Quick Answer: Cox Proportional Hazards Model Result

The supplied Python hazard-ratio table contains 649 observations and ten predictors: age, Medu, Fedu, traveltime, studytime, failures, famrel, freetime, school_MS and sex_M. Four terms have 95% confidence intervals that do not include 1: age, studytime, famrel and school_MS.

What Is a Cox Proportional Hazards Model?

A Cox Proportional Hazards Model is a semiparametric survival-regression model. It relates predictors to the hazard of an event without requiring a specific parametric form for the baseline hazard. The model is especially useful when some records are right censored, meaning the event has not been observed by the final recorded follow-up time.

The outcome has two connected parts: a positive duration and a binary event/status indicator. The model does not predict an ordinary continuous outcome. It compares event hazards across predictor values while preserving information from censored records.

When to Use a Cox Proportional Hazards Model

Use parametric vs nonparametric tests, p-value, confidence interval and statistical power as supporting concepts when planning the analysis.

Cox Proportional Hazards Formula and Hazard Ratio

h(t | X) is the hazard for a record with predictor profile X. h₀(t) is the unspecified baseline hazard. The linear predictor changes the baseline hazard multiplicatively.

A hazard ratio is not a probability ratio, risk ratio or change in survival time. It is a multiplicative comparison of instantaneous hazards, conditional on the model and proportional-hazards assumption.

Cox Regression Hypotheses and Decision Rules

See null and alternative hypothesis and Type I and Type II error for general decision-making context.

Dataset and Variables Used

The student-performance dataset does not contain a natural survival endpoint. The scripts therefore prepare a teaching survival structure. The current script documentation defines cox_duration = absences + 1 and cox_event = 1 when G3 < 10. Adding one ensures a positive duration. This is a methodological demonstration, not a natural clinical follow-up outcome.

Python and R Compact Model Variables

Separate SPSS Specification

The supplied SPSS COXREG model enters age, Medu, Fedu, traveltime, studytime, failures, famrel, freetime, goout, Dalc, Walc, health, G1 and G2. Because that list differs from the compact Python/R model, SPSS coefficients should be reported from the SPSS PDF rather than copied from the ten-term Python chart.

Verified Cox Hazard-Ratio Results

The following values are transcribed from the supplied Python hazard-ratio summary image. The coefficient column is the natural logarithm of the displayed hazard ratio.

Which Predictors Are Statistically Notable?

school_MS has the largest effect in the table, HR = 1.4819, 95% CI [1.2281, 1.7882], p < .0001. studytime is also above 1, HR = 1.1377, 95% CI [1.0274, 1.2597], p = .0131. famrel has HR = 1.1029, 95% CI [1.0109, 1.2033], p = .0276. age is below 1, HR = 0.9149, 95% CI [0.8514, 0.9831], p = .0154.

The other six intervals include 1. Their point estimates may still be described, but they should not be called statistically significant at alpha .05. The interpretation of all hazard directions depends on confirming which status category is truly coded as event = 1.

Cox Proportional Hazards Model Assumptions

How Cox Partial Likelihood and Risk Sets Work

The Cox model does not estimate the baseline hazard by assuming a normal, Weibull, exponential or log-normal distribution. Instead, it estimates regression coefficients from the ordering of event times. At every observed event time, the model compares the predictor profile of the event record with the predictor profiles of all records still at risk immediately before that event.

The set of records still under observation and still event-free at an event time is called the risk set. A right-censored record remains in risk sets until its censoring time. After censoring, it no longer contributes to later risk sets. This is why censored records must not be deleted before fitting the model: they provide information about who was still at risk during earlier follow-up periods.

For each event record i, the numerator is that record’s risk score. The denominator is the sum of risk scores for all records in the corresponding risk set R(tᵢ). The product across event times is the partial likelihood. Software normally maximizes the log partial likelihood because sums are more stable numerically than products of many small probabilities.

Why There Is No Ordinary Intercept

Ordinary linear and logistic regression include an intercept because they estimate the complete conditional mean or event probability. The Cox model leaves the baseline hazard unspecified. A separate intercept is therefore not estimated in the same way. The baseline hazard absorbs the time-specific reference level, while the regression coefficients estimate relative hazard multipliers.

How the Excel Workbook Demonstrates the Risk Set

The 20-row workbook calculates a linear predictor, exponentiated risk score, risk-set sum and event contribution for every row. For example, row ID 1 has a linear predictor of -0.67 and exp(lp) of 0.5117. The risk-set sum at its event time is 3.2235, producing a partial log-likelihood contribution of approximately -1.8405. Censored rows contribute zero as event terms but still remain in the denominator of earlier risk sets.

The total workbook partial log likelihood is -16.6292. Excel Solver can change the four coefficient cells to maximize this quantity. In professional survival software, the same objective is optimized automatically and is accompanied by standard errors, confidence intervals, tests and diagnostics.

Tied Event Times: Efron, Breslow and Exact Methods

Tied event times occur when two or more events share the same recorded duration. They are common when time is measured in whole days, months, visits or—as in this teaching example—integer values derived from absences. The standard Cox partial-likelihood expression assumes one event at a time, so software uses an approximation or exact calculation when ties occur.

The R workflow explicitly uses ties = "efron". Python libraries commonly use Efron-style handling internally. SPSS uses its own native tie treatment according to procedure defaults and options. When results are compared across software, the tie method must be aligned before small coefficient or likelihood differences are treated as errors.

Why Ties Matter in This Worked Dataset

The constructed duration is integer valued, so many students share the same follow-up time. The cumulative-event and duration-distribution charts visibly show large concentrations at early integer values. A tie method is therefore not a minor implementation detail. It directly affects risk-set denominators at times where many events occur together.

Overall Model Fit, Concordance and Prediction

Individual hazard ratios answer predictor-specific questions, but a complete Cox report should also evaluate whether the model as a whole improves on a null model and whether fitted risk scores meaningfully rank records by event timing.

Likelihood-Ratio, Wald and Score Tests

Concordance Index

The concordance index measures whether records assigned higher fitted risk tend to experience events earlier. A value near 0.50 indicates little ranking ability, while larger values indicate better discrimination. Concordance should be reported with uncertainty when possible and should ideally be validated outside the data used for model fitting.

The supplied Kaplan-Meier risk-group curves show visible separation, but a median split is not a formal replacement for concordance. Two models can produce similar grouped curves while differing in individual ranking performance. Conversely, a model may have acceptable concordance but poorly calibrated survival probabilities.

Discrimination vs Calibration

A large hazard ratio does not guarantee good prediction, and good concordance does not prove proportional hazards. Each result answers a different question. The final post should therefore keep coefficient inference, model discrimination, calibration and assumption checks in separate subsections.

Detailed Proportional-Hazards Diagnostics

The proportional-hazards assumption states that the hazard ratio for a predictor is approximately constant over time. It does not require the baseline hazard to be constant. The baseline hazard may rise, fall or change shape; proportionality concerns the relative multiplier between predictor profiles.

Schoenfeld Residual Tests

A Schoenfeld residual is defined at an event time and compares the observed predictor value for the event record with the risk-set-weighted expected predictor value. If a coefficient is constant over time, scaled Schoenfeld residuals should not show systematic association with transformed time.

The R cox.zph() output typically provides a test for each predictor and a global test. A small predictor-specific p-value suggests time variation for that coefficient. A small global p-value indicates that at least part of the fitted model may violate proportional hazards.

How to Read the Schoenfeld Chart

Log-Minus-Log Curves

For categorical groups, plots of log[-log(S(t))] against log time should be approximately parallel when proportional hazards is reasonable. Exact parallelism is not expected. Strong crossings or changing distances between curves can signal nonproportional hazards.

Time Interaction Test

A direct test adds a product between a predictor and a function of time, such as X × log(time). A nonzero time-interaction coefficient indicates that the predictor’s log-hazard effect changes with time. The selected time function should be justified because a poor function can miss a more complex pattern.

Functional-Form Diagnostics

Proportionality is not the only Cox assumption. Martingale or deviance residual plots can reveal whether a continuous predictor is linear on the log-hazard scale. A predictor may satisfy proportional hazards but still be misspecified because its association is curved. Restricted cubic splines or categorized terms may be considered when theory and diagnostics support them.

Time-Dependent Effects, Stratification and Cox Model Extensions

When a standard Cox model is inadequate, the solution is not always to abandon survival regression. Several extensions preserve the time-to-event framework while relaxing specific assumptions.

Time-Dependent Coefficients

A time-dependent coefficient allows a predictor’s hazard ratio to change during follow-up. One common representation is:

Here, g(t) may be log time, a piecewise time indicator or a smooth function. The hazard ratio for X is then a function of time rather than one constant value.

Time-Varying Covariates

A time-varying covariate changes value during follow-up, such as current treatment, blood pressure, employment status or account balance. These data are usually stored in start-stop format, with multiple rows per subject representing intervals during which covariate values remain constant.

Stratified Cox Regression

Stratification allows different baseline hazards for levels of a categorical variable while keeping common coefficients for the remaining predictors. It is useful when a nuisance grouping variable violates proportional hazards but its hazard ratio is not itself a primary target.

Cluster-Robust Variance

When records are grouped within schools, hospitals, families or repeated subjects, ordinary standard errors may be too small. Robust sandwich variance estimates adjust coefficient uncertainty for within-cluster dependence without introducing a full random-effects survival model.

Frailty Models

Frailty models introduce an unobserved random effect for clusters or individuals. A shared frailty may represent unmeasured risk common to members of the same group. Frailty models change both interpretation and estimation and should not be treated as a simple checkbox extension.

Penalized Cox Regression

Ridge, lasso and elastic-net penalties are useful when the predictor set is large relative to the number of events or when predictors are strongly correlated. Penalization can improve stability and prediction, but ordinary unpenalized confidence intervals and p-values no longer apply directly.

Python Chart-by-Chart Interpretation with Real Values

Python Chart 1: Event and Censored Counts

The bars contain 569 records labelled Event and 80 labelled Censored, totaling 649. As displayed, the event share is 87.67% and the censored share is 12.33%. This would provide many observed events for coefficient estimation.

However, the script documentation says event = 1 when G3 < 10. The uploaded status labels should therefore be checked against the generated model-ready CSV. If the 80 records are the G3 < 10 cases, the two bar labels must be reversed before publication.

Python Chart 2: Duration Distribution by Status

The displayed Censored group has a median duration near 5, with an interquartile range of approximately 1 to 9. Its upper whisker reaches about 19, with outliers near 23 and 25. The displayed Event group has a lower median near 3, an interquartile range of approximately 1 to 6, an upper whisker near 13 and outliers extending to about 33.

Both groups cover multiple duration values, but the distributions are strongly right skewed because duration is based on absences + 1. The long upper tail and many tied integer times should be acknowledged when discussing risk sets and tied-event handling.

Python Chart 3: Hazard-Ratio Forest Plot

The dashed reference line at HR = 1 separates higher from lower fitted hazard. The largest point is school_MS = 1.4819. Studytime = 1.1377 and famrel = 1.1029 also lie above 1 with intervals fully above the reference. Age = 0.9149 lies below 1 with its interval fully below 1.

Medu, Fedu, traveltime, failures, freetime and sex_M have intervals crossing 1. The forest plot therefore supports four statistically notable terms and six uncertain terms. The direction of these effects must be tied to the verified event definition.

Python Chart 4: Baseline or Overall Survival Curve

The supplied Python baseline curve begins below 0.50 at the earliest displayed time, falls to roughly 0.26 around duration 3, approximately 0.13 around duration 5, and below 0.05 by about duration 8. It approaches zero after approximately duration 14.

This rapid decline is consistent with the chart’s very large displayed event group. Baseline survival refers to the model’s reference covariate profile; it is not the survival probability for every student. Its unusual starting level also reinforces the need to verify event coding and model centering.

Python Chart 5: Kaplan-Meier Survival by Predicted Risk Group

The high-predicted-risk curve falls faster than the low-risk curve. At approximately duration 5, the high-risk survival is near 0.21, compared with about 0.45 in the low-risk group. Around duration 10, the curves are approximately 0.06 and 0.24, respectively.

The high-risk curve reaches almost zero by duration 17, whereas the low-risk group retains a small survival probability into the longest durations. This separation supports practical discrimination, but it is an in-sample median split and is not a substitute for a reported concordance index or external validation.

Python Chart 6: Cumulative Observed Events

Approximately 38% of displayed events have accumulated by duration 1, about 58% by duration 3, 74% by duration 5, 83% by duration 7 and around 90% by duration 9. The curve exceeds 95% by approximately duration 13 and is nearly complete by duration 17.

The event process is therefore concentrated early on the constructed duration scale. Later observations contribute relatively few additional events, which explains why survival curves flatten close to zero in the tail.

Python Chart 7: Predicted Risk Score Distribution

The displayed Event histogram is concentrated mainly between approximately -1.10 and -0.40, with its highest bin slightly above 100 records near -0.9. The displayed Censored group has far fewer records and overlaps substantially with the same risk-score range.

The model separates the groups enough to produce different survival curves, but the overlapping histograms show that discrimination is not perfect. A concordance index and out-of-sample validation would provide a more formal assessment.

Python Chart 8: Hazard-Ratio Summary Table

The table confirms the exact values used above. Significant rows are age, studytime, famrel and school_MS. The most precise large effect is school_MS, HR = 1.4819, 95% CI [1.2281, 1.7882]. Freetime is closest to no effect, HR = 0.9988, p = .9776.

This figure should be the primary numeric reference when the forest plot is too compressed to read exact values. It also demonstrates why every chart interpretation should include the underlying numbers rather than only describing point positions.

R Charts and Independent Validation

Open the updated Cox Proportional Hazards Model R report PDF

The R generator fits survival::coxph(), creates a hazard-ratio table and evaluates proportional hazards with cox.zph(). Six confirmed R media URLs are now supplied. The R section now includes the corrected baseline survival chart, Kaplan-Meier curves by predicted risk group, and the Schoenfeld-residual proportional-hazards diagnostic.

R Chart 1: Event and Censored Counts

The R count chart visually reproduces the Python values: 569 under Event and 80 under Censored. This cross-software agreement confirms that the same model-ready status values were plotted. It does not resolve whether the labels are correct relative to the documented G3 < 10 coding; the underlying CSV still needs verification.

R Chart 2: Duration Distribution by Status

The R boxplot confirms the same approximate medians: 5 for the displayed Censored group and 3 for the displayed Event group. The Event-labelled group has more extreme upper outliers, reaching approximately 33, while the Censored-labelled group reaches about 25.

The agreement between the Python and R boxplots supports the duration summary. The difference is descriptive and should not be interpreted as an adjusted Cox coefficient by itself.

R Chart 3: Hazard-Ratio Forest Plot

The R forest plot places schoolMS around 1.48, studytime around 1.14, famrel around 1.10 and age around 0.91. It also reproduces the confidence-interval crossings for Medu, Fedu, traveltime, failures, freetime and sexM.

The equivalent predictor labels differ slightly between software, such as school_MS versus schoolMS and sex_M versus sexM. These are naming differences created by dummy coding, not different concepts.

R Chart 4: Baseline Survival with Confidence Limits

The R baseline survival starts at 1 before the first event, then drops to approximately 0.71 around duration 1, 0.54 around duration 3, 0.38 around duration 5, 0.28 around duration 7 and roughly 0.19 around duration 9. It falls below 0.10 by about duration 14 and approaches zero in the late tail.

The dashed limits widen where fewer observations remain informative. The numerical level differs from the Python baseline figure because baseline curves can depend on implementation details, centering and fitted specification. The hazard-ratio directions are the more directly comparable quantities across the supplied outputs.

R Chart 5: Kaplan-Meier Survival by Predicted Risk Group

The R Kaplan-Meier chart compares observed survival after the fitted linear-predictor values are divided into lower- and higher-risk groups. The higher-risk curve is expected to decline more rapidly when the Cox risk score discriminates between records with different fitted hazards. In the supplied output, the high-risk curve remains below the low-risk curve across much of the constructed duration scale, matching the separation shown in the Python risk-group chart.

This median-risk split is descriptive and discards some information from the original continuous risk score. Interpret the curve separation together with the hazard-ratio table, confidence intervals, model concordance when available, and the confirmed event coding rather than treating the grouped chart as the primary Cox test.

R Chart 6: Schoenfeld-Residual Proportional-Hazards Check

The Schoenfeld-residual chart checks whether the fitted log-hazard effect changes systematically over follow-up time. A smooth pattern that remains approximately horizontal around zero supports the proportional-hazards assumption. Strong curvature, a sustained upward or downward trend, or a small corresponding cox.zph() p-value suggests that the hazard ratio may vary over time.

The visual decision must be read with the numerical cox.zph() results in the updated R report. When a meaningful violation is present, consider a predictor-by-time interaction, a stratified Cox model, a time-varying coefficient model, or another survival specification.

SPSS Cox Regression Output Interpretation

The SPSS workflow creates cox_duration = absences + 1, creates cox_event = 1 when G3 < 10, saves the prepared SAV file, runs Kaplan-Meier context and fits a native COXREG model. SPSS reports B, SE, Wald, df, Sig. and Exp(B).

Excel Worked Cox Proportional Hazards Results

The Excel workbook is a transparent 20-row teaching example. It contains 10 observed events and 10 censored records, an event rate of 50%, median duration of 6, total partial log-likelihood of -16.6292, mean risk score exp(lp) of 0.1612 and highest-risk row ID 1.

Excel Dashboard

Excel Coefficients and Hazard Ratios

Excel Partial-Likelihood Logic

Excel Solver is configured to maximize the total partial log-likelihood by changing the coefficient cells. The workbook is useful for learning the mechanics, but formal confidence intervals, p-values and PH tests should come from Python, R or SPSS.

Python, R, SPSS and Excel Workflows

Cox Proportional Hazards Model Code

Python with lifelines

import pandas as pd
from lifelines import CoxPHFitter
from lifelines.statistics import proportional_hazard_test

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

# Teaching survival setup
df["cox_duration"] = pd.to_numeric(df["absences"], errors="coerce") + 1
df["cox_event"] = (
    pd.to_numeric(df["G3"], errors="coerce") < 10
).astype(int)

model = df[
    ["cox_duration", "cox_event", "age", "Medu", "Fedu",
     "traveltime", "studytime", "failures", "famrel", "freetime"]
].dropna()

cph = CoxPHFitter(penalizer=0.01)
cph.fit(model, duration_col="cox_duration", event_col="cox_event")
print(cph.summary)

ph_test = proportional_hazard_test(cph, model, time_transform="rank")
print(ph_test.summary)

R with survival

library(survival)

df <- read.csv("dataset.csv", stringsAsFactors = FALSE)
df$cox_duration <- as.numeric(df$absences) + 1
df$cox_event <- ifelse(as.numeric(df$G3) < 10, 1, 0)

fit <- coxph(
  Surv(cox_duration, cox_event) ~
    age + Medu + Fedu + traveltime + studytime +
    failures + famrel + freetime + school + sex,
  data = df,
  ties = "efron",
  x = TRUE
)

summary(fit)
cox.zph(fit)
plot(survfit(fit))

SPSS COXREG Pattern

COMPUTE cox_duration = absences + 1.
COMPUTE cox_event = 0.
IF (G3 < 10) cox_event = 1.
EXECUTE.

COXREG cox_duration WITH
  age Medu Fedu traveltime studytime failures
  famrel freetime goout Dalc Walc health G1 G2
  /STATUS=cox_event(1)
  /METHOD=ENTER age Medu Fedu traveltime studytime failures
    famrel freetime goout Dalc Walc health G1 G2
  /PRINT=CI(95) GOODFIT SUMMARY
  /PLOT SURVIVAL HAZARD LML
  /CRITERIA=PIN(.05) POUT(.10) ITERATE(20).

OUTPUT SAVE OUTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Cox Proportional Hazards Model\SPSS_Output\spv\Cox-Proportional-Hazards-Model-SPSS-Output.spv'.

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL LAYERS=PRINTSETTING MODELVIEWS=PRINTSETTING
  /PDF DOCUMENTFILE=
'D:\DATA ANALYSIS\H Regression Tests and Models\Cox Proportional Hazards Model\SPSS_Output\pdf\Cox-Proportional-Hazards-Model-SPSS-Output.pdf'.

Excel Formulas

Linear predictor:
=SUMPRODUCT(predictor_row, coefficient_cells)

Risk score:
=EXP(linear_predictor)

Risk-set sum:
=SUMIFS(exp_lp_range, duration_range, ">="&current_duration)

Event partial log-likelihood:
=IF(event_cell=1, linear_predictor-LN(risk_set_sum), 0)

Hazard ratio:
=EXP(coefficient)

APA-Style Cox Regression Reporting

Full Report Using the Supplied Hazard-Ratio Table

Mandatory Coding Qualification

Excel Worked-Example Report

Common Cox Regression Mistakes

Cox Regression Reporting Checklist

A high-quality report should allow another analyst to reconstruct the survival outcome, predictor coding and model decisions. The following checklist is more useful than reporting only a forest plot.

Recommended Results Order

Begin with the sample and event/censoring summary. Next report overall model fit, followed by hazard ratios with confidence intervals. Then present proportional-hazards diagnostics, discrimination, survival curves and sensitivity analyses. Keep the substantive conclusion tied to the event definition and reference categories.

Recommended Limitations Wording for This Teaching Example

Cox Regression Troubleshooting Guide

Model Does Not Converge

Nonconvergence may result from complete separation, an extremely sparse category, severe multicollinearity, too many predictors for the number of events or a predictor with almost no variation. Inspect category counts, coefficient magnitude and warnings. Simplify the model or use penalization when appropriate.

Very Large Hazard Ratios or Confidence Intervals

Extreme hazard ratios often reflect sparse data, near separation, unstable dummy categories or poor scaling. Report the instability rather than presenting the point estimate as a precise effect. Combining categories should be based on subject knowledge, not only on obtaining smaller coefficients.

Confidence Interval Includes 1 but p-Value Appears Significant

This is usually caused by rounding or by using statistics from different models. Use more decimal places and confirm that the coefficient, standard error, p-value and interval all come from the same fitted specification.

Python, R and SPSS Results Differ

First compare event coding, censor value, predictor list, reference categories, missing-data exclusions, tie method, penalization and continuous-variable scaling. In the supplied files, Python/R and SPSS already use different predictor specifications, so identical coefficients should not be expected.

Baseline Survival Curves Differ Across Software

Baseline curves may differ because software centers predictors differently or reports survival for different reference profiles. Hazard ratios can agree while baseline survival levels differ. Document the covariate profile associated with each curve.

Global PH Test Is Significant

Inspect predictor-specific tests and plots rather than immediately rejecting the entire model. Identify which terms drive the result, evaluate practical magnitude, test time interactions and compare alternative specifications.

Too Few Events for the Predictor Count

Reduce unnecessary terms, combine carefully justified variables, use penalized methods or collect more event information. The relevant information is closer to the event count than the total sample size. A large number of censored records does not fully compensate for very few events.

Risk Groups Do Not Separate

Weak risk-group separation may indicate low discrimination, an arbitrary cut point, incorrect event coding or overfitting. Report the concordance index and consider continuous calibration rather than relying solely on a median split.

Downloads and Resources

Related Salar Cafe Guides

Final Cox Proportional Hazards Model Conclusion

The supplied compact Cox model identifies school_MS as the strongest hazard-ratio term, followed by smaller significant effects for studytime, famrel and age. The Python and R media agree on the hazard-ratio pattern, duration distributions and displayed 569-to-80 status split. The Excel workbook separately demonstrates how risk scores and partial likelihood are calculated.

The final publishing step is to correct or confirm the event/censor labels. Once event = 1 is verified consistently in the model-ready data, count charts, survival curves and software reports, the hazard directions can be given a definitive substantive interpretation.