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Table 1 Intervention effectiveness measures, as adapted from [14]

From: Design of trials for interrupting the transmission of endemic pathogens

Measure Intervention Mean outcome Comparator group Time-dependent effectiveness measure
number group intervention group   
Baseline comparison groups  
1 Intervened \( \frac { \sum _{i} y(i,t)I(i,t)} {N(t)} \) Baseline \( \hat {e}_{1}(t) ~=~ 1 - \frac { \sum _{i} y(i,t) I(i,t)} { \overline {Y}_{b} N(T)} \)
2 Naive \( \frac { \sum _{i} y(i,t)(1-I(i,t))} {N(T)- N(t)} \) Baseline \( \hat {e}_{2}(t)~=~ 1 - \frac { \sum _{i} y(i,t)(1-I(i,t))} {\overline {Y}_{b}(N(T)- N(t))} \)
3 Trial population \( \frac { \sum _{i} y(i,t)} {N(T)} \) Baseline \( \hat {e}_{3}(t) ~=~ 1 - \frac { \sum _{i} y(i,t)} { \overline {Y}_{b} N(T)} \)
Contemporaneous comparison groups   
4 Intervened \(\frac { \sum _{i} y(i,t)I(i,t)} {N(t)}\) Naive \( \hat {e}_{4}(t)~=~1 - \frac {(N(T)- N(t)) \sum _{i} y(i,t)I(i,t)} {N(t) \sum _{i} y(i,t) (1-I(i,t)) }\)
5 Intervened \( \frac { \sum _{i} y(i,t)I(i,t)} {N(t)} \) Naive remote from intervention \( \hat {e}_{5}(t) ~= ~1 - \frac {\sum _{i}y(i,t)I(i,t) \sum _{i} (1-I^{*}(i,t))} { N(t) \sum _{i} y(i,t)(1-I^{*}(i,t))} \)
6 Naive close to \( \frac { \sum _{i} y(i,t)(1-I(i,t))I^{*}(i,t)} { \sum _{i} (1-I(i,t))I^{*}(i,t)} \) Naive remote from intervention \( \hat {e}_{5}(t) ~= ~1 - \frac {\sum _{i} y(i,t)(1-I(i,t))I^{*}(i,t) \sum _{i} (1-I^{*}(i,t))} {\sum _{i} y(i,t)(1-I(i,t))I^{*}(i,t) \sum _{i} y(i,t)(1-I^{*}(i,t))} \)
  1. y(i,t): Outcome measured for individual i at time t. \(\overline {Y}_{b}\): Mean outcome at baseline. N(t): Total number of individuals in intervened clusters at time t. I(i,t): Indicator taking value 1 if individual i is in an intervened cluster at time t, 0 otherwise. I (i,t): indicator taking value 1 if individual i is intervened or less than distance r from the nearest intervened cluster at time t, 0 otherwise, so that (1−I(i,t))I (i,t) indicates those individuals in the naive close to intervention category Following Halloran [6], effectiveness measures \(\hat {e}_{1}\), \(\hat {e}_{2}\), and \(\hat {e}_{4}\) are direct measurements of intervention effectiveness, of which only \(\hat {e}_{4}\) is contemporaneous. \(\hat {e}_{3}\) is the overall measure of effectiveness for a before-and-after study. \(\hat {e}_{5} \) and \(\hat {e}_{6} \) are novel contemporaneous measurements that separate the direct and indirect effects during intervention rollout, avoiding the bias caused by contamination of the comparator group