Estimating and Comparing Cumulative Incidences

Often it is of interest to estimate (and compare) the cumulative incidences between two or more groups. For example, in the FA data set, it may be of interest to estimate the incidence of HM in the various complementation groups. Likewise, in the breast cancer data, it may be of interest to estimate breast cancer-specific mortality for those with and without a BRCA mutation. This is carried out by first dividing the sample into the subgroups of interest. The cumulative incidences of the event of interest are then calculated for each group separately as outlined above. Table 3Table 3

		Overall (%)	A (%)	C (%)	G (%)	O (%)
	N (HEM)	755 (120)	207 (30)	78 (19)	46 (10)	414 (60)
% Censored	KM	84	86	76	78	86
	CR	58	67	44	57	55
10 year	KM	6.3	5	15	11	5
	CR	5.9	4	11	7	5
20 year	KM	22.6	20	42	32	19
	CR	17.8	15	27	21	14
30 year	KM	39.0	45	53	45	34
	CR	27.8	31	32	31	23
40 year	KM	47.8	52	68	45	44
	CR	31.8	34	35	31	27

Column 3 (‘Overall’) shows the cumulative incidence for all the 755 patients. Columns 4, 5, 6 and 7 show the cumulative incidence estimates for patients in complementation groups A, C, G and O (mostly nontyped patients and a small number of patients in uncommon complementation groups). The sample size is denoted N. The number of haematologic malignancy events is denoted by HEM and is given in parentheses in the second row.

provides the cumulative incidences of HM using the Kaplan–Meier approach as well as the competing risks approach, separately for patients in complementation groups FA-A, -C, -G and other patients (O=mostly nontyped patients and a small number of patients in uncommon complementation groups). Likewise, Table 4Table 4

		Mortality due to breast cancer
		Overall (%)	BRCA mutation (%)	No mutation (%)
	N (BCSS)	305 (43)	28 (8)	276 (35)
% Censored	KM	86	71	87
	CR	77	64	79
1 year	KM	0.3	3.6	0.0
	CR	0.3	3.6	0.0
5 year	KM	4.4	10.9	3.7
	CR	4.3	10.7	3.7
10 year	KM	14.2	28.4	12.8
	CR	13.6	27.0	12.3
15 year	KM	18.6	37.3	16.7
	CR	17.6	35.2	15.9

Column 3 (‘Overall’) shows the cumulative incidence for all the 305 breast cancer patients. Columns 4 and 5 show the cumulative incidence estimates for patients with and without a BRCA mutation. The sample size is denoted N. The number of breast cancer-specific deaths is denoted BCSS and is given in parentheses in the second row. One patient without a BRCA mutation had missing death status and hence was excluded from the analysis.

provides breast cancer-specific mortality for patients with and without a BRCA mutation using the two methods.
The cumulative incidences in the various groups can be compared using nonparametric tests, namely the log-rank test (Kalbfleisch and Prentice 1980 ) when calculating incidences based on the Kaplan–Meier approach or a modified χ² test (Gray, 1988 ) when calculating incidences in the presence of competing risks. The cumulative incidence estimation methods outlined above are nonparametric, that is, these estimates are not based upon any specific model. Alternative model-based approaches can also be utilized to estimate cumulative incidences of specific events, adjusting for prognostic factors of interest. Under the assumption of noninformative censoring, the Cox proportional hazards model (Cox, 1972 ) can be used. In the presence of competing risk events, a modified Cox proportional hazards model or the competing risk regression approach has been developed by Fine and Gray (1999) . We do not detail these methods here, but refer the reader to the references provided above.