An Age-Stratified Poisson Model for Comparing Trends in Cancer Rates Across Overlapping Regions

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1 bimj header will be provided by the publisher An Age-Stratified Poisson Model for Comparing Trends in Cancer Rates Across Overlapping Regions Yi Li, Ram C. Tiwari 2, and Zhaohui Zou 3 Harvard University, 44 Binney Street, Boston, MA 025, USA 2 National Cancer Institute, Bethesda, MD Information Management Services, Silver Spring, MD Received 5 November 2007, revised 28 February 2008, accepted 25 April 2008 Published online Summary The annual percent change (APC) has been used as a measure to describe the trend in the age-adjusted cancer incidence or mortality rate over relatively short time intervals. The yearly data on these age-adjusted rates are available from the Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute. The traditional methods to estimate the APC is to fit a linear regression of logarithm of age-adjusted rates on time using the least squares method or the weighted least squares method, and use the estimate of the slope parameter to define the APC as the percent change in the rates between two consecutive years. For comparing the APC for two regions, one uses a t-test which assumes that the two datasets on the logarithm of the age-adjusted rates are independent and normally distributed with a common variance. Two modifications of this test, when there is an overlap between the two regions or between the time intervals for the two datasets have been recently developed. The first modification relaxes the assumption of the independence of the two datasets but still assumes the common variance. The second modification relaxes the assumption of the common variance also, but assumes that the variances of the age-adjusted rates are obtained using Poisson distributions for the mortality or incidence counts. In this paper, a unified approach to the problem of estimating the APC is undertaken by modeling the counts to follow an age-stratified Poisson regression model, and by deriving a corrected Z-test for testing the equality of two APCs. A simulation study is carried out to assess the performance of the test and an application of the test to compare the trends, for a selected number of cancer sites, for two overlapping regions and with varied degree of overlapping time intervals is presented. Key words: Age-adjusted incidence/mortality rates, age-stratified Poisson Regression, annual percent change (APC), surveillance, trends, hypothesis testing. Introduction The American Cancer Society (ACS) in its annual publication Cancer ACS 2007 ( reports that in 2007 about.5 million new cancer cases are expected to be diagnosed, and approximately 560,000 Americans are expected to die of cancer. Cancer is the most common cause of death in US, exceeded only by heart disease, and accounts for of every 4 deaths. The same report also reveals that, for a number of cancer sites (such as breast, stomach, colon and rectum, lung and bronchus and leukemia), the age-adjusted cancer mortality rates have been steadly decreasing in recent years. In addition, the National Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute (NCI) periodically publishes similar reports on trends of cancer incidence at see Ries et al. (2003) So much has been at stake in terms of human life and cost - for example, the government agencies such as the National Health of Institutes (NIH), and many private sectors spend billions of dollars every year on cancer research, health insurance and medical and other costs - that there is an urgent need for new methods that produce more accurate and reliable estimates of measures of cancer trends. Corresponding author: yili@jimmy.harvard.edu, Phone: , Fax:

2 2 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance The annual percent change (APC) has been used as a measure of cancer trends over short time periods, and to compare the recent cancer trends by gender or by geographic regions, one compares their APC values using the two-sample pooled t-test ( Kleinbaum et al., 988) that assumes that the datasets on ageadjusted rates under the study are independent. However, a fundamental statistical difficulty arises when such comparisons, largely for policy making purposes, have to be made for regions or time intervals that overlap, e.g. comparing the most recent changes in trends of cancer rates in a local area (e.g. the mortality rate of breast cancer in California) with a more global level (i.e. the national mortality rate) over two overlapping time periods. For example, as detailed in the data analysis section, it is of substantial interest to compare the changes in California cancer mortality rates with the national cancer mortality rates in the last 5 years. Recently, Li and Tiwari (2007) and Li et al. (2007) developed Z-tests which adjust for the dependence between the two APCs, and are more efficient than the naive test which assumes independence. However, these tests are based on the logarithmic transformation of the age-adjusted rates, and fits a simple linear regression model of the transformed data on time using either the ordinary least squares (OLS) or the other weighted least squares (WLS) procedures. The proposed test procedure is based on the natural assumption that the age-specific mortality or incidence counts are results of underlying Poisson processes (Brillinger, 986), and hence are realizations of independent Poisson random variables. The age-specific instantaneous hazards are modeled by a log-link function, thus leading to an age-stratified Poisson model. The estimation of the parameters is then carried out using a likelihood-based approach. The rest of the paper is organized as follows. In Section 2, we briefly review the existing tests, and derive the new test in Section 3. To compare the performance of the proposed test with respect to the above mentioned tests, a simulation study is carried out in Section 4. In this section, we also give application to breast cancer mortality data from California (CA) and the US extracted from the SEER*STAT software of the SEER Program. Section 5 ends this paper with a short discussion. 2 A Brief Review of Existing Tests Consider two regions, and let d kji denote the number of counts (deaths or new cancer cases) from the population at risk n kji observed in Region k (k, 2) in age-group j (j,..., J) and at times T,..., T m for Region and T s+,..., T s+n for Region 2, where T T s+ < T m T s+n, with 0 s < m leading to overlapping time intervals. Note that this formulation is general and allows one region to have fewer time points than the other. In the SEER program, it is common to choose n kji (at year T i ) to be the mid-year population representing the total person-years in one year, with the assumption of drop-outs being uniform over the unit-intervals. The age-adjusted rates are defined as r ki J j w j d kji n kji, where w j > 0, j,..., J, are the known standards for the age group j so that J j w j. For the SEER analysis, there are J 9 standard age-groups consisting of 0-, -4, 5-9,..., 85+, and w j are chosen to be the year 2000 population standards (Fay et al. 2006). Let y ki log(r ki ), be the logarithmic transformations of the age-adjusted rates. Consider the linear regression models y ki β 0k + β k t ki + e ki, i,...,, () for k, 2, flagging Regions and 2, respectively. Here e ki are random errors with mean 0, and t ki corresponds to the calendar times of data collection in region k with I m and I 2 n. More specifically, (t,..., t I ) (T,..., T m ), while (t 2,..., t 2I2 ) (T s+,..., T s+n ). For the two regions, the

3 bimj header will be provided by the publisher 3 annual percent change (APC) are defined as AP C k 00(e β k ). 00β k, for a small β k, e.g. in the order of 0 2 (Kim et al., 2000; Fay et al., 2006; Tiwari et al., 2006). Then under the assumptions that e ki are independent and have common variance σ 2, the two-sample pooled t-test (Kleinbaum et al., 988) for testing the null hypothesis H 0 : AP C AP C 2 versus the alternative H a : AP C AP C 2 is given by ˆβ T t ˆβ 2 ( ˆσ 2 ( I (t i t ) 2 ) + ( ) t (I +I 2 4), (2) I 2 (t 2i t 2 ) 2 ) where t k t ki / for k, 2, and ˆσ 2 is the pooled unbiased estimate of σ 2 given by I ˆσ 2 (y i ŷ i ) 2 + I 2 (y 2i ŷ 2i ) 2, I + I 2 4 where ŷ ki ˆβ 0k + ˆβ k t ki are the predictions for k, 2. Here, ˆβ 0k and ˆβ k are obtained from the least squares estimation. That is, Ik ˆβ k (t ki t k )(y ki ȳ k ) ik, ˆβ0k ȳ k (t ki t ˆβ k t k k ) 2 where ȳ k y ki /. The above test is not appropriate, however, when there is an overlap between the two regions or the two time periods. For this case, Li and Tiwari (2007) proposed the following corrected Z-test ˆβ Z CT ˆβ 2 ( {ˆσ 2 σ 2 + σ2 2 2σ 2 σ 2 σ 2 2 )} /2, (3) (n (O) ) 2 where σk 2 (t ki t k ) 2, σ 2 m s+ (T i t )(T i t 2 ), n k m J is+ j n kji for k, 2, n (O) m J is+ j n(o) ji and and n (O) ji are the numbers of at-risk population in the overlapping region. Note that there is no suffix k in n (O) ji. The sign of σ 2 determines, whether the covariance between ˆβ and ˆβ 2 is positive or negative, and when there is no overlap in time intervals, σ 2 0. Under the lognormal model, the corrected Z CT test was shown to follow a standard normal distribution under the null hypothesis, and to be more efficient than the pooled t-test; see Li and Tiwari (2007). However, one assumption in Li and Tiwari (2007) is the equal variance in both regression models, which may not be realistic, especially for rare cancers. A further refinement has been made to derive the variance of y ki by using the Poisson assumptions on the first two moments of the counts d kji, i.e. E(d kji ) var(d kji ). Under these assumptions, the consistent estimate of the error variance of e ki is given by vki 2 J rki 2 j w2 d kji j, leading to the following weighted least squares test proposed by Li et n 2 kji al. (2007), referred to as Z W LS : β Z W LS β 2 { σ 2 + σ σ 2 σ 2 σ 2 2 with σ 2 m (T i t ) 2 /v 2 i and σ2 2 s+n is+ (T i t 2 ) 2 /v 2 2i, σ 2 m is+ (T i t )(T i t 2 ) v(o) 2i v 2 i v2 2i, } /2. (4) (n (O) ) 2

4 4 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance where and d (o) ji v (o) 2i r i r 2i J j w 2 j d (o) ji (n (o) ji )2, are the counts in the overlapping region and during the overlapping period. Here, m t T i/vi 2 m, t 2 /v2 i s+n is+ T i/v2i 2 s+n, is+ /v2 2i and β, β 2 are weighted least square estimates of β, β 2. Under the null hypothesis, because of the normal approximation, Z W LS approximately follows a standard normal distribution. The Z W LS has been shown to be more conservative than Z CT in retaining the size of the test, but is more powerful for the common cancer sites; see Li et al. (2007). However, there are several disadvantages of the existing methods. First, one key step of Li et al. (2007) is the normal approximation of the age-adjusted rates. Secondly, both Li et al. (2007) and Li and Tiwari (2007) need adjustments for zero counts. 3 Age-stratified Poisson Regression Model As the existing approaches to dealing with age-adjusted cancer rates were all based on the normal approximation, we take a more natural route in the sequel by considering the Poisson nature of the underlying count data and propose an age-stratified Poisson regression to describe the change trend of incident (or death) counts on time. Based on this model, a proper test that accounts for overlapping is proposed. Specifically, since d kji, the number counts (deaths or new cancer cases) observed in Region k (k, 2) in age-group j, is a count, we assume that with d kji ind P ois(n kji λ kji ), log λ kji β 0kj + β k t ki, (5) which is referred to as the Age-stratified Poisson Regression Model as the age-specific intercept β 0kj is assumed for age-group j. The common slope β k is of particular importance as it transcribes the trends of mortality or incidence and, in particular, determines the APC value. Again let AP C and AP C 2 be the corresponding APC values for these two Poisson regressions. A natural test for the null hypothesis H 0 : AP C AP C 2 versus the alternative hypothesis H : AP C AP C 2 would be ˆβ Z P OIS ˆβ 2 V ar{ ˆβ ˆβ, 2 } where ˆβ and ˆβ 2 are the maximum likelihood estimates of β and β 2 derived in the Appendix. Because of the possible overlapping of Regions and 2, ˆβ and ˆβ 2 may be correlated. Thus the key to the derivation of the test lies in a correct evaluation of Cov( ˆβ, ˆβ 2 ).

5 bimj header will be provided by the publisher 5 3. Derivation of the Test To proceed, we let β k (β k, β 0k,..., β 0kJ ), k, 2, whose estimates ˆβ k can be obtained by solving the score equations [based on (5)] U k (β k ) 0, for k, 2, where U k (U k,, U 0k,,..., U 0k,J ) and U k, U 0k,j j J n kji t ki exp(β 0kj + β k t ki ) n kji exp(β 0kj + β k t ki ) d kji, j J d kji t ki, for j,..., J. As U k ( ˆβ k ) 0, expanding it around the true value β k, and ignoring the higher order terms yields 0 U k ( ˆβ k ) U k (β k ) + { ( )} U () k Ik Ik I (β k) Ik ˆβk β k + o p (), k where U () k U k (β k )/ β k are (J +) (J +) matrices with its st row as (U () k,, U () k,0,..., U () k,0j ) and the (j + ) th row (j,..., J) as (U () 0k,j, U () 0k,0j,..., U () 0k,0jJ ), for k, 2. Here U () k, U k, β k U () k,0j U k, β 0kj U () 0k,j U 0k,j β k U () 0k,0jj U 0k,j β 0kj j δ jj J n kji t 2 ki exp(β 0kj + β k t ki ), n kji t ki exp(β 0kj + β k t ki ), n kji t ki exp(β 0kj + β k t ki ), n kji exp(β 0kj + β k t ki ), and δ jj if j j and 0 otherwise. Denote by A k plim Ik U () k (β k)/ for k, 2, where plim denotes the limit in probability. Then for large I ( m) and I 2 ( n), standard probabilistic arguments yield { I ( ˆβ β ), { } I 2 ( ˆβ d 2 β 2 )} U (β ), A U 2 (β 2 ). I I2 A 2 Here, d denotes approximate equivalence in joint distribution functions. Hence, V ar( m( ˆβ β ), m( ˆβ β )) V ar( n( ˆβ 2 β 2 ), n( ˆβ 2 β 2 ) ) Cov( m( ˆβ β ), n( ˆβ 2 β 2 ) ). A m Cov(U (β ), U (β ))A T,. A 2 n Cov(U 2(β 2 ), U 2 (β 2 ))A T 2,. A Cov(U (β ), U 2 (β 2 ))A T mn 2.

6 6 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance Let V m Cov(U (β ), U (β )), W n Cov(U 2(β 2 ), U 2 (β 2 )), Σ mn Cov(U (β ), U 2 (β 2 )), and ˆV, Ŵ, ˆΣ be the corresponding estimates, whose derivations are given in the Appendix. Then, Ĉov( ˆβ, ˆβ ) m{u () ( ˆβ )} ˆV {U () ( ˆβ )} T ; Ĉov( ˆβ 2, ˆβ 2 ) n{u () 2 ( ˆβ 2 )} Ŵ {U () 2 ( ˆβ 2 )} T ; Ĉov( ˆβ, ˆβ 2 ) mn{u () ( ˆβ )} ˆΣ{U () 2 ( ˆβ 2 )} T. These are three (J +) (J +) matrices, the (, ) entries of which are ˆσ 2 ˆV ar( ˆβ ), ˆσ 2 2 ˆV ar( ˆβ 2 ) and ˆσ 2 Ĉov( ˆβ, ˆβ 2 ), respectively. From this we compute ˆV ar( ˆβ ˆβ 2 ) ˆV ar( ˆβ )+ ˆV ar( ˆβ 2 ) 2Ĉov( ˆβ, ˆβ 2 ). Hence, the Z-test for comparing APC values is given by Z P OIS ˆβ ˆβ 2 (ˆσ 2 + ˆσ2 2 2ˆσ 2 )/2, (6) which follows the standard normal distribution under H 0 : β β 2. The computation of ˆβ, ˆβ 2 is given in the Appendix. 3.2 ARE Comparison with the WLS test It is of substantial interest to evaluate the gains in efficiency of the proposed test compared with the WLS test. First note (5) implies that E(r ki ) E( j w j d kji n kji ) ( j w je β 0kj )e β k Ti. This in turn implies that E(log r ki ). log E(r ki ) log( j w j e β 0kj ) + β k t ki, (7) when V ar(r ki ) is small, which is often the case for the cancer incidence and mortality data (Kim et al., 2000). A comparison between () and (7) reveals that models () and (5) approximately specify the same first moment of the age-adjusted cancer rates, making it possible to compare the efficiency of the tests based on these two models via the measure of the Pitman Asymptotic Relative Efficiency. Specifically, standard asymptotic analysis will yield ˆβ ˆβ 2 N(β β 2, ˆσ 2 + ˆσ 2 2 2ˆσ 2 ), while, for the WLS estimates, β β 2 N(β β 2, σ 2 + σ σ 2 σ 2 σ 2 2 (n (O) ) 2 n n 2 ). Hence, the Pitman Asymptotic Relative Efficiency (ARE) comparing tests (6) and (4), which is the ratio of the noncentralities of the above two normal distributions, is given by ARE σ 2 + σ σ 2 σ 2 ˆσ 2 + ˆσ2 2 2ˆσ 2 σ 2 2 (n (O) ) 2 n n 2 The evaluation of (8) typically involves numerical computations.. (8)

7 bimj header will be provided by the publisher 7 4 SEER mortality data analysis and Simulations Li et al. (2007) demonstrated that Z W LS performs better than Z CT, via the calculation of ARE, as Z CT relies on the common variance assumption, which may not be realistic. Hence, we focus this paper on comparing Z P OIS with Z W LS. To comprehensively evaluate these tests, we consider several scenarios of overlap in two regions and in two different time intervals. Specifically, we assume that Region consists of Georgia (GA), South Carolina (SC), and North Carolina (NC), and that Region 2 consists of NC, Virginia (VA) and Maryland (MD); with NC as the overlapping state between the two regions. The three different time intervals, with varying degree of overlap in the intervals, are taken to be : (a) [980,989] for Region, and [990,999] for Region 2 so that there is no overlap between the two time intervals and, hence, σ 2 0; (b) [980,989] for Region, and [983,992] for Region 2 so that there a considerable overlap of six years between the two intervals and σ ; (c) [980,989] for Region, and [987, 996] for Region 2 so that there is a little overlap of three years between the two intervals and σ The counts, d kji were generated based on model (5) with t ki taking values in the intervals corresponding to the two regions stated above. More specifically, the t i take values of {0,,..., 9}, while the t 2i take values of {0,..., 9}, {3,..., 2}, and {7,..., 6}, respectively for cases (a)-(c). In order to fully specify λ kji in (5), we assume that β 0kj log(d kj /n kj ) β k t k, where d kj and n kj are respectively the observed number of deaths and the number of at-risk population at the beginning of the intervals considered, and take β k log(ap C k /00 + ), based on the specified values of AP C k. The age-specific counts for the overlapping state, NC, are generated from Poisson distributions with means, n (o) ji 2 (λ ji + λ 2ji ), where n (o) ji denotes the at-risk population in the overlapping region. When λ ji λ 2ji, this reduces to the situation specified by the null hypothesis. The number of at risk population and the observed number of deaths were obtained from the SEER database for all malignant male cancers and prostate cancer. The values of APC were assumed to range from -0.3% to 3.0%. For each parameter configuration, a total of 000 simulated data were obtained. The results for the three time-overlapping cases are summarized in Tables -3. We remark that that, even though both Z W LS and Z P OIS are derived under different model assumptions, they are both valid tests for testing the equality of two APCs and hence the ARE defined in (8) is valid. The tables show that the ARE of Z P OIS with respect to Z W LS is greater than for all the three cases, meaning that Z P OIS would be more powerful than Z W LS when the alternative hypothesis is true. The tables also show that in most situations Z P OIS outperforms the Z W LS in retaining the Type I error probabilities and, hence, yields a more valid test. Also the powers of both WLS and Poisson-based tests are sensitive to the delta values (the differences of APC values). The larger the delta values are, the more powerful the tests are. The larger delta values also lead to slightly larger AREs, though the differences are not so obvious. It is of substantial interest to compare the changes in cancer mortality rates in California with the national levels starting late 980 s as a California law (Health and Safety Code, Section 03885) was passed then, which mandated the reporting of malignancies diagnosed throughout the state. For this purpose, we applied the proposed methodology to compare the annual percent change (APC) in the age-adjusted mortality rates for the United States (US) for the period from to that of California (CA) for the period from 990 to We fitted the weighted linear models as well as the age-stratified Poisson model, and applied both tests to compare the age-adjusted mortality rates of female breast cancer in CA for the 6-year period from to that of US for the 6-year period from , for which the national mortality data were available. The observed values of the log-transformed annual age-adjusted rates and fitted regression lines from the Z P OIS test procedure are plotted in Figure. The parameter estimates and the values of the test statistics are summarized in Table 4. The results indicated the mortality rates of Breast cancer for California and the US have decreased. Both tests reject the null hypothesis of equality of the two APCs, indicating that the annual percent change (APC) of California, is significantly different from the national level. However, the p-value for Z P OIS is much smaller than that of Z W LS, rendering more evidence again the null hypothesis.

8 8 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance 5 Discussion In this paper, we have considered an important problem where comparisons have to be made for regions or time intervals that overlap. As opposed to the existing work, e.g Li and Tiwari (2007) and Li et al. (2007), this project advances this area in two distinct ways. First, the developed test does not rely on the normal approximation of the cancer rate, but directly model the counts to follow a Poisson regression model. The parameters are then estimated using their maximum likelihood estimates, and the Z-test is derived for testing the equality of two APCs. Secondly, the developed Poisson regression model can easily accommodate 0 count data (for the rare cancers), as opposed to the log normal model (Li and Tiwari, 2007; Li et al., 2007) which needs to involve extra zero-corrected adjustment. We have applied the developed methodology to the analysis of the major cancer sites from the SEER Program and have found that the corrected Z-test renders more power than the existing tests. A Bayesian Poisson regression would be a useful approach. However, choice of priors is always difficult and computation may not be so straightforward compared to this current work, wherein analytical solutions have been derived. Hence, we envision that the proposed method would be preferable because of simple interpretation of the model parameters, natural choice of the model and computational readiness. In our technical development, we have modelled the logarithm transformation of the age-adjusted rates as a linear regression on time in (5) and have indeed explicitly assumed parallelism across age groups. That is, the growth curves of the cancer rates for various age groups share the same slope, which carries the information for the APC. Indeed, linearity parallelism for the cancer rates could be a debatable issue in cancer surveillance, which is likely to be violated for some cancers. One alternative, along the line of generalized mixed models, is to assume a random slope (as opposed to a constant slope) across age groups. This ongoing work will be reported in a subsequent communication. Acknowledgements The authors thank the editor, an AE and two referees for their insightful suggestion, which led to a better version of this manuscript. References American Cancer Society (2007) Cancer Facts & Figures. Atlanta, Georgia. Brillinger, D.R. (986). The natural variability of vital rates and associated statistics (with discussion). Biometrics 42, Fay, M., Tiwari, R., Feuer, E. and Zou, Z. (2006). Estimating Average Annual Percent Change for Disease Rates without Assuming Constant Change. Biometrics 62, Kim, H., Fay, M., Feuer, E., Midthune, D. (2000) Permutation tests for joinpoint regression with applications to cancer rates. Statistics in Medicine 9, Kleinbaum, D., Kupper, and Muller, P. (988). Applied Regression Analysis and Other Multivariable Methods. PWS-Kent, Boston, Mass., 2nd edition. Li, Y. and Tiwari, R. (2007). Comparing Trends in Age-Adjusted Cancer Rates Across Overlapping Regions or Time Intervals for the NCI SEER Program. Technical Report, Li, Y., Tiwari, R., Walters, K. and Zou, Z. (2007) A Weighted-Least-Squares Estimation Approach to Comparing Trends in Age-Adjusted Cancer Rates Across Overlapping Regions. Technical Report, yili/apc2.pdf Ries, L., Eisner, M., Kosary, C., Hankey, B., Miller, B., Clegg, L., Mariotto, A., Feuer, E. and Edwards, B. (2003). SEER Cancer Statistics Review, , National Cancer Institute. Bethesda, MD,

9 bimj header will be provided by the publisher 9 Tiwari, R., Clegg, L. and Zou, Z. (2006). Efficient interval estimation for age-adjusted cancer rates. Statistical Methods in Medical Research 5, Write where Appendix A: Derivation of ˆV, Ŵ, ˆΣ V ( V V 2 V 2 V 22 ) V m V ar(u,) m ( ) ( ) W W ; W 2 Σ Σ W ; Σ 2 (J+) (J+) 2 W 22 Σ (J+) (J+) 2 Σ 22 (J+) (J+) m j J Ti 2 V ar(d ji ) m Hence a consistent estimate is ˆV m m J j T 2 i d ji. Similarly, ˆV 2 ( ˆV 2,,..., ˆV 2,J ) m j J Ti 2 E(d ji ). where ˆV 2,j Ĉov(U,, U 0,j ) m m T i d ji. Also, V 22 ((V 22,jj )) with ˆV 22,jj Ĉov(U 0,j, U 0,j ) 0, j j ; m m d ji, j j. Next compute the estimate of W: W n V ar(u 2,) n s+n s+n is+ j J Ti 2 V ar(d 2ji ) n s+n is+ j J Ti 2 E(d 2ji ) so that Ŵ J n is+ j T i 2d 2ji Similarly, Ŵ2 (Ŵ2,,..., Ŵ2,J ) where Ŵ2,j Ĉov(U 2,, U 02,j ) n n is+ T id 2ji. Also, Ŵ 22 ((Ŵ22,jj )) with Ŵ22,jj 0, j j ; s+n n is+ d 2ji, j j Finally, the estimate of Σ is computed as follows: Σ Cov(U,, U 2, ) mn m J Cov T i d ji, mn j m J Cov mn j s+n T i ( d (NO) ji is+ j J T i d 2ij ) + d (O) ji, s+n J is+ j T i ( d (NO) 2ji ) + d (O) ji where d kji d (NO) jki +d (O) ji and the superscripts NO and O denote the non-overlapping and overlapping regions, respectively. Thus, ˆΣ m J mn is+ j T i 2d(O) ji. Let ˆΣ 2 (ˆΣ2,,..., ˆΣ ) 2,J where ˆΣ 2,j m mn is+ T id (O) ji. Also, ˆΣ 22 ((ˆΣ 22,jj )) with ˆΣ 22,j,j 0, j j m ; mn is+ d(o) ji, j j.

10 0 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance Appendix B: Computation of the MLEs Note that the MLEs of β 0jk and β k satisfy: e β 0jk i d kji ; i eβ k Ti n kji Ũ(β k ) ( ) i n kji T d jki i e β k Ti d j,i i eβ k Ti n kji T i kji j,i ( i d kji T ) ie β k Ti kj. d j i eβ k Ti n k.i T i kji i j d kj. [A kj ()A kj (0)] i d k.i T i 0; k, 2, where d kj. i d kji, d k.i j d kji, and A kj (a) i n kji T a i eβ k Ti for a 0,, 2. Since Ũ(β k ) is a monotonic function of β k, there is a unique solution to this equation. Let ˆβ k be the solution. This is the MLE of β k. We can use a Newton-Raphson method to obtain ˆβ k as follows. Let ˆβ k (l) be the estimate at the l th iteration, then the estimate at the (l + ) th iteration is given by ˆβ k (l + ) ˆβ k (l) [Ũ () ( ˆβ k (l))] Ũ( ˆβ k (l)), where Ũ () (β) is the first (partial) derivative of Ũ(b) with respect to b evaluated at b β. We stop iterating wh en ˆβ k (l + ) ˆβ k (l) < ε for some prespecified value of ε. Note that Ũ () (β k ) j,i j n jki T 2 i e β k Ti ( i d kji i eβ k Ti n kji ) ( ) ( i n kji t2 i eβ k ti i kji) d i eβ k ti n kji j,i j n jki t i e β k ti i n kji t ie β k ti ( 2 i eβ k ti n kji) ( i n kji t ie β k ti ) 2 ( i eβ k ti n kji) 2 j j ( i d n ) kji t2 i eβ k ti kj. i eβ k ti n kji j d kj. [A kj (2)(A kj (0)) ] j ( i n kji t ie β k ti ) 2 ( i eβ k ti n kji) 2 [(A kj ()) 2 (A kj (0)) ], k, 2. Substituting the MLE ˆβ k in place for β k in e β 0kj i d kji i eβ k t i gives the MLE ˆβ ( ) n 0kj log i d kji ( kji ) log i e ˆβ k t i n kji.

11 bimj header will be provided by the publisher Table Comparison of the power functions under various hypotheses between the Poisson-based test Z P OIS and the weighted-least-squares based test Z W LS for two overlapping regions over disjoint time intervals, Region ( ) vs Region 2 ( ). APC and APC2 are the annual percent changes in Regions and 2, respectively. Cancer Sites AP C AP C 2 ARE Z W LS Z P OIS All Malignant Prostate Table 2 Comparison of the power functions between the Poisson based Z P OIS and the weighted-least-squares based Z W LS for two overlapping regions over roughly the same time intervals, Region ( ) vs Region 2 ( ). APC and APC2 are the annual percent changes in Regions and 2, respectively. Cancer Sites AP C AP C 2 ARE Z W LS Z P OIS All Malignant Prostate

12 2 Li, Tiwari, and Zou: Age-Stratified Poisson Model for Cancer Surveillance Table 3 Comparison of the power functions between the Poisson based Z P OIS and the weighted-least-squares based Z W LS for two overlapping regions over slightly overlapping time intervals, Region ( ) vs Region 2 ( ). APC and APC2 are the annual percent changes in Regions and 2, respectively. Cancer Sites AP C AP C 2 ARE Z W LS Z P OIS All Malignant Prostate Table 4 Comparing APC of Breast Cancer Mortality Between CA and the US CA US APC W LS SE W LS Z W LS (p ) APC P OIS SE P OIS Z P OIS (p )

13 bimj header will be provided by the publisher 3 Fig. Observed and fitted log-transformed age-adjusted breast cancer mortality rates in CA [ ] and US [ ]