deletions | additions       diff --git a/subsubsection_Results_from_national_TB__.tex b/subsubsection_Results_from_national_TB__.tex  index 9c3c7ca..e05b4c4 100644  --- a/subsubsection_Results_from_national_TB__.tex  +++ b/subsubsection_Results_from_national_TB__.tex 
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In countries with high-level HIV epidemics that completed a prevalence survey, the prevalence of HIV among prevalent TB cases was found to be systematically lower than the prevalence of HIV among newly notified TB cases, with an HIV rate ratio among prevalent TB over notified cases ranging from 0.07 in Rwanda (2012) to 0.5 in Malawi (2013). The HIV rate ratio was predicted from a random-effects model fitting data from 5 countries (Malawi, Rwanda, Tanzania, Uganda, Zambia) using a restricted maximum likelihood estimator and setting HIV among notified cases as an effect modifier\cite{van_Houwelingen_2002}, using the R package metafor\cite{Viechtbauer2010-vn} (Figure \ref{fig:hivratio}). The model was then used to predict HIV prevalence in prevalent cases from HIV prevalence in notified cases in African countries that were not able to measure the prevalence of HIV among survey cases.  The above two methods to derive incidence from prevalence are compared below in Table \ref{tab:2methods}.  \begin{table}   \begin{tabular}{ c c c c }  \hline  & Prevalence & Incidence - method 1 & Incidence - method 2 \\   & $(10^{-3})$ & $(10^{-3} y^{-1})$ & $(10^{-3} y^{-1})$ \\  \hline  Cambodia 2002 & 12 (10-15) & 4 (2.5-5.8) & 2.2 (1.5 – 2.9) \\   Cambodia 2011 & 8.3 (7.1-9.8) & 6.7 (4.5-9.3) & 3.8 (2.2 – 5.8) \\   Myanmar 2009 & 6.1 (5-7.5) & 3.3 (2-4.8) & 3.4 (2 – 5.1) \\   Thailand 2012 & 2.5 (1.9-3.5) & 2.3 (1-3.5) & 1.1 (0.7 – 1.6) \\   \hline  \end{tabular}   \caption{Estimates of incidence derived from prevalence survey results, based on two estimation methods.}   \label{tab:2methods}  \end{table}  It is not clear which method will perform better, validation would require a measurement of incidence. The second method requires a sufficient number of cases on treatment at the time of the survey (as a rule of thumb, at least 30 cases) to generate stable estimates. When the number of cases on treatment is too small, the amount of propagated uncertainty renders estimates of incidence unusable and only the first approach is used.   If both methods can be applied, results from two methods may be combined in a statistical ensemble approach as follows:  The incidence rate obtained using method $i$ is assumed distributed Beta with shape and scale parameters $a_i + 1$ and $b_i + 1$, respectively, and determined using the method of moments based on equation \ref{eqn:betamoments}: $I_i \sim B(a_i + 1. b_i + 1)$ so that   \begin{align*}  Prob(x = \textrm{TB})= \int_{0}^{1} x B(a_i,b_i) dx = \frac{a_i+1}{a_i+b_i+2}  \end{align*}  The combined probability is then expressed as   \begin{align*}  Prob(x = \textrm{TB}) = \frac{\sum{a_i}+1}{\sum{a_i}+\sum{b_i}+2}   \end{align*}