Brenner II age-standardized relative survival ratios
The standard errors (SEs) of the Brenner II ARSRs were computed for the 1953–1972 and 1973–2002 patient population using the new Formula [3.5.4d] (cf. Section 5.5.4.1).
These SEs were compared to those with the traditional Greenwood formula with weighted counts [3.5.4e]. Coefficients of Variations (CVs) of the ARSRs were calculated as the ratio between SEs to the estimates of ARSRs (expressed as a percentage).
The 10-, 20- and 30-year ARSRs and the corresponding SEs computed with the new method and the Greenwood method (with weighted counts) for both the patient populations are given in Tables 15 and 16. SEs of ARSRs were higher when the traditional Greenwood method (with weighted counts) was used in each cancer site of the 1953–1972 patient population (Table 15). The differences were slight at 10-year follow-up, but increased at 20 and 30 years. In 1973–2002 patients, smaller SEs were observed when the traditional Greenwood method (with weighted counts) was used up to 10- and 20-year follow-up, but these errors were greater at 30-year follow-up (Table 16).
CVs of Brenner II ARSRs plotted by method at each follow-up year for colon, lung and thyroid cancer sites showed that in 1953–1972 patients, (considering 1973–
2002 as standard), CVs were similar at shorter follow-up, whereas for longer follow-up, the CVs with Greenwood formula (weighted counts) increased at each cancer site (Figure 12a).
In 1973-2002 patients (considering 1953–1972 as standard), CVs were also similar by method at shorter follow-up years, but the new method CVs became larger with increased follow-up years in colon and lung cancer patients. For thyroid, Greenwood CVs were greater after 25-year follow-up (Figure 12b).
051015CV (New method)%
0 5 10 15
CV (Greenwood, weighted counts) % Colon
051015CV (New method)%
0 5 10 15
CV (Greenwood, weighted counts) % Lung
4681012CV (New method)%
4 6 8 10 12
CV (Greenwood, weighted counts) % CVs by follow-up Equal CVs
Thyroid
Figure 12a. Coefficients of variation (%) of ARSR (Brenner II) using the traditional Greenwood method (weighted counts) and the new method by follow-up year in male colon, lung and thyroid cancer patients diagnosed in Finland in 1953–1972 and followed-up to end of 2002 (standard population:1973–2002). The points, from left to right, represent the follow-up lengths from 1 to 30 years.
02468CV (New method)%
0 2 4 6 8
CV (Greenwood, weighted counts) % Colon
0246810CV (New method)%
0 2 4 6 8
CV (Greenwood, weighted counts) % Lung
02468CV (New method)%
0 2 4 6 8
CV (Greenwood, weighted counts) % CVs by follow-up Equal CVs
Thyroid
Figure 12b. Coefficients of variation (%) of ARSR (Brenner II) using the traditional Greenwood method (weighted counts) and the new method by follow-up year in male colon,
Table 15. Age-standardized 10-, 20- and 30-year relative survival ratios (ARSR in %) using the Brenner II method by site and the comparison of standard errors of the ARSR using the new (NEW) and Greenwood method with weighted counts (GWc) in male cancer patients diagnosed in Finland in 1953–1972 (standard population: 1973–2002). The larger standard errors have been underlined.
10-year 20-year 30-year
SE (ARSR) SE (ARSR) SE (ARSR)
Sites
ARSR
NEW GWc ARSR
NEW GWc ARSR
NEW GWc
Salivary glands 46.0 4.39 4.56 46.8 6.04 6.82 57.1 8.90 10.75
Stomach 7.2 0.27 0.29 7.4 0.42 0.48 6.6 0.63 0.76
Colon 25.9 1.34 1.36 25.8 1.98 2.19 26.8 2.90 3.54
Rectum 22.8 1.25 1.31 23.8 2.03 2.25 29.4 3.63 4.40
Lung 3.8 0.11 0.12 3.0 0.15 0.17 2.3 0.21 0.30
Bone 32.1 1.95 2.02 32.6 2.26 2.37 36.1 2.71 2.85
Skin melanoma 33.5 2.12 2.05 33.7 2.73 2.79 36.9 3.65 3.98
Prostate 25.2 0.91 0.95 18.2 1.50 1.70 24.4 3.96 5.01
Testis 53.8 2.65 2.67 54.0 3.06 3.09 55.7 3.57 3.66
Kidney 16.9 1.20 1.23 14.3 1.52 1.72 17.6 2.36 2.99
Urinary bladder 33.8 1.42 1.43 34.3 2.22 2.50 43.6 4.00 4.98
Eye 60.5 4.29 4.12 52.9 5.28 5.75 58.2 7.01 8.14
Nervous system 17.8 0.91 0.94 17.0 1.06 1.12 17.6 1.25 1.39
Thyroid gland 51.9 3.02 3.36 53.0 4.18 4.61 59.8 5.85 6.45
Non-Hodgkin l * 19.0 1.22 1.25 17.8 1.51 1.67 18.7 1.96 2.30
Leukemia 5.7 0.59 0.59 2.0 0.39 0.49 1.6 0.38 0.56
*Non-Hodgkin lymphoma
Table 16. Age-standardized 10-, 20- and 30-year relative survival ratios (ARSR in %) using the Brenner II method by site and the comparison of standard errors (SE) of the ARSR (in %) using the new (NEW) and Greenwood method with weighted counts (GWc) in male cancer patients diagnosed in Finland in 1973–2002 (standard population: 1953–1972). The larger standard errors have been underlined.
10-year 20- year 30- year
SE (ARSR) SE (ARSR) SE (ARSR)
Sites
ARSR
NEW GWc ARSR
NEW GWc ARSR
NEW GWc Salivary glands 58.4 2.83 2.88 60.3 4.79 4.88 69.6 10.64 14.73
Stomach 20.1 0.46 0.44 19.5 0.86 0.82 12.3 3.32 3.79
Colon 48.8 0.77 0.76 50.0 1.56 1.50 47.9 9.28 7.91
Rectum 42.6 0.85 0.86 43.4 1.79 1.81 51.4 5.94 6.12
Lung 7.6 0.12 0.11 4.8 0.17 0.15 2.6 0.92 0.88
Bone 48.4 2.25 2.23 48.0 2.98 3.01 55.5 4.98 5.40
Skin melanoma 71.3 0.89 0.87 69.6 1.55 1.55 79.7 3.83 3.85
Prostate 50.9 0.58 0.58 32.5 1.57 1.59 20.8 6.40 10.32
Testis 84.0 1.33 1.22 83.0 2.20 2.09 79.1 4.82 5.21
Kidney 41.4 0.77 0.74 39.9 1.47 1.38 42.0 4.62 4.61
Urinary bladder 64.2 0.79 0.78 58.0 1.80 1.68 39.7 12.24 13.15
Eye 73.4 2.49 2.45 66.9 4.10 4.22 68.1 7.02 7.67
Nervous system 47.8 0.72 0.68 42.1 1.06 0.99 44.3 2.00 1.89
Thyroid gland 78.3 1.71 1.72 87.7 3.25 3.19 100.2 6.48 7.24
Non-Hodgkin l * 44.7 0.79 0.77 41.7 1.37 1.33 23.8 6.59 7.76
Leukaemia 29.3 0.77 0.74 24.8 1.08 1.07 28.0 1.45 1.95
*Non-Hodgkin lymphoma
4681012CV (New method)%
4 6 8 10 12
CV (Greenwood, weighted counts) % 15 - year
51015CV (New method)%
6 8 10 12 14
CV (Greenwood, weighted counts) % 20 - year
5101520CV (New method)%
5 10 15 20
CV (Greenwood, weighted counts) % CVs by site Equal CVs
30 - year
Figure 13a. Coefficients of variation (%) of ARSR (Brenner II) using the traditional Greenwood method (weighted counts) and the new method at 15, 20 and 30 years follow-up by site in male cancer patients diagnosed in Finland in 1953–1972 (standard population: 1973–2002)
234567CV (New method)%
2 3 4 5 6 7
CV (Greenwood, weighted counts) % 15 - year
2468CV (New method)%
2 4 6 8
CV (Greenwood, weighted counts) % 20 - year
01020304050CV (New method)%
0 10 20 30 40 50
CV (Greenwood, weighted counts) % CVs by site Equal CVs
30 - year
Figure13b. Coefficients of variation (%) of ARSR (Brenner II) using the traditional Greenwood method (weighted counts) and the new method at 15, 20 and 30 years follow-up by site in male cancer patients diagnosed in Finland in 1973-2002 (standard population: 1953-1972)
CVs of 15-, 20- and 30-year Brenner II ARSRs plotted by method for 16 male cancer sites showed that CVs were higher in each cancer site when the Greenwood method was used for 1953–1972 patient population considering 1973–2002 population as a standard (Figure 13a), whereas for 1973–2002, CVs of 15- and 20- year ARSRs were higher with the new method for most of the cancer sites (Figure 13b, 15- year follow up). However, CVs of age-standardized 30-year relative survival ratios were higher when the Greenwood method was used except colon and lung in 1973-2002 patient population (Figure 13b, 30-year follow-up).
The Brenner II method has a characteristic that the ARSRs are same as the RSR when the study population itself is considered as standard. The Brenner II ARSRs, in general, increase with follow-up years and are not comparable between follow-up years within the same population. Nevertheless, they are useful for the comparison of RSRs between populations at a particular follow-up year (especially long-term). The ARSRs by the Brenner II method may be interpreted in terms of the conditional probabilities (c.f Section 3.7). Otherwise, ratio interpretation (cf. Section 3.7) can be applied.
The Greenwood Formula [3.4.1] has been used for the estimation of SEs of the RSRs (or observed survival proportions) of the group of patients assuming that the death probabilities are the same in each age stratum. This assumption does not hold in practice as the death probabilities are, in general, higher in older patients. Moreover, when age-specific weights are also applied to each individual, the Greenwood formula with weighted counts [3.5.4e] does not give the correct estimates of SE of Brenner II ARSRs. The new Formula [3.5.4d] has been developed for this purpose.
The comparison of CVs of ARSRs between two methods was performed with an empirical example in such a way that the effect of censoring in the patients’ follow-up years was also taken into account in one patient population (i.e., 1973–2002) whereas in 1953–1972, censoring was absent up to 30 years of follow-up (except for the very few lost cases). The SEs of the Brenner II ARSRs estimated with the traditional Greenwood method (with weighted counts) and the new method were closer at shorter follow-up years regardless of the weights. For longer follow-up years, SEs depended on how the weights were applied. SEs of long-term ARSRs by the Brenner II method were generally larger (smaller) with the traditional Greenwood method when the smaller (larger) weights were applied to the youngest patients in the study population.
When no weights were applied and observed survival probabilities were considered to be different by age, the SEs were higher when the traditional Greenwood method was used compared to the new method (Tables 17 and 18, Figures 14a, 14b, 15a and 15b). However, the differences of CVs between methods were smaller for
non-246810CV (New method)%
2 4 6 8 10
CV (Greenwood method) % Colon
02468CV (New method)%
0 2 4 6 8
CV (Greenwood method) % Lung
4681012CV (New method)%
4 6 8 10 12
CV (Greenwood method) % CVs by follow-up Equal CVs
Thyroid
Figure14a. Coefficients of variation (%) of observed survival proportions using the traditional Greenwood method and the new method by follow-up year in male colon, lung and thyroid cancer patients diagnosed in Finland in 1953–1972 and followed-up to the end of 2002. The points, from left to right, represent the follow-up lengths from 1 to 30 years.
0246810CV (New method)%
0 2 4 6 8 10
CV (Greenwood method) % Colon
0246810CV (New method)%
0 2 4 6 8 10
CV (Greenwood method) % Lung
0246CV (New method)%
0 2 4 6
CV (Greenwood method) % CVs by follow-up Equal CVs
Thyroid
Figure14b. Coefficients of variation (%) of observed survival proportions using the traditional Greenwood method and the new method by follow-up year in male colon, lung and thyroid cancer patients diagnosed in Finland in 1973–2002 and followed-up to the end of 2002. The points, from left to right, represent the follow-up lengths from 1 to
4681012CV (New method)%
4 6 8 10 12
CV (Greenwood method) % 15 - year
4681012CV (New method)%
4 6 8 10 12
CV (Greenwood method) % 20 - year
6810121416CV (New method)%
6 8 10 12 14 16
CV (Greenwood method) % CVs by site Equal CVs
30 - year
Figure15a. Coefficients of variation (%) of observed survival proportions using the traditional Greenwood method and the new method at 15, 20 and 30 years of follow-up by site in male cancer patients diagnosed in Finland in 1953-1972 and followed-up to the end of 2002.
2468CV (New method)%
2 4 6 8
CV (Greenwood method) % 15 - year
246810CV (New method)%
2 4 6 8 10
CV (Greenwood methd) % 20 - year
010203040CV (New method)%
0 10 20 30 40
CV (Greenwood method) % CVs by site Equal CVs
30 - year
Figure15b. Coefficients of variation (%) of observed survival proportions using the traditional Greenwood method and the new method at 15, 20 and 30 years of follow-up
Table 17. Observed survival proportions (OSP in %) and a comparison of standard errors of OSP calculated with the new (NEW) and traditional Greenwood (GW) methods by site based on male cancer patients diagnosed in 1953–1972 followed-up to the end of 2002. The larger standard errors have been underlined.
10-year 20-year 30-year
SE(OSP) SE(OSP) SE(OSP)
Sites
OSP
NEW GW OSP
NEW GW OSP
NEW GW Salivary glands 30.3 2.62 2.70 19.0 2.19 2.30 13.4 1.87 2.00
Stomach 3.9 0.14 0.14 1.6 0.09 0.09 0.5 0.05 0.05
Colon 15.1 0.66 0.68 7.6 0.49 0.50 3.8 0.35 0.36
Rectum 11.9 0.61 0.62 5.4 0.42 0.43 2.6 0.29 0.30
Lung 2.6 0.07 0.07 0.9 0.04 0.04 0.2 0.02 0.02
Bone 23.5 1.48 1.54 18.6 1.36 1.42 16.3 1.29 1.35
Skin melanoma 25.3 1.35 1.37 17.3 1.17 1.20 12.1 1.00 1.04
Prostate 9.9 0.34 0.34 2.0 0.16 0.16 0.5 0.08 0.08
Testis 43.3 2.26 2.34 37.0 2.21 2.28 30.8 2.10 2.19
Kidney 12.0 0.72 0.73 5.8 0.51 0.53 3.6 0.40 0.42
Urinary bladder 18.6 0.66 0.67 8.0 0.46 0.47 3.7 0.31 0.33
Eye 44.1 2.58 2.66 26.6 2.22 2.37 19.3 1.97 2.12
Nervous system 16.3 0.72 0.73 11.5 0.62 0.63 8.7 0.55 0.56 Thyroid gland 33.6 2.08 2.26 21.9 1.85 1.98 15.5 1.64 1.73 Non-Hodgkin l* 13.5 0.75 0.77 7.9 0.59 0.61 5.1 0.48 0.50 Leukaemia 3.6 0.33 0.33 0.9 0.16 0.17 0.5 0.12 0.13 * Non-Hodgkin lymphoma
Table 18. Observed survival proportions (%) and the comparison of standard errors of OSPs calculated with the new (NEW) and the traditional Greenwood (GW) methods by site based on male cancer patients diagnosed in 1973–2002 followed-up to the end of 2002. The larger standard errors have been underlined
10-year 20-year 30-year
SE(OSP) SE(OSP) SE(OSP)
Sites
OSP
NEW GW OSP
NEW GW OSP
NEW GW Salivary glands 36.9 1.99 2.06 23.9 2.09 2.17 16.3 2.74 3.80
Stomach 10.3 0.24 0.25 4.7 0.21 0.22 1.2 0.30 0.33
Colon 26.1 0.46 0.46 13.4 0.49 0.50 6.2 1.18 1.22
Rectum 22.4 0.48 0.48 10.1 0.45 0.47 4.7 0.58 0.61
Lung 3.8 0.06 0.06 1.1 0.04 0.04 0.2 0.07 0.08
Bone 42.2 1.81 1.86 34.3 1.91 1.98 29.7 2.52 2.65
Skin melanoma 51.6 0.71 0.72 35.1 0.92 0.95 25.8 1.55 1.62
Prostate 21.6 0.26 0.26 3.9 0.20 0.20 0.6 0.19 0.21
Testis 76.3 1.01 1.04 66.4 1.48 1.54 53.8 2.97 3.29
Kidney 25.0 0.51 0.51 13.5 0.57 0.58 6.8 0.90 0.98
Urinary bladder 32.2 0.44 0.45 12.9 0.44 0.45 3.3 1.01 1.09
Eye 50.9 1.96 2.00 30.6 2.18 2.34 21.2 2.61 2.96
Nervous system 35.1 0.56 0.58 24.5 0.66 0.67 19.8 0.94 0.99 Thyroid gland 58.7 1.21 1.29 46.7 1.54 1.61 37.5 2.13 2.42 Non-Hodgkin l* 27.8 0.54 0.55 16.6 0.61 0.63 5.6 1.56 1.94 Leukaemia 16.4 0.48 0.50 9.0 0.46 0.49 7.2 0.47 0.68 * Non-Hodgkin lymphoma
In summary, it is inappropriate to use the traditional Greenwood method (with weighted counts) for the estimation of SEs of RSR (or the observed survival proportions) when the individuals in the study population are weighted. When no weights are applied, the proposed new method gives a smaller SE of RSR (or observed survival proportions) compared to the Greenwood method. It has been shown theoretically that the new method is the same as the Greenwood method when no weights are applied and death probabilities are assumed to be the same in each stratum.
The new method is a generalization of the traditional Greenwood method that accounts for the patients’ different death probabilities by age and when weights are applied to the individual counts.
As a drawback, stratum specific relative (or observed) survival estimates are needed in the new method for the calculation of SE of the Brenner II ARSR, which were not needed for the estimation of ARSR itself.