# Analytical Studies: Methods and ReferencesZeno: A Tool for Calculating Confidence Intervals of Rates in HealthAnalytical Studies: Methods and ReferencesZeno: A Tool for Calculating Confidence Intervals of Rates in Health

by Philippe Finès and Gisèle Carriere

Release date: January 19, 2017

## 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 health-care service use. The variety of methods for calculating confidence intervals for these and other health-related 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 health-care 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 Chi-square. A second issue, in addition to rare events, is that analyses of recurrent events (such as hospitalizations) as opposed to non-recurrent 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 over-dispersion or zero-inflated 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 up-to-date 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.

Table 1
Formulas used in Zeno Tool
Table summary
This table displays the results of Formulas used in Zeno Tool. The information is grouped by Formula (appearing as row headers), Appropriate for one event per person or for all events?, Initially devised for rates or for numbers of events? and References (equations) (appearing as column headers).
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 age-standardized outcomes, the symbols become $ASR$ and $AS#E$. Within an age-stratum $i$, 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 (stratum-specific) size of the population under study; and $#NR\left(#NR_i\right)$ is the total (stratum-specific) size of the reference population, from which weights $w_i=#NR_i/#NR$ are computed.

Table 2
Symbols used for concepts in Zeno Tool
Table summary
This table displays the results of Symbols used for concepts in Zeno Tool. The information is grouped by Concept (appearing as row headers), Sizes and weights, Number of events and Rates (appearing as column headers).
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
Age-standardized 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]
Age-standardized outcomes Note ...: not applicable [AS#E] [ASR]

In cases where figures represent both single events for some persons and recurrent, non-independent 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,  reads “the size $\left[#\right]$ of the set  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 $j\text{ }\text{ }\text{ }\text{ }\left[C_k\left(n\right)=j\right]$.” 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 ; 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:

which, using the notations of Table 2, gives, for example:

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 age-standardized rate is a weighted sum of the age stratum-specific rates.

(where the weights $w_i$ are equal to $#NR_i/#NR$ ). If in this formula, instead of weights $w_i$, “pseudo-weights” $w’_i$ (each equal to $#N_i/#N$ ), are used, the result is:

This shows that $r$ (crude rate) can be expressed as a weighted sum of stratum-specific rates; in other words, $r$ is the age-standardized rate obtained when using “pseudo-weights.” 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, stratum-specific 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 3-1 to 3-4, 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:  for $j$ =1 up to 14 (because up to 14 events per person are allowed in the Tool).
Table 3-1
Essential components of the Zeno Tool main Excel sheet, by row and column, with color codes — Part I
Table summary
This table displays the results of Essential components of the Zeno Tool main Excel sheet. The information is grouped by Rows (appearing as row headers), Columns A-C, Column D, Column E, Column F, Columns G-T, Column V, Columns W-Z, Columns AI-AL, Column AM and Column AW (appearing as column headers).
Rows Columns A-C Column D Column E Column F Columns G-T Column V Columns W-Z Columns AI-AL 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=k-hat (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])
Table 3-2
Essential components of the Zeno Tool main Excel sheet, by row and column, with color codes — Part 2
Table summary
This table displays the results of Essential components of the Zeno Tool main Excel sheet. The information is grouped by Rows (appearing as row headers), Columns AX-BA, Columns BB-BE, Columns BF-BI, Columns BM-BP, Columns BQ-BT and Columns BU-BX (appearing as column headers).
Rows Columns AX-BA Columns BB-BE Columns BF-BI Columns BM-BP Columns BQ-BT Columns BU-BX
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 AX44-BA44=CI limits, CI width, CV for r using Poisson distribution, exact formula BB44-BE44=CI limits, CI width, CV for r using Poisson distribution, normal approximation BF44-BL44=estimate of r, CI limits of ln(r), CI limits, CI width, CV for r using Poisson distribution, Lognormal transformation formula BM44-BP44=CI limits, CI width, CV for r using Binomial distribution BQ44-BT44=CI limits, CI width, CV for r using Binomial distribution, normal approximation BU44-BX44=CI limits, CI width, CV for r using Binomial distribution, normal approximation
45 Empty Empty Empty Empty Empty Empty
47 AX47-BA47=CI limits, CI width, CV for ASR using Poisson distribution, exact formula BB47-BE47=CI limits, CI width, CV for ASR using Poisson distribution, normal approximation BF47-BL47=estimate of ASR, CI limits of ln(ASR), CI limits, CI width, CV for ASR using Poisson distribution, Lognormal transformation formula BM47-BP47=CI limits, CI width, CV for ASR using Binomial distribution BQ47-BT47=CI limits, CI width, CV for ASR using Binomial distribution, normal approximation BU47-BX47=CI limits, CI width, CV for ASR using Binomial distribution, normal approximation
50 AX50-BA50=CI limits, CI width, CV for [r] using Poisson distribution, exact formula BB50-BE50=CI limits, CI width, CV for [r] using Poisson distribution, normal approximation BF50-BL50=estimate of [r], CI limits of ln([r]), CI limits, CI width, CV for [r] using Poisson distribution, Lognormal transformation formula BM50-BP50=CI limits, CI width, CV for [r] using Binomial distribution BQ50-BT50=CI limits, CI width, CV for [r] using Binomial distribution, normal approximation BU50-BX50=CI limits, CI width, CV for [r] using Binomial distribution, normal approximation
53 AX53-BA53=CI limits, CI width, CV for [ASR] using Poisson distribution, exact formula BB53-BE53=CI limits, CI width, CV for [ASR] using Poisson distribution, normal approximation BF53-BL53=estimate of [ASR], CI limits of ln([ASR]), CI limits, CI width, CV for [ASR] using Poisson distribution, Lognormal transformation formula BM53-BP53=CI limits, CI width, CV for [ASR] using Binomial distribution BQ53-BT53=CI limits, CI width, CV for [ASR] using Binomial distribution, normal approximation BU53-BX53=CI limits, CI width, CV for [ASR] using Binomial distribution, normal approximation
Table 3-3
Essential components of the Zeno Tool main Excel sheet, by row and column, with color codes — Part 3
Table summary
This table displays the results of Essential components of the Zeno Tool main Excel sheet. The information is grouped by Rows (appearing as row headers), Columns BY-CF, Columns CG-CJ, Columns CM-CO, Columns CP-CR, Columns CS-CU and Columns CV-CX (appearing as column headers).
Rows Columns BY-CF Columns CG-CJ Columns CM-CO Columns CP-CR Columns CS-CU Columns CV-CX
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 BY44-CF44=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for r using Compound Poisson distribution CG44-CJ44=CI limits, CI width, CV for r using Negative Binomial assumption CM44-CO44=CI limits, CI width for #E using Poisson distribution, exact formula CP44-CR44=CI limits, CI width for #E using Poisson distribution, normal approximation CS44-CU44=CI limits, CI width for #E using Poisson distribution, Lognormal transformation formula CV44-CX44=CI limits, CI width for #E using Binomial distribution
45 Empty Empty Empty Empty Empty Empty
47 BY47-CF47=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for ASR using Compound Poisson distribution CG47-CJ47=CI limits, CI width, CV for ASR using Negative Binomial assumption CM47-CO47=CI limits, CI width for AS#E using Poisson distribution, exact formula CP47-CR47=CI limits, CI width for AS#E using Poisson distribution, normal approximation CS47-CU47=CI limits, CI width for AS#E using Poisson distribution, Lognormal transformation formula CV47-CX47=CI limits, CI width for AS#E using Binomial distribution
50 BY50-CF50=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for [r] using Compound Poisson distribution Empty CM50-CO50=CI limits, CI width for [#E] using Poisson distribution, exact formula CP50-CR50=CI limits, CI width for [#E] using Poisson distribution, normal approximation CS50-CU50=CI limits, CI width for [#E] using Poisson distribution, Lognormal transformation formula CV50-CX50=CI limits, CI width for [#E] using Binomial distribution
53 BY53-CF53=Calculations related to Compound Poisson distribution, CI limits, CI width, CV for [ASR] using Compound Poisson distribution Empty CM53-CO53=CI limits, CI width for [AS#E] using Poisson distribution, exact formula CP53-CR53=CI limits, CI width for [AS#E] using Poisson distribution, normal approximation CS53-CU53=CI limits, CI width for [AS#E] using Poisson distribution, Lognormal transformation formula CV53-CX53=CI limits, CI width for [AS#E] using Binomial distribution
Table 3-4
Essential components of the Zeno Tool main Excel sheet, by row and column, with color codes — Part 4
Table summary
This table displays the results of Essential components of the Zeno Tool main Excel sheet. The information is grouped by Rows (appearing as row headers), Columns CY-DA, Columns DB-DD, Columns DE-DG, Columns DH-DJ and Column DK (appearing as column headers).
Rows Columns CY-DA Columns DB-DD Columns DE-DG Columns DH-DJ 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*(1-p_i)): criterion used for validity of test based on Binomial distributionThis cell is for verification.
33 Empty Empty Empty Empty Empty
44 CY44-DA44=CI limits, CI width for #E using Binomial distribution, normal approximation DB44-DD44=CI limits, CI width for #E using Binomial distribution, normal approximation DE44-DG44=CI limits, CI width for #E using Compound Poisson distribution DH44-DJ44=CI limits, CI width for #E using Negative Binomial assumption DK44= min(np,n(1-p)): criterion used for validity of test based on Binomial distribution
45 Empty Empty Empty Empty Empty
47 CY47-DA47=CI limits, CI width for AS#E using Binomial distribution, normal approximation DB47-DD47=CI limits, CI width for AS#E using Binomial distribution, normal approximation DE47-DG47=CI limits, CI width for AS#E using Compound Poisson distribution DH47-DJ47=CI limits, CI width for AS#E using Negative Binomial assumption DK47= criterion used for validity of test based on Binomial distribution
50 CY50-DA50=CI limits, CI width for [#E] using Binomial distribution, normal approximation DB50-DD50=CI limits, CI width for [#E] using Binomial distribution, normal approximation DE50-DG50=CI limits, CI width for [#E] using Compound Poisson distribution Empty DK50= criterion used for validity of test based on Binomial distribution
53 CY53-DA53=CI limits, CI width for [AS#E] using Binomial distribution, normal approximation DB53-DD53=CI limits, CI width for [AS#E] using Binomial distribution, normal approximation DE53-DG53=CI limits, CI width for [AS#E] using Compound Poisson distribution Empty DK53= criterion used for validity of test based on Binomial distribution

## 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.

Table 4
Pivot table extracted from Zeno Tool
Table summary
This table displays the results of Pivot table extracted from Zeno Tool. The information is grouped by Row labels (appearing as row headers), Estimate, Exact CI (Poisson), CI (Poisson Lognormal) and CI (Compound Poisson) (appearing as column headers).
Row labels Estimate Exact CI (PoissonTable 4 Note 1) CI (Poisson LognormalTable 4 Note 2) CI (Compound PoissonTable 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

## 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 over-dispersion or zero-inflated 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 Low-Incidence 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 injury-related mortality rates.” Epidemiologic Perspectives & Innovations 4 (1). DOI: 10.1186/1742-5573-4-1.

van den Broek, J. 1995. “A score test for zero inflation in a Poisson distribution.” Biometrics 51 (2): 738–743.

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