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					Calculation of Standardised Mortality Ratios (SMRs).

This workbook contains examples of the methods of calculation used by ONS to produce Standardised Mortality Ratios for war
where the age at death was under 85.

Contents:

SMR Template          The method used to calculate an SMR for each ward.

95% confidence intervals were also calculated for each SMR using a method described by Goldblatt.1

The confidence intervals are derived from an assumption that the Poisson distribution of the observed number of deaths has a
to the expected number.

Where the number of deaths is less than 100 the calculation of the upper and lower limits are based on a table of exact confide
which is also included in this workbook.

For larger numbers of deaths little accuracy is lost by using a method which approximates the calculation of the exact limits.
This method of calculation differs slightly if the observed number of deaths is greater than 900.
Three examples are therefore included for the calculation of confidence intervals which vary depending on the observed numbe

CIs - Example 1       Calculation of SMRs and 95% confidence intervals where observed number of deaths is less than 100.

CIs - Example 2       Calculation of SMRs and 95% confidence intervals where observed number of deaths is 100 or greater but

CIs - Example 3       Calculation of SMRs and 95% confidence intervals where observed number of deaths is 900 or greater.

From these examples it can be noted that although the SMR is the same in each calculation, the width of the confidence interva
as the number of deaths increase.

The table of exact 95% confidence intervals used for the calculations in Example 1 is also included.
99% confidence intervals are also inlcuded in the table for reference:

Exact CIs             Exact 95 and 99 per cent confidence intervals when observed numbers are less than 100


1
    Goldblatt P. Longitudinal Study, Mortality and social organisation. Series LS no 6, Chapter 3. HMSO London, 1990.
dised Mortality Ratios for wards in England & Wales,




ved number of deaths has a mean which is equal


 d on a table of exact confidence intervals


ulation of the exact limits.

 ding on the observed numbers of deaths:

deaths is less than 100.

deaths is 100 or greater but less than 900.

deaths is 900 or greater.

 idth of the confidence intervals decrease




SO London, 1990.
Template for the calculation of Standardised Mortality Ratios (SMRs).

This spreadsheet illustrates the method used by ONS to calculate SMRs for ward in England and Wales
where the age at death was less than 85 years.
The figures highlighted in blue are the data needed to allow the SMR to be calculated.
Figures highlighted in red are then calculated within the spreadsheet.



                      Standard Population e.g. England & Wales                                   Example Ward
                                                          Deaths per
                       Population           Deaths          1000,000              Population
                                                          population
    Age group

             0             297,256                1,449                487                 555
           1-4           1,247,768                  272                 22               2,087
           5-9           1,663,285                  198                 12               2,985
         10-14           1,666,353                  171                 10               2,509
         15-19           1,568,759                  386                 25               2,136
         20-24           1,525,390                  472                 31               2,491
         25-29           1,789,723                  632                 35               4,096
         30-34           2,047,096                1,033                 50               3,889
         35-39           2,098,035                1,629                 78               3,564
         40-44           1,822,329                2,229                122               2,764
         45-49           1,669,145                3,300                198               2,314
         50-54           1,813,517                5,684                313               2,392
         55-59           1,453,409                7,352                506               1,918
         60-64           1,298,083               10,256                790               1,653
         65-69           1,188,619               15,230              1,281               1,443
         70-74           1,131,035               25,409              2,247               1,380
         75-79           1,042,657               39,762              3,814               1,262
         80-84             710,306               49,274              6,937                 791

 All ages 0-84
for ward in England and Wales




                     Example Ward

                    Observed deaths Expected deaths




                                                 3
                                                 0
                                                 0
                                                 0
                                                 1
                                                 1
                                                 1
                                                 2
                                                 3
                                                 3
                                                 5
                                                 7
                                                10
                                                13
                                                18
                                                31
                                                48
                                                55

                                300            202

                                SMR            149
                                                                                                 1
Calculation of Standardised Mortality Ratios with 95% confidence intervals.
Example 1 - where number of deaths is less than 100.

This spreadsheet illustrates the method used by ONS to calculate SMRs for ward in England and Wales
where the age at death was less than 85 years.
The figures highlighted in blue are the data needed to allow the SMR to be calculated.
Figures highlighted in red are then calculated within the spreadsheet.

In this example, as there are fewer than 100 deaths, the 95% confidence intervals for the SMR are calculated using
the exact method. The figures for the exact upper and lower limits are taken from the table of exact 95%
confidence limits which is available on the following spreadsheet within this workbook ->      Exact CIs


                     Standard Population e.g. England & Wales                                  Ward Example 1
                                                         Deaths per
                      Population           Deaths          1000,000               Population Observed deaths
                                                         population
  Age group

           0             297,256               1,449                  487                 55
         1-4           1,247,768                 272                   22                208
         5-9           1,663,285                 198                   12                298
       10-14           1,666,353                 171                   10                250
       15-19           1,568,759                 386                   25                213
       20-24           1,525,390                 472                   31                249
       25-29           1,789,723                 632                   35                409
       30-34           2,047,096               1,033                   50                388
       35-39           2,098,035               1,629                   78                356
       40-44           1,822,329               2,229                  122                276
       45-49           1,669,145               3,300                  198                231
       50-54           1,813,517               5,684                  313                239
       55-59           1,453,409               7,352                  506                191
       60-64           1,298,083              10,256                  790                165
       65-69           1,188,619              15,230                1,281                144
       70-74           1,131,035              25,409                2,247                138
       75-79           1,042,657              39,762                3,814                126
       80-84             710,306              49,274                6,937                 79

All ages 0-84                                                                                               30

                                                                                                          SMR



                                                                                 95% Confidence intervals
                                                 SMR EL                     EU               Lower limit
                                                  149            20.2409             42.8269              100
1
    The methods used for calculating the confidence intervals is described in more detail in:
1
    Goldblatt P. Longitudinal Study, Mortality and social organisation. Series LS no 6, Chapter 3. HMSO London, 1990.
alculated using




Example 1

             Expected deaths



                           0
                           0
                           0
                           0
                           0
                           0
                           0
                           0
                           0
                           0
                           0
                           1
                           1
                           1
                           2
                           3
                           5
                           5

                          20

                          149



ervals
            Upper limit
                          212
O London, 1990.
Calculation of Standardised Mortality Ratios with 95% confidence intervals. 1
Example 2 - where number of deaths is 100 or greater but less than 900.

This spreadsheet illustrates the method used by ONS to calculate SMRs for ward in England and Wales
where the age at death was less than 85 years.
The figures highlighted in blue are the data needed to allow the SMR to be calculated.
Figures highlighted in red are then calculated within the spreadsheet.

In this example as there are more than 100 deaths an approximation to the exact method illustrated in Example 1 is used.


                              Standard Population e.g. England & Wales                                 Ward Example 2
                                                                 Deaths per
                               Population           Deaths          1000,000              Population
                                                                  population
           Age group

                     0            297,256               1,449                 487                555
                   1-4          1,247,768                 272                  22              2,087
                   5-9          1,663,285                 198                  12              2,985
                 10-14          1,666,353                 171                  10              2,509
                 15-19          1,568,759                 386                  25              2,136
                 20-24          1,525,390                 472                  31              2,491
                 25-29          1,789,723                 632                  35              4,096
                 30-34          2,047,096               1,033                  50              3,889
                 35-39          2,098,035               1,629                  78              3,564
                 40-44          1,822,329               2,229                 122              2,764
                 45-49          1,669,145               3,300                 198              2,314
                 50-54          1,813,517               5,684                 313              2,392
                 55-59          1,453,409               7,352                 506              1,918
                 60-64          1,298,083              10,256                 790              1,653
                 65-69          1,188,619              15,230               1,281              1,443
                 70-74          1,131,035              25,409               2,247              1,380
                 75-79          1,042,657              39,762               3,814              1,262
                 80-84            710,306              49,274               6,937                791

         All ages 0-84




                                                                                         95% Confidence intervals
                                                         SMR EL                     EL
                                                          149                 267                336


1
The methods used for calculating the confidence intervals is described in more detail in:
1
Goldblatt P. Longitudinal Study, Mortality and social organisation. Series LS no 6, Chapter 3. HMSO London, 1990.
d in England and Wales




 method illustrated in Example 1 is used.


                      Ward Example 2

                      Observed deaths Expected deaths



                                                           3
                                                           0
                                                           0
                                                           0
                                                           1
                                                           1
                                                           1
                                                           2
                                                           3
                                                           3
                                                           5
                                                           7
                                                          10
                                                          13
                                                          18
                                                          31
                                                          48
                                                          55

                                    300                   202

                                   SMR                    149


       95% Confidence intervals
                  Lower limit               Upper limit
                                    132                   166




6, Chapter 3. HMSO London, 1990.
                                                                                                    1
Calculation of Standardised Mortality Ratios with 95% confidence intervals.
Example 3 - where number of deaths is 900 or greater.

This spreadsheet illustrates the method used by ONS to calculate SMRs for ward in England and Wales
where the age at death was less than 85 years.
The figures highlighted in blue are the data needed to allow the SMR to be calculated.
Figures highlighted in red are then calculated within the spreadsheet.


In this example an approximation to the exact method illustrated in Example 1 is used.
As there are more than 900 deaths the method of approximation differs slightly from the calculation illustrated in Example 2.



                        Standard Population e.g. England & Wales                                  Ward Example 3
                                                            Deaths per
                         Population           Deaths          1000,000              Population Observed deaths
                                                            population
      Age group

               0             297,256              1,449                 487               2,220
             1-4           1,247,768                272                  22               8,348
             5-9           1,663,285                198                  12              11,940
           10-14           1,666,353                171                  10              10,036
           15-19           1,568,759                386                  25               8,544
           20-24           1,525,390                472                  31               9,964
           25-29           1,789,723                632                  35              16,384
           30-34           2,047,096              1,033                  50              15,556
           35-39           2,098,035              1,629                  78              14,256
           40-44           1,822,329              2,229                 122              11,056
           45-49           1,669,145              3,300                 198               9,256
           50-54           1,813,517              5,684                 313               9,568
           55-59           1,453,409              7,352                 506               7,672
           60-64           1,298,083             10,256                 790               6,612
           65-69           1,188,619             15,230               1,281               5,772
           70-74           1,131,035             25,409               2,247               5,520
           75-79           1,042,657             39,762               3,814               5,048
           80-84             710,306             49,274               6,937               3,164

    All ages 0-84                                                                                            1200

                                                                                                             SMR


                                                                                   95% Confidence intervals
                                                   SMR EL                     EU              Lower limit
                                                    149               1133               1270               140


1
    The methods used for calculating the confidence intervals is described in more detail in:
1
    Goldblatt P. Longitudinal Study, Mortality and social organisation. Series LS no 6, Chapter 3. HMSO London, 1990.
ustrated in Example 2.



Example 3

             Expected deaths



                           11
                            2
                            1
                            1
                            2
                            3
                            6
                            8
                           11
                           14
                           18
                           30
                           39
                           52
                           74
                          124
                          193
                          219

                          808

                          149


ervals
            Upper limit
                          157




London, 1990.
Exact 95 and 99 per cent confidence intervals when observed numbers are less than 100

                       95 per cent confidence interval
Observed number             Lower limit        Upper limit   Observed number
                   0             0.0000             3.6889                  0
                   1             0.0253             5.5716                  1
                   2             0.2422             7.2247                  2
                   3             0.6187             8.7673                  3
                   4             1.0899            10.2416                  4
                   5             1.6235            11.6683                  5
                   6             2.2019            13.0595                  6
                   7             2.8144            14.4227                  7
                   8             3.4538            15.7632                  8
                   9             4.1154            17.0848                  9
                  10             4.7954            18.3904                 10
                  11             5.4912            19.6820                 11
                  12             6.2006            20.9616                 12
                  13             6.9220            22.2304                 13
                  14             7.6539            23.4896                 14
                  15             8.3954            24.7402                 15
                  16             9.1454            25.9830                 16
                  17             9.9031            27.2186                 17
                  18            10.6679            28.4478                 18
                  19            11.4392            29.6709                 19
                  20            12.2165            30.8884                 20
                  21            12.9993            32.1007                 21
                  22            13.7873            33.3083                 22
                  23            14.5800            34.5113                 23
                  24            15.3773            35.7101                 24
                  25            16.1787            36.9049                 25
                  26            16.9841            38.0960                 26
                  27            17.7932            39.2836                 27
                  28            18.6058            40.4678                 28
                  29            19.4218            41.6488                 29
                  30            20.2409            42.8269                 30
                  31            21.0630            44.0020                 31
                  32            21.8880            45.1745                 32
                  33            22.7157            46.3443                 33
                  34            23.5460            47.5116                 34
                  35            24.3788            48.6765                 35
                  36            25.2140            49.8392                 36
                  37            26.0514            50.9996                 37
                  38            26.8911            52.1580                 38
                  39            27.7328            53.3143                 39
                  40            28.5766            54.4686                 40
                  41            29.4223            55.6211                 41
                  42            30.2699            56.7718                 42
                  43            31.1193            57.9207                 43
                  44            31.9705            59.0679                 44
                  45            32.8233            60.2135                 45
                  46            33.6778            61.3576                 46
                  47            34.5338            62.5000                 47
                  48            35.3914            63.6410                 48
                  49            36.2505            64.7806                 49
                  50            37.1110            65.9188                 50
                  51            37.9728            67.0556                 51
                    52           38.8361            68.1911                                52
                    53           39.7006            69.3253                                53
                    54           40.5665            70.4583                                54
                    55           41.4335            71.5901                                55
                    56           42.3018            72.7207                                56
                    57           43.1712            73.8501                                57
                    58           44.0418            74.9784                                58
                    59           44.9135            76.1057                                59
                    60           45.7863            77.2319                                60
                    61           46.6602            78.3571                                61
                    62           47.5350            79.4812                                62
                    63           48.4109            80.6044                                63
                    64           49.2878            81.7266                                64
                    65           50.1656            82.8478                                65
                    66           51.0444            83.9682                                66
                    67           51.9241            85.0876                                67
                    68           52.8047            86.2062                                68
                    69           53.6861            87.3239                                69
                    70           54.5684            88.4408                                70
                    71           55.4516            89.5568                                71
                    72           56.3356            90.6721                                72
                    73           57.2203            91.7865                                73
                    74           58.1059            92.9002                                74
                    75           58.9923            94.0131                                75
                    76           59.8794            95.1253                                76
                    77           60.7672            96.2368                                77
                    78           61.6558            97.3475                                78
                    79           62.5450            98.4576                                79
                    80           63.4350            99.5669                                80
                    81           64.3257           100.6756                                81
                    82           65.2170           101.7836                                82
                    83           66.1090           102.8910                                83
                    84           67.0017           103.9977                                84
                    85           67.8950           105.1038                                85
                    86           68.7889           106.2093                                86
                    87           69.6834           107.3142                                87
                    88           70.5786           108.4185                                88
                    89           71.4743           109.5222                                89
                    90           72.3706           110.6253                                90
                    91           73.2675           111.7278                                91
                    92           74.1650           112.8298                                92
                    93           75.0630           113.9313                                93
                    94           75.9616           115.0322                                94
                    95           76.8607           116.1326                                95
                    96           77.7603           117.2324                                96
                    97           78.6605           118.3318                                97
                    98           79.5611           119.4360                                98
                    99           80.4623           120.5289                                99


Goldblatt P. Longitudinal Study, Mortality and social organisation. Series LS no 6.HMSO London, 1990. Table 3.7, p58.
umbers are less than 100

                 99 per cent confidence interval
                       Lower limit         Upper limit
                           0.0000              5.2983
                           0.0050              7.4301
                           0.1035              9.2738
                           0.3379             10.9775
                           0.6722             12.5941
                           1.0779             14.1498
                           1.5369             15.6597
                           2.0373             17.1336
                           2.5711             18.5782
                           3.1324             19.9984
                           3.7169             21.3978
                           4.3214             22.7793
                           4.9431             24.1449
                           5.5801             25.4967
                           6.2307             26.8360
                           6.8934             28.1641
                           7.5670             29.4820
                           8.2506             30.7906
                           8.9434             32.0907
                           9.6445             33.3830
                          10.3533             34.6680
                          11.0692             35.9463
                          11.7918             37.2183
                          12.5207             38.4844
                          13.2553             39.7450
                          13.9954             41.0004
                          14.7406             42.2510
                          15.4906             43.4969
                          16.2452             44.7384
                          17.0042             45.9758
                          11.7672             47.2093
                          18.5342             48.4391
                          19.3049             49.6652
                          20.0791             50.8880
                          20.8567             52.1074
                          21.6376             53.3238
                          22.4215             54.5372
                          23.2085             55.7477
                          23.9983             56.9554
                          24.7908             58.1605
                          25.5860             59.3631
                          26.3837             60.5631
                          27.1838             61.7609
                          27.9864             62.9563
                          28.7912             64.1495
                          29.5982             65.3405
                          30.4073             66.5295
                          31.2185             67.7165
                          32.0317             68.9016
                          32.8468             70.0847
                          33.6638             71.2661
                          34.4826             72.4457
                             35.3032         73.6235
                             36.1255         14.7997
                             36.9494         75.9742
                             37.7750         77.1472
                             38.6022         78.3186
                             39.4309         79.4886
                             40.2611         80.6570
                             41.0927         81.8241
                             41.9258         82.9898
                             42.7602         84.1541
                             43.5960         85.3170
                             44.4332         86.4787
                             45.2716         87.6392
                             46.1112         88.7984
                             46.9521         89.9564
                             47.7942         91.1132
                             48.6375         92.2689
                             49.4819         93.4234
                             50.3274         94.5769
                             51.1741         95.7292
                             52.0218         96.8806
                             52.8705         98.0308
                             53.7203         99.1801
                             54.5711        100.3284
                             55.4229        101.4757
                             56.2757        102.6220
                             57.1294        103.7674
                             57.9841        104.9119
                             58.8396        106.0555
                             59.6961        107.1982
                             60.5535        108.3401
                             61.4117        109.4811
                             62.2707        110.6212
                             63.1307        111.7605
                             63.9914        112.8991
                             64.8529        114.0368
                             65.7152        115.1737
                             66.5783        116.3099
                             67.4422        117.4453
                             68.3069        118.5800
                             69.1722        119.7139
                             70.0383        120.8472
                             70.9051        121.9797
                             71.7727        123.1115
                             72.6409        124.2427
                             73.5098        125.3731
                             74.3794        126.5029
                             75.2496        127.6321


S no 6.HMSO London, 1990. Table 3.7, p58.

				
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