Chapter 4: Confidence Intervals for the Common Quantile of Several Normal Populations

Justin Dunnam

1. Introduction

In this chapter, we address the problem of estimating a common quantile of several normal populations. Suppose that we have a sample of size $n_{i}$ from a $N(\mu_{i},\sigma_{i}^{2})$ distribution, $i=1,...,k$ . The $p$ th quantile of the $i$ th population is given by $\xi_{i}=\mu_{i}+z_{p}\sigma_{i}$ , $i=1,...,k$ . If a hypothesis test for $H_{0}:\xi_{1}=...=\xi_{k}$ vs. $H_{a}:\xi_{i}\neq\xi_{j}$ for some $i\neq j$ , accepts the $H_{0}$ , then it is desired to estimate the unknown common quantile of all $k$ populations. Estimation of such common parameter of several populations is a special research topic in the area of meta-analysis. Meta-analysis is a statistical technique used to combine results from multiple studies addressing the same hypothesis. Once we decide that the population quantiles are equal and the common quantile $\xi$ is unknown, then we like to test

(1)

H_{0}:\xi=\xi_{0}\quad{\rm vs.}\quad H_{a}:\xi\neq\xi_{0},

where $\xi_{0}$ is a specified value. Let $P_{i}$ denote the p-value of a one-sample test based on the $i$ th sample, $i=1,...,k$ . Note that the $P_{i}$ ’s are independent because the samples are independent.

Fisher’s ([fisher1930statistical]) method combines log-transformed p-values, so that the combination of the p-values follows a $\chi^{2}$ distribution with degrees of freedom (df) depending on the number of independent samples. Since the Fisher paper, several papers appeared suggesting other alternative methods of combining the p-values $P_{i}$ ’s to arrive at a single exact test. Recent studies by [krishnamoorthy2024combining] and [krishnamoorthy2024confidence] indicated that there is no clear-cut winner among all the combined tests proposed in the literature.

Even though there are several papers that have addressed the problem of testing a common parameter of interest, only a few papers considered the problems of interval estimating a common parameter in various setups. In this chapter, we use the general method by [krishnamoorthy2024confidence] to find CI for a common quantile of several normal distributions. While several authors have addressed interval estimation for the difference of quantiles or confidence intervals for ratios of quantiles (see, for example, [krishnamoorthy2024conf_sap] and [huang2006confidence]), relatively little work has focused on inference for a common quantile shared by multiple normal populations. This problem is particularly relevant when a hypothesis test fails to reject the null hypothesis that two or more quantiles are equal, in which case estimation of the common quantile becomes of primary interest. The remainder of this chapter is organized as follows. In the next section, we outline a one-sample test for quantile based on the noncentral $t$ (NCT) distribution and an approximate test based on a normal approximation to the NCT distribution. In Section 3, we describe some combined tests obtained by combining the individual tests. In particular, we describe Fisher’s test, inverse normal test and inverse $\chi^{2}$ test. An algorithm is provided to obtain CIs by inverting the combined tests. In addition, we propose a fiducial approach to find a simple closed-form CI for a common quantile. Finally, we present simulation studies to evaluate coverage probabilities and interval precision (Section 4), followed by two practical examples illustrating the proposed methods (Section 5).

2. One-Sample Tests for a Quantile

Let $X_{1},...,X_{n}$ be a sample from a normal distribution with parameters $\mu$ and $\sigma^{2}$ , say, $N(\mu,\sigma^{2})$ . For $0<p<1$ , the $p$ th quantile of the normal distribution is given by $\xi=\mu+z_{p}\sigma$ , where $z_{p}$ is the standard normal quantile. Consider testing

(2)

H_{0}:\xi=\xi_{0}\ \ {\rm vs.}\ \ H_{a}:\xi>\xi_{0}.

The following tests for $\xi$ can be easily obtained using the results in Section LABEL:quant_ci_chp2 of Chapter LABEL:ch2.

Noncentral t Test

It is well-known that $\frac{\sqrt{n}(\bar{X}-\xi_{0})}{S}$ is a pivotal quantity and it is easy check that it has a noncentral $t$ distribution with degrees of freedom (df) $m=n-1$ and the noncentrality parameter $-z_{p}\sigma$ , say, $t_{m}(-z_{p}\sqrt{n})$ . For example, see [owen1968survey] and Section 5.3.1.1 of [lawless2011statistical]. So, under $H_{0}$ , the statistic

(3)

T=\frac{\sqrt{n}(\bar{X}-\xi_{0})}{S}\sim t_{m}(-z_{p}\sqrt{n}).

For an observed value $T_{0}=\frac{\sqrt{n}(\bar{x}-\xi_{0})}{s}$ , the NCT test rejects the null hypothesis in (2) at the level $\alpha$ whenever the p-value

(4)

P\left(t_{m}(-z_{p}\sqrt{n})>\frac{\sqrt{n}(\bar{x}-\xi_{0})}{s}\right)<\alpha% \ \Longleftrightarrow\ P\left(t_{m}(z_{p}\sqrt{n})<\frac{\sqrt{n}(\xi_{0}-\bar% {x})}{s}\right)<\alpha.

Normal Approximate Test

An approximate test for a quantile can be found using the normal approximation to the NCT distribution in Section LABEL:NA_hz. Using this approximation, we find

(5)

\frac{c_{m}T-z_{p}}{\sqrt{\frac{1}{n}+\frac{T^{2}}{2m}}}\sim N(0,1)

approximately, where $c_{m}=1$ if $0.15\leq p\leq 0.8$ and is $1+\dfrac{1}{4m}$ otherwise. Let $T_{0}$ be an observed value of the $T$ statistic in (3), and let

\widetilde{T}_{0}=\frac{c_{m}T_{0}-z_{p}}{\sqrt{\frac{1}{n}+\frac{T_{0}^{2}}{2% m}}}.

Then the normal approximate test rejects the null-hypothesis in (2) if $P(Z>\widetilde{T}_{0})<\alpha$ , where $Z$ is the standard normal random variable.

3. Combined Tests and Confidence Intervals for a Common Quantile

Let $X_{i1},...,X_{in_{i}}$ be a sample from a normal distribution with mean $\mu_{i}$ and variance $\sigma_{i}^{2}$ , say, $N(\mu_{i},\sigma_{i}^{2})$ , $i=1,...,k$ . The $p$ th quantile of the $i$ th normal distribution is given by $\xi_{i}=\mu_{i}+z_{p}\sigma_{i}$ , where $z_{p}$ is the standard normal quantile. Let us denote the common quantile by $\xi=\xi_{1}=...=\xi_{k}$ . The problem of interest is to test

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H_{0}:\xi=\xi_{0}\ \ {\rm vs.}\ \ H_{a}:\xi\neq\xi_{0}.

Let $(\bar{X}_{i},S_{i}^{2})$ denote the (mean, variance) based on a sample of size $n_{i}$ from the $i$ th normal population, and let $(\bar{x}_{i},s_{i}^{2})$ be an observed value of $(\bar{X}_{i},S_{i}^{2})$ , $i=1,...,k$ . Let $m_{i}=n_{i}-1$ .

Some Combined Tests

Let $P_{i}$ denote the p-value of a test based on the $i$ th sample. The test could be either the exact NCT test or the normal approximate test in the preceding section. We consider three combined tests for $H_{0}:\xi=\xi_{0}$ that are based on some combinations of p-values $P_{1}$ ,…, $P_{k}$ . Let $p_{i}$ be an observed value of $P_{i}$ , $i=1,...,k$ .

Fisher’s test: [fisher1930statistical] has proposed this test which is based on the combination of p-values given by $-2\sum_{i=1}^{k}\ln P_{i}$ . Since $P_{i}$ ’s are independent uniform $(0,1)$ random variables, $-2\sum_{i=1}^{k}\ln P_{i}$ a $\chi^{2}_{2k}$ distribution. The p-value of the Fisher test is given by $P\left(\chi^{2}_{2k}>-2\sum_{i=1}^{k}\ln P_{i}\right)$ . For an observed set of p-values $\{p_{1},...,p_{k}\}$ , this tests reject the null-hypothesis in (6) if

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P\left(\chi^{2}_{2k}>-2\sum_{i=1}^{k}\ln p_{i}\right)<\alpha.

Inverse Normal Test: [liptak1958combination] has proposed a combined test which is based on the weighted sum of inverse normal p-values. The combined test statistic is given by

\frac{\sum_{i=1}^{k}m_{i}\Phi^{-1}(P_{i})}{\sqrt{\sum_{i=1}^{k}m_{i}^{2}}},

where $m_{i}=n_{i}-1$ , $i=1,...,k$ , and $\Phi$ is the standard normal distribution function. Since $\Phi^{-1}(P_{i})$ has a $N(0,1)$ distribution, the above statistic follows a standard normal distribution. This test rejects the null-hypothesis in (6) if

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P\left(Z>\frac{\sum_{i=1}^{k}m_{i}\Phi^{-1}(p_{i})}{\sqrt{\sum_{i=1}^{k}m_{i}^% {2}}}\right)<\alpha.

Inverse $\chi^{2}$ Test: Let $\chi^{2}_{m;\alpha}$ denote the $\alpha$ quantile of the chi-squared distribution with df = $m$ . [krishnamoorthy2024combining] have proposed a combined test statistic given by $\sum_{i=1}^{k}\chi^{2}_{n_{i};1-P_{i}},$ which has a $\chi^{2}_{N}$ distribution, where $N=\sum_{i=1}^{k}n_{i}$ . This test rejects the null-hypothesis in (6) if

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P\left(\chi^{2}_{N}>\sum_{i=1}^{k}\chi^{2}_{n_{i};1-p_{i}}\right)<\alpha.

There are some other combined tests are available in the literature. On the basis of applications of various combined tests to many other problems, [krishnamoorthy2024confidence] have found that the above three tests are in general better than the other tests.

Confidence Intervals on the Basis of Combined Tests

A confidence interval for a common quantile can be obtained by inverting a combined test following the approach of [krishnamoorthy2024confidence]. Let $\xi$ denote a common quantile of $k$ normal populations. Let $P_{i}$ denote the p-value for testing

H_{0}:\xi\leq\xi_{0}\ \ \ vs.\ \ \ H_{a}:\xi>\xi_{0},

based on the sample from the $i$ th population, $i=1,...,k$ . In the following algorithm, we provide a numerical method of finding a CI based on the Fisher combined test. Confidence intervals based on the other combined tests can be found similarly.

Algorithm 4.1

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Compute the p-value for testing $H_{0}:\xi=\xi_{0}$ vs. $H_{a}:\xi>\xi_{0}$ using the sample from the $i$ th population, and denote the p-value by $p_{i}(\xi_{0})$ , $i=1,...,k.$
(2)

Compute the Fisher combined statistic $-2\sum_{i=1}^{k}\ln p_{i}(\xi_{0}).$
(3)

Set the function $f(\xi_{0})=P(\chi^{2}_{2k}>-2\sum_{i=1}^{k}\ln p_{i}(\xi_{0}))-\alpha$ , where $0<\alpha<0.5$ is the confidence level.
(4)

Solve $f(\xi_{0})=0$ for $\xi_{0}.$
(5)

The root of the equation in the preceding step is a $1-\alpha$ lower confidence limit for $\xi.$
(6)

To compute the upper confidence limit for $\xi$ , compute the p-value $p_{i}(\xi_{0})$ for testing $H_{0}:\xi\geq\xi_{0}\ {\rm vs.}\ H_{a}:\xi<\xi_{0}$ in step $1$ and then follow steps $2-5$ .

The $100(1-\alpha)\%$ one-sided confidence limits form a $100(1-2\alpha)\%$ two-sided CI for the parameter $\xi$ . To solve the equation in Step 4 of the above algorithm, the endpoints of the approximate fiducial CIs in (14) can used as starting points in a root searching method.

3.1. Fiducial Confidence Interval

Let $(\bar{x},s^{2})$ be an observed value of $(\bar{X},S^{2})$ , which is based on a sample of $n$ observations from a $N(\mu,\sigma^{2})$ distribution. Let $\xi=\mu+z_{p}\sigma$ . The pivotal quantity

\frac{\sqrt{n}(\xi-\bar{X})}{S}\ {\mathrel{\mathop{\kern 0.0pt=}\limits^{d}}}% \ t_{m}(z_{p}\sqrt{n}).

Solving the above “equation” for $\xi$ , and then replacing $(\bar{X},S^{2})$ with $(\bar{x},s^{2})$ , we find the fiducial quantity for $\xi$ as

(10)

Q_{\xi}=\bar{x}+t_{m}(z_{p}\sqrt{n})\frac{s}{\sqrt{n}},

where $m=n-1$ . The conditional distribution of $Q_{\xi}$ , given $(\bar{x},s)$ , is called the fiducial distribution of $\xi$ .

A combined fiducial quantity for a common $p$ th quantile $\xi$ of $k$ populations is given by

(11)

Q_{\xi}=\sum_{i=1}^{k}w_{i}Q_{i,\xi}=\sum_{i=1}^{k}w_{i}\bar{x}_{i}+\sum_{i=1}% ^{k}w_{i}\frac{s_{i}}{\sqrt{n_{i}}}t_{m_{i}}(z_{p}\sqrt{n_{i}}),

where $Q_{i,\xi_{i}}$ is the fiducial quantity for $\xi_{i}=\mu_{i}+z_{p}\sigma_{i}$ based on the $i$ th sample and $w_{i}=({n_{i}}/{s^{2}_{i}})/\sum_{j=1}^{k}({n_{j}}/{s^{2}_{j}})$ so that $\sum_{i=1}^{k}w_{i}=1.$ .

For a given set of statistics $(\bar{x}_{1},s_{1}),\cdots,(\bar{x}_{k},s_{k})$ , the lower and upper $100\alpha$ percentiles of $Q_{\xi}$ form a $1-2\alpha$ fiducial CI for $\xi$ . To find this fiducial CI, we need to find the percentiles of $T=\sum_{i=1}^{k}w_{i}\frac{s_{i}}{\sqrt{n_{i}}}t_{m_{i}}(z_{p}\sqrt{n_{i}})$ , which can be estimated by Monte Carlo simulation, or by a modified-normal approximation given in [krishnamoorthy2016modified].

To apply the normal-based approximation, let $t_{m_{i};\alpha}(z_{p}\sqrt{n_{i}})$ denote the $\alpha$ quantile of the noncentral $t$ distribution, $i=1,...,k.$ The modified normal-based approximations for the percentiles of $T$ are expressed as follows. For $0<\alpha<0.5$ ,

(12)

T_{\alpha}\simeq\sum_{i=1}^{k}w_{i}\frac{s_{i}}{\sqrt{n_{i}}}\bar{x}_{i}-\sqrt% {\sum_{i=1}^{k}w^{2}_{i}\frac{s_{i}^{2}}{n_{i}}\left(t_{m_{i};0.5}(z_{p}\sqrt{% n_{i}})-t_{m_{i};\alpha}(z_{p}\sqrt{n_{i}})\right)^{2}},

and

(13)

T_{1-\alpha}\simeq\sum_{i=1}^{k}w_{i}\frac{s_{i}}{\sqrt{n_{i}}}\bar{x}_{i}+% \sqrt{\sum_{i=1}^{k}w^{2}_{i}\frac{s_{i}^{2}}{n_{i}}\left(t_{m_{i};0.5}(z_{p}% \sqrt{n_{i}})-t_{m_{i};1-\alpha}(z_{p}\sqrt{n_{i}})\right)^{2}},

The 100 $(1-2\alpha)$ % fiducial CI for the common quantile $\xi$ is given by

(14)

\left(\sum_{i=1}^{k}w_{i}\bar{x}_{i}+T_{\alpha},\ \sum_{i=1}^{k}w_{i}\bar{x}_{% i}+T_{1-\alpha}\right).

4. Coverage and Precision

The confidence intervals based on the combined tests are exact and the coverage probabilities of these CIs should be the same as the nominal level for all sample size and parameter configurations. However, the precisions of these exact CIs could be different. To compare these CIs in terms of precision, we estimated the expected widths of 95% CIs and reported them in Table 1 for the case of $k=2$ . We also estimated the coverage probabilities of the CIs based on combination of independent approximate normal tests and the fiducial test. We used simulation consisting of 100,000 runs. For each generated set of samples, we first computed the fiducial CIs, and using the endpoints of the fiducial CIs as starting values, we used a bisection method to compute CIs based on the combined tests.

In Table 1, we reported the estimated values for Fisher’s CIs (Fisher), CIs intervals based on the inverse normal combined test (Inv Norm), CIs based on the inverse $\chi^{2}$ combined test (Inv $\chi^{2}$ ) and the fiducial CIs. In Table 1, we see that the estimated coverage probabilities of all exact CIs are very close to the nominal level 0.95, which indicate that our simulation studies are quite accurate. The inverse normal CIs and the inverse $\chi^{2}$ CIs are somewhat shorter than the corresponding Fisher’s CIs in most cases. This comparison holds for the CIs of the common mean ( $p=0.50)$ or of the common 75th percentile. There is no clear-cut winner between the inverse $\chi^{2}$ and inverse normal CIs. Relation between population variances may help to choose between the inverse normal and inverse $\chi^{2}$ confidence intervals. The fiducial CIs could be liberal for small sample sizes. They perform satisfactorily by controlling confidence levels close to the nominal level. In some cases, the coverage probabilities of the fiducial CIs are very close to 0.95 with smaller expected widths smaller than the other CIs. For example, see the cases $p=0.50,0.75$ , $(n_{1},n_{2})=(6,6)$ , $(\sigma_{1}^{2},\sigma_{2}^{2})=(1,.1)$ . The fiducial CIs are not only simple to compute, but also seem to be very satisfactory for modest sample sizes.

Simulation results for the case of $k=4$ are reported in Table 2. Examination of the simulation estimates suggests that the Fisher CIs and inverse $\chi^{2}$ CIs are similar with higher precisions than the inverse normal CIs when all the sample sizes are approximately equal. If the sample sizes are quite different, then the inverse normal CIs are expected to be shorter than the others. Fiducial CIs could be liberal with coverage probabilities as low as 0.915 when the sample sizes are small. However, when all sample sizes are 10 or more, the fiducial CIs are satisfactory with practical accuracy (see Table 3). For these sample sizes, the coverage probabilities of the fiducial CIs are seldom fall below 0.940 and their expected widths are appreciably smaller than those of the other CIs.

Our overall recommendations are as follows. If the sample sizes are quite different, then inverse $\chi^{2}$ CIs and the inverse normal CIs are preferable. If the sample sizes are approximately close to each others, then the Fisher CIs are recommended for applications. If all sample sizes are 15 or more, then fiducial CIs are safe to use in applications.

Table 1. Coverage Probabilities and (Expected Widths) of 95% Confidence Intervals

	$k=2,\sigma_{1}=1,\xi=5$
	$(n_{1},n_{2})$	$\sigma_{2}$	Fisher	Inv Norm	Inv $\chi^{2}$	Fiducial
		$p$ = 0.50
NCT CIs	(6,6)	0.1	0.950 (0.242)	0.951 (0.314)	0.952 (0.256)	.948(0.198)
		0.3	0.949 (0.628)	0.953 (0.654)	0.955 (0.628)	.941(0.557)
		0.5	0.952 (0.908)	0.950 (0.879)	0.949 (0.889)	.934(0.847)
		0.8	0.950 (1.209)	0.953 (1.135)	0.948 (1.176)	.930(1.160)
		1.0	0.949 (1.359)	0.948 (1.273)	0.947 (1.316)	.931(1.309)
normal CIs	(6,6)	0.1	0.950(0.251)	0.942(0.398)	0.949(0.280)
		0.3	0.948(0.629)	0.947(0.664)	0.947(0.635)
		0.5	0.949(0.891)	0.945(0.871)	0.946(0.877)
		0.8	0.946(1.176)	0.945(1.109)	0.945(1.139)
		1.0	0.951(1.315)	0.947(1.239)	0.951(1.280)
NCT CIs	(12,14)	0.1	0.951 (0.131)	0.951(0.147)	0.951(0.137)	.951(0.113)
		0.3	0.950 (0.360)	0.952(0.364)	0.951(0.359)	.947(0.325)
		0.5	0.950 (0.541)	0.950(0.526)	0.950(0.530)	.945(0.506)
		0.8	0.950 (0.745)	0.951(0.710)	0.951(0.721)	.941(0.711)
		1.0	0.949 (0.845)	0.950(0.808)	0.949(0.818)	.942(0.811)
normal CIs	(12,14)	0.1	0.947 (0.129)	0.945 (0.148)	0.948(0.135)
		0.3	0.946 (0.352)	0.947 (0.361)	0.947(0.354)
		0.5	0.945 (0.531)	0.946 (0.518)	0.946(0.520)
		0.8	0.945 (0.729)	0.946 (0.698)	0.946(0.708)
		1.0	0.946 (0.830)	0.946 (0.795)	0.947(0.804)
		$p$ = 0.75
NCT CIs	(6,6)	0.1	0.949(0.276)	0.950(0.346)	0.950(0.288)	.949(0.228)
		0.3	0.945(0.708)	0.951(0.733)	0.949(0.706)	.942(0.641)
		0.5	0.950(1.028)	0.949(0.999)	0.950(1.006)	.936(0.974)
		0.8	0.951(1.376)	0.950(1.295)	0.950(1.335)	.933(1.334)
		1.0	0.951(1.548)	0.951(1.453)	0.950(1.501)	.932(1.506)
normal CIs	(6,6)	0.1	0.953(0.301)	0.948(0.430)	0.951(0.323)
		0.3	0.953(0.724)	0.948(0.748)	0.951(0.708)
		0.5	0.953(1.026)	0.949(0.989)	0.952(1.001)
		0.8	0.953(1.360)	0.948(1.127)	0.952(1.312)
		1.0	0.952(1.528)	0.949(1.418)	0.952(1.473)
NCT CIs	(12,14)	0.1	0.948(0.147)	0.950(0.163)	0.951(0.153)	.951(0.127)
		0.3	0.950(0.401)	0.949(0.405)	0.951(0.399)	.948(0.365)
		0.5	0.950(0.604)	0.949(0.586)	0.949(0.591)	.943(0.568)
		0.8	0.951(0.832)	0.950(0.794)	0.950(0.804)	.941(0.800)
		1.0	0.949(0.945)	0.951(0.905)	0.950(0.915)	.942(0.913)
normal CIs	(12,14)	0.1	0.948(0.147)	0.948(0.166)	0.949(0.154)
		0.3	0.950(0.399)	0.950(0.403)	0.949(0.398)
		0.5	0.950(0.598)	0.948(0.580)	0.949(0.585)
		0.8	0.951(0.823)	0.949(0.784)	0.948(0.796)
		1.0	0.949(0.934)	0.949(0.893)	0.949(0.903)

Table 1 continued.
$k=2,\sigma_{1}=1,\xi=5$ $(n_{1},n_{2})$ $\sigma_{2}$ Fisher Inv Norm Inv $\chi^{2}$ Fiducial $p$ = 0.90 NCT CIs (6,6) 0.1 .953(0.354) .952(0.423) .953(0.367) .949(0.293) 0.3 .952(0.900) .951(0.912) .953(0.891) .941(0.826) 0.5 .954(1.304) .948(1.259) .952(1.273) .934(1.255) 0.8 .952(1.745) .951(1.641) .952(1.691) .928(1.721) 1.0 .950(1.967) .950(1.833) .950(1.892) .929(1.947) normal CIs (6,6) 0.1 .947(0.339) .938(0.434) .943(0.354) 0.3 .946(0.824) .939(0.832) .944(0.812) 0.5 .944(1.179) .935(1.131) .941(1.146) 0.8 .946(1.583) .938(1.476) .943(1.527) 1.0 .944(1.781) .939(1.657) .943(1.717) NCT CIs (12,14) 0.1 .949(0.182) .949(0.200) .948(0.189) .951(0.157) 0.3 .952(0.496) .952(0.498) .952(0.494) .948(0.454) 0.5 .952(0.747) .954(0.724) .952(0.730) .941(0.708) 0.8 .951(1.026) .951(0.980) .951(0.993) .940(0.996) 1.0 .952(1.167) .951(1.115) .951(1.128) .939(1.138) normal CIs (12,14) 0.1 .948(0.178) .943(0.196) .944(0.185) 0.3 .944(0.476) .937(0.476) .941(0.473) 0.5 .949(0.718) .946(0.694) .949(0.700) 0.8 .947(0.987) .944(0.940) .947(0.953) 1.0 .947(1.126) .941(1.074) .943(1.087)

Table 2. Coverage probabilities and (Expected Widths) of 95% CIs based on combined NCT tests and combined normal tests

	$k=4,\sigma_{1}=1,\xi=5$
$(n_{1},...,n_{4})$	$(\sigma_{2},\sigma_{3},\sigma_{4})$	Fisher	Inv Norm	Inv $\chi^{2}$	Fiducial
NCT CIs	$p$ = 0.50
(6,6,6,6)	(.2,.3,.4)	.950(0.313)	.949(0.305)	.951(0.305)	.921(0.275)
	(.2,.8,.5)	.945(0.401)	.948(0.411)	.946(0.397)	.929(0.333)
	(.1,.9,.1)	.953(0.162)	.953(0.181)	.953(0.163)	.930(0.130)
	(.1,1,1)	.950(0.285)	.948(0.378)	.950(0.305)	.945(0.194)
	(.5,.8,1)	.948(0.718)	.946(0.672)	.949(0.690)	.915(0.651)
normal CIs
(6,6,6,6)	(.2,.3,.4)	.952(0.307)	.949(0.300)	.948(0.299)
	(.2,.8,.5)	.950(0.403)	.947(0.412)	.947(0.399)
	(.1,.9,.1)	.953(0.160)	.945(0.188)	.951(0.163)
	(.1,1,1)	.951(0.328)	.953(0.431)	.947(0.357)
	(.5,.8,1)	.949(0.699)	.951(0.655)	.947(0.672)
NCT CIs
(6,16,5,10)	(.2,.3,.4)	.952(0.215)	.954(0.188)	.952(0.198)	.929(0.184)
	(.2,.8,.5)	.952(0.242)	.951(0.205)	.953(0.222)	.935(0.196)
	(.1,.9,.1)	.951(0.100)	.949(0.085)	.952(0.091)	.940(0.081)
	(.1,1,1)	.951(0.142)	.951(0.125)	.952(0.135)	.947(0.104)
	(.5,.8,1)	.951(0.506)	.952(0.448)	.951(0.467)	.925(0.448)
normal CIs
(6,16,5,10)	(.2,.3,.4)	.951(0.209)	.950(0.185)	.950(0.194)
	(.2,.8,.5)	.951(0.237)	.949(0.203)	.949(0.218)
	(.1,.9,.1)	.949(0.098)	.950(0.083)	.948(0.089)
	(.1,1,1)	.948(0.140)	.948(0.124)	.946(0.133)
	(.5,.8,1)	.948(0.494)	.949(0.441)	.946(0.458)
NCT CIs	$p=0.75$
(6,6,6,6)	(.2,.3,.4)	.950(.352)	.950(.341)	.950(.342)	.920(.317)
	(.2,.8,.5)	.952(.450)	.951(.457)	.952(.443)	.929(.383)
	(.1,.9,.1)	.948(.183)	.948(.201)	.948(.184)	.930(.149)
	(.1,1,1)	.950(.317)	.947(.409)	.950(.334)	.947(.223)
	(.5,.8,1)	.952(.806)	.950(.753)	.951(.774)	.914(.749)
normal CIs
(6,6,6,6)	(.2,.3,.4)	.954(0.349)	.947(0.335)	.950(0.337)
	(.2,.8,.5)	.955(0.450)	.948(0.455)	.952(0.442)
	(.1,.9,.1)	.954(0.184)	.948(0.205)	.953(0.185)
	(.1,1,1)	.954(0.355)	.949(0.468)	.952(0.380)
	(.5,.8,1)	.956(0.792)	.950(0.734)	.954(0.757)

Table 2 continued.

$(n_{1},...,n_{4})$	$(\sigma_{2},\sigma_{3},\sigma_{4})$	Fisher	Inv Norm	Inv $\chi^{2}$	Fiducial
	$k=4,\sigma_{1}=1,\xi=5$
NCT CIs	$p=0.75$
(6,16,5,10)	(.2,.3,.4)	.953(.238)	.949(.208)	.952(.219)	.930(.209)
	(.2,.8,.5)	.952(.269)	.952(.228)	.954(.247)	.935(.221)
	(.1,.9,.1)	.951(.111)	.951(.095)	.949(.101)	.939(.091)
	(.1,1,1)	.950(.158)	.951(.139)	.951(.150)	.948(.117)
	(.5,.8,1)	.954(.564)	.950(.499)	.952(.521)	.924(.508)
normal CIs
(6,16,5,10)	(.2,.3,.4)	.953(0.235)	.949(.206)	.950(0.217)
	(.2,.8,.5)	.954(0.267)	.948(.225)	.951(0.244)
	(.1,.9,.1)	.948(0.110)	.945(.093)	.946(0.100)
	(.1,1,1)	.952(0.159)	.952(.138)	.951(0.149)
	(.5,.8,1)	.954(0.556)	.950(.492)	.948(0.512)

Table 3. Coverage Probabilities and (Expected Widths) of 95% CIs based on Combined NCT tests and Fiducial

$(n_{1},...,n_{4})$	$(\sigma_{2},\sigma_{3},\sigma_{4})$	Fisher	Inv Norm	Inv $\chi^{2}$	Fiducial
	$k=4,\sigma_{1}=1,\xi=5$
$p=0.50$
(10,10,10,10)	(.2,.3,.4)	.955(.228)	.953(.225)	.954(.223)	.934(.201)
	(.2,.8,.5)	.954(.285)	.952(.297)	.954(.285)	.939(.240)
	(.1,.9,.1)	.953(.115)	.953(.127)	.953(.117)	.938(.094)
	(.1,1,1)	.954(.187)	.952(.250)	.953(.206)	.948(.136)
	(.5,.8,1)	.952(.526)	.952(.500)	.952(.505)	.933(.478)
(10,15,10,15)	(.2,.3,.4)	.952(.193)	.951(.183)	.951(.185)	.939(.169)
	(.2,.8,.5)	.951(.230)	.951(.223)	.953(.225)	.945(.193)
	(.1,.9,.1)	.951(.090)	.953(.086)	.949(.088)	.943(.075)
	(.1,1,1)	.950(.144)	.950(.167)	.951(.153)	.949(.107)
	(.5,.8,1)	.951(.451)	.950(.425)	.949(.430)	.936(.408)
(15,15,15,15)	(.2,.3,.4)	.949(.180)	.951(.179)	.951(.176)	.940(.160)
	(.2,.8,.5)	.947(.223)	.949(.235)	.948(.225)	.944(.190)
	(.1,.9,.1)	.947(.090)	.949(.099)	.948(.092)	.942(.074)
	(.1,1,1)	.948(.141)	.949(.187)	.949(.158)	.948(.107)
	(.5,.8,1)	.947(.417)	.950(.400)	.948(.401)	.939(.381)
(15,20,15,20)	(.2,.3,.4)	.950(.161)	.954(.156)	.952(.156)	.944(.142)
	(.2,.8,.5)	.950(.194)	.953(.192)	.951(.192)	.947(.164)
	(.1,.9,.1)	.951(.077)	.953(.075)	.951(.076)	.945(.064)
	(.1,1,1)	.952(.120)	.952(.143)	.952(.130)	.949(.091)
	(.5,.8,1)	.951(.377)	.953(.358)	.950(.361)	.942(.343)
(20,20,20,20)	(.2,.3,.4)	.948(.154)	.946(.153)	.947(.151)	.943(.137)
	(.2,.8,.5)	.946(.189)	.946(.200)	.949(.192)	.946(.162)
	(.1,.9,.1)	.948(.077)	.946(.084)	.948(.079)	.949(.064)
	(.1,1,1)	.946(.118)	.946(.155)	.946(.133)	.950(.091)
	(.5,.8,1)	.946(.356)	.945(.343)	.946(.343)	.943(.327)

Table 3 continued.
$k=4,\sigma_{1}=1,\xi=5$ $(n_{1},...,n_{4})$ $(\sigma_{2},\sigma_{3},\sigma_{4})$ Fisher Inv Norm Inv $\chi^{2}$ Fiducial $p=0.75$ (10,10,10,10) (.2,.3,.4) .950(.254) .950(.250) .950(.248) .935(.227) (.2,.8,.5) .952(.317) .949(.329) .951(.316) .941(.271) (.1,.9,.1) .949(.128) .948(.141) .947(.131) .939(.106) (.1,1,1) .948(.209) .948(.272) .948(.227) .951(.154) (.5,.8,1) .950(.583) .952(.556) .950(.561) .932(.540) (10,15,10,15) (.2,.3,.4) .951(.214) .950(.203) .952(.206) .938(.190) (.2,.8,.5) .949(.256) .950(.247) .950(.249) .944(.216) (.1,.9,.1) .949(.101) .949(.095) .950(.098) .942(.084) (.1,1,1) .951(.160) .947(.183) .949(.169) .950(.120) (.5,.8,1) .954(.503) .950(.473) .952(.479) .934(.459) (15,15,15,15) (.2,.3,.4) .947(.200) .950(.199) .948(.196) .940(.179) (.2,.8,.5) .947(.247) .951(.259) .950(.249) .945(.213) (.1,.9,.1) .948(.100) .949(.110) .949(.103) .942(.084) (.1,1,1) .949(.157) .951(.205) .951(.174) .951(.120) (.5,.8,1) .950(.463) .948(.444) .949(.445) .940(.428) (15,20,15,20) (.2,.3,.4) .951(.179) .951(.173) .951(.174) .942(.160) (.2,.8,.5) .955(.214) .955(.212) .954(.212) .947(.183) (.1,.9,.1) .952(.085) .953(.084) .952(.085) .944(.071) (.1,1,1) .952(.134) .951(.157) .952(.144) .951(.102) (.5,.8,1) .952(.418) .953(.397) .952(.401) .939(.384) (20,20,20,20) (.2,.3,.4) .953(.171) .954(.170) .953(.168) .943(.153) (.2,.8,.5) .955(.211) .952(.220) .953(.212) .946(.181) (.1,.9,.1) .950(.085) .949(.093) .949(.088) .943(.071) (.1,1,1) .955(.132) .952(.169) .952(.147) .951(.101) (.5,.8,1) .950(.397) .952(.381) .953(.381) .941(.366)

5. Examples

Example 4.5.1 The relationship between Vitamin D and colorectal cancer (CRC) has been a topic of interest since the hypothesis proposed by [garland1980sunlight]. While initial studies suggested a potential protective effect of Vitamin D against CRC, subsequent research, including a study conducted at Roswell Park Cancer Center, aimed to clarify the impact of Vitamin D supplementation on CRC incidence and mortality. Participants in the study were divided into three groups based on their baseline serum 25-D3 levels: (i) Vitamin D3 Deficient: Serum 25-D3 levels less than 20 ng/ml, (ii) Vitamin D3 Insufficient: Serum 25-D3 levels between 20 ng/ml or more and less than 32 ng/ml and (iii) Vitamin D3 Sufficient: Serum 25-D3 levels greater than or equal to 32 ng/ml. Subjects within each group received vitamin D supplements for 6 months, and serum levels of 1,25-D3 and 24,25-D3 were measured at the end of this period. The data were analyzed by [li2012comparison] and their tests for normality indicated that the data fit normal distributions. This finding is crucial for selecting appropriate statistical tests for analyzing the effects of vitamin D supplementation.

Table 4. Descriptive statistics for the data from a vitamin D study

Group	Variable	Size ( $n_{i}$ )	Mean ( $\bar{x}_{i}$ )	SD ( $s_{i}$ )
Vitamin D3 sufficient	1, 25-D3	16	62.39	17.99
	24, 25-D3	17	4.65	1.98
Vitamin D3 insufficient	1, 25-D3	22	72.60	23.52
	24, 25-D3	22	3.62	1.17
Vitamin D3 deficient	1, 25-D3	9	70.13	19.67
	24, 25-D3	9	2.66	1.35

A hypothesis of interest here is that if all of these three vitamin D groups would reach the same serum 1, 25-D3 and 24, 25-D3 vitamin levels. [li2012comparison] have noted that a thorough understanding about how the distributions differ across three groups is to compare the quantiles, especially the common quantiles such as 1st quartile, median, and third quartile. [li2012comparison] have provided the summary statistics of serum 1, 25-D3 and 24, 25-D3 vitamin D3 metabolites in Table 6 of their paper, and the statistics are reproduced here in Table 4.

The tests by [li2012comparison] and Section LABEL:ch3_s5 indicated that no significant differences among the groups in Serum 1 25-D3 in terms of quartiles and medians. So it is desired to estimate the common quartiles and the common mean (median) of these three different groups. Table 5 shows 95% CIs for obtained using different methods as indicated in percentiles, $p$ , using the combination of the noncentral $t$ tests and normal approximate tests.

Table 5. Confidence intervals for common quantiles in the vitamin D study

methods	$p$	Fisher	Inv Norm	Inv $\chi^{2}$	Fiducial
	95% Confidence Intervals
NCT CIs	0.25	$(44.89,60.53)$	$(46.23,60.37)$	$(45.86,60.24)$	(45.26, 59.67)
normal CIs		$(45.28,60.64)$	$(46.56,60.49)$	$(46.23,60.36)$
NCT CIs	0.50	$(61.24,74.57)$	$(61.91,75.04)$	$(61.71,74.54)$	(61.19, 74.02)
normal CIs		$(61.38,74.46)$	$(62.00,74.95)$	$(61.83,74.45)$
NCT CIs	0.75	$(76.39,90.60)$	$(76.55,91.32)$	$(76.48,90.59)$	$(75.53,89.94)$
normal CIs		$(76.28,90.39)$	$(76.41,91.10)$	$(76.35,90.36)$

NOTE: NCT CIs – CIs obtained by inverting a combination of individual NCT tests; normal CIs – CIs obtained by inverting a combination of individual normal approximate tests

The NCT CIs and the normal CIs based on all three methods are practically the same. The fiducial CIs are also in agreement with other CIs.

Example 4.5.2 This example and data were taken from [eberhardt1989minimax]. Four different analytical methods were used to estimate the mean selenium content in non-fat milk powder. The means and variances of samples of measurements are reported in Table 6.

Table 6. Selenium content in non-fat milk powder using four methods.

Method	Sample Size ( $n_{i}$ )	Mean ( $\bar{x}_{i}$ )	Variance ( $s^{2}_{i}$ )
.Atomic absorption Spectrometry	8	105.00	85.711
Neutron activation: Instrumental	12	109.75	20.748
Neutron activation: Radiochemical	14	109.50	2.729
Isotope dilution mass spectrometry	8	113.25	33.640

We first conducted a hypothesis test to assess whether quantiles of measurements by different analytical methods are the same. We applied the parametric bootstrap test and a modified likelihood ratio test (MLRT) proposed in Section LABEL:sec-LRT to test the hypotheses

(15)

H_{0}:\xi_{1}=...=\xi_{4}\ \ {\rm vs.}\ \ H_{a}:\xi_{i}\neq\xi_{j}\ \mbox{for % some }i\neq j.

Equality of the quantiles is tenable when $p=0.4,0.5$ and $0.6$ . We report the confidence intervals for the common quantiles in Table 7. We focus on the combined-test confidence interval based on the noncentral $t$ -distribution, as the confidence intervals obtained using the normal approximation were nearly identical. We notice in Table 7 that CIs based all the methods are in good agreement, except the fiducial CIs. Even though, fiducial CIs are somewhat different from other exact CIs, they are practically equivalent to other CIs.

As noted previously, in the normal case with $p=0.5$ , the common quantile coincides with the common mean, and the corresponding confidence interval reduces to a confidence interval for the common mean. The data analyzed here were also used by [krishnamoorthy2024confidence], and for $p=0.5$ our results agree with those reported in Table 4 of their paper.

Table 7. 95% Confidence intervals for the percentiles of selenium content in non-fat milk powder

Method	$p=0.4$	$p=0.5$	$p=0.6$
Approx. Fiducial	$(108.06,\,109.86)$	$(108.72,\,110.49)$	$(109.34,\,111.14)$
Fisher	$(107.77,\,109.82)$	$(108.60,\,110.65)$	$(109.44,\,111.51)$
Inv Normal	$(107.73,\,109.79)$	$(108.63,\,110.63)$	$(109.43,\,111.58)$
Inverse $\chi^{2}$	$(107.75,\,109.79)$	$(108.63,\,110.63)$	$(109.46,\,111.55)$