STATISTICS IN MEDICINE Statist. Med. 2007; 00:1 22 [Version: 2002/09/18 v1.11] Analysis of recurrent event data under the case-crossover design with applications to elderly falls Xianghua Luo 1,, and Gary S. Sorock 2 1 Division of Biostatistics, School of Public Health, University of Minnesota, 420 Delaware Street SE, Minneapolis, MN 55455, U.S.A. 2 Geriatric Research Services, 312 Central Ave. Box 280, Glyndon, MD 21071, U.S.A. SUMMARY The case-crossover design is useful for studying the effects of transient exposures on short-term risk of diseases or injuries when only data on cases are available. The crossover nature of this design allows each subject to serve as his own control. While the original design was proposed for univariate event data, in many applications recurrent events are often encountered (e.g., elderly falls, gout attacks, and sexually transmitted infections). In such situations, the within-subject dependence among recurrent events needs to be taken into account in the analysis. We review three existing conditional logistic regression-based approaches for analyzing recurrent event data under the case-crossover design. A simple approach is to use only one (e.g. the first) event for each subject, such that no assumption on the correlation among multiple events is needed, while we would expect loss of efficiency in estimation. The validity of the other two reviewed approaches rely on independence assumptions for the recurrent Correspondence to: Division of Biostatistics, School of Public Health, University of Minnesota, 420 Delaware Street SE, Minneapolis, MN 55455, U.S.A. E-mail: luox0054@umn.edu. Telephone: (612) 612-2158. Fax: (612) 626-0660. Copyright c 2007 John Wiley & Sons, Ltd.
2 X. LUO AND G. S. SOROCK events, conditionally on a subject-level latent variable and a set of observed time-varying covariates. In this paper, we propose to adjust the conditional logistic regression using either a within-subject pairwise resampling technique or a weighted estimating equation. No specific dependency structure among the recurrent events is needed for these two methods. We also propose a weighted Mantel- Haenszel estimator for situations with a binary exposure. Simulation studies are conducted to evaluate the performance of the discussed methods. We present the analysis of a study of the effect of medication changes on falls among the elderly. Copyright c 2007 John Wiley & Sons, Ltd. key words: case-crossover; conditional logistic regression; Mantel-Haenszel; recurrent events; weighted estimating equation; within-cluster resampling 1. INTRODUCTION The case-crossover design was introduced by Maclure [1] as an analogue to the matched casecontrol design for studying the effects of transient exposures on the risk of acute-onset diseases or injuries. Only data on people who have the disease or injury are required (cases only). The association between disease onset and risk factors is estimated by comparing the exposure of risk factors prior to the disease onset (case time) with that in a reference time (control time) or multiple reference times [2]. The case and control times have a prefixed relation to the onset time of the disease. For example, the association between fall risk and medication change can be estimated by comparing the medication change record during 1-2 days before (case time) versus 8-9 days before (control time) each fall date. By using the person as his own matched control, potential between-subject confounding variables are readily controlled. As a consequence, subject-level covariates effects are not estimable. A case-crossover study is
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 3 intended to identify the short-term, transicent triggers of disease, rather than identifying who is at highest risk of disease. As for a matched case-control study, standard methods such as Mantel-Haenszel (MH) method and conditional logistic regression (CLR) [3] can be used for analyzing case-crossover data, as suggested by Maclure [1] and Mittleman et al [2]. Greenland [4] and Navidi [5] pointed out that the case-crossover design is subject to bias from time trends in exposures. Vines and Farrington [6] proved that a sufficient condition is needed to avoid this bias, namely the global exchangeability of exposure distribution. If M control times are matched to each case time, this condition requires the distribution of exposures in the M + 1 consecutive time periods is exchangeable. Recurrent events are frequently encountered in case-crossover studies, examples include studies of falls [7], gout attacks [8], and sexually transmitted infections [9]. Maclure [1] was aware of the threat of inappropriate statistical analyses for repeated events to the validity of the design, but no specific methods were suggested. The present paper concerns how recurrent events in case-crossover studies can be analyzed by accounting for the within-subject correlation among recurrent events. A simple approach is to use only one (e.g. the first) event for each person, such that no assumption on the correlation among multiple events is needed. Another simple approach is to assume that within-subject correlation is completely accounted for by subject-specific variables (observed or unobserved), hence the recurrent events are independent conditionally on these subject-specific variables and other observed time-varying covariates. Under this assumption, the usual conditional likelihood method can be implemented on the pooled events data without modification. It should be noted that the unit of analysis in the conditional likelihood of this approach is the
4 X. LUO AND G. S. SOROCK event rather than person. Navidi [5] proposed the full-stratum design, which deviates from the original form of the case-crossover design by requiring that control information being collected for all possible time intervals. In that sense, the sampling mechanism of the fullstratum design is more similar to cohort rather than case-control studies. Navidi s conditional logistic regression method for recurrent events requires the same conditional independence assumption, but treats the recurrent events within each person as a cluster and constructs the conditional likelihood on a person/cluster level. Though this method was proposed for the full-stratum design, it can be used for analyzing recurrent events under the case-crossover design. This is because the same form of conditional likelihood can be obtained whether the data are regarded as from a cohort or a case-control study (Reference [3], p248). In Section 2 of this paper, we describe a case-crossover study of the transient effects of medication changes on recurrent falls conducted in three nursing homes. Next, in Section 3, we introduce notations and summarize the above existing methods. Then, we propose two other approaches for analyzing recurrent events. The first one adopts the within-cluster resampling technique, proposed by Hoffman, Sen, and Weinberg [10] and Follmann, Proschan, and Leifer [11], while our resampling unit is matched pairs or sets (of observations in case and control periods) rather than individual observations. As pointed out in Reference [10], the withincluster resampling-based methods are robust in the situations where the cluster size is nonignorable. For example, the number of recurrent events per subject is informative to the effect of exposures on the risk of disease. The second proposed method is based on a weighted estimating equation, which was introduced by Williamson, Datta, and Satten [12], and the similar idea has been widely used for different types of correlated data. Connection between within-cluster resampling and weighted estimating equation approaches was also made in Reference [12]. Pros
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 5 and cons of the above methods are discussed in Section 3 and then compared in a simulation study in Section 4. We present the applications of the above methods to the elderly falls data in Section 5. Some discussion is in Section 6. 2. ELDERLY FALLS DATA Falls in the elderly often occur recurrently. When the research interest is to determine the risk factors for falling rather than for being a faller, the case-crossover design can be useful. In this paper, we consider a data set with a total of 158 nursing home residents who were at least 65 years old and fell at least once at one of three study sites (the Johns Hopkins Bayview Care Center in Baltimore, MD and the V.A. nursing homes in Tampa and Orlando, FL) during 2002-2003. The selected residents were on average 81 years old, mainly white (68%), male (66%), and had a diagnosis of dementia (51%). There were 419 falls in total or 2.7 falls per person observed among the 158 residents, the number of falls per resident ranged from 1 to 37 (median 1). Data were collected from medical records on medication changes, including medication name, date of change, and type of change (i.e. new start, dose change, an as-needed dose given, or discontinuation), over a nine-day period prior to each fall. These data were originally presented in Sorock et al. (submitted manuscript). The main research question in this study was to estimate the association between fall risk and medication change by comparing the medication change record during 1-2 days before versus 8-9 days before each fall date. A one-week lapse between case and control periods was chosen to minimize any changes of potentially unobserved time-varying confounding variables, e.g. health status.
6 X. LUO AND G. S. SOROCK 3. METHODOLOGY 3.1. Notation Consider a sample of subjects, i = 1,, n, who have at least one event during a fixed time interval, where time is assumed to be discrete, i.e. t {τ 1, τ 2,, τ T }. For ease of presentation, we consider the case-crossover design with one control time matched for each event time, analogous to one-to-one matched case-control studies. However, the discussed methods can be easily extended to deal with the setting with more than one control (1:M matching) and/or more than one event in each matched set. For the ith subject, m i (m i 1) events occur at t i1,, t imi and for each event time t ij, a control time s ij is chosen, j = 1,, m i, i = 1,, n. Let X 1ij denote the covariate of subject i at the event time t ij and X 0ij be this subject s covariate at the corresponding control time s ij. Let O ij = {t ij, X 1ij, s ij, X 0ij } denote the matched set for the jth event of the ith subject. We assume that the log odds of event for subject i at time t are given by log p it 1 p it = λ i + βx it, (1) where X it represents the time-varying covariate of subject i at time t and β is the difference in log odds of event associated with one unit increases in X. The subject-specific λ i is timeinvarying and could be a linear combination of a set of baseline covariates, including observed or unobserved variables. 3.2. Single event analysis approach To avoid modelling the dependence among multiple matched sets from the same subject, one simple approach is to use only one event (e.g. first event) and its matched control per subject, as
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 7 mentioned in Reference [7]. Hence, a sample of n independent matched sets, {O i1 ; i = 1,, n}, can be constructed and then the conditional logistic regression can be used for the estimation of β. The conditional likelihood (Reference [13], p190) of the exposure status at the case time given it being one of the two observed exposures in the matched set is n i=1 and the conditional likelihood score function is S 1 (β) = exp(βx 1i1 ) exp(βx 1i1 ) + exp(βx 0i1 ), (2) { n } 1 l=0 X 1i1 X li1 exp(βx li1 ) 1 l=0 exp(βx. (3) li1) i=1 A sufficient condition for the validity of this conditional likelihood is that the distribution of X in any two successive time periods are exchangeable within a subject [6]. The conditional maximum likelihood estimate (CMLE), ˆβ solves S1 (β) = 0 and is asymptotically normal with mean equal to the true parameter and variance (or variance-covariance matrix) equal to the negative inverse of S 1 (β)/ β [14]. The variance can be estimated by replacing the true parameter with the estimated parameter. Although the estimate of β from this approach is valid, we would expect loss of efficiency since this method ignores the recurrent nature of the data and truncates all second and later events from the analysis. 3.3. Analysis of pooled recurrent events Another frequently used CLR-based approach is to assume that the occurrence of events at distinct times are independent given the subject-specific effect λ i and the time-varying covariate X it, so that the pooled matched sets from different subjects, {O ij ; i = 1,, n, j = 1,, m i } are conditionally independent and identically distributed (iid). The conditional
8 X. LUO AND G. S. SOROCK likelihood is, hence, n m i i=1 j=1 exp(βx 1ij ) exp(βx 1ij ) + exp(βx 0ij ), (4) and the conditional likelihood score function is { n m i } 1 l=0 X 1ij X lij exp(βx lij ) 1 l=0 exp(βx. (5) lij) i=1 j=1 Applications of this approach can be found in Reference [7] when falls are the unit of analysis rather than persons. 3.4. Analysis of clustered data Navidi [5] proposed the full-stratum design, which requires that control information being collected for all possible time intervals. As pointed out by Whitaker et al [15], the full-stratum design is based on cohort sampling, while the original case-crossover design is based on casecontrol sampling. It was discussed in Reference [5] that the conditional logistic regression can be used not only for clusters with a single case, but also for clusters with multiple cases. The proposed method for analyzing recurrent events in that paper treats the recurrent events within each person as a cluster and constructs the conditional likelihood on a person/cluster level. Though this method was proposed for the full-stratum design, it can be used for analyzing recurrent events under the case-crossover design. We can argue that the same form of conditional likelihood can be obtained whether the data are regarded as from a cohort or a case-control study (Reference [3]). For subject i, let the set of covariates observed at the m i event times and the m i control times be E i = {X 1ij, X 0ij ; j = 1,, m i }. The set E i can be treated as a cluster. The conditional probability that events covariate values are precisely in {X 1ij = x 1ij ; j = 1,, m i }, given that they lie in E i, is exp ( βσ mi j=1 X 1ij) /ΣSi exp(βσ l Si X l ), where S i is a subset of E i of size
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 9 m i and S i is the summation over all such subsets. A sufficient condition for the validity of this probability requires the joint distribution of covariates in any 2m i successive time periods being exchangeable within subject i [6]. Note that the exchangeability condition for the clustered data analysis method is stronger than that for the single event analysis or the pooled data analysis methods because 2m i could be bigger than 2. The conditional likelihood is ( n exp β ) m i j=1 X 1ij S i exp(β l S i X l ), (6) i=1 and the conditional likelihood score function is n m i ) ( )} S X 1ij i {( l S i X l exp β l S i X l S i exp(β. (7) l S i X l ) i=1 j=1 This method requires the same conditional iid assumption as the pooled data analysis method described in Section 3.3. The difference is that the conditional likelihood in the pooled analysis is based on events rather persons, while the method for clustered data treats the recurrent events within a person as a cluster and constructs the conditional likelihood on a person/cluster level. An advantage of this property is that events failed to be matched with reference time(s) by design do not need to be excluded from the clustered data analysis method. However, as the number of events per subject grows large, this method can become computationally infeasible. Another disadvantage of this method is that the natural bound between the case and control times within each matched set is broken, hence can be vulnerable to biases caused by potentially unobserved time-varying confounding variables. 3.5. Within-subject pairwise resampling Hoffman, Sen, and Weinberg [10] proposed a within-cluster resampling (WCR) procedure for clustered data, where cluster size could be nonignorable. Within the framework of WCR,
10 X. LUO AND G. S. SOROCK an observation is randomly sampled from each cluster with replacement. The resulting (sub)sample consists of independent data and then can be analyzed using existing univariate methods. By repeating the resampling a large number of times, the parameter can be estimated by averaging the estimates from each univariate analysis. The variance estimate and weak consistency are provided in that paper. Follmann, Proschan, and Leifer [11] discussed how to choose the number of resampling and proposed an estimator, referred as exhaustive multiple outputation or EMO, obtained by finding all possible subsamples and averaging the estimates from all these subsamples. Rieger and Weinberg [16] adopted the within-cluster resmapling method in the framework of CLR for clustered binary outcome data. In their method, a resampled data set is constructed by randomly selecting one case and one control from each cluster, so that the resulting data set consists of pairs of observations. The essence of this method is to transform clustered data into multiple 1:1 matched data sets and then conduct the matched-pair case-control data analysis repeatedly. The advantage is that the withincluster conditional independence assumption can be dropped. In the case-crossover data that we described in Section 3.1, the case period and control period are matched pairwisely by design, so that the resampling can be based on matched sets rather than on individual observations as in Reference [16]. The matched sets-based resampling should be more robust in situations with unmeasured time-varying confounding variables. The rest of the estimation procedure is the same as other typical WCR methods. The whole procedure is outlined as follows. Step 1. Sample one matched pair, O ij, randomly from each subject. The resulting data set consists of n independent pairs of case and control periods. Step 2. Conduct conditional logistic regression on the resampled data set and record the
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 11 parameter estimate and its variance (or variance-covariance matrix) estimate as ˆβ(q) and ˆΣ(q) (estimation method is the same as the one described in Section 3.2). The same exchangeability assumption on the exposure distribution as for the single event analysis method is needed. Step 3. Repeat steps 1 and 2 for a large number of times, say Q. Step 4. The formulas to calculate the parameter estimate and its variance estimate were provided in Reference [10] and are repeated as follows: ˆβ WCR = 1 Q ˆβ(q), (8) Q q=1 { } 1 Q ˆΣ WCR = ˆΣ(q) 1 Q ( Q Q ˆβ(q) ˆβ WCR )( ˆβ(q) ˆβ WCR ) T. (9) q=1 For the EMO estimate, Q = n i=1 m i, the total number of all possible resamples. q=1 The proposed within-subject pairwise resampling or WSPR method inherits the advantages of a general WCR method, namely that (1) the correlation structure among multiple matched sets within each subject is left unspecified, and (2) the number of matched sets or events per subject can be nonignorable [10]. 3.6. Weighted estimating equation Considering the intensiveness of computing for the WCR method, Williamson, Datta, and Satten [12] proposed a weighted estimating equation (WEE) method for clustered data. In that paper, it was observed that in each round of resampling in the WCR procedure, ˆβ(q) solves the score equation S (q) (β) = 0, where n m i S (q) (β) = U ij (β)i[(i, j) r q ], (10) i=1 j=1
12 X. LUO AND G. S. SOROCK with r q being the set of indices (i, j) that are sampled in the qth resampling. For our casecrossover data, U ij (β) = X 1ij 1 l=0 X lij exp(βx lij )/ 1 l=0 exp(βx lij). They, further, argued that the resampling distribution is a discrete uniform distribution with a probability mass of 1/m i on each observation within subject i, hence, the indicator I[(i, j) r q ] in (10) can be replace by 1/m i. The resulting WEE for our case-crossover data is then { n 1 m i } 1 l=0 S W EE (β) = X 1ij X lij exp(βx lij ) m 1 i l=0 exp(βx. (11) lij) i=1 j=1 It was also proved in Reference [12] that EMO estimator and WEE estimator are asymptotically equivalent, and the WEE estimator has weak convergence, namely that, n( ˆβ β) converges to a normal distribution with mean 0 and variance that can be estimated by a sandwich form ˆν = Ĥ 1 ˆV Ĥ 1, where Ĥ = n 1 n 1 m i m i=1 i j=1 U ij (β) β β= ˆβ and ˆV = n 1 n i=1 1 m i m i j=1 U ij ( ˆβ) 1 m i m i j=1 U ij ( ˆβ) Applications of the WEE method can also be found in the analysis of clustered survival data [17]. T. 3.5 Mantel-Haenszel estimator revisited It is noteworthy to mention that in applications, if only one dichotomous variable, e.g., exposure or non-exposure status, is of interest, the Mantel-Haenszel method provide an easyto-implement alternative to the conditional likelihood based methods. Under the conditional
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 13 iid assumption, the Mantel-Haenszel estimator of the odds ratio for 1 : M matched data is M m=1 (M m + 1)n 1,m 1 M m=1 mn, (12) 0,m where n 1,m and n 0,m are the total number of matched sets with, respectively, the case exposed and the case not exposed, and exactly m controls also exposed. When M = 1 (1 : 1 matching), it is simply the ratio of the discordant pair, n 10 /n 01 of a two-by-two table. We note that the Mantel-Haenszel estimator is a special case of the pooled data analysis method that we described in Section 3.3 when only one binary covariate is present in the model. Analogous to the WEE method, we can also propose a weighted Mantel-Haenszel (WMH) estimator for odds ratio, namely M m=1 (M m + 1)ñ 1,m 1 M m=1 mñ, (13) 0,m where ñ i,m and ñ 0,m have similar definitions as n 1,m and n 0,m, but with each matched set O ij being counted as 1/m i instead of 1. When M = 1, the WMH estimator also has a simple form, ñ 10 /ñ 01. It can be proved that the WMH estimator is the solution of the WEE when only one binary covariate is present in the model. 4. A SIMULATION STUDY We conduct a simulation study to asses the performance of the existing and proposed methods. The simulated data has a similar structure as the elderly falls data that we describe in Section 2. For each simulation study, 1000 replicate data sets were generated, with a sample size n = 200. For the WSPR method, each data set is resampled Q = 1000 times. A fixed time period with discrete time points 1, 2,, T (T = 100) is assumed for each subject. We genrate the data under the model logit(p it ) = λ i + βx it, where the subject-specific baseline risk λ i
14 X. LUO AND G. S. SOROCK is a logged Beta (a = 1, b = 100) distributed variable, and X it is generated from Bernoulli distribution with probability of 1/20. Table I displays the mean of the estimated regression coefficient, the sample standard deviation over the 1000 simulations, and the mean of the estimated standard error, for five different methods. The regression coefficient is chosen β = 1, 2, or 3, the corresponding average number of events per subject is m i = 2.027, 2.211, 2.570, respectively. The results show that all methods yield reasonably unbiased estimates of the regression coefficient. The estimated regression coefficient from the method which uses only the first event has the biggest variation, while that from the method treating all events and controls from the same subject as a cluster (clustered data analysis method) has the least variation. However, when the number of events per cluster increases, the clustered data analysis method requires longer computing time. The second smallest variation of estimation is from the pooled data analysis method, which is comparable to the clustered data analysis method. The WSPR and WEE methods produce similar variation of estimation that is about midway among all methods. However, as we discussed earlier, these two methods can accommodate flexibility in correlation structure in recurrent events. The WEE method is more efficient, in terms of computing time, than the WSPR method especially when the number of resampling is large. In summary, when the conditional iid assumption holds, the pooled data analysis approach is the most efficient method, taken into account both efficiency of the estimator and the computing time. We also recommend using the WEE method when the dependence structure of the recurrent events is unclear.
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 15 5. APPLICATION TO THE ELDERLY FALLS DATA We illustrate the application of the existing and proposed methods in the elderly falls data discussed in Section 2. All the previously discussed methods, except for the method treating the data as clustered data, require at least one control period matched to each case period prior to a fall. For the elderly falls study, a nine day window before a fall is used to define the case (1-2 days before the fall date) and the control (8-9 days before the fall date) periods. Therefore, some falls have to be excluded due to the difficulties in defining controls described below. First, falls that occur within 9 days from the date of admission are excluded from the analysis due to lacking of information on exposure status. Second, if consecutive falls are separated by less than 9 days, only the first one (called primary fall ) is included in the analysis because the control period of later falls (called secondary falls ) could have overlap with preceding falls case period. As a consequence, 311 out of 419 falls are left for analysis and no statistically significant difference between the remaining 148 residents and the 10 excluded residents is observed, in terms of age, gender, race, and dementia status. If we use the method which treats the data as clustered data as discussed in Section 3.4, it is not necessary to find a control period for each individual fall. Hence, the falls, being failed to be matched to controls can still be included in the analysis. However, we still need a two-day window before the fall date to define the case period. Therefore, we exclude the falls that occur within 2 days from the date of admission and the secondary falls that are separated by less than 2 days from proceeding falls. This leaves us 376 falls (150 residents) for analysis. Again, no significant difference between the included and excluded residents is found in terms of patient characteristics.
16 X. LUO AND G. S. SOROCK Two types of exposure are of interest: changes in Central Nervous System (CNS) medication use and changes in non-cns medication use. We summarize both the estimated odds ratio and the 95% confidence intervals in Table II. All methods find that significantly elevated fall risk follows CNS medication changes and no significant effect of non-cns medication changes on fall risk is observed. From the method which uses only the first fall for each resident, we can see that the odds ratio estimate for CNS changes is higher than the other methods which use recurrent falls, while there is loss of efficiency, i.e. wider 95% confidence interval, compared with other methods. The relatively large odds ratio estimate could be due to the bigger variation of the estimate. The limitation of the application of the clustered data analysis method on this data set is that the computing time is not affordable because a certain number of residents fell more than 10 times (up to 31) during the study period and the computing time is exponential in the number of falls per resident. We choose to use the Breslow [18] s and Efron [19] s methods to approximate the exact conditional likelihood in estimation. Both methods yield odds ratio estimates for CNS changes that are substantially smaller in magnitude than the other methods. Considering the fact that approximation methods are used in estimation, the estimates from this analysis are not considered reliable. The pooled data analysis method and the proposed WSPR method and WEE method produce consistent results in terms of both point and interval estimates for both types of medication changes. Based on the estimates from these three methods, the change in CNS medication use resulted in a three to four folds increase in the fall risk among the studied nursing home residents. The change in non-cns medication use did not significantly change their fall risk.
ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 17 6. DISCUSSION In this article, we consider different methods for estimating the effect of exposures on risk of disease for recurrent event data under the case-crossover design. Commonly used existing methods are reviewed and two new methods are proposed (WSPR and WEE methods), which enrich the available tools for analyzing recurrent event data under the case-crossover design. It is also discussed in this paper that the two proposed methods have more flexibility than the existing methods in the situations with unknown correlation structures among recurrent events. The Mantel-Haenszel estimator is revisited and a weighted Mantel-Haenszel (WMH) estimator is proposed. Both estimators are easy to compute and implement in applications. In all the previously discussed conditional likelihood-based approaches, subject-level covariates, observed or unobserved, are readily controlled. However, none of these methods can deal with subject-specific slopes in the model, for example, log p it 1 p it = λ i + (β + α i )X it, where α i is the subject-specific effect of exposures on event risks (or slope). Future research can proceed in this direction. In application, we can still use the previously discussed models, but including interactions of subject-level covariate (e.g. race, gender, and different diagnosis of disease) with the time-varying exposure status to capture the subject-specific exposure effect as much as possible.
18 X. LUO AND G. S. SOROCK ACKNOWLEDGEMENTS Dr. Gary Sorock s research is supported by Centers for Disease Control, National Center for Injury Prevention and Control, grant # H400-888-2152. This material is also the result of work supported with resources and the use of facilities at the James A. Haley Veterans Hospital. The authors thank Dr. Chiung-Yu Huang for her careful reading of the manuscript and insightful comments and valuable suggestions.
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ANALYSIS OF RECURRENT EVENT DATA UNDER THE CASE-CROSSOVER DESIGN 21 Table I. Mean of estimated regression coefficients, sample standard deviation of estimated regression coefficients, and mean of estimated standard errors, for five different methods. β = 1 β = 2 β = 3 Method Est SD SE Est SD SE Est SD SE First event method 1.046 0.427 0.416 2.082 0.454 0.435 3.104 0.538 0.534 Pooled data method 1.018 0.302 0.291 2.035 0.302 0.289 3.044 0.347 0.328 Clustered data method 1.016 0.296 0.286 2.026 0.281 0.270 3.019 0.278 0.274 WSPR method 1.044 0.356 0.341 2.076 0.370 0.346 3.108 0.415 0.416 WEE method 1.029 0.347 0.337 2.048 0.359 0.338 3.067 0.427 0.385 Est = mean of the estimated regression coefficients. SD = sample standard deviation of the estimated regression coefficients. SE = mean of the estimated standard errors.
22 X. LUO AND G. S. SOROCK Table II. The estimated odds ratio and 95% confidence interval for the elderly falls data based on different methods. CNS medication change Non-CNS medication change First event method 5.50 (1.22-24.81) 1.00 (0.52-1.92) Pooled data method 4.00 (1.34-11.96) 1.19 (0.71-2.01) Clustered data method Breslow s approximation 1.86 (1.08-3.21) 1.13 (0.78-1.64) Efron s approximation 2.35 (1.34-4.12) 1.21 (0.83-1.77) WSPR method 3.61 (1.05-12.37) 1.00 (0.54-1.84) WEE method 3.51 (1.05-11.68) 1.00 (0.55-1.84)