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Repeated events analyses
- 2. Outline for Talk Recurrent Events Analyses for Count Outcomes Survival Analysis Approaches for Repeated Events Parametric Analyses for Counts Poisson Regression Negative Binomial Non-parametric Analyses for Counts Mean of Ratios (Q) Ratio of Means (R) 2
- 3. Recurrent Events Usually in survival studies subjects are followed until they experience a single index event Death Cancer Return to employment But there are many events of interest that are repeatable Seizures Infections Hospitalizations 3
- 4. Recurrent Events Most of the methods employed in survival analysis are set up for a single event But, there have been methods developed for use with repeated events Some of these methods come out of the area of statistical methods developed for count data – these focus on the count of the number of recurrent event Some methods come directly out of survival analysis methods – these focus on the time duration between the recurrent events 4
- 5. Recurrent Events Methods coming out of either area need to address the domain of the other methods The count methods also need to address the time component The survival methods need to address the number of events as well as the gap times between events 5
- 6. Diabetes Control and Complications Trial (DCCT) NIH-funded trial Launched in 1981 To evaluate the effect of intensive blood glucose- lowering on risk of albuminuria in diabetic subjects Subjects were randomized to either intensive blood glucose lowering or conventional treatment Intensive glucose lowering used self-monitoring 4 or 5 times daily, multiple daily insulin injections or a pump, diet and exercise 6
- 7. Diabetes Control and Complications Trial (DCCT) A concern was that the intensive glucose lowering could lead to hypoglycemia Dizzy spells Possible comas Seizures Hypoglycemia events were tracked as a secondary outcome (DCCT 1997) 7
- 8. Diabetes Control and Complications Trial (DCCT) Intensive Treatment Conventional Treatment Subjects 363 352 Hypoglycemia Events 1723 543 Person-Years of Follow-Up 2598.5 2480.8 Rate per 100 Person-Years 66.3 21.9 8
- 9. Count Outcome Approaches The most basic analysis for count outcomes comes from the Poisson distribution 𝑃𝑃 𝑦𝑦|𝜆𝜆 = 𝑒𝑒−𝜆𝜆 𝜆𝜆 𝑦𝑦 𝑦𝑦! Here 𝜆𝜆 is the mean count y is the observed count (non-negative integer) One feature of the Poisson distribution is that Variance = Mean = 𝜆𝜆 9
- 11. Count Outcome Approaches The maximum likelihood estimator of the mean is provided by the sample average ̂𝜆𝜆 = 𝑦𝑦1+𝑦𝑦2+⋯+𝑦𝑦𝑛𝑛 𝑛𝑛 Often people are followed for somewhat different lengths of time, giving some people more opportunity to get larger event counts In this case 𝜆𝜆 is standardized to be per-unit of time 11
- 12. Count Outcome Approaches With unequal follow-up times 𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛 𝑃𝑃 𝑦𝑦|𝑡𝑡, 𝜆𝜆 = 𝑒𝑒−𝜆𝜆𝑡𝑡(𝜆𝜆𝑡𝑡)𝑦𝑦 𝑦𝑦! Where 𝑡𝑡 is the length of follow-up With unequal follow-up, the MLE for 𝜆𝜆 is provided by ̂𝜆𝜆 = 𝑦𝑦1+𝑦𝑦2+⋯+𝑦𝑦𝑛𝑛 𝑡𝑡1+𝑡𝑡2+⋯+𝑡𝑡𝑛𝑛 = �𝑦𝑦 ̅𝑡𝑡 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓−𝑢𝑢𝑢𝑢 12
- 13. Count Outcome Approaches With unequal follow-up times 𝑡𝑡1, 𝑡𝑡2, … , 𝑡𝑡𝑛𝑛 𝑃𝑃 𝑦𝑦|𝑡𝑡, 𝜆𝜆 = 𝑒𝑒−𝜆𝜆𝑡𝑡(𝜆𝜆𝑡𝑡)𝑦𝑦 𝑦𝑦! Where 𝑡𝑡 is the length of follow-up With unequal follow-up, the MLE for 𝜆𝜆 is provided by ̂𝜆𝜆 = 𝑦𝑦1+𝑦𝑦2+⋯+𝑦𝑦𝑛𝑛 𝑡𝑡1+𝑡𝑡2+⋯+𝑡𝑡𝑛𝑛 = �𝑦𝑦 ̅𝑡𝑡 = 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 𝑜𝑜𝑜𝑜 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑡𝑡𝑡𝑡 𝑡𝑡𝑡𝑡 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑓𝑓−𝑢𝑢𝑢𝑢 We will see this again 13
- 14. Link From Poisson to Exponential If the count of events follows a Poisson distribution, then the times between events follow an exponential distribution 𝑓𝑓 𝑡𝑡 𝜆𝜆 = 𝜆𝜆𝑒𝑒−𝜆𝜆𝜆𝜆 A feature of exponential survival times is that the hazard event rate is 𝜆𝜆 and is constant over time The mean time to an event is 1 𝜆𝜆 Gives a way to estimate 𝜆𝜆 in a survival analysis setting (rather than a count data setting) 14
- 16. Poisson Regression In Poisson regression we model the expected count for an observation as a function of predictor variables Poisson regression is a log-linear model so that the log of the mean is set equal to a linear combination of the predictors 16
- 17. Poisson Regression 𝑙𝑙𝑙𝑙 𝑙𝑙 𝜆𝜆𝜆𝜆 = 𝛽𝛽0 + 𝛽𝛽1 𝑥𝑥 This leads to 𝑙𝑙𝑙𝑙 𝑙𝑙 𝜆𝜆 + 𝑙𝑙𝑙𝑙 𝑙𝑙 𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1 𝑥𝑥 So, we get 𝑙𝑙𝑙𝑙 𝑙𝑙 𝜆𝜆 = 𝛽𝛽0 + 𝛽𝛽1 𝑥𝑥 − 𝑙𝑙𝑙𝑙 𝑙𝑙 𝑡𝑡 The −𝑙𝑙𝑙𝑙 𝑙𝑙 𝑡𝑡 is called the offset and corrects for the differences in follow-up time between subjects The model can be expressed 𝜆𝜆 = 𝑒𝑒 𝛽𝛽0+𝛽𝛽1 𝑥𝑥−𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡 17
- 18. Poisson Regression This model is fit using maximum likelihood SAS GENMOD The 𝛽𝛽 estimates for predictors are interpreted as logs of incidence rate ratios A one-unit increase in 𝑥𝑥 gives a ratio of 𝑒𝑒𝑒𝑒𝑒𝑒 𝛽𝛽1 in the expected count If 𝑥𝑥 codes for a group difference (0 = control, 1 = treatment), then 𝑒𝑒𝑒𝑒𝑒𝑒 𝛽𝛽1 corresponds to the ratio of counts in the treated group divided by counts in the control group 18
- 19. Poisson Regression in DCCT proc genmod; model nevents = group / dist = poisson link = log offset = lnyears; TITLE1 'Poisson regression models of risk of hypoglycemia'; title2 'unadjusted treatment group effect'; 19
- 20. Poisson Regression in DCCT 20 Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -1.5190 0.0429 -1.6031 -1.4349 1252.90 <.0001 group Exp 1 1.1081 0.0492 1.0117 1.2046 507.01 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 1.0000 0.0000 1.0000 1.0000
- 21. Poisson Regression in DCCT 21 Highly significant result! But, is it based on a reasonable model for the data? Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -1.5190 0.0429 -1.6031 -1.4349 1252.90 <.0001 group Exp 1 1.1081 0.0492 1.0117 1.2046 507.01 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . Scale 0 1.0000 0.0000 1.0000 1.0000
- 22. Poisson Regression The big limitation for Poisson regression is overdispersion The data are overdispersed when the variation in the observed counts is greater that the mean value Under Poisson model these should match Variance in the counts much greater than the means Goodness of fit testing for overdispersion is typically done in Poisson regression 22
- 23. Overdispersion in DCCT 23 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 713 3928.7828 5.5102 Scaled Deviance 713 3928.7828 5.5102 Pearson Chi-Square 713 5131.3429 7.1968 Scaled Pearson X2 713 5131.3429 7.1968 Log Likelihood 775.1804
- 24. Overdispersion in DCCT 24 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 713 3928.7828 5.5102 Scaled Deviance 713 3928.7828 5.5102 Pearson Chi-Square 713 5131.3429 7.1968 Scaled Pearson X2 713 5131.3429 7.1968 Log Likelihood 775.1804 If model fits well then Value/DF should be close to 1.0 • Large ratios indicate overdispersion or model misspecification
- 25. Poisson Regression For a clinical trial, our primary goal is a good test of treatment effect, so wouldn’t normally need to include covariates But, could try to remove excess variation by controlling for covariates 25
- 26. Poisson Regression in DCCT proc genmod; class group; model nevents = group insulin duration female adult bcval5 hbael hxcoma / dist = poisson link = log offset = lnyears covb; title2 'covariate adjusted treatment group effect'; 26
- 27. Poisson Regression in DCCT 27 Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -0.9568 0.2174 -1.3829 -0.5308 19.38 <.0001 group Exp 1 1.0845 0.0493 0.9879 1.1812 483.91 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . insulin 1 0.0051 0.0995 -0.1898 0.2000 0.00 0.9593 duration 1 0.0015 0.0006 0.0004 0.0026 6.79 0.0092 female 1 0.1794 0.0424 0.0963 0.2624 17.93 <.0001 adult 1 -0.5980 0.0656 -0.7265 -0.4694 83.13 <.0001 bcval5 1 -0.5283 0.3630 -1.2398 0.1833 2.12 0.1456 hbael 1 -0.0335 0.0151 -0.0631 -0.0038 4.89 0.0271 hxcoma 1 0.6010 0.0685 0.4669 0.7352 77.09 <.0001 Scale 0 1.0000 0.0000 1.0000 1.0000
- 28. Poisson Regression in DCCT 28 Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -0.9568 0.2174 -1.3829 -0.5308 19.38 <.0001 group Exp 1 1.0845 0.0493 0.9879 1.1812 483.91 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . insulin 1 0.0051 0.0995 -0.1898 0.2000 0.00 0.9593 duration 1 0.0015 0.0006 0.0004 0.0026 6.79 0.0092 female 1 0.1794 0.0424 0.0963 0.2624 17.93 <.0001 adult 1 -0.5980 0.0656 -0.7265 -0.4694 83.13 <.0001 bcval5 1 -0.5283 0.3630 -1.2398 0.1833 2.12 0.1456 hbael 1 -0.0335 0.0151 -0.0631 -0.0038 4.89 0.0271 hxcoma 1 0.6010 0.0685 0.4669 0.7352 77.09 <.0001 Scale 0 1.0000 0.0000 1.0000 1.0000 Still very significant, but does the model fit better?
- 29. Poisson Regression in DCCT 29 Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 706 3707.7027 5.2517 Scaled Deviance 706 3707.7027 5.2517 Pearson Chi-Square 706 4792.7876 6.7887 Scaled Pearson X2 706 4792.7876 6.7887 Log Likelihood 885.7204 The fit is better with the covariates included, but still very overdispersed.
- 30. Poisson Regression Another approach that can be used to remove overdispersion is to use a variance correction Pearson correction Standard approach – has the actual variance equal the modeled variance multiplied by an estimated overdispersion parameter Corrects the standard errors and test results to account for the overdispersion GEE correction based on robust sandwich variance estimator Based on a robust sandwich-variance estimator 30
- 31. GEE Poisson Regression in DCCT proc genmod; class group subnum; model nevents = group / dist = poisson link = log offset = lnyears type3; repeated subject=subnum / type=unstr; title2 ‘GEE unadjusted treatment group effect'; 31
- 32. GEE Poisson Regression in DCCT 32 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Standard Error 95% Confidence Limits Z Pr > |Z| Intercept -1.5190 0.1003 -1.7155 -1.3225 -15.15 <.0001 group Exp 1.1081 0.1256 0.8620 1.3543 8.82 <.0001 group Std 0.0000 0.0000 0.0000 0.0000 . .
- 33. GEE Poisson Regression in DCCT 33 Analysis Of GEE Parameter Estimates Empirical Standard Error Estimates Parameter Estimate Standard Error 95% Confidence Limits Z Pr > |Z| Intercept -1.5190 0.1003 -1.7155 -1.3225 -15.15 <.0001 group Exp 1.1081 0.1256 0.8620 1.3543 8.82 <.0001 group Std 0.0000 0.0000 0.0000 0.0000 . . Still very significant treatment effect, but note that standard error is much larger in this model. Standard error was 0.0492 in Poisson model with only treatment.
- 34. Negative Binomial Regression The negative binomial distribution is another count data distribution with more variance than the Poisson Can be used to give the probability that n trials are required in order to get m successes (m≤n) Model takes the form 𝜆𝜆 = 𝑒𝑒 𝛽𝛽0+𝛽𝛽1 𝑥𝑥−𝑙𝑙𝑙𝑙𝑙𝑙 𝑡𝑡 +𝜀𝜀 Where 𝑒𝑒𝑒𝑒𝑒𝑒 𝜀𝜀 has a gamma distribution Can be considered to be a Poisson model with gamma- distributed random effects Parametric approach to overdispersion 34
- 35. Negative Binomial Regression in DCCT proc genmod; class group; model nevents = group / dist = negbin link = log offset = lnyears; title2 ‘Neg Binomial unadjusted treatment group effect'; 35
- 36. Negative Binomial Regression in DCCT36 Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -1.5510 0.0902 -1.728 -1.374 295.60 <.0001 group Exp 1 1.1173 0.1215 0.8791 1.3555 84.52 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . Dispersion 1 2.1863 0.1649 1.8631 2.5095
- 37. Negative Binomial Regression in DCCT37 Analysis Of Parameter Estimates Parameter DF Estimate Standard Error Wald 95% Confidence Limits Chi-Square Pr > ChiSq Intercept 1 -1.5510 0.0902 -1.728 -1.374 295.60 <.0001 group Exp 1 1.1173 0.1215 0.8791 1.3555 84.52 <.0001 group Std 0 0.0000 0.0000 0.0000 0.0000 . . Dispersion 1 2.1863 0.1649 1.8631 2.5095 The results here are quite similar to those of the GEE Poisson model, with almost the same standard errors.
- 39. Survival Analyses Repeated events Cox model on time to first event Use Andersen-Gill approach and partition follow-up time according to which event it applies Restart the follow-up time clock after an event Multiple observations per subject Need to be concerned about correlation Unobserved heterogeneity (due to the correlation) can bias estimates downward while significance is overstated 39
- 40. Survival Analysis in DCCT proc phreg data=three; model stopday*event(0) = intgroup/ risklimits; title1 'Cox Model for First Event'; run; 40
- 41. Survival Analysis in DCCT 41 Analysis of Maximum Likelihood Estimates Variable DF Parameter Estimate Standard Error Chi-Square Pr > ChiSq Hazard Ratio 95% Hazard Ratio Confidence Limits INTGROUP 1 0.77252 0.10354 55.6711 <.0001 2.165 1.768 2.652
- 42. Survival Analysis in DCCT proc phreg data=four; model gaptime*event(0) = intgroup priorgap / risklimits; title1 'Cox Model for Second Event - Correlation'; run; 42
- 43. Survival Analysis in DCCT 43 Analysis of Maximum Likelihood Estimates Variable DF Parameter Estimate Standard Error Chi-Square Pr > ChiSq Hazard Ratio 95% Hazard Ratio Confidence Limits INTGROUP 1 0.27790 0.12451 4.9814 0.0256 1.320 1.034 1.685 priorgap 1 -0.000523 0.000114 20.9317 <.0001 0.999 0.999 1.000 So, having a longer time to the first event reduces risk of the second event. So correlation in these times to events is quite substantial.
- 44. Survival Analysis in DCCT proc phreg data=six covsandwich(aggregate); model gaptime*event(0) = intgroup / risklimits; id patient; title1 'GEE Cox Model for all Events'; run; 44
- 45. Survival Analysis in DCCT 45 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 74.8679 1 <.0001 Score (Model-Based) 72.4448 1 <.0001 Score (Sandwich) 51.8412 1 <.0001 Wald (Model-Based) 71.4253 1 <.0001 Wald (Sandwich) 40.2564 1 <.0001
- 46. Survival Analysis in DCCT 46 Analysis of Maximum Likelihood Estimates Variable DF Parameter Estimate Standard Error StdErr Ratio Chi-Square Pr > ChiSq Hazard Ratio 95% Hazard Ratio Confidence Limits INTGROUP 1 0.67526 0.08988 1.808 56.4435 <.0001 1.965 1.647 2.343 So, accounting for the correlation increases the standard error by about 80%. However, the resulting hazard ratio is much less than the other estimates so far.
- 47. Nonparametric Analyses Work done with Jing Xu Motivated by his experiences with analysis of repeated events at The SDAC for the AIDS Clinical Trials Group Later experience in the pharmaceutical industry 47
- 48. Nonparametric Analyses In follow-up studies of recurrent events, there were studies taking a non-parametric approach to analysis of two-arm trials by 𝑞𝑞𝑖𝑖𝑖𝑖 = 𝑦𝑦𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖 𝑦𝑦𝑖𝑖𝑖𝑖 is the number of events for subject i in arm j (j is 0 or 1) 𝑡𝑡𝑖𝑖𝑖𝑖 is the total length of follow-up for subject i in group j 𝑞𝑞𝑖𝑖𝑖𝑖 represents a subject-specific event rate 48
- 49. Nonparametric Analyses 𝑞𝑞𝑖𝑖𝑖𝑖 were analyzed without assuming any particular distribution by standard two-sample tests Student t-test Wilcoxon test Van der Waerden test 49
- 50. Nonparametric Analyses For estimation, the means of the subject-specific event rates were calculated for each group �𝑄𝑄𝑗𝑗 = 1 𝑛𝑛𝑗𝑗 ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑞𝑞𝑖𝑖𝑖𝑖 = 1 𝑛𝑛𝑗𝑗 ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑦𝑦𝑖𝑖𝑖𝑖 𝑡𝑡𝑖𝑖𝑖𝑖 These �𝑄𝑄𝑗𝑗 could then be used to estimate ratios of the rates and differences in the rates by group �𝑅𝑅 𝑅𝑅𝑄𝑄 = �𝑄𝑄1 �𝑄𝑄0 �𝑅𝑅 𝑅𝑅𝑄𝑄 = �𝑄𝑄1 − �𝑄𝑄0 50
- 51. Nonparametric Analyses Our concern about this 𝑞𝑞𝑖𝑖𝑖𝑖 approach was that There could be outliers – subjects having a large number of events in a short amount of time who then drop out of the study early The asymptotic consistency of the �𝑄𝑄𝑗𝑗 estimators depends on both the numbers of events and the amount of follow-up increasing within all subjects 51
- 52. Nonparametric Analyses A different approach had been suggested by L.J. Wei �𝑅𝑅𝑗𝑗 = 𝑥𝑥𝑗𝑗 𝑡𝑡𝑗𝑗 = � ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑥𝑥𝑖𝑖𝑖𝑖 ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑡𝑡𝑖𝑖𝑖𝑖 = 𝑥𝑥𝑗𝑗 �𝑡𝑡𝑗𝑗 Ratio of the means, not the mean of the ratios 52
- 53. Nonparametric Analyses A different approach had been suggested by L.J. Wei �𝑅𝑅𝑗𝑗 = 𝑥𝑥𝑗𝑗 𝑡𝑡𝑗𝑗 = � ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑥𝑥𝑖𝑖𝑖𝑖 ∑𝑖𝑖=1 𝑛𝑛𝑗𝑗 𝑡𝑡𝑖𝑖𝑖𝑖 = 𝑥𝑥𝑗𝑗 �𝑡𝑡𝑗𝑗 Ratio of the means, not the mean of the ratios This is just the standard Poisson events per person- time without assuming the distribution 53
- 54. Nonparametric Analyses Can use the �𝑅𝑅𝑗𝑗 to estimate rate ratio and difference in rates between groups �𝑅𝑅𝑅𝑅𝑅𝑅 = �𝑅𝑅1 �𝑅𝑅0 �𝑅𝑅 𝑅𝑅𝑅𝑅 = �𝑅𝑅1 − �𝑅𝑅0 54
- 55. Nonparametric Analyses �𝑅𝑅𝑗𝑗 advantages Average out any outliers Makes consistency more reasonable as it uses the group totals for number of events and total follow-up time �𝑅𝑅𝑗𝑗 disadvantages Measure is group-specific and not subject-specific Can’t use standard 2-sample tests 55
- 56. Nonparametric Analyses Asymptotic forms of the variance for the �𝑅𝑅 measures are in the Xu and LaValley paper in the references Confidence intervals based on Fieller’s method and resampling In this talk, I’ll use resampling methods to evaluate the �𝑅𝑅𝑗𝑗 measures Permutation tests Bootstrap confidence intervals 56
- 57. 57
- 58. Nonparametric Analyses In simulations, we found that the �𝑄𝑄𝑗𝑗 with a non- parametric test and the �𝑅𝑅𝑗𝑗 methods maintained the type-1 error and had comparable power over a range of sample sizes The �𝑅𝑅𝑗𝑗 methods provided reasonable confidence interval coverage if the sample sizes were at least 100 per group 58
- 59. Nonparametric Analyses In the paper we use the nonparametric methods on a dataset for recurrence of bladder cancer in a two-arm clinical trial of the drug thiotepa 59
- 60. Nonparametric Analyses Thiotepa trial results Results are uniformly non-significant, with rate ratios from 0.71 to 0.84 60
- 61. Nonparametric Analyses Thiotepa trial results These are the results that I trust the most out of these 61
- 62. Nonparametric Analyses in DCCT 62 Intensive Treatment Conventional Treatment 𝑛𝑛 Subjects 363 352 �𝑦𝑦 Hypoglycemia Events/Subject 4.75 1.54 ̅𝑡𝑡 Person-Years of Follow-Up/Subject 7.16 7.05 � �𝑦𝑦 ̅𝑡𝑡 �𝑅𝑅𝑗𝑗 0.66 0.22 �𝑄𝑄𝑗𝑗 0.65 0.21
- 63. Nonparametric Analyses in DCCT 63 Distribution of qij by group
- 64. Nonparametric Analyses in DCCT 64 Test Test Statistic P-value t-test of qij 8.28 P < 0.0001 Van der Waerden test of qij 8.77 P < 0.0001 Wilcoxon test of qij 8.57 P < 0.0001
- 65. Nonparametric Analyses in DCCT 65 Estimator Estimate Permutation Test P-value* Rate Difference (Q) 0.43681 P < 0.0005 Rate Difference (R) 0.44415 P < 0.0005 Rate Ratio (Q) 3.08059 P < 0.0005 Rate Ratio (R) 3.02872 P < 0.0005 *Based on 2000 permutations
- 66. Nonparametric Analyses in DCCT 66 Q versus R measures of rate difference across 2000 permuted datasets
- 67. Nonparametric Analyses in DCCT 67 Estimator Estimate Bootstrap Confidence Interval Rate Difference (Q) 0.43681 (0.329, 0.549) Rate Difference (R) 0.44415 (0.329, 0.561) Rate Ratio (Q) 3.08059 (2.399, 3.979) Rate Ratio (R) 3.02872 (2.333, 3.951)
- 68. Nonparametric Analyses in DCCT 68 Q versus R measures of rate difference across 2000 bootstrap samples
- 69. Nonparametric Analyses Both Q and R measures work well for reasonable sized datasets In these DCCT data, both are fine In the bladder cancer data, the R (ratio of means) seems to have a slight edge 69
- 70. Conclusions There are a lot of good options for the analysis of repeated events GEE Survival models Gee Poisson Regression Negative binomial regression Q and R measures – especially in clinical trial setting Worthwhile to work with several as secondary analyses to verify consistency 70
- 71. Main References Allison PD. Survival Analysis Using SAS: a Practical Guide, second edition. SAS Publishing, 2010. Lachin JM. Biostatistical Methods: the Assessment of Relative Risks. Wiley, 2000. Xu J, LaValley M. One-sample and two-sample analysis of heterogeneous person-time data in clinical trials. Pharmaceutical Statistics 2012; 11: 194 – 203. 71