Sample Size Calculation of Exact Tests for the Weak Causal Null Hypothesis in Randomized Trials with a Binary Outcome

The main purpose in many randomized trials is to make an inference about the average causal effect of a treatment. Therefore, on a binary outcome, the null hypothesis for the hypothesis test should be that the causal risks are equal in the two groups. This null hypothesis is referred to as the weak causal null hypothesis. Nevertheless, at present, hypothesis tests applied in actual randomized trials are not for this null hypothesis;Fisher’s exact test is a test for the sharp causal null hypothesis that the causal effect of treatment is the same for all subjects. In general, the rejection of the sharp causal null hypothesis does not mean that the weak causal null hypothesis is rejected. Recently, Chiba developed new exact tests for the weak causal null hypothesis: a conditional exact test, which requires that a marginal total is fixed, and an unconditional exact test, which does not require that a marginal total is fixed and depends rather on the ratio of random assignment. To apply these exact tests in actual randomized trials, it is inevitable that the sample size calculation must be performed during the study design. In this paper, we present a sample size calculation procedure for these exact tests. Given the sample size, the procedure can derive the exact test power, because it examines all the patterns that can be obtained as observed data under the alternative hypothesis without large sample theories and any assumptions.

Causes of Big Bushfires in Australia: Higher Temperatures and Rainfall or More Fuel?

The 2019-2020 bushfires in Australia caused the loss of 34 lives and an estimated 100 bn AU$ damage. This has sharpened the apparent division between Australians who believe that the increasing number of bushfires is due to climate change, and those who suggest that fuel loads must be managed more carefully. Bushfires whose area equals or exceeds 1 mHa have been analysed in this paper. The results show that the number, duration, and size have increased over the period 1850-2020, but that since 1953, there has been a downward trend in the number of big bushfires. There is a range of temperatures of about 20°C that are associated with the fires, with a modal temperature of 30°C - 32.5°C. Using an analysis of the maximum temperatures for the period 1970-2020 as a standard for comparison with bushfires for the period 1850-2020, shows that during bushfire events the standardised temperatures have a downward trend. This is most clearly shown by the application of the Fisher Exact Test. This suggests that the fuel load in forests is a key factor for bushfires. The role of pre-bushfire rainfall shows a rapid rise in the area burnt when the rainfall exceeds 150 mm month-1 which would lead to more plant growth and hence fuel load. The role of traditional burning over the whole of Australia is described based on documentary evidence. A tentative cost benefit analysis has shown that a comprehensive program of wildfire management is cost effective. The recommendations of previous research, National Inquiries and more recent management practices, have all failed to prevent the 2019-20 fire disaster in Australia.