[docs] Move crude mortality around (#46)

* move crude mortality in the docs

* up

docs/src/getting_started.md

@@ -35,24 +35,6 @@ One central hypothesis in net survival (on top of non-informative censoring) is
 
 The population hazard for each individual $\lambda_{P_i}$ is usally drawn from a reference life table, and may depend on covariates $\mathbf D_i$ such as age and date, sex, country, race, etc... See the [RateTables.jl](https://github.com/JuliaSurv/RateTables.jl) package for more details on the potential covariates. On the other hand, the excess mortality is assumed to be i.i.d. between individuals and not to depend on covariates at all. Thus, we mostly omit these covariates from our notations.
 
-## Crude Mortality
-
-The crude mortality rate is the mortality rate from all causes of death for a population. This measure does not use the cause of death information which, as we previously mentioned, can be unreliable and incomplete. For this reason, we wish to estimate:
-
-$$F_C(t) = \int_0^t S_O(u-) \lambda_E(u)du$$
-
-The Cronin-Feuer estimator [cronin2000cumulative](@cite) is thus introduced as:
-
-$$\hat{F}_C(t) = \int_0^t \hat{S}_O(u-) \hat{\lambda}_E(u)du$$
-
-where, traditionally, Kaplan-Meier is used to estimate the overall survival function $\hat{S}_O$ and Ederer II is used for the excess hazard rate $\hat{\lambda}_E$ and the population hazard rate $\hat{\lambda}_P$.
-
-Our implementation is a bit more permissive, as any net survival estimators can be used for $\hat{\lambda}_E$ and  $\hat{\lambda}_P$. Of course, the default is still the original Ederer II which provides the original Cronin-Feuer estimator. 
-
-```@docs
-CrudeMortality
-```
-
 ## Available Estimators
 
 The estimation of net survival is usually discussed in terms of the estimation of the cumulative excess hazard $\Lambda_E(t)$ and/or the instantaneous hazard $\lambda_E = \partial\Lambda_E$. To describe the estimators, we use the following counting processes notations, similar to standard survival analysis(see e.g. [FlemingHarington2013](@cite) or [ABGK1993](@cite)). 
@@ -82,6 +64,24 @@ EdererII
 Hakulinen
 ```
 
+## Crude Mortality
+
+The *crude mortality rate* is the global mortality rate (from all causes of death) for a population. This measure does not use the cause of death information which, as we previously mentioned, can be unreliable and incomplete. It is given as:
+
+$$F_C(t) = \int_0^t S_O(u-) \lambda_E(u)du$$
+
+There exists a few estimators of this quantity, the most known one being the Cronin-Feuer estimator [cronin2000cumulative](@cite), given by:
+
+$$\hat{F}_C(t) = \int_0^t \hat{S}_O(u-) \hat{\lambda}_E(u)du$$
+
+where, traditionally, Kaplan-Meier is used to estimate the overall survival function $\hat{S}_O$ and Ederer II is used for the excess hazard rate $\hat{\lambda}_E$ and the population hazard rate $\hat{\lambda}_P$.
+
+Our implementation is a bit more permissive, as any net survival estimators can be used for $\hat{\lambda}_E$. Of course, the default is still the original Ederer II which provides the original Cronin-Feuer estimator. 
+
+```@docs
+CrudeMortality
+```
+
 ## Grafféo Log-Rank Test
 
 The Grafféo Log-Rank Test [Graffeo2016](@cite) was constructed as a complement to the Pohar Perme estimator, aiming to compare the net survival functions provided by the latter. The test  is designed to compare these functions across multiple groups, including stratified covariables, and to ultimately determine, with the given p-value, which covariables are impactful to the study.

docs/src/index.md

@@ -4,7 +4,6 @@ CurrentModule = NetSurvival
 
 ## Introduction
 
-
 The `NetSurvival.jl` package provides the necessary tools to perform estimations and analysis in the Net Survival field. This specialized branch of Survival Analysis focuses on estimating the probability of survival from a specific event of interest, for example a given cancer, without considering other causes of death. This is especially relevant in the (unfortunately quite common) case where the cause of death indicatrix is either unavailable or untrustworthy. Consequently, the so-called *missing indicatrix* issue forbids the use of standard competitive risks survival analysis methods on these datasets.  For that, a few standard estimators were established in the last 50 years, backed by a wide literature.
 
 By integrating observed data from the target population with historical population mortality data (usually sourced from national census datasets), Net Survival allows the extraction of the specific mortality hazard associated with the particular disease, even under the missing indicatrix issue. The concept of relative survival analysis dates back several decades to the seminal article by Ederer, Axtell, and Cutler in 1961 [Ederer1961](@cite) and the one by Ederer and Heise in 1959 [Ederer1959](@cite).
@@ -19,6 +18,7 @@ Some key features in `NetSurvival.jl` are:
 
 - A panel of different non-parametric net survival estimators (Ederer I [Ederer1961](@cite), Ederer II [Ederer1959](@cite), Hakulinen [Hakulinen1977](@cite), Pohar Perme [PoharPerme2012](@cite)) with an interface compliant with Julia's standards. 
 - Grafféo's log-rank test [Graffeo2016](@cite) to compare net survival curves accross groups, including stratified testing.
+- Crude mortality, Expected Sample Size, and other usefull metrics in net survival field.
 - A compact, readable and efficient codebase (up to 1000x less LOC than `relsurv` for the same functionalities), ensuring long-term maintenability.
 - Significant performance improvements (up to 50x) compared to the R package `relsurv`.