Table 1.

Summary of Mathematical Functions

Ages and calendar times used as arguments in the survival functions
x	Current age in years	y	Current (calendar) time in years
x0	Minimal age observed in data (see Figure 1)	y00	Initial time in data (see Figure 1)
x̄0 = max(x0, y00 − yb)	The cohort-specific bounding (minimal) age	ȳ0 = yb + x̄0	The cohort-specific bounding (minimal) time
τ, x̄0 < τ ≤ x	Age at diagnosis	yd = y − x + τ	Time of diagnosis
x00 = y00 − y + x	The cohort specific age at y00	y0 = y − x + x0	The cohort-specific time of reaching age x0
		yb = y − x	Time of birth for a cohort
Survival analysis functions for age-specific prevalence and mortality
Pc(x, yb)		Probability of being prevalent at age x in cohort with birth time yb
Ic(τ, yb)		Incidence density function for birth cohort yb. The normalization rule for the density is ${\int }_{{\overline{x}}_0}^{\infty }{I}_c(\tau ,y_b)d\tau =1$
Mc(x, yb)		Mortality density function for birth cohort yb. The normalization rule for the density is ${\int }_{{\overline{x}}_0}^{\infty }{M}_c(x,y_b)dx=1$
$\begin{array}{c}\overline{S}(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)\\ {\overline{f}}_c(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)=-{\overline{S}}_x^{\prime }(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)\end{array}$		Survival function and respective density function of a patient group formed at age x̄0 and time ȳ0, diagnosed before x̄0, and survived to x (i.e., living x − x̄0 years after cohort forming)
$\begin{array}{c}S(x-\tau ,\tau ,y_d)\\ f_c(x-\tau ,\tau ,y_d)=-S_x^{\prime }(x-\tau ,\tau ,y_d)\end{array}$		Survival function and respective density function of a patient group diagnosed at age τ and time yd, and survived to x (i.e., living x − τ years after cohort forming)
St(x − τ, τ, yd) and μ(x, y)		Survival function and respective mortality hazard function in the general population for the cohort formed at age τ and yd, i.e., St(0, τ, yd) = 1.
Survival analysis functions for age-adjusted prevalence and mortality
P(x, y)		Prevalence at age x and time y
I(x, y)		Incidence hazard function at age x and time y
M(x, y)		Incidence based mortality hazard function at age x and time y
P(y)		Age-adjusted prevalence at time y
M(y)		Age-adjusted Incidence-based mortality hazard function at time y
$\begin{array}{c}{S}^r(x-\tau ,\tau ,y_d)\\ f^r(x-\tau ,\tau ,y_d)=-S_x^{r^{\prime }}(x-\tau ,\tau ,y_d)\end{array}$		Relative survival and respective density function of a patient group diagnosed at age τ and year yd, and reached age x (i.e., living x − τ years after diagnosis and cohort forming)
$\begin{array}{c}{\overline{S}}^r(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)\\ {\overline{f}}^r(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)=-{\overline{S}}_x^{r^{\prime }}(x-{\overline{x}}_0,{\overline{x}}_0,{\overline{y}}_0)\end{array}$		Relative survival and respective density function of a patient group function of a patient group formed at age x̄0 and time ȳ0, diagnosed before x̄0, and survived to age x (i.e., living x − x̄0 years after cohort forming)
p(x)		the density of age distribution in a standard year
T(y) and T̂(y)		Specific contributions to the time trends of age-adjusted disease prevalence and incidence-based mortality (discussed in detail in Interpretation of partitioning components)