standardized coefficients, residual diagnostics and model comparison.

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