Exercise 23. Calculating SMRs/SIRs


(a)

##      sex age     stage mmdx yydx surv_mm surv_yy       status       subsite
## 1 Female  81 Localised    2 1981    26.5     2.5  Dead: other Head and Neck
## 2 Female  75 Localised    9 1975    55.5     4.5  Dead: other Head and Neck
## 3 Female  78 Localised    2 1978   120.0    14.5  Dead: other         Limbs
## 4 Female  75 Localised    9 1975    19.5     1.5 Dead: cancer         Trunk
## 5 Female  75 Localised   10 1975    65.5     5.5  Dead: other Head and Neck
## 6 Female  80 Localised    6 1980    50.5     4.5  Dead: other Head and Neck
##          year8594         dx       exit agegrp id      ydx    yexit dead
## 1 Diagnosed 75-84 1981-02-02 1983-04-20    75+  1 1981.088 1983.298    1
## 2 Diagnosed 75-84 1975-09-21 1980-05-07    75+  2 1975.720 1980.348    1
## 3 Diagnosed 75-84 1978-02-21 1992-12-07    75+  3 1978.140 1992.934    0
## 4 Diagnosed 75-84 1975-09-03 1977-04-19    75+  6 1975.671 1977.296    1
## 5 Diagnosed 75-84 1975-10-14 1981-03-30    75+  7 1975.783 1981.241    1
## 6 Diagnosed 75-84 1980-06-11 1984-08-27    75+  8 1980.444 1984.654    1
##   id astart    astop dead
## 1  1   81.5 82.00000    0
## 2  1   82.0 83.00000    0
## 3  1   83.0 83.70833    1
## 4  2   75.5 76.00000    0
## 5  2   76.0 77.00000    0
## 6  2   77.0 78.00000    0
## 7  2   78.0 79.00000    0
## 8  2   79.0 80.00000    0
## 9  2   80.0 80.12500    1

(b)

##   id  adx astart    astop dead      ydx   ystart    ystop
## 1  1 81.5   81.5 82.00000    0 1981.088 1981.088 1981.588
## 2  1 81.5   82.0 83.00000    0 1981.088 1981.588 1982.588
## 3  1 81.5   83.0 83.70833    1 1981.088 1982.588 1983.296
## 4  2 75.5   75.5 76.00000    0 1975.720 1975.720 1976.220
## 5  2 75.5   76.0 77.00000    0 1975.720 1976.220 1977.220
## 6  2 75.5   77.0 78.00000    0 1975.720 1977.220 1978.220
## 7  2 75.5   78.0 79.00000    0 1975.720 1978.220 1979.220
## 8  2 75.5   79.0 80.00000    0 1975.720 1979.220 1980.220
## 9  2 75.5   80.0 80.12500    1 1975.720 1980.220 1980.345
##    id   ystart    ystop   astart    astop dead
## 1   1 1981.088 1981.588 81.50000 82.00000    0
## 2   1 1981.588 1982.000 82.00000 82.41239    0
## 3   1 1982.000 1982.588 82.41239 83.00000    0
## 4   1 1982.588 1983.000 83.00000 83.41239    0
## 5   1 1983.000 1983.296 83.41239 83.70833    1
## 6   2 1975.720 1976.000 75.50000 75.77993    0
## 7   2 1976.000 1976.220 75.77993 76.00000    0
## 8   2 1976.220 1977.000 76.00000 76.77993    0
## 9   2 1977.000 1977.220 76.77993 77.00000    0
## 10  2 1977.220 1978.000 77.00000 77.77993    0
## 11  2 1978.000 1978.220 77.77993 78.00000    0
## 12  2 1978.220 1979.000 78.00000 78.77993    0
## 13  2 1979.000 1979.220 78.77993 79.00000    0
## 14  2 1979.220 1980.000 79.00000 79.77993    0
## 15  2 1980.000 1980.220 79.77993 80.00000    0
## 16  2 1980.220 1980.345 80.00000 80.12500    1

(c)

##    id   ystart    ystop   astart    astop dead age year        pt
## 1   1 1981.088 1981.588 81.50000 82.00000    0  81 1981 0.5000000
## 2   1 1981.588 1982.000 82.00000 82.41239    0  82 1981 0.4123864
## 3   1 1982.000 1982.588 82.41239 83.00000    0  82 1982 0.5876136
## 4   1 1982.588 1983.000 83.00000 83.41239    0  83 1982 0.4123864
## 5   1 1983.000 1983.296 83.41239 83.70833    1  83 1983 0.2959470
## 6   2 1975.720 1976.000 75.50000 75.77993    0  75 1975 0.2799255
## 7   2 1976.000 1976.220 75.77993 76.00000    0  75 1976 0.2200745
## 8   2 1976.220 1977.000 76.00000 76.77993    0  75 1976 0.7799255
## 9   2 1977.000 1977.220 76.77993 77.00000    0  76 1977 0.2200745
## 10  2 1977.220 1978.000 77.00000 77.77993    0  76 1977 0.7799255
## 11  2 1978.000 1978.220 77.77993 78.00000    0  77 1978 0.2200745
## 12  2 1978.220 1979.000 78.00000 78.77993    0  77 1978 0.7799255
## 13  2 1979.000 1979.220 78.77993 79.00000    0  78 1979 0.2200745
## 14  2 1979.220 1980.000 79.00000 79.77993    0  78 1979 0.7799255
## 15  2 1980.000 1980.220 79.77993 80.00000    0  79 1980 0.2200745
## 16  2 1980.220 1980.345 80.00000 80.12500    1  79 1980 0.1250000
##     year
## age      1980     1981     1982     1983     1984     1985     1986     1987     1988
##   50 17.75739 16.10950 22.14116 21.53778 28.80944 35.87093 32.43368 36.35429 43.11127
##   51 17.84611 21.10601 19.73652 25.74929 25.34206 33.81678 36.15502 36.99009 39.73722
##   52 22.66005 23.20135 26.14500 23.00520 27.86951 26.86491 34.68995 38.43219 37.43768
##   53 17.68110 27.56296 28.56208 30.58475 27.26166 31.79285 26.89462 36.09376 42.05711
##   54 16.85111 19.77104 32.88876 33.04496 35.43436 30.98549 33.83041 28.67146 42.20236
##   55 15.34311 23.06560 24.02609 35.78792 38.21690 37.88843 33.36533 37.75103 33.63208
##   56 19.62087 17.07697 29.49875 29.69993 40.84652 40.83995 40.13706 38.25580 45.57863
##   57 20.87159 23.91900 21.89273 33.33116 35.31722 44.38920 43.96677 41.94074 42.72208
##   58 22.71497 23.32395 30.11496 28.08525 36.47228 38.15391 45.32161 47.98965 45.11685
##   59 22.12377 29.11800 29.50430 35.09614 31.73203 40.31984 37.83583 46.55349 54.18740
##     year
## age      1989
##   50 46.04857
##   51 47.18098
##   52 40.44656
##   53 40.60234
##   54 49.13991
##   55 48.03838
##   56 38.02925
##   57 54.65228
##   58 49.34837
##   59 52.15563
##     year
## age  1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
##   50    1    1    3    0    0    2    1    1    1    1
##   51    1    0    1    1    1    1    0    0    1    0
##   52    1    1    0    0    0    0    1    1    0    0
##   53    2    1    2    1    1    2    1    2    0    0
##   54    0    1    2    0    3    0    0    1    1    0
##   55    1    0    1    2    0    1    2    1    0    2
##   56    1    2    1    1    3    2    0    1    0    1
##   57    0    3    0    1    0    3    1    2    2    0
##   58    1    1    2    1    2    3    1    0    0    0
##   59    1    0    3    2    2    2    0    3    2    3

(d)

##     sex   age10      year10      tstop event        rate        lower      upper
## 1  Male  [0,10) [1980,1985)   6.509911     0 0.000000000 0.0000000000 0.56665587
## 2  Male  [0,10) [1985,1990)   4.490089     0 0.000000000 0.0000000000 0.82156048
## 3  Male [10,20) [1975,1980)   4.458333     1 0.224299065 0.0056787607 1.24971441
## 4  Male [10,20) [1980,1985)   6.842898     0 0.000000000 0.0000000000 0.53908148
## 5  Male [10,20) [1985,1990)  19.640948     0 0.000000000 0.0000000000 0.18781575
## 6  Male [10,20) [1990,1995)  24.219691     0 0.000000000 0.0000000000 0.15230910
## 7  Male [10,20) [1995,2000)   4.171463     0 0.000000000 0.0000000000 0.88431320
## 8  Male [20,30) [1975,1980)  26.398108     1 0.037881503 0.0009590766 0.21106222
## 9  Male [20,30) [1980,1985)  57.999389     4 0.068966244 0.0187909804 0.17658098
## 10 Male [20,30) [1985,1990) 143.213782     1 0.006982568 0.0001767833 0.03890438
## 11 Male [20,30) [1990,1995) 180.307318     3 0.016638260 0.0034312092 0.04862406
## 12 Male [20,30) [1995,2000)  27.414736     1 0.036476732 0.0009235109 0.20323534
## 13 Male [30,40) [1975,1980)  84.052956     6 0.071383569 0.0261965118 0.15537198
## 14 Male [30,40) [1980,1985) 292.510103     8 0.027349483 0.0118075654 0.05388938
## 15 Male [30,40) [1985,1990) 412.675104    15 0.036348207 0.0203438154 0.05995084
## 16 Male [30,40) [1990,1995) 434.040225     9 0.020735405 0.0094815477 0.03936226
## 17 Male [30,40) [1995,2000)  70.554946     3 0.042520053 0.0087686571 0.12426164
## 18 Male [40,50) [1975,1980) 160.375465    11 0.068589045 0.0342394041 0.12272475
## 19 Male [40,50) [1980,1985) 470.379381    15 0.031889153 0.0178481168 0.05259631
## 20 Male [40,50) [1985,1990) 846.907111    26 0.030699943 0.0200542220 0.04498253

(e)

## `summarise()` has grouped output by 'sex', 'age'. You can override using the `.groups`
## argument.
## # A tibble: 6 × 5
##     sex   age  year    pt observed
##   <int> <dbl> <dbl> <dbl>    <dbl>
## 1     1     4  1980 0.5          0
## 2     1     4  1983 0.5          0
## 3     1     5  1980 0.213        0
## 4     1     5  1981 0.787        0
## 5     1     5  1983 0.297        0
## 6     1     5  1984 0.703        0
##   sex    prob        rate age year
## 1   1 0.96429 0.036363177   0 1951
## 2   1 0.99639 0.003616547   1 1951
## 3   1 0.99783 0.002172384   2 1951
## 4   1 0.99842 0.001581249   3 1951
## 5   1 0.99882 0.001180690   4 1951
## 6   1 0.99893 0.001070595   5 1951
##       sex           prob             rate               age             year     
##  Min.   :1.0   Min.   :0.5238   Min.   :0.000090   Min.   :  0.0   Min.   :1951  
##  1st Qu.:1.0   1st Qu.:0.9055   1st Qu.:0.001181   1st Qu.: 26.0   1st Qu.:1963  
##  Median :1.5   Median :0.9926   Median :0.007468   Median : 52.5   Median :1976  
##  Mean   :1.5   Mean   :0.9278   Mean   :0.084049   Mean   : 52.5   Mean   :1976  
##  3rd Qu.:2.0   3rd Qu.:0.9988   3rd Qu.:0.099218   3rd Qu.: 79.0   3rd Qu.:1988  
##  Max.   :2.0   Max.   :0.9999   Max.   :0.646626   Max.   :105.0   Max.   :2000
## Joining, by = c("sex", "age", "year")
## # A tibble: 6 × 7
##     sex   age  year    pt observed  prob     rate
##   <int> <dbl> <dbl> <dbl>    <dbl> <dbl>    <dbl>
## 1     1     4  1980 0.5          0  1.00 0.000410
## 2     1     4  1983 0.5          0  1.00 0.000280
## 3     1     5  1980 0.213        0  1.00 0.000350
## 4     1     5  1981 0.787        0  1.00 0.000250
## 5     1     5  1983 0.297        0  1.00 0.000250
## 6     1     5  1984 0.703        0  1.00 0.000250
## # A tibble: 6 × 8
##     sex   age  year    pt observed  prob     rate  expected
##   <int> <dbl> <dbl> <dbl>    <dbl> <dbl>    <dbl>     <dbl>
## 1     1     4  1980 0.5          0  1.00 0.000410 0.000205 
## 2     1     4  1983 0.5          0  1.00 0.000280 0.000140 
## 3     1     5  1980 0.213        0  1.00 0.000350 0.0000744
## 4     1     5  1981 0.787        0  1.00 0.000250 0.000197 
## 5     1     5  1983 0.297        0  1.00 0.000250 0.0000744
## 6     1     5  1984 0.703        0  1.00 0.000250 0.000176

(f)

## # A tibble: 1 × 1
##     SMR
##   <dbl>
## 1  2.04
## # A tibble: 2 × 2
##     sex   SMR
##   <int> <dbl>
## 1     1  2.08
## 2     2  2.00
## # A tibble: 21 × 2
##     year   SMR
##    <dbl> <dbl>
##  1  1975  1.07
##  2  1976  2.91
##  3  1977  4.43
##  4  1978  3.15
##  5  1979  2.85
##  6  1980  2.72
##  7  1981  2.52
##  8  1982  3.00
##  9  1983  2.39
## 10  1984  2.28
## # … with 11 more rows

## # A tibble: 11 × 2
##    age_group   SMR
##    <fct>     <dbl>
##  1 [0,10)     0   
##  2 [10,20)   29.0 
##  3 [20,30)   16.2 
##  4 [30,40)   14.9 
##  5 [40,50)    7.51
##  6 [50,60)    4.15
##  7 [60,70)    2.50
##  8 [70,80)    1.51
##  9 [80,90)    1.36
## 10 [90,100)   1.23
## 11 [100,110)  1.25

## joint$sex: 1
## 
##  Exact Poisson test
## 
## data:  sum(data$observed) time base: sum(data$expected)
## number of events = 818, time base = 394.2, p-value < 2.2e-16
## alternative hypothesis: true event rate is not equal to 1
## 95 percent confidence interval:
##  1.935316 2.222317
## sample estimates:
## event rate 
##   2.075102 
## 
## ------------------------------------------------------------------- 
## joint$sex: 2
## 
##  Exact Poisson test
## 
## data:  sum(data$observed) time base: sum(data$expected)
## number of events = 762, time base = 381.47, p-value < 2.2e-16
## alternative hypothesis: true event rate is not equal to 1
## 95 percent confidence interval:
##  1.858221 2.144564
## sample estimates:
## event rate 
##   1.997553

(g)

## 
## Call:
## glm(formula = observed ~ sex + year + age_group + offset(log(expected)), 
##     family = poisson, data = joint2)
## 
## Deviance Residuals: 
##     Min       1Q   Median       3Q      Max  
## -2.3618  -0.7668  -0.3192   0.2020   3.7834  
## 
## Coefficients:
##                     Estimate Std. Error z value Pr(>|z|)    
## (Intercept)          0.24931    0.12399   2.011 0.044354 *  
## sexf                 0.19732    0.05232   3.771 0.000162 ***
## year1975            -0.73554    0.71610  -1.027 0.304348    
## year1976             0.29944    0.29010   1.032 0.301975    
## year1977             0.69624    0.21205   3.283 0.001026 ** 
## year1978             0.39484    0.20517   1.925 0.054291 .  
## year1979             0.32616    0.18814   1.734 0.082996 .  
## year1980             0.28623    0.17860   1.603 0.109023    
## year1981             0.22373    0.17758   1.260 0.207711    
## year1982             0.40014    0.15913   2.515 0.011917 *  
## year1983             0.17359    0.16124   1.077 0.281660    
## year1984             0.13835    0.15719   0.880 0.378769    
## year1986            -0.04542    0.15913  -0.285 0.775291    
## year1987            -0.07071    0.15586  -0.454 0.650058    
## year1988             0.07989    0.14809   0.539 0.589558    
## year1989            -0.05832    0.15090  -0.386 0.699130    
## year1990            -0.05248    0.14929  -0.352 0.725201    
## year1991            -0.04901    0.15068  -0.325 0.744989    
## year1992            -0.03529    0.14910  -0.237 0.812920    
## year1993             0.10860    0.14366   0.756 0.449680    
## year1994            -0.22877    0.15316  -1.494 0.135246    
## year1995             0.16654    0.14207   1.172 0.241115    
## age_group[0,10)     -7.66192  317.84697  -0.024 0.980768    
## age_group[10,20)     2.95588    0.70911   4.168 3.07e-05 ***
## age_group[20,30)     2.38059    0.25515   9.330  < 2e-16 ***
## age_group[30,40)     2.29433    0.12768  17.969  < 2e-16 ***
## age_group[40,50)     1.63236    0.09788  16.677  < 2e-16 ***
## age_group[50,60)     1.03105    0.08554  12.053  < 2e-16 ***
## age_group[60,70)     0.52924    0.07571   6.990 2.74e-12 ***
## age_group[80,90)    -0.11656    0.07472  -1.560 0.118759    
## age_group[90,100)   -0.21350    0.12769  -1.672 0.094519 .  
## age_group[100,110)  -0.18214    1.00359  -0.181 0.855982    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for poisson family taken to be 1)
## 
##     Null deviance: 3205.1  on 3268  degrees of freedom
## Residual deviance: 2540.6  on 3237  degrees of freedom
## AIC: 5037.3
## 
## Number of Fisher Scoring iterations: 14
##                       exp(beta)         2.5 %        97.5 %
## (Intercept)        1.283137e+00  1.006314e+00  1.636109e+00
## sexf               1.218128e+00  1.099401e+00  1.349677e+00
## year1975           4.792457e-01  1.177646e-01  1.950300e+00
## year1976           1.349103e+00  7.640366e-01  2.382189e+00
## year1977           2.006194e+00  1.323953e+00  3.039998e+00
## year1978           1.484151e+00  9.927525e-01  2.218786e+00
## year1979           1.385637e+00  9.582975e-01  2.003542e+00
## year1980           1.331396e+00  9.381684e-01  1.889442e+00
## year1981           1.250729e+00  8.830989e-01  1.771403e+00
## year1982           1.492038e+00  1.092268e+00  2.038124e+00
## year1983           1.189572e+00  8.672427e-01  1.631702e+00
## year1984           1.148381e+00  8.438903e-01  1.562737e+00
## year1986           9.555917e-01  6.995579e-01  1.305332e+00
## year1987           9.317305e-01  6.864679e-01  1.264621e+00
## year1988           1.083166e+00  8.102976e-01  1.447924e+00
## year1989           9.433463e-01  7.018197e-01  1.267993e+00
## year1990           9.488769e-01  7.081704e-01  1.271399e+00
## year1991           9.521713e-01  7.086851e-01  1.279313e+00
## year1992           9.653299e-01  7.207160e-01  1.292967e+00
## year1993           1.114719e+00  8.411598e-01  1.477245e+00
## year1994           7.955094e-01  5.892228e-01  1.074017e+00
## year1995           1.181209e+00  8.941165e-01  1.560483e+00
## age_group[0,10)    4.704029e-04 1.320197e-274 1.676105e+267
## age_group[10,20)   1.921857e+01  4.787698e+00  7.714632e+01
## age_group[20,30)   1.081125e+01  6.556804e+00  1.782625e+01
## age_group[30,40)   9.917758e+00  7.722012e+00  1.273786e+01
## age_group[40,50)   5.115943e+00  4.222904e+00  6.197837e+00
## age_group[50,60)   2.804008e+00  2.371198e+00  3.315818e+00
## age_group[60,70)   1.697641e+00  1.463531e+00  1.969199e+00
## age_group[80,90)   8.899779e-01  7.687412e-01  1.030335e+00
## age_group[90,100)  8.077550e-01  6.289151e-01  1.037450e+00
## age_group[100,110) 8.334822e-01  1.165831e-01  5.958774e+00
## Single term deletions
## 
## Model:
## observed ~ sex + year + age_group + offset(log(expected))
##           Df Deviance    AIC    LRT  Pr(>Chi)    
## <none>         2540.6 5037.3                     
## sex        1   2554.8 5049.5  14.17 0.0001673 ***
## year      20   2590.8 5047.5  50.19 0.0002079 ***
## age_group 10   3135.6 5612.3 594.95 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1