Content

 Rates

 Rates

 Poisson rate confidence interval

 Compare two crude rates

 Standardize and compare two rates

 Direct standardization

 Indirect standardization and SMR

Rates. 1

Poisson rate confidence interval. 2

Compare two crude rates. 3

Standardize and compare two rates. 5

Direct standardization. 10

Indirect standardization and SMR.. 13

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Rates

·Confidenceinterval for a Poisson rate or count

·Comparetwo crude rates

·Standardizeand compare two rates

·Direct standardization

·Indirect standardization

Menu location: Analysis_Rates.

This section constructsconfidence intervals, tests and standardization methods for event rates such asdisease incidence or mortality rates.

For uncommon events, the Poissondistribution is usually appropriate, and most of the methods in this section useexact approaches to the Poissondistribution.

For common events, specificallywere the event rate r can not be considered small in comparison with 1-r, youshould use binomialdistribution methods. The direct standardization and comparison ofstandardized rates functions give you alternative binomial results. For asingle large rate you can use the singleproportion method as an alternative to the Poisson rate or count methodabove. For the comparison of two large rates you can use the prospective riskfunction as an alternative to the comparison of two crude rates given above.

See also incidence ratemeta-analysis.

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Poisson rateconfidence interval

Menu location: Analysis_Rates_Poisson Rate CI.

Uncommon events in populations,such as the occurrence of specific diseases, are usefully modelledusing a Poissondistribution. A common application of Poisson confidence intervals is to incidencerates of diseases (Gail and Benichou,2000; Rothman and Greenland, 1998; Selvin, 1996).

The incidence rate is estimatedas the number of events observed divided by the time at risk of event duringthe observation period.

Technical validation

Exact Poisson confidence limitsfor the estimated rate are found as the Poisson means, for distributions withthe observed number of events and probabilities relevant to the chosenconfidence level, divided by time at risk. The relationship between the Poissonand chi-square distributions is employed here (Ulm, 1990):

- where Y is the observed numberof events, Yl and Yu are lower and upper confidencelimits for Y respectively, c²n,a is the chi-square quantile for upper tailprobability a on n degrees of freedom.

Example

Say that 14 events are observedin 200 people studied for 1 year and 100 people studies for 2 years.

The person time at risk is 200 +100 x 2 = 400 person years

For this example:

Events observed = 14

Time at risk of event = 400

Poisson (e.g. incidence) rateestimate = 0.035

Exact 95% confidence interval =0.019135 to 0.058724

Here we can say with 95%confidence that the true population incidence rate for this event lies between0.02 and 0.06 events per person year.

See also incidence ratecomparisons

confidenceintervals

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Compare twocrude rates

Menu location: Analysis_Rates_Compare Two Crude Rates.

Person-time data from prospectivestudies of two groups with different exposures may be expressed as a differencebetween incidencerates or as a ratio of incidence rates.

This function constructsconfidence intervals for incidence rate differences and ratios where there aretwo exposures (i.e. exposed or not exposed, defined according to certain riskfactors) (Sahaiand Kurshid, 1996).

Data input

The use of person-time as opposedto just "time" enables you to handle situations where there aredrop-outs in a study or where you have not been able to follow an entire cohortat risk to watch for the development of the outcome under investigation. Thus,the follow up period does not have to be uniform for all participants.Person-time for a group is the sum of the times of follow up for eachparticipant in that group.

Technical validation

Poisson distribution andtest-based methods are used to construct the confidence intervals (Sahai and Kurshid,1996):

- where IRD hat and IRR hat arepoint estimates of incidence rate difference and ratio respectively, m is thetotal number of events observed, PT is the total person-time observed, Z is a quantile of the standard normal distribution and F is a quantile of the F distribution (denominator degrees offreedom are quoted last).

The optional conditional maximumlikelihood analysis for the rate ratio employs the polynomial multiplicationmethod described by Martin and Austin(1996); this also provides mid-P estimates.

Example

From Stampfer et al.(1985).

For this example:

Exposed incidence rate = 0.000552

Non-exposed incidence rate =0.001166

Incidence rate difference =-0.000613

Approximate 95% confidenceinterval = -0.000965 to -0.000261

Chi-square = 11.678635 P = .0006

Incidence rate ratio = 0.473934

Exact 95% confidence interval =0.295128 to 0.746416

Conditional maximum likelihoodestimate of rate ratio = 0.473934

Exact Fisher 95% confidenceinterval = 0.295128 to 0.746416

Exact Fisher one sided P =0.0004, two sided P = 0.0007

Exact mid-P 95% confidenceinterval = 0.302362 to 0.730939

Exact mid-P one sided P = 0.0003,two sided P = 0.0006

Here we may conclude with 95%confidence that the true population value for the difference between the twoincidence rates lies somewhere between -0.001 and 0.0003. We may also concludewith 95% confidence that the incidence rate for those who used postmenopausalhormones in the circumstances of the study was between 0.30 and 0.75 of thatfor those who did not take post- menopausal hormones.

P values

confidenceintervals

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Standardizeand compare two rates

Menu location: Analysis_Rates_Standardize & Compare TwoRates

This function calculates directlystandardized rates (DSR) for two study populations, and then compares the DSRs as a rate ratio. Stratum-specific rates are comparedalso.

DSR is simply a weighted meanevent rate for a study population, using the group/stratum sizes of a referencepopulation as the weighting scheme. Standardized or adjusted rates are summaryindex measures for the purpose of comparison only; their magnitude has nointrinsic value.

The choice of a reference orstandard population is important; it must relate to the population under studynaturally.

Please note that standardizationis not a substitute for individual comparisons of stratum-specificrates. This function produces a plot of stratum-specific rate ratios inaddition to comparing the standardized rates.

Direct standardization is notappropriate if there is not a consistent relationship between stratum-specificrates in different populations being compared. There are pitfalls in usingdirectly standardized rates; if you have any doubts then please consult with anEpidemiologist and/or Statistician.

Some of the methods used here areunreliable with small numbers; generally, there should be at least 25 eventsobserved overall and at least one event in each stratum. If the number ofevents is small, consider aggregating strata.

Note than an alternative binomialmethod is provided for situations where your observed rates are too large forthe Poisson distribution to be used, namely one or more rates r are notso small that 1-r can be considered almost equal to 1.

See also:

Directstandardization

Comparison oftwo crude rates

Poissonrate confidence interval

- whichprovide some of the calculations given here either in more detail or with moreoptions.

Data input

·Number of events for each group from index/study populations a and b

·Person-time for each group from index/study populations a and b(e.g. size of each group if just one year observed and all subjects followedup)

·Group sizes or weights from a reference/standard population

·Group/stratum labels, e.g. age bands

Technical validation

Exact Poisson confidence limitsfor the crude rates in both study populations are found as the Poisson means,for distributions with the observed number of events and probabilities relevantto the chosen confidence level, divided by time at risk. The relationshipbetween the Poisson and chi-square distributions is employed here (Ulm, 1990):

- where Y is the observed numberof events, Yl and Yu are lower and upper confidencelimits for Y respectively, c²n,a is the chi-square quantile for upper tailprobability a on n degrees of freedom.

The two crude rates are comparedas a ratio using Poisson distribution and test-based methods (Sahai and Kurshid,1996):

- where IRD hat and IRR hat arepoint estimates of incidence rate difference and ratio respectively, m is thetotal number of events observed, PT is the total person-time observed, Z is a quantile of the standard normal distribution and F is a quantile of the F distribution (denominator degrees offreedom are quoted last).

Approximate confidence intervalsfor the DSR are calculated firstly by Chiang's normal approximation to Poissonrate sums (Chiang,1961; Keyfitz, 1966; Breslow and Day, 1987; Armitage and Berry, 1994) andsecondly by an improved approximation adjusted for the total number of observedevents (Dobson etal., 1991).

- where v is theapproximate (Chiang) variance, wi is thereference weight for the ith stratum, ri is the observed study rate for the ith stratum, Ni is the reference population size forthe ith stratum, yiis the number of events observed in the ith stratumof the study population, ni is the person-timefor the ith stratum of the study population, za/2 is the (100 * a/2) the centile of the standard normaldistribution, Y is the total number of events observed, Yl and Yu are the exact lower and upperconfidence limits for the Poisson count Y and ICI l to u is theimproved confidence interval due to Dobson et al. For large rates, the binomialvariance is used, where r(1-r) issubstituted for r in the variance formula above.

Approximate confidence intervalsfor standardized rate ratios are calculated as follows (Newman, 2001;Armitage et al., 2002):

- where SRR is thestandardized rate ratio, var(log SRR) is theapproximate variance of the natural logarithm of SRR, DSR and v are thedirectly standardized rate and its variance as above, za/2 is the (100 * a/2) the centile of the standard normaldistribution, and CI is the approximate confidence interval for SRR. Forlarge rates, the binomial variance is used, where r(1-r)is substituted for r in the variance formulae above.

Example

From Newman (2001) p 254:

Test workbook (Rates worksheet:d1, pt2, d2, pt2, ref, age strata).

The following data relate to aretrospective cohort study of 2122 males who received treatment forschizophrenia in the province of Alberta, Canada during 1976-1985. Thestandard/reference population was taken as the Alberta general population in 1981.

To analysethese data in StatsDirect you must select Standardize& Compare Two Rates from the rates section of the analysis menu. Note thatannual mortality rates are often expressed as rates per 100,000 population orunits of person time (i.e. 100,000 person years); .

For this example:

Comparisonof two directly standardized rates

Analysis model for rates: Poisson(small rates)

Rates are expressed per 1,000units of person time:

Crude rate exposed = 15.430094

Exact 95% CI = 13.313979 to17.786968

Crude rate not exposed = 7.608293

Exact 95% CI = 7.434549 to7.785072

Standardized rate exposed =17.616898

Approximate 95% CI = 14.217636 to21.016159

Standardized rate not exposed =7.608293

Approximate 95% CI = 7.433557 to7.783028

Standardized rate ratio =2.315486

Approximate 95% CI = 1.906565 to2.812113

confidenceintervals

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Directstandardization

Menu location: Analysis_Rates_Direct Standardization.

This function calculates directlystandardized rates (DSR) with approximate confidence intervals.

DSR is simply a weighted meanevent rate for a study population, using the group/stratum sizes of a referencepopulation as the weighting scheme. Standardized or adjusted rates are summaryindex measures for the purpose of comparison only; their magnitude has nointrinsic value.

The choice of a reference orstandard population is important; it must relate to the population under studynaturally.

Please note that standardizationis not a substitute for individual comparisons of stratum-specificrates.

This method is unreliable withsmall numbers; there should be at least 25 events observed overall and at leastone event in each stratum. If the number of events is small, consideraggregating strata.

Direct standardization is notappropriate if there is not a consistent relationship between stratum-specificrates in different populations being compared.

There are a lot of pitfalls inusing directly standardized rates; if you have any doubts then please consultwith an Epidemiologist and/or Statistician.

Data input

·Number of events for each group from the index/study population

·Person-time for each group from the index/study population (e.g.size of each group if just one year and all subjects were followed up)

·Group sizes or weights from a reference/standard population

·Group/stratum labels, e.g. age bands

Note than an alternative binomialmethod is provided for situations where your observed rates are too large forthe Poisson distribution to be used, namely one or more rates r are notso small that 1-r can be considered almost equal to 1.

Technical validation

Approximate confidence intervalsfor the DSR are calculated firstly by Chiang's normal approximation to Poissonrate sums (Chiang,1961; Keyfitz, 1966; Breslow and Day, 1987; Armitage and Berry, 1994) andsecondly by an improved approximation adjusted for the total number of observedevents (Dobson etal., 1991).

- where v is the approximate(Chiang) variance, wi is the reference weight for theith stratum, ri is theobserved study rate for the ith stratum, Ni is thereference population size for the ith stratum, yi is the number of events observed in the ith stratum of the study population, niis the person-time for the ith stratum of the studypopulation, Za/2 is the (100 * a/2) the centile of the standard normaldistribution, Y is the total number of events observed, Yland Yu are the exact lower and upper confidence limits for the Poisson count Yand ICI l to u is the improved confidence interval due to Dobson et al. Forlarge rates, the binomial variance is used, where r(1-r)is substituted for r in the variance formula above.

Example

From Curtin and Klein(1995):

Test workbook (Rates worksheet:Age Bands, Index Events, Index Group Sizes, Reference Sizes).

The following data relate tostroke deaths for males from a hypothetical medium-size US State. The referencepopulation is the 1940 US Standard Million.

To analysethese data in StatsDirect you must select directstandardization from the rates section of the analysis menu. Note that annualmortality rates are often expressed as rates per 100,000 population or units ofperson time (i.e. 100,000 person years); so a multiplier of 100,000 should beselected for the scaling of rates in the output - you are prompted to providethis.

For this example:

Directly Standardized Rates

Rates are expressed per 100,000units of person time:

Total events = 1019

Adjusted events = 766.55342

Rates are expressed per 100,000units of person time:

Crude rate = 43.809114

Adjusted rate R = 32.955865

Any rates (binomial model)

Approximate standard error of R =1.050864

Approximate 95% confidenceinterval = 30.89621 to 35.01552

Small rates (Poisson model)

Approximate standard error of R =1.053213

Approximate 95% confidenceinterval = 30.891605 to 35.020125

Improved approximate (Dobson) 95%confidence interval = 30.923031 to 35.085216

confidenceintervals

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Indirectstandardization and SMR

Menu location: Analysis_Rates_Indirect Standardization & SMR.

This function calculatesstandardized mortality ratios (SMR) with exact confidence intervals.

Indirect standardization is usedto calculate the expected mortality rate for the index population, given agespecific mortality rates from a reference population.

The method applies not only tomortality rates but also to any rates of uncommon events (i.e. the Poissondistribution can be applied). An example of a different application iscalculation of standardized incidence rate ratio (SIR) for a disease in apopulation.

If you want to standardize byboth age and sex then enter two sets of age groups (i.e. 10 rows instead of 5for the example below) split into male and female consecutively. Provided thatyou have sex and age specific mortality rates for the reference population andage and sex specific population sizes for the index population then you canproduce an age and sex standardized SMR.

SMRs from different index/study populations are not strictly comparablebecause they are calculated using different weighting schemes that depend uponthe age structures of the index/study populations. SMRscan, however, be compared if you make the assumption that the ratio of ratesbetween index and reference populations is constant; this is similar to theassumption of proportional hazards in Cox regression (Armitage and Berry,1994). Directstandardization is an alternative to indirect standardization that doesprovide comparable measures. Direct standardization, however, has otherweaknesses, such as greater susceptibility than the indirect method to errorwith small numbers, which make the choice of method a matter for carefulstatistical consideration. If you are in doubt, please consult with aStatistician and/or Epidemiologist.

Please note that methods ofstand医学招聘网ardization can mask differences in rates between populations; in order toavoid this you should supplement SMR analysis with individual comparisons ofgroup (e.g. age) specific rates.

Data input

·Groups/strata, e.g. age bands for indirect age standardization

·Rates for each group from a reference population

·Person-time for each group from the index/study population (e.g.size of each group if just one year and all subjects were followed up)

·Multiplier for reference rates, e.g. 10000 if mortality entered asdeaths per 10000, 1 if mortality entered as a decimal fraction

The SMR is expressed in ratio andinteger (ratio * 100) formats with a confidence interval.

A test based on the nullhypothesis that the number of observed and expected deaths is equal is alsogiven. This test uses a Poisson distribution to calculate probability (Armitage and Berry,1994, Bland, 2000; Gardner and Altman, 1989).

Technical validation

The confidence intervals arecalculated by the exact Poisson method of Owen, this gives better coverage thanthe frequently quoted Vandenbroucke approximation orother asymptotic methods (Ulm, 1990;Greenland, 1990).

- where LL and UL are lower andupper confidence limits respectively for the SMR, c² n,a is the (100*a)th chi-square centilewith n degrees of freedom, d is the number of observed deaths, e is thenumber of expected deaths, ni is the person-time forthe ith study group stratum and Riis the reference population rate for the ith stratum.

Example

From Bland (2000).

Test workbook (Rates worksheet:Reference Mortality, Index Group Sizes, Age Groups).

The following data represent theage-specific mortality rates for liver cirrhosis in men and the number of maledoctors in each age stratum:

To analysethese data in StatsDirect you must select indirectstandardization and SMR from the rates section of the analysis menu. Enter themortality rate and group size for each age group. Note that group size refersto the study sample of doctors and not to the male population used to derivemortality data. Enter the mortality denominator as 1000000. Enter the observeddeaths as 14.

For this example:

Indirectly Standardized Ratesand SMR

Standardized Mortality Ratio(SMR) = 3.113463

SMR (*100 as integer) = 311

Exact 95% confidence interval =1.702159 to 5.223862 (170 to 522)

Probability of observing 14 ormore deaths by chance P = .0002

Probability of observing 14 orfewer deaths by chance P > .9999

Here we can see that the totalexpected deaths from liver cirrhosis in male doctors is4.5 per year. The observed number, 14, was statistically highly significantlygreater than expected. With 95% confidence we can state that male doctors inthis country exhibit between 1.7 and 5.2 times the number of deaths from livercirrhosis than expected from the general male population of a similar agedistribution.

P values

confidenceintervals