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- Abstract
- 1. Introduction
- 2. Confidence intervals for standard error of inverse prediction
- 3. Hypothesis testing and power computation
- 4. Confidence interval for the individual inverse prediction
- 5. Example: electron paramagnetic resonance tooth dosimetry
- 6. Discussion and summary points
- References

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Stat Med. Author manuscript; available in PMC 2013 June 21.

Published in final edited form as:

PMCID: PMC3689210

NIHMSID: NIHMS468181

Dartmouth Physically Based Biodosimetry Center for Medical Countermeasures Against Radiation (Dart-Dose CMCR), Geisel School of Medicine at Dartmouth, Hanover, NH 03755, U.S.A

This paper develops a new metric, the standard error of inverse prediction (SEIP), for a dose–response relationship (calibration curve) when dose is estimated from response via inverse regression. SEIP can be viewed as a generalization of the coefficient of variation to regression problem when *x* is predicted using *y*-value. We employ nonstandard statistical methods to treat the inverse prediction, which has an infinite mean and variance due to the presence of a normally distributed variable in the denominator. We develop confidence intervals and hypothesis testing for SEIP on the basis of the normal approximation and using the exact statistical inference based on the noncentral *t*-distribution. We derive the power functions for both approaches and test them via statistical simulations. The theoretical SEIP, as the ratio of the regression standard error to the slope, is viewed as reciprocal of the signal-to-noise ratio, a popular measure of signal processing. The SEIP, as a figure of merit for inverse prediction, can be used for comparison of calibration curves with different dependent variables and slopes. We illustrate our theory with electron paramagnetic resonance tooth dosimetry for a rapid estimation of the radiation dose received in the event of nuclear terrorism.

Dose–response relationship, or calibration curve, is an established quantitative tool of science and technology. Typically, response *y* is related to dose/stimulus *x* via simple linear regression

(1)

where *y _{i}* is the

In many studies, regression (1) is used to predict *y* given *x*, generally referred to as *direct prediction*. For example, in toxicology, one wants to predict the mortality of pest (*y*) given pesticide concentration (*x*). In this case, regression standard error *σ* serves as the metric of the accuracy prediction *ŷ* = *a* + *bx* given *x* because

(2)

where *a* and *b* are the least squares estimates for *α* and *β*, respectively. Consequently, in direct prediction problems, we are looking for the best fit that is equivalent to minimizing the residual sum of squares, so that the respective metric that reflects the quality of the dose–response relationship, the standard error of prediction (SEP), is *σ*.

In other studies, the inverse of this problem is of interest, referred to as *inverse prediction*; that is, the response *y* is known, and the variable to be predicted is *x* [1, p. 172]. The example used in this paper (see more detail in Section 5) is that we measure the reaction of tooth enamel to radiation and want to estimate the dose received from accidental or terrorist-related exposures. Thus, in inverse prediction, we predict dose *x* given response value *y* as

(3)

sometimes called as the method of inverse regression or classical calibration problem [2]. An alternative approach, when *x* is predicted from regression of *x* on *y*, is called inverse estimator approach [3]. The discussion of the pros and cons of the two approaches is deferred to the end of the paper.

Statistical inference for prediction (3) is not trivial because of the presence of a normally distributed random variable (*b*) in the denominator. Because *$\widehat{x}$* is the ratio of two normally distributed random variables, its distribution follows the Cauchy distribution and consequently has no finite expectation and variance (i.e., the respective integrals diverge).

The goal of the present paper is to develop statistical inference for the standard error of inverse prediction (SEIP) in the framework of simple linear regression (1) with normally distributed errors.

The motivation of our metric is as follows. Let *$\widehat{x}$ _{i}* be the prediction of

The choice of *n* − 2 is explained by the fact that ^{2} is an unbiased estimator of *σ*^{2}. This representation justifies the following definition of the empirical SEIP for inverse prediction, denoted , as

(4)

where

the estimate of the regression standard error *σ*. Obviously, the theoretical SEIP is defined as

(5)

As in the case of *x*-prediction (3), the empirical SEIP (4) does not have a finite mean and variance. We emphasize the difference between the direct (SEP) and indirect (SEIP) regression prediction. SEIP is the metric of the dose–response relationship regardless of the dose design points *x*_{1}, *x*_{2}, …, *x _{n}* similarly as

Prediction of *x* given *y* in the framework of linear regression model, sometimes referred to as linear calibration problem, has a long history (see more detail in Section 4). The emphasis of the present paper is different—to develop a metric (figure of merit) for calibration curve with inverse prediction that reflects the overall quality of calibration. In a way, the SEIP is more fundamental than individual prediction (3) because it allows comparison between regressions with different slopes, whereas individual predictions assume the same model (1). Consequently, the SEIP can be used for quality comparison across models with different dependent variables and different slopes in the way the signal-to-noise ratio is used in signal processing in technological applications.

From a statistical perspective, SEIP is a generalization of the coefficient of variation (CV) defined as the ratio of the standard deviation to the mean when observations *y _{i}* are independent and identically normally distributed as
(

The aims of this paper are to (1) generalize CV to include the concept of SEIP and (2) develop new approximate and exact confidence intervals (CIs) with coverage probability close to the nominal even for a small sample size.

The goal of this section is to develop two-sided and one-sided CIs for the true SEIP *η* using its empirical counterpart . We start with the delta method to approximate the standard error of SEIP (SE SEIP) based on the first-order approximation. Then we propose a more accurate CI by observing that the distribution of reciprocal follows the noncentral *t*-distribution. First, we construct a CI based on the normal approximation to the noncentral *t*-distribution, and second, we construct the exact CI, that is, the CI with the exact coverage probability. In our derivations, we assume that the slope *b* (and consequently ) is positive. In general cases, we use the absolute value of *b*, but we assume that with probability close to 1, *b* is either positive or negative. This assumption holds when the standard error of *b* is fairly small, so that the CI does not contain zero. This is a fair assumption in many practical situations. At the end of this section, we compare the coverage probability of these CIs via Monte Carlo simulations.

Although does not have the finite mean and variance, the delta method produces a fairly satisfactory CI for *η* as follows from our simulations; see details in Section 2.2.3. The following facts are used:

where

is a fixed number. Using the preceding formulas and because of the independence of *s* and *b*, from the delta method, we obtain

Thus, SE SEIP can be approximated as

(6)

On the basis of this formula, an approximate *symmetric* 100(1 − *α*)% CI for *η* is

It is more convenient to work with reciprocal SEIP when it comes to CI or hypothesis testing for *η*. Indeed, whereas has neither mean nor variance, the reciprocal has finite mean and variance. Moreover, statistic

(8)

has the noncentral *t*-distribution with *ν* = *n* − 2 degrees of freedom (DOF) and noncentrality parameter (NCP) *δ* = *S _{x}*/

The easiest way to construct a CI for *η* is to apply the following normal approximation to the cumulative distribution function (CDF) of the noncentral *t*-distribution *T* with DOF = *ν* and NCP = *δ*:

(9)

where Φ denotes the CDF of the standard normal distribution [14]. Note that this normal approximation does not use DOF and is also poor for extreme values of *x* because the right-hand side does not approach either 1 or 0 when *x* → ±∞. We obtain the approximate equal-tail two-sided CI for *η* by inverting the inequality

which yields

(10)

with approximate coverage probability 1 − *α*. If the denominator in the upper limit is nonpositive, we set the upper limit to infinity. The one-sided CI for *η* is obvious to obtain using this normal approximation.

The method for CI with the exact coverage probability based on the inverse CDF is known for a long time. Let statistic *X* have CDF *F* (*x; θ*), where *θ* is a one-dimensional parameter. If *F*(*x; θ*), as a function of *θ*, is strictly decreasing for each *x*, the lower and the upper limits of the inverse CDF, *θ _{L}* and

Using distribution (8), we derive the exact CI for *η* as the solution to the following equations:

(11)

where *T* is the CDF of the noncentral *t*-distribution and *X* = *S _{x}*/. The CDF of the noncentral

In a special case when *y _{i}* are i.i.d. normally distributed, Lehmann and Romano [13, p. 224] prove that the one-sided test for

We compare the coverage probability of three CIs for SEIP (*η*) via *N* = 50, 000 simulations using the true slope *β* = 1, the intercept *α* = 2, and the sample size *n* = 10 with *x* = 1, 2, …, 10 in the dose–response relationship governed by Equation (1). The regression *σ* varied from 0.5 to 2.5, which corresponds to the true SEIP, *η* = 0.5 to 2.5 with the increment 0.5; see Figure 1. The delta method produces symmetric CI with the SE SEIP approximated by (6). The normal-approximation CI is given by formula (10). The lower and the upper limits of the noncentral *t*-distribution method are the solutions of Equation (11).

The coverage probability of three confidence intervals as a function of the true SEIP, *η*. The number of simulations is 50,000, and the nominal confidence level is 95%. SEIP, standard error of inverse prediction.

As seen from Figure 1, the delta method consistently underestimates the nominal confidence level. The coverage probability of the normal CI is very close to the nominal 95% but abruptly drops when SEIP approaches 2.5. To the contrary, the coverage probability of the CI based on the noncentral *t*-distribution remains very close to the nominal value over the entire range of the true SEIP.

There is a well-known duality between CI and hypothesis testing [15, 18, 19]. For example, if the two-sided null hypothesis about an unknown parameter is formulated as *H*_{0}: *η* = *η*_{0} with the alter-native *H _{A}*:

In this section, we derive the theoretical power function for testing the null hypothesis about SEIP expressed in the form *H*_{0}: *η* = *η*_{0} against the alternative *H _{A}*:

Suppose that a calibration curve is estimated at design points *x*_{1}, …, *x _{n}* with

where = /*b* is the estimated SEIP and

is the variance dependent on the alternative SEIP (*η*), the number of design points (*n*), the number of repeated measurements (*k*), and
is the sum of squares over distinct values of *x*. The null hypothesis is rejected if |*Z*| > *Z*_{1−}_{α}_{/2}, where *Z*_{1−}_{α}_{/2} is the (1 − *α*/2)th normal quantile. Therefore, the power of the normal-approximation test is given by

Now we derive the power of the exact test based on the noncentral *t*-distribution. Let *X*_{1} and *X*_{2} be the *α*th and the (1 − *α*/2)th quantiles of the noncentral *t*-distribution with DOF = *k n* − 2 and
. The null hypothesis is rejected if either of the two inequalities occur

Thus, the power of the exact test is given by

These powers may be used to determine the number of repeated measurements (*k*) to achieve a statistically significant hypothesis testing on the target and the observed SEIP.

We compare the powers of the two tests for different number of repeated measurements (*k*) and verify them using statistical simulations. In Figure 2, we display the powers *P _{N}* and

Two theoretical and empirical power functions for *k* = 1 (no repeated measurements) and *n* = 5. The normal approximation does not match well the exact test especially for large alternative values of SEIP. SEIP, standard error of inverse prediction.

In Figure 3, we repeat the analysis with *k* = 3 repeated measurements (the actual number of observations is *k n* = 15). The normal-approximation power is much closer to the exact one but still quite deviates for SEIP apart from the null value. On the other hand, simulated/empirical powers are quite close.

Theoretical and empirical power functions for three repeated measurements, *k* = 3 (the total number of measurements is 15). SEIP, standard error of inverse prediction.

In summary, we suggest using the exact test with the noncentral distribution when computing power and determining the required number of repeated measurements to achieve statistical significance.

Assume we want to compare the accuracy of two dose–response relationships in terms of their inverse prediction. This task reduces to the comparison of their SEIPs with the null hypothesis *H*_{0}: *η*_{1} = *η*_{2} versus the alternative *H _{A}*:

(12)

under *H*_{0}, where *X _{k}* =

(13)

In this section, we discuss CI and hypothesis testing for *individual* inverse prediction (3), where *y* is the individual measurement outside of the calibration data set satisfying the dose–response relationship, namely, *y* = *α* + *βx* + *ε*, where *x* is the unknown true dose, subject to interval estimation, and *ε* is the normally distributed measurement error with zero mean and variance *σ*^{2}. As was mentioned earlier, the dose estimate is given by (3). But it is worthwhile to be reminded that we cannot talk about unbiasedness or variance of *$\widehat{x}$* as an estimator of *x* because they do not exist (in fact, they are infinite). However, the problem of interval estimation is still valid: for example, we can talk about the CI for the dose (*x _{L}*,

As was mentioned in Section 1, the problem of individual prediction with linear regression model has a long history going back to the paper by Eisenhart [20]. Several authors, including Berkson [21], Brown [22], and Buonaccorsi [23], examined statistical properties of *$\widehat{x}$*.

The easiest way to construct a CI for *x* is to inverse the CI for *y*. Indeed, as we know, the probability that *y* is within the interval

(14)

is 1 − *α*, where *q*_{1−}_{α}_{/2} is the (1 − *α*/2)th quantile of the *t*-distribution with *n* − 2 DOF, so solving for *x*, we obtain CI for *x* [24, pp. 275–283; 25, pp. 22–26]. There exists an alternative approach based on the least squares estimate in regression of *x* on *y*, originally suggested by Krutchkoff [3] and further studied by Williams [26], Chow and Shao [27], Brown [25], and Srivastava [28], among others (more discussion is found in Section 6).

Note that (14) uses the equal *t*-tail symmetrical critical values, *q*_{1−}_{α}_{/2} = −*q _{α}*

(15)

where, for the brevity of the presentation, we denote *g* = (*q*_{1−}_{α}_{/2})^{2},
, and *D* = *r*^{2} + *H* [*g*(*n* + 1)/*n* − *r*^{2}], the discriminant of the quadratic equation. We cannot guarantee that the discriminant is nonnegative (usually it is positive); if it is negative, the CI is not defined by this method. The discriminant may be negative only when the empirical SEIP, , is large, or when the *x*-values do not spread enough (
is small).

From formula (15), we see that the width of CI is small when the SEIP is small and increases with increased SEIP. Naturally, the SEIP is the standard error of the *x*-prediction and as such may be used as a crude standard error of the dose estimate, *$\widehat{x}$*. This fact may be used for a ‘quick-and-dirty’ CI for *$\widehat{x}$* as it formally follows from (14): when *n* is large, the expression under the square root is close to 1 because (*x* − )^{2} is much smaller than
. Ignoring the square root and recalling that SEIP = *s*/*b*, we can obtain an approximate (1 − *α*)100% symmetric CI as

(16)

Confidence intervals (15) and (16) are two-sided. Sometimes we are interested in the lower limit only. In the example of the radiation dose estimation, our concern is that a person received a dose greater than a certain value, say, 2 Gy. Then the CI is one-sided and formally takes the form (*x _{L}*, ∞), where the lower limit

In this section, we validate the two CIs for individual prediction using simulations (*N* = 100,000) with *x*-values: 0, 1, 2, 5, 10, each repeated two times (*k* = 2) with the true intercept *α* = 0.01 and the true slope *β* = 0.02 (these values are close to the real-life estimates from our example in the next section). Our goal is to understand how the coverage probability is close to the nominal level for different values of the true SEIP. The true SEIP (*η* = *σ*/*β*) ranged from 0.25 to 2.5, so the range of *σ* is 0.25 × 0.02 to 2 × 0.02; see Figure 4. We generated 100,000 ten-dimensional normally distributed vectors **y** for each SEIP with parameters specified earlier. Also we generated 100,000 normally distributed observations *ŷ* that represent individual measurements with the mean *α* + *βx* and SD = *σ*, where *x* = 2 is the true value to predict from *y* and estimated intercept and slope as *$\widehat{x}$* = (*y* − *a*)/*b*. The first CI is computed by formula (15), and the second CI is the naive CI (16) based on the definition of SEIP as the standard error of *$\widehat{x}$*. The first interval has a coverage probability close to nominal, 95% on the entire range of the true SEIP, but the coverage probability of the naive CI is consistently smaller than 0.95. Thus, we conclude that (15) yields CI with coverage probability very close to nominal, whereas (16) may be used for a ‘quick-and-dirty’ assessment.

The availability of methods for fast and reliable measurement of absorbed ionizing radiation after events involving large numbers of exposed individuals is an essential component for preparing for possible radiation accidents such as occurred in Chernobyl and Fukushima or acts of terrorism involving ionizing radiation [29–31]. As a part of preparedness to assess dose and treat exposed individuals in such events, the Dartmouth Physically Based Biodosimetry Center for Medical Countermeasures Against Radiation (Dart-Dose CMCR) is one of seven CMCRs recently established by the National Institutes of Health. Electron paramagnetic resonance (EPR) *in vivo* measurements of tooth enamel is a promising method for fast and reliable dose estimation and is especially useful when thousands and maybe hundreds of thousands of people may need to be triaged for medical treatment [32]. A recent paper by Fattibene and Gallens [33] provides an overview of EPR dosimetry.

The potential for the use of EPR measurements in teeth for estimating radiation dose is based on the linear relationship between the amplitude of the EPR signal and the radiation dose [34]. The dose estimation, framed as a linear regression problem, is an inverse prediction problem (1), where *y* is the EPR amplitude and *x* is the corresponding dose. Rigorous evaluation of the dose–response relationship may play a crucial role in instrumental development to determine improvements of hardware (e.g., the type of resonator, bridge, or magnet; number of scans or repeated tooth measurements). Different designs lead to different calibration curves and variation around the curve. Which alternative is better? We argue that SEIP is the metric to guide such decisions in EPR tooth dosimetry.

Statistics also can play a central role in using EPR individual tooth measurement to assess the dose and its uncertainty for purposes of medical triage. Especially when very large numbers of people have potentially been exposed, it is important to quickly and accurately differentiate people whose exposure is above or below the threshold for treatment. Therefore, it is important to know, in addition to the estimate of the individual radiation dose value, the CI to assess the probability that the dose is above the threshold for medical treatment.

In this section, we illustrate the use of the SEIP with its CI for decision-making, first to improve the instrument by identifying what surfaces of incisors to measure. Second, we quantify the uncertainty of the individual dose estimate by CI given the calibration curve, to be used in medical decision-making to triage people for treatment.

Although EPR measurements can be made on any type of tooth, the use of incisors is especially attractive because these are more easily measured, are less likely to have caries or restorations, and are the first permanent teeth so even children could be measured. Williams *et al*. [35] reported comparisons of calibration curves based on measurements of the front and back surface of irradiated incisor teeth in the mouth model. We use those data to assess which surface incisor yields a more accurate dose estimate, as we know the actual dose given. Following our approach, this question reduces to computation and statistical comparison of two SEIPs. We show the tooth EPR data with the least squares estimated dose–response relationship for both surfaces in Figure 5. Here *b* is the slope of the regression line, and *s* is the regression standard error. We emphasize that we cannot compare the front and back incisor calibration curves using traditional statistical goodness-of-fit measures such as residual sum of squares or coefficient of determination because the dependent variables and slopes are different. Because the quality of inverse prediction is our concern, the SEIP is the parameter by which we decide what side of the tooth should be measured to achieve the most accurate radiation dose estimate.

Two calibration curves from front and back incisor surfaces in the mouth model experiments. Each point represents an independent EPR measurement (three different teeth per dose). EPR, electron paramagnetic resonance; SEIP, standard error of inverse prediction. **...**

We show three 95% CIs for SEIPs in Table I. These methods produce consistent results; the normal-approximation and the exact CI are especially close.

The SEIP for the back surface of incisors is larger, but is the difference statistically significant? To answer this question, we used the *Z*-test (12). The *Z*-score statistic is 1.77, which gives the *p*-value = 0.077—the data fail to reject the null hypothesis that the SEIPs are the same.

On the basis of this table and assuming for this illustration that there is no plausible reason to expect the dose estimate to differ on the front and back surfaces, we would conclude that it does not matter which surface is used in estimating dose, and so other considerations—like ease of making measurements—can determine instrument development. However, it would be important to carry out more measurements to determine whether the surfaces may be statistically significantly different in view of close proximity of the calculated *p*-value to the borderline 0.05.

In this section, we illustrate individual radiation dose prediction using the calibration curve from [36] shown in Figure 6. We performed the EPR measurements using an isolated molar that was irradiated externally and then inserted into a gap between teeth of a volunteer. We made three measurements over three consecutive days on six irradiated teeth and used the average of the EPR amplitude adjusted for the tooth size for each dose for the calibration. We assume that the amount of enamel will impact the measurement and that size of tooth is a reasonable estimate of amount of enamel. The SEIP = *s*/*b* = 0.66 Gy, where *a* = 0.1096 is the intercept estimate, *b* = 0.0195 is the slope estimate, and *s* = 0.0129 is the regression standard error. We computed the 75% CI (0.49,1.20) for the true SEIP using the normal approximation.

Tooth-size adjusted EPR signal amplitude as a function of the radiation dose received with the 75% confidence band (shaded area). If the tooth-size adjusted amplitude is 0.16, the dose estimate is 2.59 Gy with the 75% CI (1.57,3.58) as the minimum and **...**

Now we apply this calibration curve for individual dose estimation. Let the EPR amplitude adjusted for the molar tooth size of individual *X* be 0.16 V. What is his/her radiation dose estimate and 75% CI for the true dose? We show the results in Table II. We compute the dose estimate via inverse prediction as (0.16 − 0.1096)/0.0195 = 2.59 Gy. We compute the 75% SEIP-based CI for the true dose on the basis of Equation (16), which yields 2.59 ± 1.34 × 0.66, where 1.34 = *q*_{1−0.05/2}, the quantile of the central *t*-distribution with *n* − 2 = 6 − 2 = 4 DOF. We compute the inverse CI for individual dose prediction on the basis of Equation (15). As expected, the SEIP-based CI underestimates the uncertainty of the dose estimate (the interval is slightly more narrow). We can assess the probability that individual *X* received radiation more than 2 Gy as 1 − *T* ((2 − 2.59)/0.66, DOF = 4) = 0.79, where *T* is the CDF of the central *t*-distribution. In this case, we have a strong evidence that the person should be triaged for medical treatment.

We have developed a novel figure of merit for calibration curve, called the SEIP. Whereas the quality of direct prediction in the classical regression analysis is expressed by the standard error of the fit, the quality of prediction in inverse regression is expressed by the ratio of the error of the fit to regression slope.

We can view the SEIP metric as a generalization of the coefficient of variation, CV = *σ*/*μ*, an established statistical parameter that reflects the relative variation. Consequently, statistical inference developed in this paper for SEIP may be interpreted as a generalization of statistical methods previously developed for CV in the case of i.i.d. observations to regression. Whereas traditional (direct) prediction leads to the best regression fit (minimal residual standard error), the inverse prediction leads to the minimum residual standard error divided by the regression slope (SEIP).

Statistical treatment of inverse prediction requires nontraditional approaches because, even for a simple linear regression, the *x*-estimate does not have finite mean and variance while the interval estimation of *x* remains valid. Two statistical problems emerge in the context of inverse prediction: (1) developing a metric (and the associated statistical inference) that reflects the quality of the inverse prediction regardless of the design *x*-values and (2) developing accurate CIs and hypothesis tests for individual prediction when *x* is estimated from observed *y* via inverse regression.

We have developed exact and approximate CIs for the SEIP in simple linear regression with normal errors using the fact that the reciprocal SEIP follows the noncentral *t*-distribution. We have derived the power functions for the hypothesis test about SEIP for the normal approximation and the exact test based on the noncentral *t*-distribution, and we have investigated the performance of both approaches via statistical simulations.

Individual prediction, as the ratio of two normally distributed random variables, also requires a non-traditional statistical approach. We have adopted the CI for individual prediction and showed how SEIP may be used for a ‘quick-and-dirty’ interval estimation.

We have illustrated major theoretical developments with the EPR tooth dosimetry for comparison of front and back incisor measurements and individual dose prediction using exact and approximate CIs. Probabilistic assessment of individual dose prediction is crucial for medical treatment in the case of nuclear terrorism that may involve a large number of victims.

Finally, we would like to make some comments on the comparison of the classic approach, that is, regressing *y* on *x*, undertaken in this paper, with the alternative approach, that is, regressing *x* on *y*, called the inverse estimator approach. In the latter case, with the slope and the intercept estimated as *b _{I}* = Σ(

This work was supported by two funding agencies in the US Department of Health and Human Services: (1) The National Institutes of Health (NIAID, the Center for Radiation Dosimetry, Grant # U19AI091173) and (2) The Biomedical Advanced Research and Development Authority (BARDA) in the Office of the Assistant Secretary for Preparedness and Response (Contract no. HHSO100201100024C). Dr. Demidenko was also supported by NIH/NCI grants U54 CA151662 and CA130880.

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