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

Abstract

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.

INTRODUCTION

Intervals in statistics include various types such as confidence interval (CI), prediction interval, credible interval, highest density interval, likelihood interval (LI), and likelihood-based confidence interval (LBCI). The current paradigm in statistics is the Null Hypothesis Significance Testing framework, which involves type I and type II errors, alpha and beta values, and null and alternative hypotheses. Within this framework, probability-based inference, particularly Wald CI estimation, is predominantly used [1]. This probability-based inference is robust in linear models such as t-tests, chi-square tests, ANOVA, and linear regression. However, it is less robust in models based on post-Fisher likelihood methods, such as generalized linear models, mixed-effects models, and survival analysis. For example, the Wald CI does not account for the asymmetry of the likelihood profile in nonlinear regression.

The LI is derived directly from the likelihood profile and reflects the shape of the likelihood profile without assuming symmetry around the parameter estimate. Therefore, the LI is well-suited for parameters obtained by Maximum Likelihood Estimation (MLE) methods [2]. However, since the current statistical paradigm primarily relies on probability-based inference, a LBCI or LI for the normal distribution has been developed to yield results similar to the conventional CI [3]. LI is an inference based solely on the likelihood function, which is not necessarily related to probability. Thus, its coverage is only approximately similar to that of the conventional CI. In contrast, LBCI is an interval that targets the probability-based conventional CI while utilizing the likelihood function, ensuring that its coverage is the same as the conventional CI.

Due to the complexity of the calculations required for LI estimation, software implementation is essential for its practical use in interval estimation. The lack of such software has hindered its widespread application [2]. In this study, we developed the R software (R Foundation for Statistical Computing, Vienna, Austria) package to facilitate the use of LI and LBCI for variance interval estimation in a single group, in addition to conventional methods, and added the and functions. Through simulation, we evaluated the performance of the LI and LBCI in comparison to the conventional CI for variance estimation.

METHODS

Lower limit (LL) and upper limit (UL) of conventional CI are calculated according to the following formula:

$$\left(\mathrm{L}\mathrm{L},\mathrm{ }\mathrm{U}\mathrm{L}\right)\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{C}\mathrm{I}\mathrm{ }=\mathrm{ }\left(\frac{(n-1)\widehat{{\sigma }^2}}{{\chi }_{1-\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^2},\frac{(n-1)\widehat{{\sigma }^2}}{{\chi }_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^2}\right)$$

The concept of LI is shown in Fig. 1. CI has different heights of h₁ and h₂ in terms of likelihood, while LI or LBCI has the same height (h₁ = h₂). Therefore, for CI, and for LI or LBCI.

Figure 1. Concept of likelihood interval. The confidence interval bounds are determined by drawing a horizontal line at on likelihood function. Conventional confidence interval has different heights of h₁ and h₂ in terms of likelihood, while the likelihood interval or likelihood-based confidence interval has the same height (h₁ = h₂).

maxL, maximum likelihood.

The function calculates the LI for the standard deviation and variance, assuming a normal distribution in a single group. This function, as shown in Fig. 2, computes the variance of the data and then applies MLE. If the cutoff value k is not provided, it defaults to calculating the log (k) value corresponding to approximate 95% CI, and the LI is estimated accordingly. In this process, the LLs and ULs are determined using R’s basic function .

Figure 2. Bare R script of function. This function computes the variance of the data and then applies Maximum Likelihood Estimation using the log-likelihood (maxLL). If the cutoff value k is not provided, it defaults to calculating the log (k) value corresponding to an approximate 95% confidence interval, and the likelihood interval is estimated accordingly. In this process, the lower and upper limits are determined using R’s basic function, [2,3].

The formula for the MLE and the calculation of log (k) are as follows:

$$\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{L}\mathrm{L} = -\frac{n}{2}\text{·(log(2π}\widehat{{\sigma }^2}+1\text{))} \text{\# maximum log-likelihood for variance estimation}$$

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{ }\left(k\right) = \frac{n}{2}\text{·log}\left(\text{1+}\frac{F_{\alpha }\left(1, n-2\right)}{n-2}\right)\mathrm{ }\#\mathrm{ }\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{ }\left(k\right)\mathrm{ }\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{ }\mathrm{a}\mathrm{p}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{ }\mathrm{C}\mathrm{I}$$

The degree of freedom for the numerator is n−2 because there are 2 unknown parameters (µ, σ²).

In contrast, the function calculates an LBCI for the standard deviation and variance, assuming a normal distribution in a single group. As shown in Fig. 3, unlike , this function utilizes 2 objective functions and uses both F and χ² distributions, the former to find LL and UL corresponding to current temporary log (k) and the latter to find exact log (k) for exact CI [4].

Figure 3. Bare R script of function. What distinguishes from is that utilizes 2 objective functions and employs both F and χ² distribution, the former objective function is used to find LL and UL corresponding to the current temporary log (k) and the latter objective function is used to determine exact log (k) for the precise confidence interval.

LL, lower limit; UL, upper limit.

Through simulation, the performance of the LI and LBCI methods was evaluated in comparison to the conventional CI. A total of 10,000 simulations were conducted for each method to estimate the 95% CIs using the luteinizing hormone () data available with the default R installation and random small sample sizes of 10, 20, and 30 drawn from a standard normal distribution. Descriptive statistics of the intervals were computed to show the coverage percentage and width of the intervals.

These functions were developed by Kyun-Seop Bae after a thorough review of the literature that discusses the concepts of LIs and the selection of cutoff values for interval estimation [1,3,4,5,6,7]. The software implementation and data analysis presented in the results section were conducted using R version 4.4.2 (R Foundation).

RESULTS

Fig. 4 shows the application of conventional CI, LBCI, and LI to data, which consist of 48 levels. The interval width for variance is 0.2665 for CI, 0.2580 for LBCI, and 0.2602 for LI, indicating that LBCI has the narrowest interval. A simulation comparing the results of the 3 methods for estimating variance intervals, using data samples that follow a normal distribution with the given data’s mean, standard deviation, and number of data points, is presented in Fig. 5. To compare whether the intervals from each method include the actual variance and to assess the width of the intervals, the results for (CI), (LBCI), and (LI) were stored to indicate whether the actual variance was included. The differences between the upper and lower limits of the interval (interval width) were stored in (CI), (LBCI), and (LI), and the results were returned accordingly.

Figure 4. R script for comparing conventional CI, LBCI, and LI using data in R software. The results show that the ranges of LBCI and LI are narrower than those of the conventional CI, with LBCI being the narrowest.

CI, confidence interval; LBCI, likelihood-based confidence interval; LI, likelihood interval; lh, luteinizing hormone; LL, lower limit; UL, upper limit.

Figure 5. R script for simulating and comparing CI, LBCI, and LI using data in R software. The results for (CI), (LBCI), and (LI) were stored to indicate whether the actual variance was included, while the differences between the upper and lower limits of the interval (interval width) were stored in (CI), (LBCI), and (LI), and the results were returned accordingly.

CI, confidence interval; LBCI, likelihood-based confidence interval; LI, likelihood interval; lh, luteinizing hormone; LBI, Likelihood-Based Interval; LL, lower limit; UL, upper limit.

The results of the simulation using data are summarized in Fig. 6. The coverage was 95.45% for LI, 95.38% for LBCI, and 95.24% for CI, respectively. The average width (standard deviation) of the intervals 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.

Figure 6. Comparison results of the simulation of CI, LBCI, and LI using data in R software. The results show that the ranges of LBCI and LI are narrower than those of the conventional CI, with LBCI being the narrowest.

CI, confidence interval; LBCI, likelihood-based confidence interval; LI, likelihood interval; lh, luteinizing hormone; LBI, Likelihood-Based Interval; sd, standard deviation.

Additionally, when comparing cases where the CIs were wider than those from other methods, no instances of wider interval estimates were found for LBCI or LI compared to CI. The coverage rate for LI was the highest, while the range of the intervals was narrowest for LBCI. The results were consistent with different seed values and data. In simulations with random small sample sizes from a standard normal distribution, described in Fig. 7, the coverage was 94.81% for CI, 94.89% for LBCI, and 96.08% for LI at a sample size of 10; 94.93% for CI, 94.82% for LBCI, and 95.32% for LI at a sample size of 20; and 95.44% for CI, 95.34% for LBCI, and 95.69% for LI at a sample size of 30. LI and LBCI exhibited narrower widths than CI, with LBCI showing the narrowest width. LBCI and CI also showed similar interval coverages.

Figure 7. Comparison of the interval results for CI, LBCI, and LI at small sample sizes (10, 20, and 30). The results show that the ranges of LBCI and LI are narrower than those of the conventional CI, with LBCI being the narrowest. Additionally, the coverage percentages of LBCI and LI are similar to that of CI.

CI, confidence interval; LBCI, likelihood-based confidence interval; LI, likelihood interval; sd, standard deviation.

DISCUSSION

The 3 primary methodologies commonly used in statistics are ‘frequentist,’ ‘Bayesian,’ and ‘likelihoodist.’ Among these, ‘likelihoodist’ or ‘Fisherian’ is relatively less well-known compared to the other 2 schools of thought. The dominant paradigm in modern statistics is the frequentist approach, which relies on probability-based inference to calculate p-values. One of the interval estimation methods used in the frequentist approach is the Wald CI, which assumes that the population follows a normal distribution and is calculated using quadratic approximation. This method does not show significant discrepancies in interval estimation when the likelihood is symmetric. However, when the likelihood function is asymmetric, discrepancies may occur.

While the Bayesian methodology incorporates the likelihood concept, it is fundamentally a probability-based statistical technique. A drawback of the Bayesian approach is that prior probabilities are typically subjective, meaning that posterior probabilities cannot be considered fully objective. In most cases, probability-based inference is preferred, but when calculating probabilities is difficult, likelihood-based inference methods, such as MLE proposed by Fisher and the Likelihood Ratio Test (LRT) developed by Neyman, can be valuable [8]. The LI, introduced by Fisher in 1956 or earlier, is derived directly from the likelihood profile, reflecting its shape without approximation and without necessarily being symmetric around the point estimate [1]. Therefore, the LI is well-suited for parameters estimated using MLE methods, such as nonlinear regression or nonlinear mixed-effects models. Interval estimation based on the estimated likelihood rarely fails when MLE is successful [9].

In comparison to Wald CI, LBCI are generally more challenging to estimate but offer greater efficiency to model reparameterization. The accuracy of Wald CI is contingent on approaching a normal distribution and is not invariant under reparameterization or parameter transformation. For instance, the upper and lower limits of a Wald CI for variance typically do not correspond to the squares of those for the corresponding standard deviation [10]. To address these limitations of Wald CI, Neale and Miller (1997) [11] developed the LBCI by inverting the LRT. The negative of twice the logarithm of the likelihood ratio statistic has a recognizable limiting probability density function—the χ² distribution with one degree of freedom—which can be used to construct asymptotically exact CIs [11,12]. Since the LRT does not change regardless of how the parameters are set in the model, the interval produced is also unaffected by parameter settings [13]. This extended invariance property ensures more reliable conclusions across different models.

The LI presented in this study is based on the estimated likelihood rather than the profile likelihood. When focusing on a single parameter of interest among several, the other parameters are treated as nuisance parameters and replaced with their maximum likelihood estimates [2]. Despite their advantages, LI and LBCI have not been widely used for interval estimation due to the lack of available software for their calculation. In response to this gap, this study contributes the LBCIvar and LInormVar functions to the LBI R package for variance interval estimation.

Simulation results, including those with small sample sizes, demonstrated that both LBCI and LI produced narrower intervals compared to the conventional χ² CI. Additionally, the coverage of the LBCI was similar to that of the CI. Consequently, LBCI exhibited similar coverage while producing the narrowest interval compared to CI.

The limitations of this study include the assumption of a normal distribution in the development of the LInormVar and LBCIvar functions, which restricts their applicability to other distribution assumptions. However, authors opinion is that these can be used regardless of distribution when there is no better option. Furthermore, the simulation was conducted using in silico data rather than real-world data. While interval estimation of variance is not frequently used in clinical trial reports, it could be valuable in the descriptive statistics of continuous variables for real-world data. Given these limitations, further evaluation using real-world data is necessary, and future development of LI and LBCI methods applicable to distributions other than the normal distribution is needed. Despite these limitations, LBCI can be considered a useful method for variance interval estimation in single-group data when compared to conventional CI.