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Differences among the survival curves

Hypotheses:

$\begin{array}{ll} \mathcal{H}_0: & S_1(t)=S_2(t)=...=S_k(t),$\quad for all $t,\\ \mathcal{H}_1: & $not all $S_i(t)$ are equal$. \end{array}$

In calculations was used chi-square statistics form:

$\begin{displaymath} \chi^2=U'V^{-1}U \end{displaymath}$

where:

$U_i=\sum_{j=1}^{m}w_j(d_{ij}-e_{ij})$

$V$ - covariance matrix of dimensions $(k-1)\times(k-1)$

where:

diagonal: $\sum_{j=1}^{m}w_j^2\frac{n_{ij}(n_j-n_{ij})d_j(n_j-d_j)}{n^2_j(n_j-1)}$ ,

off diagonal: $\sum_{j=1}^{m}w_j^2\frac{n_{ij}n_{lj}d_j(n_j-d_j)}{n^2_j(n_j-1)}$

$m$ – number of moments in time with failure event (death),

$d_j=\sum_{i=1}^k d_{ij}$ – observed number of failure events (deaths) in the $j$ -th moment of time,

$d_{ij}$ – observed number of failure events (deaths) in the w $i$ -th group w in the $j$ -th moment of time,

$e_{ij}=\frac{n_{ij}d_j}{n_j}$ – expected number of failure events (deaths) in the w $i$ -th group w in the $j$ -th moment of time,

$n_j=\sum_{i=1}^k n_{ij}$ – the number of cases at risk in the $j$ -th moment of time.

The statistic asymptotically (for large sizes) has the Chi-square distribution with $df=k-1$ degrees of freedom.

The p-value, designated on the basis of the test statistic, is compared with the significance level $\alpha$ :

$\begin{array}{ccl} $ if $ p \le \alpha & \Longrightarrow & $ reject $ \mathcal{H}_0 $ and accept $ \mathcal{H}_1, \\ $ if $ p > \alpha & \Longrightarrow & $ there is no reason to reject $ \mathcal{H}_0. \\ \end{array}$

Hazard ratio

In the log-rank test the observed values of failure events (deaths) $O_i=\sum_{j=1}^m d_{ij}$ and the appropriate expected values $E_i=\sum_{j=1}^m e_{ij}$ are given.

The measure for describing the size of the difference between a pair of survival curves is the hazard ratio ( $HR$ ).

$\begin{displaymath} HR= \frac{O_1/E_1}{O_2/E_2} \end{displaymath}$

If the hazard ratio is greater than 1, e.g. $HR=2$ , then the degree of the risk of a failure event in the first group is twice as big as in the second group. The reverse situation takes place when $HR$ is smaller than one. When $HR$ is equal to 1 both groups are equally at risk.

Note

The confidence interval for $HR$ is calculated on the basis of the standard deviation of the $HR$ logarithm (Armitage and Berry 1994¹⁾).

EXAMPLE cont. (transplant.pqs file)

¹⁾

Armitage P., Berry G., (1994), Statistical Methods in Medical Research (3rd edition); Blackwell

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Differences among the survival curves

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