deletions | additions       diff --git a/subsection_Four_main_methods_subsubsection__.tex b/subsection_Four_main_methods_subsubsection__.tex  index 1e79c69..9e79b40 100644  --- a/subsection_Four_main_methods_subsubsection__.tex  +++ b/subsection_Four_main_methods_subsubsection__.tex 
...
The proportion of cases that were not reported in the three reference years were assumed to follow a Beta distribution, with parameters $\alpha$ and $\beta$ obtained from the expected value $E$ and variance $V$ using the method of moments\cite{Renyi2007}, as follows:   Time series for the period 1990–2014 were built according to the characteristics of the levels of under-reporting and under-diagnosis that were estimated for the three reference years. A cubic spline extrapolation of $V$ and $E$, with knots set at the reference years, was used for countries with low-level or concentrated HIV epidemics. In countries with a generalized HIV epidemic, the trajectory of incidence from 1990 to the first reference year (usually 1997) was based on the annual rate of change in HIV prevalence and time changes in the fraction $F$ of incidence attributed to HIV, determined as follows:  \begin{equation}  \begin{align}  F &= \frac{h(\rho - 1)}{h(\rho - 1) + 1} \\  &= \frac{t - h}{1 - h}  \end{align}  \end(equation}  where $h$ is the prevalence of HIV in the general population, $\rho$ is the TB incidence rate ratio among HIV-positive individuals over HIV-negative individuals and $t$ is the prevalence of HIV among new TB cases.  If there were insufficient data to determine the factors leading to time-changes in case notifications, incidence was assumed to follow a horizontal trend going through the most recent estimate of incidence.   Limitations of the method based on eliciting expert opinion about gaps in case detection and reporting included a generally small number of interviewed experts; lack of clarity about vested interests when eliciting expert opinion; lack of recognition of over-reporting (due to over-diagnosis, e.g. in some countries of the former Soviet Union implementing a large-scale systematic population screening policy that may result in many people with abnormal chest X-ray but no bacteriological confirmation of TB disease being notified and treated as new TB cases); incomplete data on laboratory quality and high proportion of patients with no bacteriological confirmation of diagnosis are a potential source of error in estimates.  \begin{equation}  \begin{align}  \alpha &= E \left(\frac{E(1-E)}{V} - 1 \right) \\