# Assess surrogacy for two survival endpoints based on information theory and a two-stage approach

### Description

The function `SurvSurv`

implements the information-theoretic approach to estimate individual-level surrogacy (i.e., *R^2_{h.ind}*) and the two-stage approach to estimate trial-level surrogacy (*R^2_{trial}*, *R^2_{ht}*) when both endpoints are time-to-event variables (Alonso & Molenberghs, 2008). See the **Details** section below.

### Usage

1 2 | ```
SurvSurv(Dataset, Surr, SurrCens, True, TrueCens, Treat,
Trial.ID, Weighted=TRUE, Alpha=.05)
``` |

### Arguments

`Dataset` |
A |

`Surr` |
The name of the variable in |

`SurrCens` |
The name of the variable in |

`True` |
The name of the variable in |

`TrueCens` |
The name of the variable in |

`Treat` |
The name of the variable in |

`Trial.ID` |
The name of the variable in |

`Weighted` |
Logical. If |

`Alpha` |
The |

### Details

*Individual-level surrogacy*

Alonso & Molenbergs (2008) proposed to redefine the surrogate endpoint *S* as a time-dependent covariate *S(t)*, taking value *0* until the surrogate endpoint occurs and *1* thereafter. Furthermore, these author considered the models

*λ [t \mid x_{ij}, β] = K_{ij}(t) λ_{0i}(t) exp(β x_{ij}),*

*λ [t \mid x_{ij}, s_{ij}, β, φ] = K_{ij}(t) λ_{0i}(t) exp(β x_{ij} + φ S_{ij}),*

where *K_{ij}(t)* is the risk function for patient *j* in trial *i*, *x_{ij}* is a p-dimensional vector of (possibly) time-dependent covariates, *β* is a p-dimensional vector of unknown coefficients, *λ_{0i}(t)* is a trial-specific baseline hazard function, *S_{ij}* is a time-dependent covariate version of the surrogate endpoint, and *φ* its associated effect.

The mutual information between *S* and *T* is estimated as *I(T,S)=\eqn{1}{n}G^2*, where *n* is the number of patients and *G^2* is the log likelihood test comparing the previous two models. Individual-level surrogacy can then be estimated as

*R^2_{h.ind} = 1 - exp ≤ft(-\frac{1}{n}G^2 \right).*

O'Quigley and Flandre (2006) pointed out that the previous estimator depends upon the censoring mechanism, even when the censoring mechanism is non-informative. For low levels of censoring this may not be an issue of much concern but for high levels it could lead to biased results. To properly cope with the censoring mechanism in time-to-event outcomes, these authors proposed to estimate the mutual information as *{I}(T,S)=\frac{1}{k}G^2*, where *k* is the total number of events experienced. Individual-level surrogacy is then estimated as

*R^2_{h.ind} = 1 - exp ≤ft(-\frac{1}{k}G^2 \right).*

*Trial-level surrogacy*

A two-stage approach is used to estimate trial-level surrogacy, following a procedure proposed by Buyse et al. (2011). In stage 1, the following trial-specific Cox proportional hazard models are fitted:

*S_{ij}(t)=S_{i0}(t) exp(α_{i}Z_{ij}),*

*T_{ij}(t)=T_{i0}(t) exp(β_{i}Z_{ij}),*

where *S_{i0}(t)* and *T_{i0}(t)* are the trial-specific baseline hazard functions, *Z_{ij}* is the treatment indicator for subject *j* in trial *i*, and *α_{i}*, *β_{i}* are the trial-specific treatment effects on S and T, respectively.

Next, the second stage of the analysis is conducted:

*\widehat{β_{i}}=λ_{0}+λ_{1}\widehat{α_{i}}+\varepsilon_{i},*

where the parameter estimates for *β_i* and *α_i* are based on the full model that was fitted in stage 1.

When the argument `Weighted=FALSE`

is used in the function call, the model that is fitted in stage 2 is an unweighted linear regression model. When a weighted model is requested (using the argument `Weighted=TRUE`

in the function call), the information that is obtained in stage 1 is weighted according to the number of patients in a trial.

The classical coefficient of determination of the fitted stage 2 model provides an estimate of *R^2_{trial}*.

### Value

An object of class `SurvSurv`

with components,

`Results.Stage.1` |
The results of stage 1 of the two-stage model fitting approach: a |

`Results.Stage.2` |
An object of class |

`R2.ht` |
A |

`R2.hind` |
A |

`R2h.ind.QF` |
A |

`R2.hInd.By.Trial.QF` |
A |

### Author(s)

Wim Van der Elst, Ariel Alonso, & Geert Molenberghs

### References

Alonso, A. A., & Molenberghs, G. (2008). Evaluating time-to-cancer recurrence as a surrogate marker for survival from an information theory perspective. *Statistical Methods in Medical Research, 17,* 497-504.

Buyse, M., Michiels, S., Squifflet, P., Lucchesi, K. J., Hellstrand, K., Brune, M. L., Castaigne, S., Rowe, J. M. (2011). Leukemia-free survival as a surrogate end point for overall survival in the evaluation of maintenance therapy for patients with acute myeloid leukemia in complete remission. *Haematologica, 96,* 1106-1112.

O'Quigly, J., & Flandre, P. (2006). Quantification of the Prentice criteria for surrogate endpoints. *Biometrics, 62,* 297-300.

### See Also

plot.SurvSurv

### Examples

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