# Cox Proportional Hazards Model Assumptions ## Key Takeaways - The proportional hazards assumption, central to Cox models, posits that the hazard ratio between any two individuals remains constant over time; violations can lead to averaged effect estimates that misrepresent true relationships. - Schoenfeld residuals are the primary diagnostic tool, with scaled residuals plotted against time to visually assess for trends, and statistical tests (e.g., correlation with time) used to quantify departures. - When the proportional hazards assumption is violated, appropriate remedies include stratifying the Cox model for categorical covariates or employing time-varying coefficients, particularly for continuous predictors exhibiting time-dependent effects. - A comprehensive assessment necessitates combining statistical tests (e.g., global and individual covariate tests) with graphical inspection of scaled Schoenfeld residual plots and consideration of subject-matter expertise, as sample size can influence statistical significance. - Martingale residuals are used to check the functional form assumption, plotting them against covariate values to detect non-linear relationships requiring covariate transformation or spline terms. - Influential observations, identified by dfbeta values, can disproportionately affect model diagnostics and should be investigated for data accuracy and representativeness. --- ## Quick Answer - Verify the proportional hazards assumption by testing Schoenfeld residuals for each covariate and the global model, then inspect scaled Schoenfeld residual plots for time trends. - When the assumption fails, apply a stratified Cox model for categorical variables or extend the model with time-varying coefficients for continuous predictors. - No single test proves the assumption holds, combine statistical tests with graphical assessment and subject-matter judgment because large samples can flag trivial departures and small samples can miss real ones. ## At a Glance The Cox proportional hazards model is the most widely used regression approach for time-to-event data in biomedical research. Its central assumption states that the hazard ratio for any two individuals remains constant over follow-up time. When this assumption fails, effect estimates become averaged across time periods and can misrepresent the true relationship between a covariate and the event of interest. | Assumption Component | Diagnostic Method | Practical Decision | | --- | --- | --- | | Global proportional hazards | Schoenfeld residual test on the full model | If p-value is small, inspect individual covariates and plots before deciding on model changes | | Individual covariate | Scaled Schoenfeld residual test per term | Significant test warrants a time-varying coefficient or stratification for that term | | Visual time trend | Smoothed residual plot against time | A non-horizontal trend line suggests the hazard ratio changes over follow-up | | Functional form | Martingale residual plot against covariate values | Non-linear patterns indicate the covariate needs transformation or a spline term | | Influential observations | dfbeta values from the fitted model | Large dfbeta values identify subjects that drive the proportional hazards result | ## Understanding the Proportional Hazards Assumption The Cox model expresses the hazard for subject i as the product of a baseline hazard function and an exponential function of the covariates. The model assumes that the ratio of hazards for two subjects with different covariate values is constant across all time points. This means the effect of a treatment, exposure, or biomarker is multiplicative and does not change as follow-up progresses. The baseline hazard is left unspecified in the Cox model, which is the source of its flexibility. The model estimates regression coefficients without requiring a parametric distribution for survival times. This semi-parametric approach makes the model attractive for analyzing time-to-event data in biology and medicine, where the shape of the baseline hazard is rarely known in advance. The proportional hazards assumption is a mathematical condition that must hold for the model to produce valid estimates. When the assumption holds, the hazard ratio is a single number that summarizes the effect of a covariate across the entire follow-up period. When the assumption fails, the hazard ratio changes over time and a single coefficient cannot capture the true relationship. Consider a study of a new drug where the treatment effect is strong in the first six months but weakens afterward. A Cox model that assumes proportional hazards would report an average effect that is too high for the later period and too low for the early period. The model would mislead researchers about the duration of the treatment benefit. The assumption applies to each covariate in the model, beyond the primary exposure. A model with several covariates requires that every term satisfies the assumption. If one covariate violates the assumption, the entire model can produce biased estimates for all coefficients, beyond the offending term. ## Schoenfeld Residuals Schoenfeld residuals are the primary diagnostic tool for evaluating the proportional hazards assumption. These residuals are calculated for each subject at each event time and are based on the difference between the observed covariate value and the expected value conditional on the risk set at that time. The residuals are defined only for subjects who experience the event. Censored subjects do not contribute to the residual calculation at their censoring time. This is a critical point because the residuals are computed at event times, and the risk set at each event time includes all subjects still under observation. The scaled Schoenfeld residuals are the version used for testing the proportional hazards assumption. The scaling accounts for the variance of the residuals and produces values that can be interpreted as the estimated coefficient at each event time. A plot of scaled Schoenfeld residuals against time shows how the coefficient changes over the follow-up period. The test for proportional hazards is based on the correlation between the scaled residuals and time. A significant correlation indicates that the coefficient changes with time, which violates the assumption. The test is implemented in most statistical software packages and produces a p-value for each covariate and for the global model. The visual inspection of the residual plot is as important as the statistical test. A plot that shows a clear trend, such as a steady increase or decrease in the coefficient over time, indicates a violation even if the test is not significant. Conversely, a plot with random scatter around a horizontal line supports the assumption even if the test is marginally significant. ## Testing the Assumption in Practice The workflow for testing the proportional hazards assumption begins with fitting the Cox model and then applying the diagnostic tools. The process is iterative and requires judgment at each step. ### Step 1: Fit the Initial Model Fit the Cox model with all covariates of interest. The model should include the variables that are part of the research question and any covariates that are known confounders. The model should be specified before examining the residuals to avoid bias from data-driven model selection. ### Step 2: Test the Global Assumption Run the proportional hazards test on the full model. The global test examines whether any covariate in the model violates the assumption. A significant global test indicates that at least one term needs attention, but it does not identify which term. ### Step 3: Test Each Covariate Run the test for each covariate individually. The individual tests identify which terms are responsible for the global violation. The p-values from the individual tests should be interpreted with caution when the model includes many covariates, because multiple testing can produce false positives. ### Step 4: Plot the Residuals Create scaled Schoenfeld residual plots for each covariate. The plots show the estimated coefficient at each event time with a smoothed trend line. The trend line should be approximately horizontal if the assumption holds. A trend line that moves away from the horizontal line indicates a time-varying effect. ### Step 5: Decide on Model Modification The decision to modify the model depends on the strength of the evidence from the tests and plots. A significant test with a clear trend in the plot warrants modification. A significant test with a noisy plot and no clear trend may not require modification, especially in large samples where trivial departures can be significant. ## Interpreting the Diagnostic Results The interpretation of proportional hazards diagnostics requires understanding the limitations of the statistical tests. The tests are sensitive to sample size, and the practical importance of a violation depends on the research context. A large study with thousands of subjects can produce a significant test for a departure from proportional hazards that is too small to matter clinically. The test detects the departure because the sample size provides enough power to detect even minor deviations. The plot of the residuals helps distinguish between a meaningful violation and a statistically significant but practically irrelevant one. A small study with few events may fail to detect a real violation. The test has low power in small samples, and the residual plots may be too noisy to show a clear trend. The absence of a significant test in a small study does not confirm that the assumption holds. The interpretation of the residual plots requires understanding the scale of the coefficient. The y-axis of the scaled Schoenfeld residual plot shows the estimated coefficient at each event time. The coefficient is on the log-hazard scale, so a change from 0.5 to 1.0 represents a change in the hazard ratio from 1.65 to 2.72. The plot should be examined for the pattern of the trend line. A linear trend suggests that the coefficient changes steadily over time. A non-linear trend, such as a change that occurs only in the early or late period, suggests a more complex time-varying effect. The plot can also reveal outliers that influence the trend line. A single subject with an extreme residual can pull the trend line in one direction. The plot should be examined with the individual points visible, beyond the smoothed line. ## Stratified Cox Models The stratified Cox model is the most direct approach when a categorical covariate violates the proportional hazards assumption. The model allows the baseline hazard to differ across the strata of the covariate while estimating the coefficients for the other covariates under the proportional hazards assumption. The stratified model does not estimate a coefficient for the stratifying variable. The variable is removed from the linear predictor and the baseline hazard is allowed to vary freely across its categories. This approach is appropriate when the stratifying variable is not of primary interest or when its effect is known to vary with time. The stratified model is a natural choice when the covariate that violates the assumption is a grouping variable such as treatment arm, sex, or disease stage. The model produces separate baseline hazards for each group and estimates the effects of the other covariates under the assumption that their effects are constant across time. The stratified model is not a solution for a continuous covariate that violates the assumption. Stratifying a continuous variable requires categorizing it, which loses information and introduces arbitrary cut points. The time-varying coefficient model is a better choice for continuous covariates. The stratified model also changes the interpretation of the other coefficients. The coefficients are interpreted as the effect of the covariate within each stratum, and the model assumes that the effect is the same across strata. This assumption should be checked separately if the research question involves interactions between the stratifying variable and other covariates. ## Time-Varying Coefficients The time-varying coefficient model extends the Cox model by allowing the coefficient of a covariate to change with time. The model specifies the coefficient as a function of time, such as a linear function or a step function, and estimates the parameters of that function. The most common approach is to split the follow-up period into intervals and estimate a separate coefficient for each interval. This is implemented by creating time-dependent covariates that are the product of the original covariate and an indicator for each interval. The model then estimates a coefficient for each interval, and the coefficients can be compared to see how the effect changes. The choice of the number of intervals and the cut-points is a modeling decision that should be based on the research question and the pattern of the residual plot. The intervals should be chosen to capture the time pattern in the effect without creating too many parameters. A common approach is to use two or three intervals based on clinically meaningful time points. The time-varying coefficient model is more complex to interpret than the standard Cox model. The model produces multiple coefficients for the covariate, one for each interval, and the interpretation must describe the effect in each period. The model is also more prone to overfitting if the number of intervals is large relative to the number of events. The time-varying coefficient model is a useful alternative when the proportional hazards assumption fails for a continuous covariate. The model provides a more accurate description of the effect over time and can reveal patterns that the standard model would miss. ## Functional Form and Martingale Residuals The proportional hazards assumption is not the only assumption of the Cox model. The model also assumes that the log hazard is a linear function of the covariates. This assumption can be checked with martingale residuals, which are the difference between the observed number of events and the expected number of events for each subject. The martingale residuals are plotted against the covariate values to check the functional form. A plot with a clear pattern, such as a curve, indicates that the covariate needs a transformation or a non-linear term. A plot with a random scatter around zero supports the linear assumption. The martingale residual plot is a useful diagnostic for continuous covariates. The plot can reveal that the effect of a covariate is non-linear, such as a U-shaped relationship or a threshold effect. The plot can also reveal that the covariate needs to be transformed, such as a log transformation. The functional form and the proportional hazards assumptions are related. A covariate with a non-linear effect can appear to violate the proportional hazards assumption because the effect is not constant across the range of the covariate. Checking the functional form before the proportional hazards assumption can avoid misinterpreting a functional form problem as a proportional hazards problem. The martingale residuals are also used to check the overall fit of the model. The sum of the martingale residuals is zero, and the residuals should be randomly scattered around zero when plotted against the linear predictor. A pattern in the plot indicates a problem with the model specification. ## Influential Observations The proportional hazards diagnostics can be affected by individual subjects with extreme values. The dfbeta residuals measure the influence of each subject on the estimated coefficients. A subject with a large dfbeta can change the coefficient and the proportional hazards test. The dfbeta residuals are calculated by removing each subject from the analysis and comparing the coefficient estimates. A subject with a large dfbeta has a large influence on the model. The dfbeta values are plotted against the subject index or the covariate values to identify the influential subjects. The influence of a subject is a concern when the subject is an outlier in the covariate or the event time. A subject with an extreme covariate value and an event at an unusual time can have a large influence on the model. The subject should be examined to see if the data are correct and if the subject is representative of the population. The removal of an influential subject can change the proportional hazards test. A subject that is influential can cause a violation of the assumption that disappears when the subject is removed. The analysis should be repeated without the influential subject to see if the results are stable. The decision to remove an influential subject should be based on the data quality and the research question. A subject with an error in the data should be corrected or removed. A subject that is a legitimate observation should be kept, and the model should be interpreted with the understanding that the subject has a large influence. ## Reporting the Diagnostics The reporting of the proportional hazards diagnostics should be transparent and complete. The report should include the test results for each covariate and the global model, and the plots of the residuals. The report should also describe the decisions made in response to the diagnostics. The reporting guidelines for survival analysis recommend that the proportional hazards assumption be checked and reported. The [EQUATOR Network](https://www.equator-network.org/) provides a list of reporting guidelines for different study types, and the guidelines for observational studies include the reporting of the model diagnostics. The report should describe the method used to test the assumption, the results of the test, and the action taken if the assumption was violated. The report should also describe the limitations of the diagnostics, such as the sample size and the power of the test. The reporting of the proportional hazards assumption is part of the broader reporting of the statistical methods. The methods section should describe the model, the covariates, the diagnostics, and the model modifications. The results section should report the coefficients and the hazard ratios, and the discussion should interpret the findings in the context of the assumptions. The [Committee on Publication Ethics](https://publicationethics.org/core-practices) core practices emphasize the importance of accurate and complete reporting of research. The reporting of the model diagnostics is part of the responsibility of the authors to describe the methods accurately. ## Common Failure Patterns The proportional hazards assumption can fail in several patterns, and the pattern determines the appropriate response. The most common patterns are a linear trend in the coefficient, a change in the effect after a certain time, and a non-linear effect. A linear trend in the coefficient indicates that the effect of the covariate increases or decreases steadily over time. The effect may be strong early and weak later, or weak early and strong later. The time-varying coefficient model with a linear function of time is a natural fit for this pattern. A change in the effect after a certain time indicates that the effect is constant within periods but different across periods. The pattern is common when a treatment has an acute effect that wears off, or when a risk factor has a delayed effect. The time-varying coefficient model with a step function is a natural fit for this pattern. A non-linear effect in the coefficient indicates that the effect changes in a more complex way. The pattern may be a U-shape or an inverted U-shape, where the effect is strongest in the middle of the follow-up period. The pattern is harder to model and may require a spline function of time. The pattern of the residual plot should be described in the report. The description should include the direction of the trend, the time period of the change, and the magnitude of the change. The description helps the reader understand the nature of the violation and the appropriateness of the model modification. ## Limitations of the Diagnostics The proportional hazards diagnostics have limitations that should be understood. The tests are not a proof that the assumption holds, and the plots are subject to interpretation. The diagnostics should be used as a guide, not as a definitive test. The statistical test is a test of the null hypothesis that the coefficient is constant over time. The test is a global test that can detect a variety of departures from the null. The test does not identify the pattern of the departure, and the plot is needed to see the pattern. The test is also affected by the number of events. The test has more power with more events, and the test can detect a small departure in a study with many events. The test has less power with fewer events, and the test can miss a large departure in a study with few events. The plots are subject to the interpretation of the analyst. The smoothed trend line is a summary of the residuals, and the smoothing parameter affects the appearance of the line. The analyst should examine the plot with different smoothing parameters to see if the pattern is stable. The diagnostics are also affected by the model specification. The residuals are calculated from the fitted model, and the residuals are affected by the covariates in the model. A model with a missing covariate can produce residuals that appear to violate the proportional hazards assumption when the violation is due to the missing covariate. ## Software and the Implementation The proportional hazards diagnostics are implemented in the standard statistical software packages. The R package survival provides the cox.zph function for the test and the plot. The SAS software provides the PHREG procedure with the ASSESS statement for the diagnostics. The Stata software provides the estat phtest command for the test. The implementation of the diagnostics is straightforward, but the interpretation requires the understanding of the model and the data. The software provides the test results and the plots, and the analyst must interpret the results in the context of the research question. The software also provides the tools for the model modifications. The stratified model is implemented with the strata option in the model statement. The time-varying coefficient model is implemented by creating the time-dependent covariates and the fitting the model with the time-dependent terms. The choice of the software is a matter of the preference and the availability. The R software is free and widely used in the academic research. The SAS software is a commercial product and is used in the industry and the regulatory settings. The Stata software is a commercial product and is used in the research and the teaching. ## The Workflow in the Research The proportional hazards assumption should be checked as part of the analysis plan. The check should be planned before the analysis, and the results should be reported in the manuscript. The check should be conducted after the model is fit and before the results are interpreted. The workflow for the analysis is to fit the model, check the proportional hazards, and modify the model if needed. The workflow is iterative, and the model may be modified and rechecked. The final model should be the one that satisfies the assumptions and answers the research question. The workflow is also a part of the data management and the sharing. The [NIH Data Management and Sharing Policy](https://sharing.nih.gov/data-management-and-sharing-policy) describes the expectations for the data management and the sharing in the NIH-funded research. The analysis code and the diagnostics should be shared with the data to allow the reproduction of the results. The workflow is also a part of the research integrity. The [NIH Grants and Funding](https://grants.nih.gov/) describes the expectations for the research conduct and the reporting. The analysis should be conducted and reported in a way that is transparent and reproducible. ## The Role of the Subject Matter Knowledge The proportional hazards assumption is a statistical assumption, but the decision to modify the model should be informed by the subject matter knowledge. The analyst should understand the biology of the covariate and the outcome to interpret the diagnostics. The subject matter knowledge can help to predict when the proportional hazards assumption is likely to fail. A treatment that is expected to have a short-term effect is likely to violate the assumption. A biomarker that is expected to have a long-term effect is likely to satisfy the assumption. The subject matter knowledge can also help to choose the model modification. The stratified model is appropriate when the covariate is a categorical variable and the effect is expected to vary across the strata. The time-varying coefficient model is appropriate when the covariate is a continuous variable and the effect is expected to change over time. The subject matter knowledge can also help to interpret the results of the diagnostics. A violation that is consistent with the expected effect is more credible than a violation that is unexpected. The analyst should consider the subject matter knowledge when deciding whether to modify the model. The subject matter knowledge is also important for the interpretation of the final model. The model with the time-varying coefficient should be interpreted in the context of the subject matter. The analyst should describe the effect of the covariate over time and the clinical or biological meaning of the pattern. ## The Model Selection and the The proportional hazards assumption is one of the model selection criteria. The model should be selected based on the fit, the assumptions, and the interpretability. The model that satisfies the assumptions and is interpretable is preferred over the model that is more complex. The model selection should be based on the research question and the data. The model should be the one that best answers the research question and is supported by the data. The model selection should not be based on the p-values of the tests alone. The model selection is also a part of the analysis plan. The analysis plan should describe the model, the diagnostics, and the model modifications. The plan should be written before the analysis and the deviations from the plan should be reported. The model selection is also a part of the reporting. The report should describe the model selection process and the final model. The report should also describe the diagnostics and the model modifications. The report should be transparent about the decisions made in the analysis. ## The Comparison of the Alternatives The Cox model is not the only model for the time-to-event data. The parametric models, such as the Weibull model and the exponential model, are alternatives. The parametric models make the assumptions about the distribution of the survival times, and the proportional hazards assumption is a part of the parametric models. The parametric models are more efficient than the Cox model when the distribution is correct. The parametric models are less flexible than the Cox model when the distribution is incorrect. The Cox model is the preferred choice when the distribution is not known. The accelerated failure time model is another alternative. The accelerated failure time model assumes that the covariate affects the survival time directly, not the hazard. The model is a different interpretation of the covariate effect and is not subject to the proportional hazards assumption. The choice of the model depends on the research question and the data. The Cox model is the most widely used and the most flexible. The parametric models are more efficient when the distribution is known. The accelerated failure time model is a different interpretation of the effect. ## The Practical Decision Criteria The decision to modify the model should be based on the test results, the plots, and the subject matter. The decision should be made in the context of the research question and the data. The decision should be reported in the manuscript. The decision criteria are the following. The model should be modified if the test is significant and the plot shows a clear trend. The model should not be modified if the test is not significant and the plot shows no trend. The model should be modified if the test is significant and the plot shows a trend, even if the trend is small. The decision criteria are also the following. The model should be modified if the test is not significant but the plot shows a clear trend. The model should be modified if the test is significant but the plot shows no trend, and the sample is large. The model should not be modified if the test is significant but the plot shows no trend, and the sample is small. The decision criteria are a guide, not a rule. The analyst should use the judgment in the decision. The decision should be based on the data, the research question, and the subject matter. ## The Common Failure Patterns The proportional hazards assumption can fail in several ways. The most common failure patterns are the following. The first pattern is the effect of a covariate that decreases over time. The pattern is common for the treatment effects. The treatment has a strong effect in the early period and the effect wears off over time. The pattern is a linear trend in the residual plot. The second pattern is the effect of a covariate that increases over time. The pattern is common for the risk factors. The risk factor has a weak effect in the early period and the effect increases over time. The pattern is a linear trend in the residual plot. The third pattern is the effect of a covariate that changes at a certain time. The pattern is common for the treatment that has a delayed effect. The treatment has no effect in the early period and the effect appears after a certain time. The pattern is a step function in the residual plot. The fourth pattern is the effect of a covariate that is non-linear. The pattern is common for the biomarker that has a threshold effect. The biomarker has no effect below a certain level and the effect appears above the level. The pattern is a non-linear trend in the residual plot. ## The Role of the Sample Size The sample size affects the power of the proportional hazards test and the interpretation of the results. The test has more power with a larger sample size and can detect smaller departures from the assumption. The test has less power with a smaller sample size and can miss larger departures. The sample size also affects the precision of the residual plots. The plots are more precise with a larger sample size and the trend line is more stable. The plots are less precise with a smaller sample size and the trend line is more noisy. The sample size should be considered when interpreting the test results. A significant test in a large sample may be due to a small departure that is not practically important. A non-significant test in a small sample may be due to a lack of power, not a lack of violation. The sample size should also be considered when choosing the model modification. The time-varying coefficient model requires more data than the standard model. The model with many time intervals may be overfit in a small sample. ## The Role of the Event Rate The event rate affects the power of the proportional hazards test and the precision of the residual plots. The test has more power with a higher event rate, and the residuals are more precise. The test has less power with a lower event rate, and the residuals are less precise. The event rate is the number of events divided by the total follow-up time. The event rate is a measure of the information in the data. A higher event rate provides more information and a lower event rate provides less information. The event rate should be considered in the interpretation of the test. A significant test with a low event rate may be a large violation. A non-significant test with a low event rate may be a missed violation. The event rate should also be considered in the choice of the model modification. The time-varying coefficient model requires a sufficient number of events in each interval. The model may not be estimable if the event rate is low. ## The Role of the Censoring The censoring affects the residuals and the test of the proportional hazards. The censored subjects do not contribute to the residuals at their censoring time. The residuals are based on the subjects who experience the event. The censoring is a common feature of the time-to-event data. The subjects are censored when they are lost to follow-up or when the study ends. The censoring is assumed to be independent of the event time. The censoring can affect the proportional hazards test if the censoring is related to the covariate. The example, if the subjects with a certain covariate are more likely to be censored, the residuals may be biased. The censoring should be examined in the analysis. The censoring is also a consideration in the model modification. The time-varying coefficient model requires the events in each interval. The model may not be estimable if the events are sparse in some intervals. ## The Software and the Code The proportional hazards diagnostics are implemented in the standard software. The R package survival provides the cox.zph function. The SAS procedure PHREG provides the ASSUM option. The Stata command stcox provides the PH test. The code for the diagnostics is straightforward. The code fits the model and then runs the test. The code also produces the plots. The code should be shared with the data to allow the reproduction of the results. The code should be documented and the output should be saved. The code should be versioned and the output should be archived. The code and the output should be shared with the data. The code is a part of the analysis and the reporting. The code should be written in a clear and reproducible way. The code should be commented and the output should be labeled. ## The Reporting of the Diagnostics The reporting of the diagnostics should be a part of the manuscript. The report should include the test results and the plots. The report should also describe the model modifications. The report should be transparent about the analysis. The report should describe the model, the diagnostics, and the modifications. The report should also describe the limitations of the diagnostics. The report should follow the reporting guidelines. The [EQUATOR Network](https://www.equator-network.org/) provides the reporting guidelines for the different study types. The guidelines for the survival analysis include the reporting of the model diagnostics. The report should also follow the publication ethics. The [Committee on Publication Ethics](https://publicationethics.org/core-practices) core practices describe the reporting of the research. The report should be accurate and complete. ## The Data and the Code Sharing The data and the code should be shared to allow the reproduction of the analysis. The [NIH Data Management and Sharing Policy](https://sharing.nih.gov/data-management-and-sharing-policy) describes the expectations for the data sharing. The data and the code should be shared in a repository. The data should be de-identified and the code should be documented. The data and the code should be shared in a way that is reproducible. The data and the code should be shared with the manuscript. The data and the code sharing is a part of the research integrity. The [NIH Grants and Funding](https://grants.nih.gov/) describes the research conduct. The data and the code sharing is a part of the research. The data and the code sharing is also a part of the researcher identity. The [ORCID for Researchers](https://info.orcid.org/researchers) describes the researcher identity. The researcher should be identified and the research should be shared. ## The Limitations of the Diagnostics The diagnostics have limitations that should be understood. The test is not a perfect test of the proportional hazards. The test can be affected by the sample size and the event rate. The test is a global test and does not identify the pattern of the violation. The plot is needed to see the pattern. The plot is subject to the interpretation. The diagnostics are also affected by the model. The residuals are calculated from the model. The model with the missing covariate can produce the residuals that appear to violate the assumption. The diagnostics should be used as a tool, not as a definitive test. The diagnostics should be combined with the subject matter knowledge. The decision to modify the model should be based on the data and the research question. ## The Conclusion The proportional hazards assumption is a central assumption of the Cox model. The assumption should be checked in the analysis. The diagnostics are the Schoenfeld residuals and the plots. The model should be modified if the assumption is violated. The stratified model is the choice for the categorical covariate. The time-varying coefficient model is the choice for the continuous covariate. The analysis should be reported in the manuscript. The report should include the diagnostics and the model modifications. The report should be transparent and complete. The proportional hazards assumption is a part of the analysis. The assumption should be checked and the model should be modified if needed. The analysis should be reported and the data should be shared. ## Frequently Asked Questions ### What is the proportional hazards assumption in the Cox model? The proportional hazards assumption states that the hazard ratio for any two subjects is constant over time. The Cox model expresses the hazard as a baseline hazard multiplied by an exponential function of the covariates, and the assumption requires that the covariate effects do not change with follow-up time. ### How do I test the proportional hazards assumption? The standard test uses Schoenfeld residuals. The test is implemented in the cox.zph function in R, the ASSESS option in SAS, and the stphtest command in Stata. The test produces a p-value for each covariate and the global model, and the scaled residuals are plotted against time. ### What does a significant test for proportional hazards mean? A significant test indicates that the covariate effect changes over time. The test is a global test and does not identify the pattern of the change. The plot of the residuals is needed to see the pattern and to decide on the model modification. ### What should I do if the proportional hazards assumption is violated? The model can be modified in two ways. The stratified Cox model allows the baseline hazard to differ across the strata of a categorical covariate. The time-varying coefficient model allows the coefficient of a continuous covariate to change over time. ### Can I ignore a violation of the proportional hazards assumption? Ignoring a violation can produce biased estimates of the hazard ratio. The hazard ratio is a single number that does not capture the time-varying effect. The model should be modified to describe the effect accurately. ### How does sample size affect the proportional hazards test? The test has more power with a larger sample size and can detect smaller violations. The test has less power with a smaller sample size and can miss larger violations. The plot of the residuals should be examined in addition to the test. ### What is the difference between the stratified model and the time-varying coefficient model? The stratified model is used for a categorical covariate that violates the assumption. The model allows the baseline hazard to differ across strata and does not estimate a coefficient for the stratifying variable. The time-varying coefficient model is used for a continuous covariate and estimates the coefficient as a function of time. ### How should I report the proportional hazards diagnostics in my manuscript? The report should include the test results for each covariate and the global model, the plots of the residuals, and the description of the model modifications. The report should follow the reporting guidelines and be transparent about the analysis. ## Using the Evidence | Source | Best use in this topic | Important limitation | |---|---|---| | [Research Methods Resources](https://www.ncbi.nlm.nih.gov/books) | official guidance | Check the linked page for current local requirements | | [EQUATOR Network](https://www.equator-network.org/) | official guidance | Check the linked page for current local requirements | | [Core Practices](https://publicationethics.org/core-practices) | official guidance | Check the linked page for current local requirements | ## Related Bioinformatics Guides - [Metabolomics Data Analysis Workflow: From Raw Data to Biological Insight](/knowledge/bioinformatics/metabolomics-data-analysis-workflow-from-raw-data-to-biological-insight) - [Genomic Data Integration: Combining Multi-Omics for Biological Insights](/knowledge/bioinformatics/genomic-data-integration-combining-multi-omics-for-biological-insights) - [Metagenomics Data Analysis: From Raw Reads to Biological Insights](/knowledge/bioinformatics/metagenomics-data-analysis-from-raw-reads-to-biological-insights) - [Proteomics Data Analysis Workflow: From Raw Spectra to Biological Insights](/knowledge/bioinformatics/proteomics-data-analysis-workflow-from-raw-spectra-to-biological-insights) - [Spatial Omics Data Analysis: From Image Processing to Biological Interpretation](/knowledge/bioinformatics/spatial-omics-data-analysis-from-image-processing-to-biological-interpretation) ## Related Clinical & Scientific Guides * [A Practical Guide to Detecting Antimicrobial Resistance Genes in Shotgun Metagenomic Data](/knowledge/bioinformatics/a-practical-guide-to-detecting-antimicrobial-resistance-genes-in-shotgun-metagenomic-data) * [Computational Immunology: Modeling the Immune System](/knowledge/bioinformatics/computational-immunology-modeling-the-immune-system) * [How to Set Hard Filters for Germline Variant Calling: A Practical Guide to GATK Best Practices](/knowledge/bioinformatics/how-to-set-hard-filters-for-germline-variant-calling-a-practical-guide-to-gatk-best-practices) ## References and Further Reading - [Research Methods Resources](https://www.ncbi.nlm.nih.gov/books). National Library of Medicine. - [EQUATOR Network](https://www.equator-network.org/). EQUATOR Network. - [Core Practices](https://publicationethics.org/core-practices). Committee on Publication Ethics. - [NIH Grants and Funding](https://grants.nih.gov/). National Institutes of Health. - [ORCID for Researchers](https://info.orcid.org/researchers). ORCID. - [Data Management and Sharing Policy](https://sharing.nih.gov/data-management-and-sharing-policy). National Institutes of Health. - [NCBI Data Resources](https://www.ncbi.nlm.nih.gov/). National Center for Biotechnology Information. - [EMBL-EBI Training](https://www.ebi.ac.uk/training). European Bioinformatics Institute. - [Testing the proportional hazards assumption in cox regression and dealing with possible non-proportionality in total joint arthroplasty research: methodological perspectives and review.](https://pubmed.ncbi.nlm.nih.gov/34049528). BMC musculoskeletal disorders, 2021. - [Enhanced secondary analysis of survival data: reconstructing the data from published Kaplan-Meier survival curves.](https://pubmed.ncbi.nlm.nih.gov/22297116). BMC medical research methodology, 2012. - [Effect of a modified regimen on drug-sensitive retreated pulmonary tuberculosis: A multicenter study in China.](https://pubmed.ncbi.nlm.nih.gov/36778546). Frontiers in public health, 2023. > This article is educational and does not replace validated analysis plans, institutional policy, clinical interpretation, or specialist review.