The relative survival algorithms of the survival output table depend on numbers obtained from the observed survival algorithms and the expected survival table. The observed survival algorithms are independent of both the relative survival algorithms and the expected survival table.

## Observed Survival Section of Output Table

*\(L_{1}\)*= total number of cases included in the survival output table.

*\(L_{x}\)*= number of cases alive at the beginning of interval x.

*\(D_{x}\)*= number of cases dying during interval x.

*\(U_{x}\)*= number of cases untraced (lost to follow-up) during interval x, where the definition for "lost to follow-up" is "alive with follow-up date prior to some prescribed date".

*\(W_{x}\)*= number of cases withdrawn alive during interval x, where the definition of "withdrawn alive" is "alive with follow-up date later than some specified date". (This is the same date as in the previous definition.)

*\(L_{x} = L_{x-1} - D_{x-1} - U_{x-1} - W_{x-1}\)*

If the Actuarial Method is used, then the Adjusted Alive formula is as follows:

*\(L^{*}_{x} = L_{x} - \frac{1}{2}(U_{x} + W_{x})\)*

If the Kaplan-Meier method is used, then the Adjusted Alive is set to the alive count for the interval. This is done so that any formula that uses Adjusted Alive would use the alive count when the Kaplan-Meier method is used. Adjusted Alive is not displayed in any Life Page when the Kaplan-Meier method is used.

*\(L^{*}_{x} = L_{x}\)*

*\(P_{x} = 1 - \frac{D_{x}}{L^{*}_{x} }\)*

*\(CP_{x} = P_{1} \cdot P_{2} \cdots P_{x}\)*

## Relative Survival Section of Output Table

\(\Large P^\ast_x = \frac{1}{L_x}\sum_{i=1}^{Lx} \widetilde{P}_i\Large\)

where

- \(\widetilde{P}_i \)= each individual's (i's) expected probability for surviving interval
*x*. - The sum is over all individuals entering interval
*x*alive.

\(\Large CP^\ast_x = \frac{1}{L_1}\sum_{i=1}^{L_1} \left( \widetilde{P}_1 \cdot \widetilde{P}_2 \cdots \widetilde{P}_x\right)_i \)

where \( \widetilde{P}_1,\widetilde{P}_2, \cdots \widetilde{P}_x \) refers to the expected probability of each individual (i) for surviving intervals 1,2,....,x and the sum is over all individuals in the table.

\(\Large R_x = \frac{P_x}{P^\ast_x}\)

\(\Large CR_x = \frac{CP_x}{CP^\ast_x}\)

\(\Large SR_x = R_x \sqrt{\frac{D_x}{L^\ast_x\left(L^\ast_x - D_x \right)}}\)

\(\Large SP_x = P_x \sqrt{\frac{D_x}{L^\ast_x\left(L^\ast_x - D_x \right)}}\)

\(\Large SCR_x = CR_x \sqrt{\sum_{j=1}^{x}\frac{D_j}{L^\ast_j\left(L^\ast_j - D_j \right)}}\)

\(\Large SCP_x = CP_x \sqrt{\sum_{j=1}^{x}\frac{D_j}{L^\ast_j\left(L^\ast_j - D_j \right)}}\)

### Crude probability of death calculated using expected survival

Details of this calculation are described in:

Cronin KA, Feuer EJ. Cumulative cause-specific mortality for cancer patient in the presence of other causes: a crude analogue to relative survival. Stat Med 2000 Jul 15;19:1729-40.

### Crude probability of death using cause of death information

Details of this calculation are found in:

Marubini E, Valsecchi MG. Analysing Survival Data from Clinical Trials and Observational Studies. John Wiley & Sons: England, 1995.

## Survival Confidence Intervals

The endpoints of a (1 - p) × 100% confidence interval are calculated as:

**Normal Approximation**

\({\large CI_{low} = \textbf{surv_prob}- (p \times SE_{\textbf{surv_prob}}) } \)

\({\large CI_{high} = \textbf{surv_prob}+ (p \times SE_{\textbf{surv_prob}}) } \)

**Log( - ( Log ) ) Transformation**

\( {\huge CI_{low} = \textbf{surv_prob}^{e^{\left(p \times abs(\frac{SE_{\textbf{surv_prob}}} {(\textbf{surv_prob}\times log(\textbf{surv_prob}))})\right)}} }\)

\( {\huge CI_{high} = \textbf{surv_prob}^{e^{\left(-p \times abs(\frac{SE_{\textbf{surv_prob}}} {(\textbf{surv_prob}\times log(\textbf{surv_prob}))})\right)}} }\)