Response to: Practical methods for incorporating summary time-to-event data into meta-analysis

A response to and comment on Practical methods for incorporating summary time-to-event data into meta-analysis, by Jayne F Tierney et al.

O−E ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Observed events research Â Observed events control p Total observed events Â z score for P value Ä 2 ð Þ In the example of the present article, the log-rank P-value of 0.075 gave a z-score of 1.78 according to the latter part of Tierney's equation 14. Nevertheless, we found 2-sided P-value of 0.075 gave a z-score of 1.78 by using the net tools [2]. But the z-score for the 2-sided P-value should be divided by 2 as described by Tierney and colleagues. Thus we can speculate that the z-score for a P-value of 0.075 should be 0.89, which was produced from 1.78/2.
We also advertently found that a right-tailed P-value of 0.0375 gave a z-score of 1.78 by using the previous mentioned net tools [2], and 0.0375 by chance equals 0.075/2. So we speculated that the latter part of equation 14 required a more accurate representation of z-score and P-value as: z-score for (P-value/2).
Tierney and colleagues reported that if a 1-sided P-value was reported, it can be used directly to calculate the z-score without dividing by 2. According to Li [3], a 1-sided P-value was used in log-rank test or Cox regression, and the exact P-value was given in the table for Chi-square (right-tailed test). However, we verified that a right-tailed P-value 0.075 gave a z-score of 1.44 by using the previous mentioned net tools [2]. Therefore, we can conclude that a 2-sided P-value can be used to directly to obtain the z-score. The latter part of equation 14 need to be modified into: z-score for P-value. Otherwise, a 1-sided P-value divided by 2 is required to obtain the z-score and the latter part of equation 14 should be expressed more exactly as: z-score for (P-value/2).

Authors' reply
Jayne F Tierney, Lesley A Stewart, Davina Ghersi, Sarah Burdett and Matthew R Sydes.
We thank the correspondents for bringing to our attention the unfortunate ambiguity in our article [1]. It is correct that the P-value that should be divided by 2 in equation 14 and not the z-score. This was the intention, and is more explicit in the original equation [4], provided in the appendix of Tierney and colleagues [1]: As suggested by the correspondents, equation 14 could be more precisely stated as:

Observed events research Â Observed events control
Total observed events r Â z−score for p Ä 2 ð Þ and the related explanatory text altered accordingly: "As well as the events on each arm and overall, the z-score for half the two-sided P-value is required" However, we disagree with the other points raised. It is suggested that one-sided P-values are used in log-rank tests and Cox regression models [3], whereas we find that it is standard practice to present two-sided (or two-tailed) P-values. Also, to our knowledge, all major statistical packages output two-sided P-values by default. It is therefore reasonable to assume that a P-value quoted in a trial publication will be two-sided unless otherwise stated, and to use this in equation 14. However, as described in the text [1]: "If a one-sided P-value is reported it can be used directly to obtain the z-score." This is justified by equation 14 being derived algebraically from the definition of the log-rank statistic as a normallydistributed random variable, and by the fact that a one-sided P-value (assuming the most extreme direction of effect) is half the magnitude of the corresponding two-sided P-value.
The correspondents go on to suggest that either a two-sided P-value or a one-sided P-value divided by 2 can be used directly in equation 14. In fact, this would produce an incorrect z-score and hazard ratio. Using the example in Tierney and colleagues, the z-score for the reported P-value of 0.075/2 (= 0.0375) is 1.78, as the correspondents themselves found using internet tools. This produces an O-E of 19.57 and hazard ratio of 0.85; the latter being identical to that reported in the trial publication [5]. If, as the correspondents suggested, we had used the reported P-value directly in equation 14, we would have obtained a z-score of 1.44, O-E of 15.82 and an incorrect hazard ratio of 0.88.
Finally, researchers wishing to estimate hazard ratios from published time-to-event data need not rely on deriving them manually using the equations provided [1], but instead can use the Excel spreadsheet that accompanies the paper. Moreover, they can use this to cross-check hazard ratios derived from different methods of estimation.