Table 3 Overview and relationships of approaches to quantify prolonged hospital stay associated with nosocomial infections
From: Estimands to quantify prolonged hospital stay associated with nosocomial infections
Approach | real data example (SIR-3 study) |
|---|---|
\(\hat {\lambda }_{01}=124/6442 \approx 0.0192\) | |
\(\hat {\lambda }_{02}=(756-124)/6442 \approx 0.0981\) | |
\(\hat {\lambda }_{12}=124/1527 \approx 0.0812\) | |
\(A_{1}=\frac {1}{\lambda _{12}}\) | 12.31 days |
\(A_{2}=\frac {1} {\lambda _{12}} - \frac {1}{\lambda _{02}}\) | 2.12 days |
\(A_{3}=\frac {\lambda _{01}}{\lambda _{01}+\lambda _{02}} \times \frac {1}{\lambda _{12}} \times \frac {(\lambda _{02}-\lambda _{12})} {\lambda _{02}}\) | 0.35 days |
\(A_{4}=(\frac {\lambda _{02}} {\lambda _{12}}-1) \times \frac {1}{\lambda _{01}+\lambda _{02}}\) | 1.77 days |
Additive relationships between approaches (differences) | |
\(A_{1}-A_{4}=\frac {1}{\lambda _{12}} \times \frac {\lambda _{01}+\lambda _{12}}{\lambda _{01}+\lambda _{02}}\) | 10.54 days |
\(A_{1}-A_{3}=\frac {1}{\lambda _{02}} \times \frac {\lambda _{02}^{2}+\lambda _{12}\lambda _{01}} {\lambda _{12}\lambda _{02}+\lambda _{12}\lambda _{01}}\) | 11.97 days |
\(A_{1} - A_{2}=\frac {1}{\lambda _{02}}\) | 10.19 days |
\(A_{4}-A_{3}=\frac {\lambda _{02}-\lambda _{01}}{\lambda _{02}} \times \frac {\lambda _{02}-\lambda _{12}}{\lambda _{12}(\lambda _{01}+\lambda _{02})} \) | 1.43 days |
\(A_{2}-A_{3}=A_{4}=\frac {\lambda _{02}-\lambda _{12}}{\lambda _{12}(\lambda _{01}+\lambda _{02})}\) | 1.77 days |
\(A_{2} - A_{4}=A_{3}=\frac {\lambda _{01}}{\lambda _{02}}\times \frac {\lambda _{02}-\lambda _{12}}{\lambda _{12}(\lambda _{01}+\lambda _{02})} \) | 0.35 days |
Following relationship holds: A3+A4=A2 | |
Multiplicative relationships between approaches (ratios) | |
\(\frac {A_{1}}{A_{4}}=\frac {\lambda _{01}+\lambda _{02}}{\lambda _{02}-\lambda _{12}} \ge 1\) | 6.94 |
\(\frac {A_{1}}{A_{3}}=\frac {\lambda _{02}(\lambda _{01}+\lambda _{02})} {\lambda _{01}(\lambda _{02}-\lambda _{12})} \ge 1\) | 35.4 |
\(\frac {A_{1}}{A_{2}}=\frac {\lambda _{02}}{\lambda _{02}-\lambda _{12}} \ge 1\) | 5.80 |
\(\frac {A_{3}}{A_{4}}=\frac {\lambda _{01}}{\lambda _{02}}= \text {odds(NI)} \) | 0.196 |
\(\frac {A_{3}}{A_{2}}=\frac {\lambda _{01}}{\lambda _{01}+\lambda _{02}}=\text {risk(NI)} \le 1\) | 0.164 |
\(\frac {A_{2}}{A_{4}}=\frac {\lambda _{01}+\lambda _{02}}{\lambda _{02}}=\frac {\text {odds(NI)}}{\text {risk(NI)}} \ge 1\) | 1.20 |
Following relationship holds if λ01≤λ02:A3≤A4≤A2≤A1 | |