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- Abstract
- 1 Introduction
- 2 A Motivating Example
- 3 Prior Effective Sample Size
- 4 Case Studies of Data analysis and Study Design
- 5 Discussion
- References

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Stat Biosci. Author manuscript; available in PMC 2010 July 27.

Published in final edited form as:

Stat Biosci. 2010 July 1; 2(1): 1–17.

doi: 10.1007/s12561-010-9018-xPMCID: PMC2910452

NIHMSID: NIHMS183751

Satoshi Morita, Department of Biostatistics and Epidemiology, Yokohama City University Medical Center, 4-57 Urafune-cho, Minami-ku, Yokohama 232-0024, Japan, Tel.: +81-45-253-5399, Fax: +81-45-253-9902;

Satoshi Morita: pj.ca.uc-amahokoy.pharu@atiroms; Peter F. Thall: gro.nosrednadm@xer; Peter Müller: gro.nosrednadm@relleump

See other articles in PMC that cite the published article.

A common concern in Bayesian data analysis is that an inappropriately informative prior may unduly influence posterior inferences. In the context of Bayesian clinical trial design, well chosen priors are important to ensure that posterior-based decision rules have good frequentist properties. However, it is difficult to quantify prior information in all but the most stylized models. This issue may be addressed by quantifying the prior information in terms of a number of hypothetical patients, i.e., a prior effective sample size (ESS). Prior ESS provides a useful tool for understanding the impact of prior assumptions. For example, the prior ESS may be used to guide calibration of prior variances and other hyperprior parameters. In this paper, we discuss such prior sensitivity analyses by using a recently proposed method to compute a prior ESS. We apply this in several typical Bayesian biomedical data analysis and clinical trial design settings. The data analyses include cross-tabulated counts, multiple correlated diagnostic tests, and ordinal outcomes using a proportional-odds model. The study designs include a phase I trial with late-onset toxicities, a phase II trial that monitors event times, and a phase I/II trial with dose-finding based on efficacy and toxicity.

Understanding the strength of prior assumptions relative to the likelihood is a fundamental issue when applying Bayesian methods. The processes of formulating a putatively non-informative prior or eliciting a prior from an area expert typically require one to make many arbitrary choices, including the choice of particular distributional forms and numerical hyperparameter values. In practice, these choices are often dictated by technical convenience. A common criticism of Bayesian analysis is that an inappropriately informative prior may unduly influence posterior inferences and decisions. However, it is difficult to quantify and critique prior information in all but the most stylized models. These concerns may be addressed by quantifying the prior information in terms of an equivalent number of hypothetical patients, i.e., a prior effective sample size (ESS). Such a summary allows one to judge the relative contributions of the prior and the data to the final conclusions. A useful property of prior ESS is that it is readily interpretable by any scientifically literate reviewer without requiring expert mathematical training. This is important, for example, for consumers of clinical trial results.

The purpose of this paper is to discuss prior sensitivity analyses in Bayesian biostatistics by computing the prior ESS for six case studies chosen from the recent literature. We apply an ESS method proposed by Morita, Thall and Müller (MTM) [7]. Some of our case studies require prior ESS values for a subvector *θ*_{1} of the parameter vector ** θ** = (

The case studies consist of three Bayesian data analyses and three study designs. The data analysis examples include small-sample cross-tabulated counts from an animal experiment to evaluate mechanical ventilator devices, bivariate normal modeling of paired data from multiple correlated diagnostic serologic tests, and proportional odds modeling of ordinal outcomes arising from a study of viral effects in chick embryos. The study design examples include a phase I trial with dose-finding using the time-to-event continual reassessment method (TITE-CRM) [2], a phase II trial with a stopping rule for monitoring event times, and a phase I/II clinical trial in which doses were assigned based on both efficacy and toxicity.

Section 2 provides a motivating example. In section 3, we briefly summarize MTM. We discuss prior sensitivity in the real examples of Bayesian data analyses and study designs in Section 4. We close with a brief discussion in Section 5.

The following example illustrates how the prior ESS may be used as an index of prior informativeness in a Bayesian sensitivity analysis and as a tool for critiquing a Bayesian data analysis when interpreting or formally reviewing the analysis.

Carlin [1] analyzed small-sample contingency table data from an experiment carried out to examine the effects of mechanical ventilator devices on lung damage in rabbits. In the experiment, the lungs of newborn rabbits were altered to simulate lung defects seen in human infants with underdeveloped lungs due to premature birth. The aim was to learn about the joint effects of different frequency and amplitude settings of the ventilators on lung damage. Six groups of six to eight animals each were compared using a factorial design with three frequency values crossed with two amplitudes. For amplitude *g* (= 1, 2 for 20 and 60, respectively) and frequency *h* (= 1, 2, 3 for 5, 10, and 15 Hz, respectively), let *Y _{g,h}* denote the number of animals with lung damage out of

$${\pi}_{g,h}(\mathit{\theta})={\mathit{logit}}^{-1}\left(\mu +{\alpha}_{g}+{\beta}_{h}+{\gamma}_{h}{I}_{(g=2)}\right),$$

where *I*_{(}_{g}_{=2)} indicates *g* = 2. Note that *α _{g}* and

$$\mu \sim N(0,{1000}^{2}),\phantom{\rule{0.16667em}{0ex}}{\alpha}_{g}\sim N(0,{\stackrel{\sim}{\sigma}}_{\alpha}^{2}),\phantom{\rule{0.16667em}{0ex}}{\beta}_{h}\sim N(0,{\stackrel{\sim}{\sigma}}_{\beta}^{2}),\phantom{\rule{0.16667em}{0ex}}{\gamma}_{h}\sim N(0,{\stackrel{\sim}{\sigma}}_{\gamma}^{2}).$$

(1)

Since the numbers of animals studied were very small, Carlin [1] explored the effect of a range of non-informative prior distributions in the analysis.

We use prior ESS to investigate sensitivity of the inferences to hyperparameter values by considering ten alternative choices that cover a range of reasonably non-informative settings. The ten hyperparameter choices, labeled N1 to N10, are shown in Table 2. We add the four priors N1 to N4 which would have smaller ESS values than those considered by Carlin [1] for priors N5 to N10. We apply MTM’s method to compute an overall ESS for *p*(** θ**|

The prior N10 with * _{α}* =

We briefly summarize the definition of ESS proposed by MTM [7]. While the discussion following this section does not require these details, we include this brief review for completeness.

Let *f*(*Y* | ** θ**) be the sampling model for a random vector

This constructive definition may be understood in terms of the simple example where *p*(** θ** |

For the general construction, denote **Y*** _{m}* = (

$${q}_{m}(\mathit{\theta}\mid {\stackrel{\sim}{\mathit{\theta}}}_{0},{\mathbf{Y}}_{m},{\mathbf{X}}_{m})\propto {q}_{0}(\mathit{\theta}\mid {\stackrel{\sim}{\mathit{\theta}}}_{0}){f}_{m}({\mathbf{Y}}_{m}\mid {\mathbf{X}}_{m},\mathit{\theta}).$$

The ESS is the interpolated value of *m* minimizing the prior-to-posterior distance *δ* between *q _{m}* (

MTM define the prior-to-posterior distance as the difference between the traces of the information matrix of *p*(** θ** |

$$\delta (m,\overline{\mathit{\theta}},p,{q}_{0})=\left|\sum _{j=1}^{d}{D}_{p,j}(\overline{\mathit{\theta}})-\sum _{j=1}^{d}\int {D}_{q,j}(m,\overline{\mathit{\theta}},{\mathbf{Y}}_{m})\text{d}{f}_{m}({\mathbf{Y}}_{m}\mid \stackrel{\sim}{\mathit{\theta}})\right|$$

(2)

where *D _{p,j}* and

If interest is focused on a subvector *θ** _{r}* of

$${f}_{m}({\mathbf{Y}}_{m}\mid \stackrel{\sim}{\mathit{\theta}},\stackrel{\sim}{\mathit{\xi}})=\int {f}_{m}({\mathbf{Y}}_{m}\mid {\mathbf{X}}_{m},\mathit{\theta}){g}_{m}({\mathbf{X}}_{m}\mid \mathit{\xi})p(\mathit{\theta}\mid \stackrel{\sim}{\mathit{\theta}})r(\mathit{\xi}\mid \stackrel{\sim}{\mathit{\xi}})d\mathit{\theta}\phantom{\rule{0.16667em}{0ex}}d\mathit{\xi}.$$

The following examples show how prior sensitivity may be evaluated using ESS in data analysis and clinical trial design settings. The first three, Examples 1 to 3, are data analyses and the latter three, Example 4 to 6, are clinical trial designs. In each example, we explain how the prior ESS can be used as a tool to calibrate prior hyperparameters. Following Gelman *et al*. [6], we write Unif(*α, β*), Be(*α, β*), Bin(*n, θ*), Ga(*α, β*), IG(*α, β*), Exp(*θ*), N(*μ, σ*^{2}), and MVN(*μ, Σ*), for the uniform, beta, binomial, gamma, exponential, normal, and multivariate normal distributions.

This example was described earlier, in Section 2. The goal of Carlin’s analysis was to examine the effects of mechanical ventilator devices on lung damage in rabbits. Because of the very small numbers of studied animals, as shown in Table 1, the effect of a range of non-informative priors was explored. Recall that *π _{g,h}* was the probability of lung damage in cell (

Assuming a binomial model, *Y _{g,h}* |

$$f({\mathbf{Y}}_{m}\mid \mathit{\theta})\propto \prod _{g=1}^{2}\prod _{h=1}^{3}{\pi}_{g,h}{(\mathit{\theta})}^{{Y}_{g,h}}{\{1-{\pi}_{g,h}(\mathit{\theta})\}}^{{n}_{g,h}-{Y}_{g,h}}.$$

Using MTM’s method, we compute an overall ESS, and also *ESS _{μ}*,

While each of the hyperparameters * _{α}*,

Choi *et al*. [3] used a bivariate normal model to analyze multiple correlated diagnostic tests. They considered the problem of comparing two serologic tests, both enzyme-linked immunosorbent assays (ELISA) for detection of antibodies to Johne’s disease in dairy cattle. Data from *n*_{1} = 88 diseased animals and *n*_{0} = 393 disease-free animals were reported. The two tests have continuous outcomes, which we denote by *Y*_{1}* _{iD}* and

A bivariate normal distribution is assumed for the test scores, (*Y*_{1}_{i}, Y_{2}* _{i}*), from the

$$({Y}_{i1},{Y}_{i2})\sim N(\mathit{\mu},\mathrm{\sum}),$$

(3)

so that ** θ** = (

$${\mu}_{j}\sim N({\stackrel{\sim}{\mu}}_{j},{\stackrel{\sim}{\tau}}_{j}^{-1})\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}{\tau}_{j}\sim \text{Ga}({\stackrel{\sim}{a}}_{j},{\stackrel{\sim}{b}}_{j}),\phantom{\rule{0.16667em}{0ex}}j=1,2$$

(4)

with * _{j}* = 0,

To show how ESS may be applied as a tool for prior elicitation in this setting, we consider the four alternative priors shown in Table 3. This serves as an informal sensitivity analysis. Also, similar to the discussion in Section 4.1 we plot ESS as a function of the hyperparameters. We compute the ESS for _{1} = _{2} and *ã*_{1} = *ã*_{2} = _{1} = _{2} each ranging from 0.001 to 10, keeping = = 1 fixed. Figure 2 gives plots of the resulting ESS values. For example, the prior ESS for _{1} = _{2} = 10, *ã*_{1} = *ã*_{2} = _{1} = _{2} = 10, = = 1 is 9.5. This may be criticized as unacceptably high, considering the sample size of *n* = 88 diseased animals. In contrast, priors with all hyperparameters less than 1 correspond to reasonably small prior ESS.

Congdon [4] (Section 10.3.2) reports a data analysis based on a proportional odds model for ordinal response data, as shown in Table 4. The data report deformity or mortality in chick embryos as a result of arbovirus injection. Two virus groups, Facey’s Paddock (*g* = 1) and Tinaroo (*g* = 2), and a control group (*g* = 0) were investigated. The control group received no virus. The two virus groups and the control group contained *n*_{1} = 75, *n*_{2} = 72, and *n*_{0} = 18 embryos, respectively. Each embryo in the Facey’s Paddock group received one of the four doses, {3, 18, 30, 90}, denoted by {*d*_{1,1}*, d*_{1,2}*, d*_{1,3}*, d*_{1,4}}. For the Tinaroo group, the doses were {3, 20, 2400, 88000}, denoted by {*d*_{2,1}*, d*_{2,2}*, d*_{2,3}*, d*_{2,4}}. The response *Y _{g,i}* for embryo

In this example, a nonzero probability of death was assumed for zero dose. This accounts for a possible background mortality effect. In fact, one death was observed among the controls. The response in the control group *Y*_{0}* _{,i}* is assumed to be binary (0 or 2) rather than trinary, with

$${\pi}_{g,i,h}=Pr({Y}_{g,i}=h)=\alpha +(1-\alpha ){P}_{g,i,h}.$$

(5)

with *P _{g,i}*

$${\gamma}_{g,i,h}={\mathit{logit}}^{-1}({\kappa}_{g,i}+{\beta}_{g}{X}_{g,i})$$

(6)

for *h* = 1, 2 with *γ _{g,i,}*

$$f({\mathbf{Z}}_{m}\mid \mathit{\theta})\propto \prod _{{i}_{0}=1}^{{m}_{0}}{(1-\alpha )}^{{Z}_{0,{i}_{0},0}}{\alpha}^{{Z}_{0,{i}_{0},2}}\prod _{{i}_{1}=1}^{{m}_{1}}{\pi}_{1,{i}_{1},0}^{{Z}_{1,{i}_{1},0}}{\pi}_{1,{i}_{1},1}^{{Z}_{1,{i}_{1},1}}{\pi}_{1,{i}_{1},2}^{{Z}_{1,{i}_{1},2}}\prod _{{i}_{2}=1}^{{m}_{2}}{\pi}_{2,{i}_{2},0}^{{Z}_{2,{i}_{2},0}}{\pi}_{2,{i}_{2},1}^{{Z}_{2,{i}_{2},1}}{\pi}_{2,{i}_{2},2}^{{Z}_{2,{i}_{2},2}}.$$

Congdon [4] assumes independent prior distributions:

$$\alpha \sim Be(\stackrel{\sim}{\phi},\stackrel{\sim}{\phi}),\phantom{\rule{0.16667em}{0ex}}{\beta}_{j}\sim N({\stackrel{\sim}{\mu}}_{\beta},{\stackrel{\sim}{\sigma}}_{\beta}^{2}),\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}{\kappa}_{j,h}\sim N({\stackrel{\sim}{\mu}}_{\kappa},{\stackrel{\sim}{\sigma}}_{\kappa}^{2}),\phantom{\rule{0.38889em}{0ex}}j=1,2,\phantom{\rule{0.38889em}{0ex}}h=1,2$$

with = 1, * _{β}* = 0 and
${\stackrel{\sim}{\sigma}}_{\beta}^{2}=10$,

We evaluate the overall ESS for ** θ**, and subvector-specific ESS values

The continual reassessment method (CRM) [9] is used for dose-finding in phase I clinical trials based on a binary indicator of toxicity. The CRM requires complete follow-up of the current patient (or cohort) before enrolling a new patient or cohort. Depending on how long it takes to evaluate toxicity, this may lead to an unduly long study duration that make the method impractical. Cheung and Chappell [2] proposed an extension, the time-to-event (TITE) CRM, that uses time to toxicity or right censoring as the outcome.

Elkind *et al.* [5] applied the TITE-CRM to determine the maximum tolerated dose (MTD) of short-term high-dose lovastatin in stroke patients treated within 24 hours of symptom onset. Each patient received one of five initial doses 1, 3, 6, 8, 10 mg/kg, on days 1–3 post onset and received 20 mg/day for the next 27 days. Toxicity was assessed up to day 30, that is, the observation window was *T _{up}* = 30 days. Denote the time-to-toxicity in patient

$$Pr({Y}_{i}=1\mid {d}_{[i]},\beta )=F({d}_{[i]},\beta )={d}_{[i]}^{\mathit{exp}(\beta )}$$

was assumed, where *d*_{[}_{i}_{]} is the standardized dose level assigned to patient *i*. A *N*(0, 1.34) distribution was assumed for the prior of *β*. The five standardized doses **d** = (*d*_{1}*, d*_{2}*, d*_{3}*, d*_{4}*, d*_{5}) in the model were assumed to be (0.02, 0.06, 0.10, 0.18, 0.30). In general, the TITE-CRM is implemented using the weighted working likelihood for *m* patients given by

$${f}_{m}({\mathbf{Y}}_{m},{\mathbf{u}}_{m}\mid {\mathbf{d}}_{m},\beta )=\prod _{i=1}^{m}F{({d}_{[i]},\beta )}^{{Y}_{i}}{\left\{1-{w}_{i}F({d}_{[i]},\beta )\right\}}^{1-{Y}_{i}},$$

(7)

where *w _{i}* is a suitable weight function. For the lovastatin trial,

In Table 5, we assume five dose-toxicity scenarios in order to assess effects of the prior ESS. Scenario (1) corresponds to toxicity probabilities equal to the standardized doses. Scenarios (2)–(5) were constructed by starting with Scenario (1) and increasing the toxicity probabilities. In Scenarios (2) and (3), the toxicity probabilities increase linearly with dose, with all doses too toxic in Scenario (3). In Scenario (4), only *d*_{1} is safe, with toxicity increasing rapidly from *d*_{2} onward. In Scenario (5), all doses are very toxic.

As Cheung and Chappell [2] do, we assume three models for the patients’ times to toxicity, including a conditionally uniform model, a Weibull model (with a fixed shape parameter 4), and a log-logistic model (with a fixed shape parameter 1). The cumulative distribution function (CDF) of the Weibull model with a scale parameter *α* is *F*(*u, α*) = 1 − *exp*{−(*u/α*)^{4}} and the CDF of the log-logistic model with a scale parameter *α* is *F*(*u, α*) = (1 + *exp*[−{*log*(*u*) − *log*(*α*)}])^{−1}.

We compute the ESS values under each model. Figure 4 gives plots of the ESS values as a function of ^{2} under the five toxicity scenarios, assuming the conditionally uniform model for time-to-toxicity and with the prior mean of *β* fixed at = 0. Since the ESS computed at ^{2} = 1.34 is less than 2 under Scenario (1), the information from the likelihood will dominate the prior after enrolling 3 patients, hence the prior specified in the lovastatin trial seems quite reasonable. The prior also makes sense under Scenarios (2) and (3). The plot of ESS under Scenarios (4) and (5) indicates that the prior may be problematic, however. Under Scenario (5), it appears that ^{2} > 2.5 may be needed to ensure an ESS < 2. The findings are similar under the Weibull and log-logistic models. This example illustrates that prior ESS computations can be a useful device to help calibrate the prior to improve the behavior of the TITE-CRM.

Thall *et al*. [11] present a series of study designs for monitoring time-to-event outcomes in early phase clinical trials. We focus on one of the study designs, which was applied to a single-arm phase II trial for advanced kidney cancer. In the trial, the plan was to enroll up to 84 patients, with each patient’s disease status evaluated up to 12 months. In this example, we focus on the mean time-to-event, *μ*. For patient *i*, let *T _{i}* denote the time to disease progression (failure), let
${T}_{i}^{o}$ be the observed value of

$${f}_{m}({\mathbf{Y}}_{m},{\mathbf{T}}_{m}^{o}\mid \mu )={\mu}^{-{\sum}_{i=1}^{m}{Y}_{i}}exp(-\sum _{i=1}^{m}{T}_{i}^{o}/\mu ).$$

(8)

Using the relationship *μ* = mean(*T*) = median(*T*)/log(2), Thall *et al*. [11] established the prior of *μ _{S}* corresponding to the historical standard treatment from elicited mean values and a 95% credible interval of median(

The prior ESS in a simple inverse gamma-exponential model with an inverse gamma prior, *μ* ~ IG(*, *), and the exponential sampling model, *T* ~ Exp(*μ*), is analytically determined to be − 2. Thus, the ESS of the IG(5.348, 30.161) prior is 3.348 under this model. This prior ESS is obtained under the assumption that *T* is observed for all accrued patients, that is, no censoring occurs. Since in general the ESS is defined as a property of a prior and likelihood pair, a given prior might have different ESS values for different likelihoods. As mentioned in Section 3, our approach defines the ESS to be the sample size that yields a posterior containing the same amount of information as the prior. It is well known that the amount of information for time-to-event data depends on the number of observed events, not the sample size. Therefore, when *T _{i}* is right censored for some patients, the prior ESS should be larger than − 2. We apply MTM’s method to compute an ESS under the inverse gamma prior

In clinical trials with Bayesian adaptive decision making, it is important to evaluate the impact of the prior on the stopping rule. In the study design of Thall *et al*. [11], the trial should be stopped early if, based on the current data,

$$Pr({\mu}_{S}+4.3<{\mu}_{E}\mid \mathit{data})<\mathrm{0.015.}$$

(9)

This rule stops the trial if it is unlikely that the mean failure time with the experimental treatment is at least a 4.3 month improvement over the historical mean with the standard treatment. The 4.3 month improvement in mean failure time corresponds to a 3.0 month improvement in median failure time, since 4.3 = 3.0/log(2). In order to evaluate the impact of the prior of *p*(*μ _{E}* |

Figures 5a, 5b, and 5c illustrate the simulation results in terms of the probability of early termination (PET), the number of patients and trial duration, respectively. Figure 5a shows plots of PET as a function of ESS for four values of
${\mu}_{E}^{\mathit{true}}$. Since the prior mean of *μ _{S}* under IG(53.477, 301.61) is 5.7, the PET values obtained under the four
${\mu}_{E}^{\mathit{true}}$ values are reasonable for ESS values up to about 10. In contrast, for ESS > 15, the prior, rather than the data, dominates early stopping decisions. With respect to the number of patients and trial duration, plots of the 50

Thall and Cook [10] use a bivariate binary regression model in a dose-finding trial where each patient is treated at one of four doses {0.25, 0.50, 0.75, 1.00} mg/m^{2}. Denoting these by *d*_{1}*, d*_{2}*, d*_{3}*, d*_{4}, the standardized doses
${X}_{(z)}=log({d}_{z})-(1/4){\sum}_{e=1}^{4}log({d}_{e})$ are used in the model. Let *Y* = (*Y _{E}, Y_{T}*) be indicators of efficacy and toxicity, and let

$${\pi}_{a,b}={\pi}_{E}^{a}{(1-{\pi}_{E})}^{1-a}{\pi}_{T}^{b}{(1-{\pi}_{T})}^{1-b}+{(-1)}^{a+b}{\pi}_{E}(1-{\pi}_{E}){\pi}_{T}(1-{\pi}_{T})\left(\frac{{e}^{\psi}-1}{{e}^{\psi}+1}\right),$$

(10)

for *a, b* {0, 1}. Thus, ** θ** = (

$$f({\mathbf{Y}}_{m}\mid {\mathbf{X}}_{m},\mathit{\theta})=\prod _{i=1}^{m}\prod _{a=0}^{1}\prod _{b=0}^{1}{\pi}_{a,b}{({X}_{i},\mathit{\theta})}^{\text{I}\{{Y}_{i}=(a,b)\}}.$$

The prior *p*(** θ**|

ESS values computed for the subvectors of the parameters, as well as the full parameter vector, are useful feedback in the prior elicitation process. We assume fixed hyperparameters * _{μE}*,

Figure 6 shows the contours of *ESS _{E}*,

We have discussed prior sensitivity analyses in Bayesian biostatistics by using prior ESS, illustrated by examples of data analysis and study design for biomedical studies. The main advantage of using ESS is practical feasibility. The definition is pragmatic and allows one to report a meaningful prior ESS summary for most problems. Another important feature is ease of communication. A user need not understand the mathematical underpinnings of the approach to interpret the final report, since the ESS is a hypothetical sample size, in terms of patients (or animals or experimental units), which is readily interpretable.

The ESS provides a numerical value for the effective sample size of a given prior. If one wishes to utilize this methodology to construct a prior having a given ESS, two important cases may be identified. When designing a small to moderate sized clinical trial using Bayesian methods, it is desirable that the prior ESS be small enough so that early decisions are dominated by the data (e.g. the first cohort of 3 patients in a dose-finding study) rather than the prior. In this case, an ESS in the range 0.5 to 2.0 may be appropriate. On the other hand, if one is eliciting a prior for analysis of a given data set of *n* observations, then a desirable ESS may be specified relative to *n*. In this case, an ESS of .10 ×*n* or smaller might be appropriate.

Morita, Thall, and Müller (MTM2) [8] develop a variation of the ESS suitable for conditionally independent hierarchical models (CIHMs). For a two-level CIHM with *K* subgroups, in the first level, **Y*** _{k}* follows distribution

Some important limitations remain. The methodology is based on comparing curvatures of the marginal prior and the posterior distribution under an *ε*-information prior. Consequently, when an analytic solution does not exist a limitation is computational complexity. While the actual computational effort is negligible, the choice of a suitable *ε*-information prior and the evaluation of the prior-posterior distance require some problem-specific input from the investigator. That is, it is difficult to completely automate the ESS evaluation. However, the examples given here are intended to provide a basis for interested readers to compute and utilize prior ESS in similar problems. A computer program, ESS_RegressionCalculator.R, to calculate the ESS for a normal linear or logistic regression model is available from the website http://biostatistics.mdanderson.org/SoftwareDownload.

Satoshi Morita’s work was supported in part by Grant H21-CLINRES-G-009 from the Ministry of Health, Labour, and Welfare in Japan. Peter Thall’s work was partially supported by Grant NIH/NCI 2R01 CA083932. Peter Müller’s work was partially supported by Grant NIH/NCI R01 CA75981.

Satoshi Morita, Department of Biostatistics and Epidemiology, Yokohama City University Medical Center, 4-57 Urafune-cho, Minami-ku, Yokohama 232-0024, Japan, Tel.: +81-45-253-5399, Fax: +81-45-253-9902.

Peter F. Thall, Department of Biostatistics, The University of Texas M. D. Anderson Cancer Center, Houston, TX, U.S.A.

Peter Müller, Department of Biostatistics, The University of Texas M. D. Anderson Cancer Center, Houston, TX, U.S.A.

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