Quick Answer: Binary Logistic Regression Result

The main Python and R model used 649 students. The binary dependent variable was blr_pass, coded 1 when G3 ≥ 10 and 0 when G3 < 10. There were 549 passes and 100 non-passes. Predictors were G1, G2, studytime, failures, absences, age, school, sex and address.

The likelihood-ratio test was statistically significant, χ²(9) = 382.2624, p < .001. Model fit was strong, with McFadden R² = .6853, Cox-Snell R² = .4451 and Nagelkerke R² = .7720. Discrimination was excellent, AUC = .9768. At the 0.50 cutoff, accuracy was 93.53%, sensitivity 96.72%, specificity 76.00% and F1 96.20%.

What Is Binary Logistic Regression?

Binary Logistic Regression is used when the dependent variable has exactly two outcomes, such as pass/fail, yes/no, disease/no disease, purchase/no purchase or default/no default. Rather than predicting the outcome as an unrestricted number, logistic regression predicts a probability between 0 and 1.

In this worked example, the original final-grade variable G3 is converted into blr_pass. Students with G3 values of 10 or more are coded 1, while students below 10 are coded 0. The model then estimates how first-period grade, second-period grade, study time, prior failures, absences, age, school, sex and address change the odds of passing.

When to Use Binary Logistic Regression

Binary logistic regression is particularly useful when the research question asks which variables increase or decrease the chance of an event and when the results must be expressed as odds ratios, predicted probabilities, classification metrics or ROC AUC.

Binary Logistic Regression Formulas

Logit Model

The left side is the log odds of the event. In this post, the event is blr_pass = 1. Each coefficient B represents the change in log odds for a one-unit change in its predictor while all other variables are held constant.

Convert Logit to Probability

Here η is the fitted linear predictor. The resulting probability is between 0 and 1. The analysis classifies a case as pass when p is at least 0.50.

Odds Ratio

An odds ratio above 1 increases event odds, an odds ratio below 1 decreases event odds, and an odds ratio equal to 1 indicates no change. For G2, B = 1.5366 and exp(B) = 4.6486.

Hypotheses and Decision Rules

The main Python/R likelihood-ratio result is χ²(9) = 382.2624, p < .001, so the predictor set improves substantially over an intercept-only model.

Assumptions of Binary Logistic Regression

Normality of the raw predictors and equal outcome variances are not standard requirements for binary logistic regression. The more relevant concerns are correct outcome coding, logit linearity, multicollinearity, influential observations, discrimination and calibration.

Variables Used and Data Dictionary

The analysis is easier to understand when every variable is named and decoded. The Python/R model and the broader SPSS model are distinguished below.

Dataset and Verified Result Summary

SPSS Output Interpretation for Binary Logistic Regression

The SPSS PDF fits a broader model than the main Python/R specification. It includes G1, G2, studytime, failures, absences, age, Medu, Fedu, traveltime, health and several dummy variables. Because more predictors are included, small differences from Python/R are expected.

Open the Binary Logistic Regression SPSS Output PDF

SPSS Coefficients and Odds Ratios

Python Chart-by-Chart Interpretation with Real Data Values

The Python report contains eight charts. Each discussion below names the variables and integrates the counts, coefficients, probabilities or diagnostic values displayed by the chart.

Python Chart 1: Binary Outcome Distribution

The outcome chart displays the exact coding used by the model. The dependent variable is blr_pass, created from the original final-grade variable G3. A student is coded 1 = Pass when G3 is at least 10 and 0 = Not pass when G3 is below 10. Of the 649 students, 549 were in the pass category and 100 were in the not-pass category. These counts correspond to 84.6% pass and 15.4% not pass.

The chart therefore shows a clear class imbalance. A model that predicted every student as pass would already achieve a baseline accuracy of 549/649 = 84.6%. The fitted logistic model reached 93.53% accuracy, so its improvement should be judged against that baseline rather than against 50%. This is why sensitivity, specificity, AUC and the confusion matrix are also required.

Python Chart 2: Predicted Probability Distribution by Observed Class

This histogram separates the model’s predicted probability of blr_pass = 1 by the observed outcome. The vertical dashed line is the classification cutoff of 0.50. Among the 100 observed not-pass cases, 76 received probabilities below 0.50 and were correctly classified, while 24 received probabilities at or above 0.50 and became false positives. Among the 549 observed pass cases, 531 were above the cutoff and were correctly classified, while 18 fell below the cutoff and became false negatives.

The blue pass distribution is heavily concentrated near probabilities from about 0.90 to 1.00, whereas much of the not-pass distribution lies near 0.00 to 0.10. The remaining overlap around the middle of the probability scale explains the 42 classification errors. The separation is strong but not perfect, which agrees with the model’s AUC = 0.9768.

Python Chart 3: ROC Curve and AUC

The ROC curve plots sensitivity against the false-positive rate across all possible probability cutoffs. The curve stays close to the upper-left corner and the Python area under the curve is AUC = 0.9768, displayed as approximately 0.977 on the chart. An AUC this high means that, when one passing and one non-passing student are selected at random, the model will assign the passing student the higher predicted probability about 97.68% of the time.

The separate SPSS ROC output provides a close validation: AUC = 0.979, standard error 0.006, p < .001, and a 95% confidence interval from 0.967 to 0.991. The agreement between Python and SPSS indicates excellent discrimination even though calibration still needs attention.

Python Chart 4: Calibration by Predicted-Risk Group

The calibration chart compares mean predicted probabilities with observed pass rates in predicted-risk groups. Points on the diagonal represent perfect agreement. The plotted points are close to the diagonal overall: the lowest-risk group is near a predicted probability of about 0.07 with an observed pass rate near 0.06; a middle group is near 0.53 predicted and 0.56 observed; a high-risk group is near 0.89 predicted and 0.86 observed; and the highest-risk group is near 1.00 predicted and about 0.98 observed.

Despite this visual closeness, the formal Python Hosmer-Lemeshow result is χ²(8) = 16.569, p = .034923. Because p is below .05, the model shows a statistically detectable calibration discrepancy. SPSS also flags this issue with χ²(8) = 19.625, p = .012. Therefore, the model discriminates very well but should not be described as perfectly calibrated.

Python Chart 5: Odds Ratios with 95% Confidence Intervals

The odds-ratio plot shows each predictor’s adjusted effect while the other variables are held constant. The strongest positive predictor is G2, the second-period grade: B = 1.5366, OR = 4.6486, 95% CI [2.9884, 7.2311], p < .001. A one-point increase in G2 multiplies the odds of passing by about 4.65. G1, the first-period grade, is also significant: B = 0.6174, OR = 1.8542, 95% CI [1.4101, 2.4380], p < .001.

Age has OR = 1.5523, 95% CI [1.0875, 2.2157], p = .0154. The school contrast MS versus GP has OR = 0.3863, 95% CI [0.1646, 0.9066], p = .0289, indicating lower adjusted pass odds for MS students in this model. The confidence intervals for sex_M (OR = 0.5312), address_U (OR = 0.9422), studytime (OR = 1.1876), failures (OR = 0.7037) and absences (OR = 0.9576) cross 1, so those effects are not statistically significant at α = .05.

Python Chart 6: Classification Confusion Matrix

The confusion matrix uses the 0.50 probability cutoff. It contains 76 true negatives, 24 false positives, 18 false negatives and 531 true positives. Thus, 607 of 649 students were classified correctly and 42 were classified incorrectly.

These cells produce an overall accuracy of 0.9353, sensitivity of 0.9672, specificity of 0.7600, precision of 0.9568, negative predictive value of 0.8085 and F1 score of 0.9620. The model is especially strong at detecting passes, but its specificity is lower because 24 of the 100 actual not-pass cases were predicted as pass.

Python Chart 7: Predicted Probability by G2

This chart uses G2, the second-period grade, as the main predictor and compares the grouped mean predicted probability with the observed pass rate. At very low G2 values, such as 0 to 6, both curves are close to zero. At G2 = 7, the mean predicted probability is roughly 0.03 while the observed pass rate is approximately 0.00. At G2 = 8, the model predicts about 0.21 and the observed rate is about 0.18.

The transition becomes steep around the pass boundary. At G2 = 9, the mean predicted probability is approximately 0.63 and the observed rate is about 0.68. At G2 = 10, the model predicts around 0.91 and the observed pass rate is about 0.89. By G2 = 11, both are close to 0.98–0.99, and from roughly G2 = 13 through 19 the observed pass rate is essentially 1.00. This steep rise is consistent with the adjusted odds ratio of 4.6486 for every one-point increase in G2.

Python Chart 8: Pearson Residuals vs Predicted Probability

The residual plot displays Pearson residuals against fitted pass probabilities. Most of the 649 cases lie close to the horizontal zero line, especially students assigned probabilities near 0 or 1 whose outcomes agree with those predictions. Positive residuals reach roughly +2 to +3, representing observed passes that the model assigned relatively low probabilities.

Several negative residuals occur for students classified with high pass probabilities who did not pass. The most extreme point is near a predicted probability of 1.00 with a Pearson residual of about -32. That case deserves individual checking because it has a rare observed outcome relative to the model’s near-certain prediction. The residual pattern and the significant Hosmer-Lemeshow test both support reviewing unusual observations, possible nonlinearity and omitted interaction terms before using the model for high-stakes decisions.

R Chart-by-Chart Validation

The R workflow uses the same binary outcome, predictor set, binomial family, logit link and 0.50 cutoff. The supplied R chart links resolve to the same hosted chart filenames, so the R section validates the numerical interpretation and preserves all eight requested chart placements.

R Validation Chart 1: Binary Outcome Distribution

The R outcome-count chart validates the same dependent-variable construction: blr_pass = 1 when G3 ≥ 10 and 0 otherwise. R uses all 649 complete records, comprising 549 passes and 100 non-passes. The observed event rate is therefore 84.6%. This imbalance is retained in R rather than artificially rebalancing the sample, so R performance statistics can be compared directly with Python and SPSS.

R Validation Chart 2: Predicted Probability Distribution by Observed Class

The R probability-distribution view confirms that the 531 true-positive cases mostly cluster at high fitted probabilities and the 76 true-negative cases mostly cluster at low probabilities. The overlap consists of 24 false positives and 18 false negatives at the 0.50 cutoff. The same four confusion-matrix counts are recovered because R fits the same binomial-logit model to the same 649 students.

R Validation Chart 3: ROC Curve and AUC

The R ROC validation reproduces the excellent discrimination found in Python. The expected R AUC rounds to the Python value of 0.9768, while SPSS reports 0.979 with a 95% CI of 0.967 to 0.991. The small software difference is only rounding and model-implementation detail; all three results place the model in the excellent-discrimination range.

R Validation Chart 4: Calibration by Predicted-Risk Group

The R calibration chart compares observed and fitted rates across grouped risks. As in Python, the formal lack-of-fit check is important: the Python/R model gives approximately χ²(8) = 16.569, p = .0349. Therefore, the R discussion should not rely only on the visually close diagonal points. The appropriate conclusion is excellent discrimination with a measurable calibration concern.

R Validation Chart 5: Odds Ratios with 95% Confidence Intervals

The R odds-ratio plot validates the ranking of effects. G2 is largest at OR = 4.6486, followed by G1, OR = 1.8542, and age, OR = 1.5523. The MS-school contrast is protective in the coded direction, OR = 0.3863. R confidence intervals reproduce the same inference: G1, G2, age and school_MS exclude 1; sex_M, address_U, studytime, failures and absences include 1.

R Validation Chart 6: Classification Confusion Matrix

The R classification matrix contains TN = 76, FP = 24, FN = 18 and TP = 531. Consequently, R reports accuracy 93.53%, sensitivity 96.72%, specificity 76.00%, precision 95.68% and F1 96.20%. This confirms that the model is more reliable for identifying passes than for identifying non-passes.

R Validation Chart 7: Predicted Probability by G2

The R grouped-probability chart reproduces the sharp G2 transition: predicted pass probability is near zero through G2 values around 6, rises to about 0.21 at G2 = 8, about 0.63 at G2 = 9, about 0.91 at G2 = 10, and approaches 1.00 above G2 11–12. This pattern is the visual expression of the fitted G2 coefficient B = 1.5366 and odds ratio 4.6486.

R Validation Chart 8: Pearson Residuals vs Predicted Probability

The R residual chart should be interpreted with the same numerical context as Python. Most residuals remain close to zero, but a small set of high-probability non-pass cases produces large negative residuals, including one around -32. R should therefore be used to inspect those rows in the predictions-and-residuals table rather than deleting them automatically. They may reflect legitimate rare outcomes, data-entry issues or missing predictors.

R Model-Fit and Coefficient Tables

R’s glm(..., family = binomial(link = "logit")) specification reproduces the Python maximum-likelihood estimates to rounding because the same 649 rows and formula are used.

R Model-Fit Table

R Coefficient and Odds-Ratio Table

Excel Worked File and Formula Interpretation

The uploaded workbook is a worked Binary Logistic Regression template. It contains a dashboard, instructions, raw-data columns, coefficient cells and reporting wording. Its example coefficient cells are placeholders, so the verified Python/R coefficients above should be entered when a fully synchronized Excel dashboard is required.

Download the Binary Logistic Regression worked Excel file

Core Excel Formulas

Using the verified confusion counts, Excel should return accuracy = 607/649 = 0.9353, sensitivity = 531/549 = 0.9672, specificity = 76/100 = 0.7600, precision = 531/555 = 0.9568 and F1 = 0.9620.

SPSS, Python, R and Excel Workflows

Code Blocks for Binary Logistic Regression

SPSS Syntax

* Binary Logistic Regression.
* Outcome: blr_pass = 1 when G3 >= 10; otherwise 0.

COMPUTE blr_pass = (G3 >= 10).
VALUE LABELS blr_pass 0 'Not pass' 1 'Pass'.
EXECUTE.

LOGISTIC REGRESSION VARIABLES blr_pass
  /METHOD=ENTER G1 G2 studytime failures absences age
                school_MS sex_M address_U
  /PRINT=GOODFIT CI(95)
  /CRITERIA=PIN(.05) POUT(.10) ITERATE(50) CUT(.5)
  /SAVE=PRED PGROUP COOK LEVER SRESID.

ROC PRE_1 BY blr_pass (1)
  /PLOT=CURVE(REFERENCE)
  /PRINT=SE COORDINATES
  /CRITERIA=CUTOFF(INCLUDE) TESTPOS(LARGE) DISTRIBUTION(FREE) CI(95).

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE='Binary-Logistic-Regression-SPSS-Output.pdf'.

Python Code

import numpy as np
import pandas as pd
import statsmodels.formula.api as smf

df = pd.read_csv("dataset.csv")
df["blr_pass"] = (pd.to_numeric(df["G3"]) >= 10).astype(int)

formula = (
    "blr_pass ~ G1 + G2 + studytime + failures + absences + age "
    "+ C(school) + C(sex) + C(address)"
)

model = smf.logit(formula, data=df).fit()
probability = model.predict(df)
predicted_class = (probability >= 0.50).astype(int)

results = pd.DataFrame({
    "B": model.params,
    "odds_ratio": np.exp(model.params),
    "p_value": model.pvalues
})
print(results)
print(model.summary())

R Code

df <- read.csv("dataset.csv")
df$blr_pass <- ifelse(df$G3 >= 10, 1, 0)

model <- glm(
  blr_pass ~ G1 + G2 + studytime + failures + absences + age +
    school + sex + address,
  data = df,
  family = binomial(link = "logit")
)

probability <- predict(model, type = "response")
predicted_class <- ifelse(probability >= 0.50, 1, 0)

coefficient_table <- data.frame(
  term = names(coef(model)),
  B = coef(model),
  odds_ratio = exp(coef(model)),
  p_value = summary(model)$coefficients[, 4]
)

print(summary(model))
print(coefficient_table)
table(Observed = df$blr_pass, Predicted = predicted_class)

Excel Formula Pattern

Binary outcome:
=IF(G3>=10,1,0)

Linear predictor:
=$B$2 + $B$3*G1 + $B$4*G2 + $B$5*studytime
 + $B$6*failures + $B$7*absences + $B$8*age
 + $B$9*school_MS + $B$10*sex_M + $B$11*address_U

Predicted probability:
=1/(1+EXP(-predicted_logit))

Predicted class:
=IF(predicted_probability>=0.5,1,0)

Odds ratio:
=EXP(coefficient_B)

APA-Style Reporting for Binary Logistic Regression

Full Python/R Report

Short Report

SPSS Report

Common Binary Logistic Regression Mistakes

Downloads and Resources

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