This paper extends the survival-adjusted Cochran-Armitage test in order to achieve improved robustness to a variety of tumour onset distributions. The Cochran-Armitage test is routinely applied for detecting a linear trend in the incidence of a tumour of interest across dose groups. To improve the robustness to the effects of differential mortality across groups, Bailer and Portier introduced the poly-3 test by a survival adjustment using a fractional weighting scheme for subjects not at full risk of tumour development. The performance of the poly-3 test depends on how closely it represents the correct specification of the time-at-risk weight in the data. Bailer and Portier further suggested that this test can be improved by using a general k reflecting the shape of the tumour onset distribution. In this paper, we propose a method to estimate k by equating the empirical lifetime tumour incidence rate obtained from the data based on the fractional weighting scheme to a separately estimated cumulative lifetime tumour incidence rate. This poly-k test with the statistically estimated k appears to perform better than the poly-3 test which is conducted without prior knowledge of the tumour onset distribution. Our simulation shows that the proposed method improves the robustness to various tumour onset distributions in addition to the robustness to the effects of mortality achieved by the poly-3 test. Large sample properties are shown via simulations to illustrate the consistency of the proposed method. The proposed methods are applied to analyse two real data sets. One is to find a dose-related linear trend on animal carcinogenicity, and the other is to test an effect of calorie restriction on experimental animals.