Lifetable 
CMF Compare Options Compare Direct Standardization Discounting Example Fieller Fraction/Proportion lived by the death in a period General Options Index (population) Indirect Standardization Input License Life table Method of discounting Mortality Adjusting/Competing causes of death Options Output Field Output options PGLE References SMR Standard (population) YPPL and LYPLL Warranty Download Input for each of the two input fields consists of rows of age categories, which at least contain (1), an indicator of the age concerned for each category and (2), the size of the population for that category. If you want to analyze the data to obtain the life expectancy and YPLL also (3) the total number of death in each age category is required. Additionally, and optionally, one may add (4) the number of death in an age category for a specific cause of death, k., (5) the average proportion of time the death in an age category lived in that age category a_{i}, and (6) the same proportion for the death of cause k a_{i.k}. The age indicator gives the age up to the highest age concerned in the category. Thus, 0 contains the age group 0, 4 contains the age groups 1 up to and including age 4, 9 contains the ages 5 up to and including age 9, etc. In the last category you give the age from, thus 85 means death aged 85 and older. Total number of death is smaller than the number of people in the population; number death due to a specific cause is smaller than the total number of death. Proportion lived by the death in an age category is a small positive number in between the values zero and one. The data is default read rowwise, the program considers that each row of data concerns an age category. The data can also be read "free", the program reads the data and tries to make sense of it. You can select between rowwise and free in the input options tab sheet.
You can cut and paste the data into the input field from a spreadsheet or word processor or type it into the input field manually. You can also read data from a *.txt file into the visible current population field using the open option in the file menu. There are a number of readymade data sets available under the "read" button in the toolbar.
The program will ignore lines of text, which contain the words "Comment" or "comment" and lines that do not contain a number. Lines which contain the word "Title" or "title" will be read into the title array and these lines will appear in appropriate places in the output. Lines that contain the words "Mortality": the program will try to read a number from that line. Lines that contain the words "varianc": the program will try to read a number from that line too. Beware to not include nonnumerical symbols such as "e" or "E" in a line containing a number (i.e. e), that will cause confusion.
Please be careful with continental European style decimal commas for the fractions, they are not accepted as valid, should be decimal dots. Any character that does not have a numerical function will be considered to be a separator between two numbers. There is a maximum of 150 age categories. Characters with a numerical function are: '0'..'9','.','e','E','','+'. All other characters are considered to be nonnumerical. The tab setting sometimes causes the data in the input box to look strange. That is not serious. For an example of data see the example help page. TOP of pageThere are five tabsheets with options: 1) Input: see the input options help page 2) General: see the general options help page 3) Analyze: see the analyze options help page 4) Compare: see the compare options help page 5) Output: see the output options help page TOP of pageThis options page allows you to set instructions for how the compare analysis is done. If the "use cause k" option is not ticked the total number of death is used in comparing the two populations; if ticked the number of death due to cause k. is used. If a number of death due to cause k is not found for one or both of the populations the total number of death is used for the population concerned whether the box is ticked or not. If you tick "explain" you get a very basic explanation of what you find in the output. If you tick "apply age limits" the compare analysis is only done for the age groups specified in the general options page. If you want tests of statistical significance for the riskratio, CMF, SMR, or the difference in life expectancy, tick the "statistical significance" box. Tick "show CMF" to get the CMF analysis. To make a CMF analysis possible the program has to find age specific mortality in the index population. Tick "Rate in Standard's Field" if you want the directly standardized rate of an analysis and it's variance to be written to the population field of the standard population. To the standard population field is written the number of death that would occur in the standard population if it had the same age specific mortality as the index population. The accompanying variance is based on the size and structure of the index population and reweigthed to consider the size of the standard population. After replacing the index population with another index population then after calculating the standardized rate for the second population the program will take the ratio (CMF) of the standardized rates of the first and the second population. For example, if you want to compare Virginia with Iowa, or Italy with Spain, first calculate the standardized rates for Virginia or Italy using the US or EU standard populations respectively, save the standardized rates and their variances to the standard population's input field, then standardize the data for Iowa or Spain and the program will automatically take the ratio of the new and the previous standardized rate. Tick "show SMR" to get the SMR analysis. To make an SMR analysis possible the program has to find age specific mortality in the standard population. Tick "compare life expectancies" to compare the life expectancies of the two populations. First, the life expectancies are calculated on the full data, which will be different for the two populations if the age distributions differ. Note that this difference can be a cause of a difference in life expectancy, whereby the life expectancy is particularly sensitive to differences in the age structure of the oldest age groups. If the "statistical significance" box is ticked the statistical significance of the difference between life expectancies is shown; otherwise the confidence interval of the difference between the life expectancies is shown. If the "use cause k" option is ticked the life expectancies excluding death from cause k are compared, if no data on cause k is available the instruction is ignored for one or both of the populations. If the populations have a different age structure, then after the two populations have been given the same (shorter) age structure the difference in life expectancy is calculated again. In this latter case, if the "use cause k" option is ticked the analysis is not done. TOP of pageDenominator sets the radix (n) for a number of calculations such as numbers of person years per n population lost in the case of the YPLL and what happens to a birth cohort sized n in the case of the life table or the number death per n of the population etc.. Age limits. Default is from age 15 included up to but not including age 65. Set to any other value. Age limits are used in calculating the YPLL. You can limit comparing two populations to the age groups included in the age limits by ticking the "apply age limits" tick box in the compare options tab sheet. By clicking the "show global parameters" button you get an overview of the value of all kinds of variables. This button is there for the programmers to check the program but it might also be handy for the user. If you believe that something is not right in the results of your analysis and that the program is the cause of the problem, this is the button to push for a first inspection of what the program is doing. Confidence Intervals. Allows you to set the width of the confidence interval. Usually you would select a 95% confidence interval; this means that in 95% of research projects such as yours the sample mean will be within the stated interval. The alpha error is than 5%. Default the, according to SISA; most appropriate confidence interval will be given for the various statistics given the situation. If you tick the "show confidence intervals" box you get many more. TOP of pageData used gives for both compare and analyze the data that is used in the output. It concerns the first table shown in the output. Untick to not get the data used. Show summaries concerns the summaries and statistics given after each table. Untick to not get summaries.
Column numbers give the column number above the output data. These are not always consecutive; this is because the column numbers indicate the position in which the data is stored in the computer. The two width boxes set the way the data is presented in the output field. This might be handy in case you have a low resolution monitor or if you want to neatly print the data. To effectuate a different tab setting the page has to be reset using the reset button in the toolbar, you will loose any output that is already on the page. TOP of pageMethod for discounting is discussed on another page. Tick one of the analyze options to determine which method you want to use after having read the discussion. Include a_{i} considers fraction of life lived in the age category of death in the calculation of the YPLL and LYPLL. The default option is "Yes". In the case of "No" it is considered that the death all die at the beginning of each period, thus, using the "No" setting the death aged 14 all die on exactly their 14th birthday. In the case of the "Yes" setting they die on average at the ath fraction into their 14th life year, or, in general, into the age category concerned. Customarily this fraction is considered in calculating the YPLL, but not in calculating the LYPLL. Note that for the (L)YPLL all causes the first fraction in column 5 is used, for the (L)YPLL due to a specific cause k the second fraction in column 6 is used. Give the annual discount rate into the box "discount at". Default set at 1.5% per annum. If you want to only partially eliminate k as a cause of death give the elimination percentage into the box "eliminate k at". Default set at 100%, or full elimination, of cause k. Show determines which tables are presented in the output. Please note that also under the output tab sheet preferences with regard to the output can be set. Show wi gives you the weight or value of each additional life year from a particular age onwards. It gives the discounted weights in a single quite long table and the mortality adjusted weights in a large number of smaller tables. TOP of pageConsidering that death does not take place at a single moment in time at the beginning of a period, for example, at peoples birthday, in each period there is a fraction a_{i} of life lived by the death in that period. In life table analysis this fraction is often set at 0.5, considering that death takes place more or less equally spaced throughout an age period. For i=0 this fraction should be set at a_{0}=0.1, considering that in the first year of life deaths occur disproportionally more often in the beginning of the year, at birth. Two values for a_{i }can be given, one for the total number of death and one for the number of death due to a specific cause of death k. a_{i} is a fractional value, inbetween '0' and '1'. The fraction lived is used in both the calculation of the life table and the YPLL. Chiang CL. The life table and its construction. In: Chiang CL. Introduction to stochastic processes in Biostatistics. New York [etc.]: John Wiley, 1968. TOP of pageThe YPLL measures for a group of individuals the total number of years these people would have additionally lived up to some point in the future, would they not have died. Mostly, as the age up to which life is "gained", the life expectancy for a population or the age of productivity, from 15 to 65 or 70, are chosen. If the life expectancy is used as the cutoff point the terminology Lifetime Years of Potential Life Lost (LYPLL) is used, in the case of an age limit being used the term YPLL is used. Sometimes the YPLL is also named as the PYLL, Potential Years of Life Lost. It concerns exactly the same concept. An important methodological problem with the YPLL, but not the LYPLL, is the fact that this measure does not consider competing causes of death. If the death of a group of people is prevented the fact that these people can die from other causes before the upper cutoff point in calculating the YPLL is reached, is not considered. Particularly in the case of populations with high mortality this can lead to an overestimation of the importance of preventing a particular cause of death. For this situation this SISA program introduces the mortality adjusted YPLL, were it is considered that survivors from a removed cause of death k have a certain probability of dying from competing causes in future years. Another methodological problem is that the YPLL is highly sensitive to population age structure and findings regarding the costs of a disease are difficult to generalize from one population to another. Gardner JW, Sanborn JS. Years of Potential Life LostWhat does it measure? Epidemiology 1990;1:322329. TOP of pageThe life table describes the process of mortality in a population. The life table can be considered to be an old and distinguished predecessor of methods such as KaplanMeyer or CoxRegression. A life table is a hypothetical construct with age specific death rates, the chance to die in a single year in a particular age category, as input (the 'M' column); and remaining years of life left, or life expectancy, at various ages as an important output (the 'e' column). What the life table represents is what a cohort of denominator (mostly a 100,000 is used) babies would experience in terms of mortality and age structure had they have the age specific death rates, which are based on the current mortality experience of a population, as given in the 'M' column. There are various interesting things one can do. The q column gives the probability for a person who is born into an age category to die in that category. Some simple methods applied to the I column can be used to determine the probability to survive from one age up to another age, this column is for example used to calculate the mortality adjusted YPLL. Lastly, life table analysis does not have to be limited to demography or epidemiology. It can be used in estimating the life expectancy after medical operations or the life expectancy of televisions or other industrial products. Similarly, death does not necessarily have to be the outcome studied, however, few examples exist were something else is used. You can compare two life tables using the compare mode. TOP of pageDiscounting is used to weigh years in the near future as more important compared with years in the far away future. Mostly all future years of life are weighed equally in calculating the YPLL and the LYPLL; for example, no difference is made between the loss of the life years 2529 compared with the life years 6064, for an individual who dies at 24. Particularly in the case of perinatal and infant death this has the consequence that a large number of years of lost life are accumulated, many in the far away future. It is usual to discount financial/money data at or slightly above the rate of inflation as time progresses. So if you use the data to do a cost benefit analysis discounting seems appropriate (at about 5%). Discounting of life is considered much more problematic and there is a philosophy that it is incorrect to consider future lifeyears of less value than current lifeyears. This is certainly not the case for many individuals; the immediate gratification of smoking a cigarette mostly weighs more heavily compared with the possibility of a future early death. However, at the level of life in a society discounting might not be right and it is considered that no discounting or only a modest level of discounting is appropriate (at about 1.5%). Health economists in the U.S. advice to discount at 3%. Torgerson RJ, Raftery J.. Economic Notes: Discounting Br Med J 1999;319:914915. Weinstein MC, Siegel JE, Gold MR, Kamlet MS, Russell LB. Recommendations of the panel on costeffectiveness in health and medicine. JAMA 1996;276:12538. TOP of pageThe spreadsheet calculates the Potential Gains in Life Expectancy (PGLE) after removing cause k, considering competing causes of death. Adjusting and following the examples by Chiang (1991). It gives the increase in the life expectancy should people no longer die from a particular cause of death. The PGLE is actuarially "correct", it allows for easy comparison between populations, and it does consider competing causes of death. However, gains in life expectancy are difficult to apply in a wider macro economic analysis. Chiang CL. Competing risks in mortality analysis. Ann Rev Publ Hlth 1991;12:281307. TOP of pageAn important methodological problem with the YPLL is the fact that this measure does not consider competing causes of death. If the death of a group of people is prevented the fact that these people can die from other causes before the upper cutoff point in calculating the YPLL is reached, is not considered. Particularly in the case of populations with high mortality this can lead to an overestimation of the importance of preventing a particular cause of death. For this situation this SISA program introduces the mortality adjusted YPLL, were it is considered that survivors from a removed cause of death k have a certain probability of dying from competing causes in future years.
Note that the LYPLL, which is based on the lifetable and not on the difference between age of death and a future cutoff point, does consider competing causes of death. TOP of pageThis is free software. It can be freely distributed and installed. This program is provided to you by Quantitative Skills Research and Statistical Consultancy. Although this program has been tested extensively, no program is ever bug or error free, and you should always check your results carefully. This software and the accompanying files are sold "as is" and without warranties as to performance or merchantability or fitness for a particular purpose. The entire risk arising out of use or performance of the software remains with you. Copyright: Quantitative Skills and Daan Uitenbroek PhD, 2006. TOP of pageThe output reads easier if you make it full scree first, tick the appropriate button in the top right corner of your screen. The Output Field is fully editable. The Edit menu in the task bar is operational for this field and the usual short cuts to the clipboard can be used: CtrlC=Copy to clipboard; CtrlV=Paste from clipboard; CtrlX=copy & cut, CtrlZ=Undo. The content of the output field can be printed and/or saved as a text file using the file menu in the task bar. The output options settings will determine the format of the output field. TOP of pageThe data below gives in the first column the age categories for this data, 0, 124, 2544 ... 85+. In the second column is the total number of death in males in Amsterdam for the period 19962000, in the third column the death with alcohol use as the primary cause of death, fourth column the total number of person years lived in Amsterdam between 1996 and 2000 for each age category. In the last two columns the fractions lived by the death in respective age categories by all death and by the death due to the use of alcohol. These two fractions have been made equal in each age category (and note that these fractions are optional, you do not need to give them or both of them or all of them). You can copy the data below from this help page and paste it into one of the population fields. 0 175 0 24051 0.1 0.1 24 197 0 466069 0.39 0.39 44 1051 21 734859 0.46 0.46 54 1206 63 231110 0.54 0.54 64 1795 40 141725 0.57 0.57 69 1468 20 54562 0.49 0.49 74 2028 11 47565 0.5 0.5 79 2630 7 36809 0.52 0.52 84 2462 3 21901 0.54 0.54 85 2824 4 13195 Not changing any options you will get the following important results. Life expectancy at birth in Amsterdam for males for the period 19962000 equals 73.8. This would increase to 73.98 if alcohol use were eliminated as a primary cause of death. PGLE equals 73.9873.8=0.17. 1738.6 productive life years were lost in the age categories 2565 due to the use of alcohol. Per 100,000 person years lived in Amsterdam in these age categories approximately 157 life years is lost. Discounted at 1.5% per additional year gained these numbers equal 1523 and 138 respectively. Total LYPLL for eliminating alcohol use among males equals 3596, 203 per 100,000 person years lived. Discounted at 1.5% these numbers equal 3013 and 170 respectively. Please note that these are figures for alcohol use as a primary cause of death, it will primarily concern death caused by alcohol poisoning and alcoholic liver disease. YPLL lost due to alcohol use as a secondary, or contributory cause of death, will of course be much higher. TOP of pageSMR basically compares the mortality observed in the indexpopulation with the mortality that could be expected, if the indexpopulation had an age specific mortality pattern, which is comparable with the age specific mortality of the standard population. The SMR uses indirect standardization. Contrary to the CMF, the SMR reports the data in real empirical numbers, the number of death observed is divided by the number of death expected if the population studied the index would have had the same age specific death rates as the standard, the population we compare our data with. For calculating the SMR it is not necessary to have the age specific mortality figures for the index population, the age structure and the total number of death is adequate. If you only have the age structure for the index population in the input field you can enter the total number of death for the index population in the separate data box. Exact confidence intervals are given when the number of cases is relatively low and approximations are given when the number of cases is larger. Poisson based confidence intervals are given considering error free expectations, this would be the case if the standard would be a hypothetical mortality schedule. We do not know of the existence of such a schedule. Other confidence intervals relate to the situation that the standard is an empirical population with observed mortality, for example, the mean population for the U.S. for the year 2000 and the observed numbers of death per five year age category in that year. Lastly, if the program believes that the index population might be a subpopulation of the standard population the Fieller confidence interval is given which considers this case. Note that in the subpopulation scenario the expectation is never errorfree. TOP of pageAnderson RN, Rosenberg HM. Age standardization of death rates: implementation of the year 2000 Standard. National Vital Statistics Reports, Centers for Disease Control and Prevention NCHS. 1998(47),3. Breslow NE, Day NE. Statistical methods in cancer research. Vol 2: The design and analysis of cohort studies. Lyon: International Agency for Research on Cancer 1980. Chiang CL. The life table and its construction. In: Chiang CL. Introduction to stochastic processes in Biostatistics. New York [etc.]: John Wiley 1968. Chiang CL. Competing risks in mortality analysis. Annual Revue of Public Health 1991;12:281307. Fay MP, Feuer EJ. Confidence intervals for directly standardized rates: a method based on the gamma distribution. Statistics in Medicine 1997;16:791801. Feinlieb M, Zarate AO. Reconsidering age adjustment procedures.Vital and Health Statistics Reports, Series 4, No 29. Centers for Disease Control and Prevention NCHS. 1992. Fleiss JL. Statistical methods for rates and proportions, 2nd edition. New York [etc.]: John Wiley 1982. Gardner JW, Sanborn JS. Years of Potential Life LostWhat does it measure? Epidemiology 1990;1:322329. Lai D, Hardy RJ. Potential gains in life expectancy or years of potential life lost: impact of competing risks of death. International Journal of Epidemiology 1999;28:894898. Liddell FD. Simple exact analysis of the standardised mortality ratio. Journal of Epidemiology and Community Health 1984;38:8588.
Newcombe RG. Two sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine 1998;17:85787. Silcocks P. Estimating confidence limits on a standardised mortality ratio when the expected number is not error free. Journal of Epidemiology and Community Health 1994;48:313317. Torgerson RJ, Raftery J.. Economic Notes: Discounting British Medical Journal 1999;319:914915. TOP of pageThe Comparative Mortality Figure (CMF) is primarily used to compare two directly standardized rates obtained by standardizing the mortality rates of two different populations on a single standard. Direct standardization is used in calculating the two rates. Calculations for the CMF come from the book by Breslow and Day. For the CMF the data is compared in numbers per 100.000 (or some other denominator) of the population. This number can be set on the General Options tab sheet. Note that the program will also calculate the CMF in case age specific mortality is found in the standard population. In this particular case the standard population death rates are standardized to its own population. The program will also calculate the CMF if it finds a "mortality" number in the standard population data field, it considers that this number is the total number of death in the standard population. In principle this number of death can be based on any rate standardized to the population in the population field concerned. The program will automatically put a number of deaths based on the standardized rate in the standard population input field if you request the program to do that. This will enable you to compare the standard rate from an analysis with the standardized rate from a subsequent analysis. Please consult the compare options page to learn more about this technique. TOP of pageDirect standardization is the case were the age specific mortality rates from the index, which is mostly the smaller population in which we are interested, is multiplied with the age structure of a standard. The results is the standardized number of death relative to a certain fixed denominator, mostly 100000. This denominator number can be set on the General Options tab sheet. You can compare the standardized rate with another standardized rate that has been standardized in the same way. This other rate can be (1) data from another population; or (2) data from the same population from different time periods; or even (3) data from different groups in the same population. The two rates can be compared directly or the CMF, the ratio of the two rates, can be used. The standard can be theoretical, such as the world standard or the US 2000 standard, but often the age distribution of a larger existing population is used. Age specific death rates for the standard population are not required to do direct standardization. The age standardized rate and its variance, after being reweighted to be representative for the standard population, can be written to the standard population input field for subsequent analysis using the "Rate in Standard's Field" tick box in the compare options tabsheet. TOP of pageIndirect standardization is the case were the age specific mortality rates from the standard are multiplied with the age structure of the index. The end result is mostly an expected number of death for the index population which can be compared with the observed number of death using the SMR. Age specific death rates for the index are not required to do indirect standardization. TOP of pageThe standard concerns the data of the population to which you compare your specific population. Sometimes it concerns a hypothetical population, such as the U.S. standard 2000, and other times a larger population, the actual U.S. population on the 1st of July 2000. However, a standard population might sometimes be relatively small, for example, if we would want to directly compare Los Angeles with San Francisco. However, in this latter case we might prefer to use a method whereby we first compare both cities with a larger or a hypothetical standard population and then compare the standardized death rates of the two cities using the CMF. A discussion on the statistics of this second comparison can be found in Breslow and Day (1980). In the compare procedure the standard is presumed to be the population which is not in the visible field, i.e., under the other tabsheet. TOP of pageThe index concerns the data of the population in which you are interested, i.e., your country, city, or neighborhood. In the compare procedure the index is presumed to be the population which is visible at the moment you push the compare button. TOP of pageThe compare procedure compares two data sets, an index population, in the input field that is visible the moment you push the compare button. The other population is the standard population, which is not visible the moment you push the compare button. The SMR and the CMF are calculated to compare both populations and various intermediate results and statistics are given pertaining to these two measures. Below three situations of comparing are discussed. First, you compare the data with a hypothetical not really existing standard population, such as the world standard or the US standard for the year 2000. Please note that applicable standard mortality schedules are the topic of theory but do not seem to exist in reality. In a comparison with a standard population therefore indirect standardization and the calculation of the Standardized Mortality Ratio (SMR) is not possible. The procedure using the direct method produces a standardized rate, which can be compared with other standardized rates that were calculated to the same standard. You can do this comparison using the program by including the expected mortality associated with the standardized rate and optionally the associated variance in the input field of the standard population. See the compare options page for technical details. Second, the comparison that is probably most often made is to compare a smaller population with a larger population and this larger population is considered to be the standard. Mostly it concerns the comparison of local data with data for a larger population and often the local population is a part sub set of this larger population. The standard concerns a real population with mortality by age and this mortality fluctuates year on year. It is possible to apply indirect standardization and calculate the SMR. In this particular case one should assume that the expectation on which the SMR is based is not error free. Another problem is that mortality in the index and in the standard is correlated if the index population is a (substantial) part of the standard population. One solution is to subtract the index from the standard before the analysis. Another solution is to use the Fieller confidence interval for the SMR, which considers correlation between the standard and the index. Third, the comparison of two similar populations, for example, to compare New York and Los Angeles. In theory the best way to do this is to compare both populations with a hypothetical standard population (i.e. the US standard population) and to then compare the two resulting standardized rates using the CMF. A discussion on the statistics of this method can be found in Breslow and Day (1980) and in a number of excellent CDC/NCHS publications. Technically this is also quite easy to do in the program, you put the standardized rate and optionally also the associated variance you found for one of the two populations into the standard's input tab sheet and then apply the second index to the standard population. See the compare options page for a further discussion. However, in practice it might be more attractive to use the SMR because of the easy "what if" method of presentation: "what would the number of death in New York be if the city had the same age specific mortality as Los Angeles?" Also, you need the age specific mortality figures for only one of the two populations in using the SMR, to compare two populations using standardized rates and the CMF you need both. If you do this SMR analysis you should use the not error free expectation statistics. TOP of pageThree methods of discounting are available.
One, the exact method: Considering that the weight for the first year of potential life saved equals 1, the value of the weight for the second year at a discount rate of r equals approximately (1r), the value for the second year (1r)*(1r) and the value of the jth year saved equals wj=(1r) ^(j1); and j is in the range 1..xi1 (whereby i is the age at which the deaths at interest took place and j is the jth year of potential life saved for a death at age i). For example, the weight w10 for the 10th year saved at a discount rate of 1.5% (r=0.015) equals 0.873. This method is the programs default method. Method two, the usual method, as advised by some economist: Considering that the weight for the first year of potential life saved equals 1, the value of the weight for the second year at a discount rate of r is set at 1/(1+r) and the value of the jth year saved at wj=(1+r) ^(j1); whereby and j is in the range 1..xi1 (and whereby i is the age at which the deaths at interest took place and j is the jth year of potential life saved for a death at age i). For example, the weight w10 for the 10th year saved at a discount rate of 1.5% (r=0.015) equals 0.875. To obtain the total number of years of life lost W(i) for an individual who dies at age i you will have to sum the individual w(j)'s for each year of life lost. The calculation of the weight Wi for the discounted YPLLi for age i up to x would be for method one: W(i)= Sum (j=i...x) wj =1+Sum (j=2...xi1) (1r) ^(j1) + (1 ai) * (1r) ^(xi1) and for method two and for method two W(i)= Sum (j=i...x) wj =1+Sum (j=2...xi1) (1+r) ^(j1) + (1 ai) * (1+r) ^(xi1) Method three determines the W(i) directly. If n years of life is saved at age i at a discount rate of r per additional year saved this gives W(i)=(1exp(r*i))/r. In general, the number of years of life saved for a death in age category i equals W(i)=(1exp(r*(Liai))/r; whereby Li is the life expectancy for an individuals at the start of age category i or it is approximately the upper age limit minus the age of death as specified for the YPLL calculation. Note that this method discounts the first year of life lost by a bit, as discounting starts from day one. The program shows the Wi's and the wj's used in the calculation if one ticks the show wi box in the analyze options page. For discounting it is the first table of wi's given. Note: Y^x = raise Y to the power x Drummond MF, O'Brien B, Stoddart GL, Torrance GW. Methods for the economic evaluation of health care programs. Oxford: Oxford Medical Publications, 1997. TOP of pageThe Fieller Confidence Interval has two applications: 1) The Fieller estimate is a method to estimate the approximate confidence interval around an Standardize Mortality Ratio (SMR) when there is error in both the observed and the expected value and the number of cases is relatively large. For large numbers, the Fieller provides a good method to approximate the results of the exact Binomial/Incomplete Beta method, which is also presented in the module. The Incomplete Beta method is the more suitable exact method for estimating the confidence interval around an SMR with error in both the observed and the expected values, but the method may not work well for a very large number of observations.
2) The Fieller method makes it possible to take covariance between the observation and the expectation into account. This would be relevant if the observations are a subset of the data on the basis of which the expectation was calculated. For example, the SMR is often calculated by applying data of the national population to the local population. The mortality in Hampshire could be compared with the national mortality by using an expectation calculated by applying the national agespecific death rates (the standard) to the population of Hampshire (the index). A major crash on the motorway in Hampshire would show up in both the observation of the number of deaths in Hampshire and in the national death rates, which are used to calculate the expectation. Silcocks (1994) proposes the correlation 'q' parameter for the Fieller in such a case to be: q=Sum(d(i)*n(i)/N(i))/dtot whereby, d(i) number of deaths in the index population in the ith age band, n(i) number of individuals in the ith age band of the index population which are part of the standard population, N(i) number of individuals in the ith age band in the standard population, dtot, total number of deaths in the index population. The Fieller will give an estimate only if the expected value is larger than one. Thus, real cases or a mean of real cases is considered and not a proportion. Silcocks P. Estimating confidence limits on a standardized mortality ratio when the expected number is not error free. Journal of Epidemiology and Community Health 1994; 48:313317. TOP of pageAlthough this program has been tested extensively, no program is ever bug or error free, and you should always check your results carefully. This software is provided "as is" and without warranties as to performance or merchantability or fitness for a particular purpose. The entire risk arising out of use or performance of the software remains with you. TOP of pageDownload the program here by double clicking this link and saving the program to a directory of your choice. By double clicking the installation programs name it will be installed. 
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