Analytical Studies: Methods and References
Zeno: A Tool for Calculating Confidence Intervals of Rates in Health
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by Philippe Finès and Gisèle Carriere
Health Analysis Division, Statistics Canada
Acknowledgements
The authors thank Claude Nadeau for his comments on a preliminary version of this paper.
Abstract
Hospitalization rates are among commonly reported statistics related to healthcare service use. The variety of methods for calculating confidence intervals for these and other healthrelated rates suggests a need to classify, compare and evaluate these methods. Zeno is a tool developed to calculate confidence intervals of rates based on several formulas available in the literature. This report describes the contents of the main sheet of the Zeno Tool and indicates which formulas are appropriate, based on users’ assumptions and scope of analysis.
Keywords: Confidence intervals, health, hospitalization, rates
1. Introduction
Hospitalization rates are among commonly reported statistics related to healthcare service use. A recurrent problem encountered by health analysts is the choice of a method to compute confidence intervals (CIs) for these rates.
When hospital events are independent of one another (only one event is allowed per person; for example, a discharge) and when the rate is neither extremely low nor high, the CI can be computed using an approximation based on the normal distribution. However, circumstances different from these conditions often exist. For example, unlike vital events such as births or deaths, for a given person, repeat events may occur, even if they have a low probability of occurrence. These events are not independent, a situation that has implications for the validity of using approximations based on the normal distribution to calculate. Also, an examination of an individual’s hospital visits might consider all visits, or only the first visit for a specific disease. Depending on the scope of analysis, different assumptions are made, and therefore, different methods are needed to calculate CIs.
When only one event per person is examined and the probability of occurrence of the event is low, exact calculations based on specific distributions are recommended. Frequently, a Binomial or Poisson distribution is assumed. Other techniques have been proposed (Glynn et al. 1993; Carriere and Roos 1997; Fay and Feuer 1997; Kegler 2007), which assume specific distributions such as Gamma or Chisquare. A second issue, in addition to rare events, is that analyses of recurrent events (such as hospitalizations) as opposed to nonrecurrent events (such as death) violate the Poisson assumption of independence (Carriere and Roos 1997).
The literature on CIs for rates that measure these different circumstances has expanded rapidly and can be daunting for researchers who must decide which formula to use in a specific context. However, few comparative studies of methods of calculating CIs for rates have been conducted. Typically, authors report their work (with their idiosyncratic notations) with little discussion of compatibility and comparability with the work of other researchers.
The objective of a previous version of this article was to catalogue all the methods available to compute CIs for rates. The challenges were numerous and included the reconciliation of notations between authors and the need to ensure that all pertinent literature had been taken into account. The task rapidly became unwieldy. Some papers described exact methods, while others presented approximations; some described how to compare rates (with rate differences or rate ratios); some focused on one event per person, whereas others allowed for multiple occurrences; some described formulas for crude rates, while others focused on standardized rates, etc. A systematic review of published formulas was needed. It became apparent that regrouping a family of formulas related to the general problem of calculating CIs for rates into a single tool would be useful.
Therefore, the original objective shifted from the development of a catalogue of formulas to the development of a tool that enables researchers to view the effects of applying one or another of several different methods. It addresses the calculation of CIs for rates, but omits formulas for hypothesis testing on rates (such as comparison between groups, or assessment of overdispersion or zeroinflated distributions [van den Broek 1995]). This tool (worksheet), Zeno, is now available upon request. Since rates and number of events are related, CIs for both metrics are available. Although users have access to several formulas for CIs, the most appropriate formula is not necessarily the one that yields the narrowest CI. On the contrary, the most appropriate formula is the one that satisfies the conditions of assumptions used to describe event distribution; calculations are derived based on these assumptions.
The objective of this article is to describe the Zeno Tool. The next three sections present: the references for the original formulas (Section 2); notations used in the Tool and in this article (Section 3); and the contents of the main sheet of the Tool (Section 4). The Data and results section (Section 5) contains a pivot table extracted from the Tool. The Conclusion (Section 6) summarizes the description and suggests potential enhancements.
2. Literature review
Although a systematic review of all papers on this domain was not conducted, the references used are uptodate and pertinent for the purpose of the Tool. The corollary is that, as mentioned before, notation varies in all references. To appreciate the intricacies of a given method and compare it with another, readers must mentally translate the notation used by one author into the terminology used by the other. For example, the concept of weight has surprisingly diverse definitions. The assumptions of the formulas selected for inclusion in the Tool are listed in Table 1. When the proportion of recurrent cases is relatively small, all the measures identified as “One event per person” can be used for the “All events” analyses.
Formula  Appropriate for one event per person or for all events?  Initially devised for rates or for numbers of events?  References (equations) 

Based on Poisson distribution  
Exact formula  One event per person  Numbers of events  (5) in Fay and Feuer (1997)^{Table 1 Note 1} (also [14] [15], [20], [21] in Daly [1992]^{Table 1 Note 2}) 
Normal approximation  One event per person  Numbers of events  (7) in Daly (1992) 
Lognormal transformation  One event per person  Rates  (1a) in Kegler (2007)^{Table 1 Note 3} 
Based on Binomial distribution  
Exact formula  One event per person  Rates  (4) and (5) in Daly (1992) 
Normal approximation  One event per person  Rates  (3) in Daly (1992) 
Normal approximation for small proportions  One event per person  Rates  (1.26) and (1.27) in Fleiss (1981)^{Table 1 Note 4} 
For analysis of all events  
Compound Poisson distribution  All events  Rates  (1b) in Kegler (2007) 
Negative Binomial assumption  All events  Rates  Glynn et al. (1993, p. 780)^{Table 1 Note 5} 

3. Notations and formulas
The main concepts and their symbols are listed in Table 2. When all events are considered, rates are denoted by $r$, and numbers of events are denoted by $\#E$. For agestandardized outcomes, the symbols become $ASR$ and $AS\#E$. Within an agestratum $i$, suffix $\_i$ is appended to the symbol. When only one event per person is considered (for example, the first event), the same notations are bracketed by “[“ and “]”, so that $r$ becomes $\left[r\right]$, $ASR$ becomes $\left[ASR\right]$, etc. In addition, other symbols are needed: $\#N\left(\#N\_i\right)$ is the total (stratumspecific) size of the population under study; and $\#NR\left(\#NR\_i\right)$ is the total (stratumspecific) size of the reference population, from which weights $w\_i=\#NR\_i/\#NR$ are computed.
Concept  Sizes and weights  Number of events  Rates 

Size of population  
By age stratum  #N_i  Note ...: not applicable  Note ...: not applicable 
Total over age strata  #N  Note ...: not applicable  Note ...: not applicable 
Size of reference population  
By age stratum  #NR_i  Note ...: not applicable  Note ...: not applicable 
Total over age strata  #NR  Note ...: not applicable  Note ...: not applicable 
Weights used for age standardization  
In each stratum  w_i = #NR_i /#NR  Note ...: not applicable  Note ...: not applicable 
Scope of events examined — method  
All events  
Crude outcomes (by age stratum i)  Note ...: not applicable  #E_i  r_i 
Crude outcomes (for all age strata combined)  Note ...: not applicable  #E  r 
Agestandardized outcomes  Note ...: not applicable  AS#E  ASR 
One event per person only  
Crude outcomes (by age stratum i)  Note ...: not applicable  [#E_i]  [r_i] 
Crude outcomes (by age strata combined)  Note ...: not applicable  [#E]  [r] 
Agestandardized outcomes  Note ...: not applicable  [AS#E]  [ASR] 
... not applicable 
In cases where figures represent both single events for some persons and recurrent, nonindependent events for others, a more complicated notation is needed. The notation must be able to describe the contents of cells that contain values that encode the number of persons, and also, the number of events experienced per given person. Using notation from Kegler (2007), $C\_k=h$ is the number of persons for whom the number of events was equal to $h$. In Kegler (2007), $h$ ranges from 1 to 2; in the Tool presented here, $h$ ranges from 1 to 14. Thus, $\#\left\{n:\text{}n\text{}in\text{}stratum\text{}i\text{}and\text{}C\_k\left(n\right)=j\right\}$ reads “the size $[\#]$ of the set $\left[\left\{\text{}\mathrm{..}\text{}\right\}\right]$ made up of all the persons $n$ who $\left[n:\right]$ belong to stratum $i$ and for whom the number of events is equal to $j\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\left[C\_k\left(n\right)=j\right]$.” In other words, the formula expresses the number of persons in stratum $i$ who had $j$ events. For consistency with the other notations in Table 2, $\#E\_j\_i$ denotes $\#\left\{n:\text{}n\text{}in\text{}stratum\text{}i\text{}and\text{}C\_k\left(n\right)=j\right\}$; by extension, $\#E\_0\_i$ denotes the number of persons in stratum $i$ with no event, and $\#E\_0$ denotes the number of persons overall with no event.
Results can be defined according to the intended metrics, namely, the rates or the number of events based on the population (specific age strata or totals) and the scope (all events or only one event per person). Rates and number of events are related according to the general formula:
$$Rate\text{}=\text{}Number\text{}of\text{}events\text{}/\text{}Size\text{}of\text{}population$$
which, using the notations of Table 2, gives, for example:
$$r\text{}=\text{}\#E/\#N$$
and
$$ASR=AS\#E/\#N.$$
Throughout, population sizes are considered fixed, as this is standard for several authors. Therefore, the rate or number of events can be calculated when the other metric is known. In fact, some of the formulas implemented in the Tool were initially introduced in referenced material to calculate the CIs for rates only (or number of events). The formulas corresponding to the other metric were then developed.
Although several of the referenced formulas focus on a specific calculation, it is possible to expand the formulas to apply to other cases. For example, from references that present the formulas for multiple visits, simplification to one visit per person is straightforward―only minor modifications of the original formulas are needed.
Also, an agestandardized rate is a weighted sum of the age stratumspecific rates.
$$ASR=\sum i\text{}w\_i*r\_i$$
(where the weights $w\_i$ are equal to $\#NR\_i/\#NR$ ). If in this formula, instead of weights $w\_i$, “pseudoweights” $w\u2019\_i$ (each equal to $\#N\_i/\#N$ ), are used, the result is:
$$\sum i\text{}w\u2019\_i*r\_i=\sum i\text{}\#N\_i/\#N\text{}*\text{}\#E\_i/\#N\_i=\sum i\text{}\#E\_i\text{}/\#N=\#E/\#N\text{}=r.$$
This shows that $r$ (crude rate) can be expressed as a weighted sum of stratumspecific rates; in other words, $r$ is the agestandardized rate obtained when using “pseudoweights.” This property has been used to convert the formula for CI of $ASR$ into a formula for CI of $r$ when the latter was not directly provided in the references.
To summarize, the Tool contains a complete series of formulas for the six metrics in the last two columns of Table 2, using the different assumptions in Table 1.
4. Contents of main sheet
The Zeno Tool is essentially a sheet (“main sheet”) that contains the data and the results. In fact, there may be as many “main sheets” as desired; the label of the sheet being analyzed must be indicated in sheet “Prep,” which reformats the results to produce the appropriate pivot charts and tables.
Cells of a main sheet are denoted by concatenation of column (letter) and row (number). The Excel sheet allows for 21 strata, for which data will be entered by the user on rows $i*$ =11 to 31.
As mentioned earlier, stratumspecific symbols are denoted by concatenation of symbol, “ $\_$ ” and row $i$ =1 to 21. Rows and strata are linked by this relation: $i*=i+10$. The essential components of any main sheet are described in Tables 31 to 34, where the identifier of the cell is described by what follows the “=” sign. For example: cell D11 contains the number of events for stratum 1; $r\_1$ is the rate for stratum 1. The only cells to be modified by users are:
 C1 and D1 to T1 (keyword and title)
 C2: unit of reference (denominator) for rates
 C3: Alpha level of CIs (for example, 0.05)
 area D11 to T31: respectively, for each age stratum $i$ =1 to 21: $\#E\_i,\text{}\#N\_i,\text{}\#NR\_i,\text{}\#E\_j\_i$ for $j$ =1 up to 14 (because up to 14 events per person are allowed in the Tool).
Rows  Columns AC  Column D  Column E  Column F  Columns GT  Column V  Columns WZ  Columns AIAL  Column AM  Column AW 

1..10  Labels  Labels  Labels  Labels  Labels  Labels  Labels  Labels  Labels  Labels 
i*=11..31  Labels  Di*= #E_iThis cell contains data.  Ei*= #N_iThis cell contains data.  Fi*= #NR_iThis cell contains data.  Gi*= #E_1_i, up to Ti*=#E_14_i (from Kegler [2007])This cell contains data.  Vi*=r_iThis cell contains estimates and intermediate results per stratum.  Wi*=w_iThis cell contains estimates and intermediate results per stratum.  ALi*=Gi + 2*Hi + 3*Ii + ... + 14*Ti (from Kegler [2007])This cell contains estimates and intermediate results per stratum.  AMi*= Gi + 4*Hi + 9*Ii + ... + 196*Ti (from Kegler [2007])This cell contains estimates and intermediate results per stratum.  AWi*= ln(r_i)This cell contains confidence intervals for rates per stratum. 
33  Labels  D33=#E  E33=#N  F33=#NR  G33=$\Sigma $ over i of #E_1_i, up to T33=$\Sigma $ over i of #E_14_i  Empty  W33=$\Sigma $ over i of w_i =1, Y33=ASR  AI33=[#E], AK33=[ASR]  Empty  Empty 
44  Labels  Empty  Empty  Empty  G44=G33*1, up to T44=T33*14  V44=r  Y44=AS#E  AJ44=[r], AK44=[AS#E]  Empty  AW44= ln(r) 
45  Labels  Empty  Empty  Empty  G45=G33*1^2, up to T45=G33*14^2  Empty  Z45=s^2 (from Glynn et al. [1993])  Empty  Empty  Empty 
47  Labels  Empty  Empty  Empty  Empty  Empty  Z47=khat (from Glynn et al. [1993])  Empty  Empty  AW47= ln(ASR) 
50  Labels  Empty  Empty  Empty  Empty  Empty  Empty  Empty  Empty  AW50= ln([R]) 
53  Labels  Empty  Empty  Empty  Empty  Empty  Empty  Empty  Empty  AW53= ln([ASR]) 
The notes are located at the bottom of Table 34. 
Rows  Columns AXBA  Columns BBBE  Columns BFBI  Columns BMBP  Columns BQBT  Columns BUBX 

1..10  Labels  Labels  Labels  Labels  Labels  Labels 
i*=11..31  AXi*BAi*=CI limits, CI width, CV for r_i using Poisson distribution, exact formulaThis cell contains confidence intervals for rates per stratum.  BBi*BEi*=CI limits, CI width, CV for r_i using Poisson distribution, normal approximationThis cell contains confidence intervals for rates per stratum.  BFi*BLi*=estimate of r_i, CI limits of ln(r_i), CI limits, CI width, CV for r_i using Poisson distribution, Lognormal transformation formulaThis cell contains confidence intervals for rates per stratum.  BMi*BPi*=CI limits, CI width, CV for r_i using Binomial distributionThis cell contains confidence intervals for rates per stratum.  BQi*BTi*=CI limits, CI width, CV for r_i using Binomial distribution, normal approximationThis cell contains confidence intervals for rates per stratum.  BUi*BXi*=CI limits, CI width, CV for r_i using Binomial distribution, normal approximationThis cell contains confidence intervals for rates per stratum. 
33  Empty  Empty  Empty  Empty  Empty  Empty 
44  AX44BA44=CI limits, CI width, CV for r using Poisson distribution, exact formula  BB44BE44=CI limits, CI width, CV for r using Poisson distribution, normal approximation  BF44BL44=estimate of r, CI limits of ln(r), CI limits, CI width, CV for r using Poisson distribution, Lognormal transformation formula  BM44BP44=CI limits, CI width, CV for r using Binomial distribution  BQ44BT44=CI limits, CI width, CV for r using Binomial distribution, normal approximation  BU44BX44=CI limits, CI width, CV for r using Binomial distribution, normal approximation 
45  Empty  Empty  Empty  Empty  Empty  Empty 
47  AX47BA47=CI limits, CI width, CV for ASR using Poisson distribution, exact formula  BB47BE47=CI limits, CI width, CV for ASR using Poisson distribution, normal approximation  BF47BL47=estimate of ASR, CI limits of ln(ASR), CI limits, CI width, CV for ASR using Poisson distribution, Lognormal transformation formula  BM47BP47=CI limits, CI width, CV for ASR using Binomial distribution  BQ47BT47=CI limits, CI width, CV for ASR using Binomial distribution, normal approximation  BU47BX47=CI limits, CI width, CV for ASR using Binomial distribution, normal approximation 
50  AX50BA50=CI limits, CI width, CV for [r] using Poisson distribution, exact formula  BB50BE50=CI limits, CI width, CV for [r] using Poisson distribution, normal approximation  BF50BL50=estimate of [r], CI limits of ln([r]), CI limits, CI width, CV for [r] using Poisson distribution, Lognormal transformation formula  BM50BP50=CI limits, CI width, CV for [r] using Binomial distribution  BQ50BT50=CI limits, CI width, CV for [r] using Binomial distribution, normal approximation  BU50BX50=CI limits, CI width, CV for [r] using Binomial distribution, normal approximation 
53  AX53BA53=CI limits, CI width, CV for [ASR] using Poisson distribution, exact formula  BB53BE53=CI limits, CI width, CV for [ASR] using Poisson distribution, normal approximation  BF53BL53=estimate of [ASR], CI limits of ln([ASR]), CI limits, CI width, CV for [ASR] using Poisson distribution, Lognormal transformation formula  BM53BP53=CI limits, CI width, CV for [ASR] using Binomial distribution  BQ53BT53=CI limits, CI width, CV for [ASR] using Binomial distribution, normal approximation  BU53BX53=CI limits, CI width, CV for [ASR] using Binomial distribution, normal approximation 
The notes are located at the bottom of Table 34. 
Rows  Columns BYCF  Columns CGCJ  Columns CMCO  Columns CPCR  Columns CSCU  Columns CVCX 

1..10  Labels  Labels  Labels  Labels  Labels  Labels 
i*=11..31  BYi*CFi*=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for r_i using Compound Poisson distributionThis cell contains confidence intervals for rates per stratum.  CGi*CJi*=CI limits, CI width, CV for r_i using Negative Binomial assumptionThis cell contains confidence intervals for rates per stratum.  CMi*COi*=CI limits, CI width for #E_i using Poisson distribution, exact formulaThis cell contains confidence intervals for the number of events per stratum.  CPi*CRi*=CI limits, CI width for #E_i using Poisson distribution, normal approximationThis cell contains confidence intervals for the number of events per stratum.  CSi*CUi*=CI limits, CI width for #E_i using Poisson distribution, Lognormal transformation formulaThis cell contains confidence intervals for the number of events per stratum.  CVi*CXi*=CI limits, CI width for #E_i using Binomial distributionThis cell contains confidence intervals for the number of events per stratum. 
33  Empty  Empty  Empty  Empty  Empty  Empty 
44  BY44CF44=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for r using Compound Poisson distribution  CG44CJ44=CI limits, CI width, CV for r using Negative Binomial assumption  CM44CO44=CI limits, CI width for #E using Poisson distribution, exact formula  CP44CR44=CI limits, CI width for #E using Poisson distribution, normal approximation  CS44CU44=CI limits, CI width for #E using Poisson distribution, Lognormal transformation formula  CV44CX44=CI limits, CI width for #E using Binomial distribution 
45  Empty  Empty  Empty  Empty  Empty  Empty 
47  BY47CF47=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for ASR using Compound Poisson distribution  CG47CJ47=CI limits, CI width, CV for ASR using Negative Binomial assumption  CM47CO47=CI limits, CI width for AS#E using Poisson distribution, exact formula  CP47CR47=CI limits, CI width for AS#E using Poisson distribution, normal approximation  CS47CU47=CI limits, CI width for AS#E using Poisson distribution, Lognormal transformation formula  CV47CX47=CI limits, CI width for AS#E using Binomial distribution 
50  BY50CF50=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for [r] using Compound Poisson distribution  Empty  CM50CO50=CI limits, CI width for [#E] using Poisson distribution, exact formula  CP50CR50=CI limits, CI width for [#E] using Poisson distribution, normal approximation  CS50CU50=CI limits, CI width for [#E] using Poisson distribution, Lognormal transformation formula  CV50CX50=CI limits, CI width for [#E] using Binomial distribution 
53  BY53CF53=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for [ASR] using Compound Poisson distribution  Empty  CM53CO53=CI limits, CI width for [AS#E] using Poisson distribution, exact formula  CP53CR53=CI limits, CI width for [AS#E] using Poisson distribution, normal approximation  CS53CU53=CI limits, CI width for [AS#E] using Poisson distribution, Lognormal transformation formula  CV53CX53=CI limits, CI width for [AS#E] using Binomial distribution 
The notes are located at the bottom of Table 34. 
Rows  Columns CYDA  Columns DBDD  Columns DEDG  Columns DHDJ  Column DK 

1..10  Labels  Labels  Labels  Labels  Labels 
i*=11..31  CYi*DAi*=CI limits, CI width for #E_i using Binomial distribution, normal approximationThis cell contains confidence intervals for the number of events per stratum.  DBi*DDi*=CI limits, CI width for #E_i using Binomial distribution, normal approximationThis cell contains confidence intervals for the number of events per stratum.  DEi*DGi*=CI limits, CI width for #E_i using Compound Poisson distributionThis cell contains confidence intervals for the number of events per stratum.  DHi*DJi*=CI limits, CI width for #E_i using Negative Binomial assumptionThis cell contains confidence intervals for the number of events per stratum.  DKi*= min(n_i*p_i,n_i*(1p_i)): criterion used for validity of test based on Binomial distributionThis cell is for verification. 
33  Empty  Empty  Empty  Empty  Empty 
44  CY44DA44=CI limits, CI width for #E using Binomial distribution, normal approximation  DB44DD44=CI limits, CI width for #E using Binomial distribution, normal approximation  DE44DG44=CI limits, CI width for #E using Compound Poisson distribution  DH44DJ44=CI limits, CI width for #E using Negative Binomial assumption  DK44= min(np,n(1p)): criterion used for validity of test based on Binomial distribution 
45  Empty  Empty  Empty  Empty  Empty 
47  CY47DA47=CI limits, CI width for AS#E using Binomial distribution, normal approximation  DB47DD47=CI limits, CI width for AS#E using Binomial distribution, normal approximation  DE47DG47=CI limits, CI width for AS#E using Compound Poisson distribution  DH47DJ47=CI limits, CI width for AS#E using Negative Binomial assumption  DK47= criterion used for validity of test based on Binomial distribution 
50  CY50DA50=CI limits, CI width for [#E] using Binomial distribution, normal approximation  DB50DD50=CI limits, CI width for [#E] using Binomial distribution, normal approximation  DE50DG50=CI limits, CI width for [#E] using Compound Poisson distribution  Empty  DK50= criterion used for validity of test based on Binomial distribution 
53  CY53DA53=CI limits, CI width for [AS#E] using Binomial distribution, normal approximation  DB53DD53=CI limits, CI width for [AS#E] using Binomial distribution, normal approximation  DE53DG53=CI limits, CI width for [AS#E] using Compound Poisson distribution  Empty  DK53= criterion used for validity of test based on Binomial distribution 
Notes: The meaning of the color codes used in the Excel sheet is as follows: orange: labels; light purple: data; light green: estimates and intermediate results per stratum; dark green: confidence intervals (CIs) for rates per stratum; pink: CI for number of events per stratum; dark purple: verification. The formulas mentioned in this table are explained in Table 1. The references for the citations indicated in this table are in Table 1. The meaning of the symbols is given in Table 2. CV: coefficient of variation. 
5. Data and results
The data used to test the Tool were originally gathered from administrative cancer databases in Manitoba. These data were chosen to illustrate the value of the Tool and to demonstrate a situation in which interest may focus on all the events (total rate of hospitalization for cancer including recurrences) or on only one event per person (rate of initial hospitalization for cancer among residents of the province).
Table 4 gives a sample of the results. Users can choose the significance threshold (for example, 95%, 90%)^{Note 1} of the CIs and specify the metrics (for example, $r$, $ASR$, $\left[r\right]$, $\left[ASR\right]$ ), the statistics (estimate, lower [L] and upper [U] limits of CIs), the methods (formulas in Table 1), as well as whether these are required globally or for specific strata. Users must be aware of the appropriateness of the method based on the scope, as shown in Table 1.
Row labels  Estimate  Exact CI (Poisson^{Table 4 Note 1})  CI (Poisson Lognormal^{Table 4 Note 2})  CI (Compound Poisson^{Table 4 Note 3}) 

r  
Estimate  1860.2  Note ...: not applicable  Note ...: not applicable  Note ...: not applicable 
L  Note ...: not applicable  1714.3  1851.9  1845.0 
U  Note ...: not applicable  2083.5  1868.5  1875.5 
ASR  
Estimate  1885.5  Note ...: not applicable  Note ...: not applicable  Note ...: not applicable 
L  Note ...: not applicable  1737.6  1877.1  1869.4 
U  Note ...: not applicable  2111.7  1893.8  1901.6 
[r]  
Estimate  923.7  Note ...: not applicable  Note ...: not applicable  Note ...: not applicable 
L  Note ...: not applicable  851.3  917.9  917.9 
U  Note ...: not applicable  1034.6  929.6  929.6 
[ASR]  
Estimate  964.9  Note ...: not applicable  Note ...: not applicable  Note ...: not applicable 
L  Note ...: not applicable  889.3  959.0  958.0 
U  Note ...: not applicable  1080.7  970.9  971.9 
... not applicable
Source: Statistics Canada, authors' calculations. Data originally from administrative cancer databases in Manitoba, but modified for illustrative purposes. 
6. Conclusion
The Zeno Tool regroups and extends formulas from several published sources. It presents, on a single sheet, the calculations proposed by different authors and allows users to compare the impact of using various methods on the resulting confidence intervals (CIs). The Tool also expands the referenced formulas. For example, in one reference, the CI for $ASR$ might be present, but not the CI for $r$; in another reference, the CI for $r$ might be present, but not the CI for $\left[r\right]$; or formulas for $r$ may exist, but not those for $\#E$. Thus, the Zeno Tool completes the set of formulas available for use in a broader range of circumstances that may exist in the data.
The Tool could be expanded to include features such as comparisons between groups, or tests on overdispersion or zeroinflated distributions. Implementation of modelling of rates could also be considered. However, caution should be exercised before introducing additional features. While these features would be useful, ease of use could be compromised. The Tool might become cumbersome because additional analyses, such as hypothesis tests, would require more columns or rows, which might not be needed in all situations. As presented, this generalized tool is applicable to a broad array of circumstances, yet flexible and adaptable to suit other specific needs.
References
Carriere, K.C., and L.L. Roos. 1997. “A Method of Comparison for Standardized Rates of LowIncidence Events.” Medical Care 35 (1): 57–69.
Daly, L. 1992. “Simple SAS macros for the calculation of exact binomial and Poisson confidence limits.” Computers in Biology and Medicine 22 (5): 351–361.
Fay, M.P., and E.J. Feuer. 1997. “Confidence intervals for directly standardized rates: A method based on the Gamma distribution.” Statistics in Medicine 16 (7): 791–801.
Fleiss, J.L. 1981. Statistical Methods for Rates and Proportions, 2nd Edition. New York: John Wiley and Sons Ltd.
Glynn, R.J., T.A. Stukel, S.M. Sharp, T.A. Bubolz, J.A. Freeman, and E.S. Fisher. 1993. “Estimating the Variance of Standardized Rates of Recurrent Events, with Application to Hospitalizations Among the Elderly in New England.” American Journal of Epidemiology 137 (7): 776–786.
Kegler, S.R. 2007. “Applying the compound Poisson process model to the reporting of injuryrelated mortality rates.” Epidemiologic Perspectives & Innovations 4 (1). DOI: 10.1186/1742557341.
van den Broek, J. 1995. “A score test for zero inflation in a Poisson distribution.” Biometrics 51 (2): 738–743.
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