Terminology on Heritability Analysis. Hypothesis testing is a fundamental statistical tool. It is based on some statement of the population under a default state, referred to as the null hypothesis and denoted as H0; the test then measures the evidence against this hypothesis, allowing one to (possibly) reject H0 in favor of an alternate hypothesis H1, the negation of H0. The evidence against H0 is measured with a test statistic, which is a numerical summary extracting the interested information from the sample for comparison purposes, in order to distinguish the null hypothesis in favor of the alternative hypothesis. For a single test, the result of rejecting the null hypothesis at a pre-determined significance level α (e.g., α = 0.05), which gives rise to the critical value that is a threshold for the test statistic deciding the rejection of the null hypothesis, is called statistically significant and is unlikely to have happened by chance. The rejection of the null hypothesis at a given level α can also be determined by the p-value, which is defined as the conditional probability of obtaining the test statistic at least as extreme as the observed value under the null hypothesis, and the result of p-value less than the significance level indicates that the observed result is not arisen by chance and the null hypothesis should be rejected. Conventionally, the sampling distribution of the test statistic assuming the null hypothesis is true, which is called null distribution, is exactly or approximately computable. When the null distribution is not tractable or accurate, permutation inference can be utilised (described below). When testing a null hypothesis, we may arrive at a wrong decision if the result does not correspond with the reality. If the true null hypothesis is incorrectly rejected at a given level α, i.e., the value of the test statistic lies within the rejection region: [Tα, ∞] for a one-tailed test or [−∞, T 1 α] ∪ [T1− 1 α, ∞] for a two-tailed test, where Tα is the α-level critical value, or the corresponding p-value is smaller than α while the null hypothesis is actually true, we call a type I error or false positive occurs; if we fail to reject the false null hypothesis, i.e., the value of the test statistic is outside the rejection region or the corresponding p-value exceeds the level α while the null hypothesis is invalid, we call a type II error or false negative occurs. The occurrence of both type I and type II errors should be controlled, but there is a trade-off betwee...
