Analysis methods for data with repeated events or counts with connections to survival analysis methods.

1. REPEATED EVENTS ANALYSES
12/17/2012
1

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

10. Poisson Densities
10

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

15. Exponential Densities
15

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.

38. DCCT Treatment Results
38

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