The superiority of likelihood-based confidence interval for variance estimation in a single group

The χ² distribution is commonly used for estimating confidence interval (CI) for variance. However, the validity of the CIs from this method is highly dependent on the assumption that the population follows a normal distribution. Additionally, the Wald CI used in this method does not account for the asymmetry. To address this limitation and provide more accurate interval estimates, especially with relatively small sample sizes, a likelihood interval (LI) approach was adopted. The Likelihood-Based Interval R software package was developed to implement this approach. We conducted a simulation to compare 3 methods for interval estimation of variance in a single group, using the luteinizing hormone () data available with the default R installation and random small sample sizes of 10, 20, and 30 from a standard normal distribution: the conventional χ² interval method, the LI method, and the likelihood-based confidence interval (LBCI) method. The average width (standard deviation) of the CIs from the simulation with data was 0.2582 (0.0534) for LBCI, 0.2604 (0.0538) for LI, and 0.2667 (0.0551) for CI, indicating that LBCI produced the narrowest CIs. The interval coverage was 95.24% for CI, 95.38% for LBCI, and 95.45% for LI. In simulations with small sample sizes, LBCI and LI exhibited narrower widths than CI, while the coverage was similar. Therefore, LBCI or LI for variance estimation can be considered a more efficient option than the conventional method.