STATISTICS IN MEDICINE Statist. Med. 2003; 22:1977–1988 (DOI: 10.1002/sim.1368)

Con
dence intervals for the 50 per cent response dose D. Faraggi∗ , P. Izikson and B. Reiser Department of Statistics; University of Haifa; Israel

SUMMARY Con
dence intervals for the 50 per cent response dose are usually computed using either the Delta method or Fieller’s procedure. Recently, con
dence intervals computed by inverting the asymptotic likelihood ratio test have also been recommended. There is some controversy as to which of these methods should be used. By means of an extensive simulation study we examine these methods as well as con
dence intervals obtained by the approximate bootstrap con
dence (ABC) procedure and an adjusted form of the likelihood ratio based con
dence intervals. Copyright ? 2003 John Wiley & Sons, Ltd. KEY WORDS:

bioassay; bootstrap; delta method; Fieller’s method; likelihood ratio; logistic regression

1. INTRODUCTION In the typical biological assay experiment [1] subjects are given a measurable stimulus at dose level x, usually on a transformed scale and respond with probability p(x) = F(x;
). Here
is a vector of unknown parameters and F(:) is a speci
ed cumulative distribution function. Observations are taken on k groups of ni , i = 1; : : : ; k subjects with each subject in the ith group being given dose xi and the number of responses ri being noted. Consequently, r1 ; : : : ; rk are mutually independent binomial variates, ri ∼ bin(ni ; p(xi )). For the standard logistic model logit[p(xi )] = log

p(xi ) = 0 + 1 xi = 1 (xi − ) 1 − p(xi )

(1)

where  = −0 =1 represents the 50 per cent response dose (variously termed the ED50 , LD50 , LC50 ). The 50 per cent response dose is frequently used as a summary measure of the data, especially in toxicity experiments. Three methods are frequently used for obtaining approximate con
dence intervals for : the delta; Fieller, and likelihood ratio procedures. There is a controversy in the literature as to which is to be preferred [1]. In addition to the three standard methods we decided to consider two additional procedures for con
dence intervals. We examined the popular approximate bootstrap con
dence (ABC)

∗

Correspondence to: D. Faraggi, Department of Statistics, University of Haifa, Israel.

Copyright ? 2003 John Wiley & Sons, Ltd.

Received June 2001 Accepted July 2002

1978

D. FARAGGI, P. IZIKSON AND B. REISER

[2] interval which is readily computable for logistic regression. Likelihood ratio con
dence intervals are obtained by inverting the standard asymptotic likelihood ratio tests. Various adjustments to improve the small sample properties of these procedures have been suggested. Severeni [3] provides a review of the theory. Such adjustments do not appear to be often considered in practice; however, for some recent applications see Jeng and Meeker [4]. We use the DiCiccio and Tibshirani [5] adjustment which can easily be applied to logistic regression. In this paper we provide an extensive simulation study in order to compare the competing con
dence interval procedures. First in Section 2 we briey describe these
ve con
dence interval procedures. Section 3 presents the simulation study and a numerical example. We conclude in Section 4 with a discussion of our simulation results along with a review of previous comparisons which have appeared in the literature.

2. CONFIDENCE INTERVAL PROCEDURES 2.1. Background and notation The likelihood function for the logistic model described in Section 1 can be written as L(0 ; 1 ) =

k


p(xi )ri (1 − p(xi ))ni −ri

(2)

i=1

where p(xi ) = (1 + e−0 −1 xi )−1 . Alternatively, if we use the 1 ,  parameterization and write p(xi ) = (1 + e−1 (xi −) )−1 we will denote the likelihood function as L(; 1 ). Let ˆ0 , ˆ1 represent the maximum likelihood estimators obtained by maximizing (1) and let ˆ =( VV1121 VV1222 ) denote the estimated covariance matrix of (ˆ0 ; ˆ1 ) which is obtained by inverting the Fisher information matrix. Let ˆ = −ˆ0 = ˆ1 represent the maximum likelihood estimate of the 50 per cent response dose. 2.2. Delta method con
dence interval for  Using the delta method the asymptotic variance of ˆ can be estimated ([1], p. 61) by ˆ = ˆ2 ()

1 (V11 + 2V ˆ 12 + ˆ2 V22 ) 2 ˆ 1

Cox [6] shows that this estimate of the variance is the same as the one that arises from asymptotic theory if the model is parameterized in terms of 1 and  (see (1)). Consequently the delta method 1 −  con
dence interval for  is ˆ ) ˆ ˆ ± z1−=2 (

(3)

where z  is the standard normal  quantile. Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

CONFIDENCE INTERVALS FOR THE 50 PER CENT RESPONSE DOSE

1979

2.3. Fieller interval for  Following Morgan ([1], p. 62) we write the Fieller interval as      z1−=2 V12 V2 C ± ˆ + V11 + 2V ˆ 12 + ˆ 2 V22 − C V11 − 12 ˆ + 1−C V22 V22 ˆ1 (1 − C)

(4)

2 for C = z1−=2 V22 = ˆ12 ¡1. For C ¿1 the Fieller interval is the real line excluding the segment given in (4). For C suciently large so that the expression in the square root is negative, the Fieller interval is the entire real line [7]. Sitter and Wu [8] argue that for C ¿1, making inferences on  ‘makes no sense’ since in this situation the standard Wald test does not reject the hypothesis that 1 = 0.

2.4. Likelihood ratio con
dence interval for  The likelihood ratio con
dence interval for  is obtained by inverting the usual asymptotic likelihood ratio test. Set l(; 1 ) = log L(; 1 ) and denote by ˆ1 () the value of 1 which maximizes the likelihood (2) for a
xed value of . Then the pro
le log-likelihood for  is lp () = l(; ˆ1 ())

(5)

Inverting the likelihood ratio test provides an approximate 1 −  con
dence interval whose endpoints are the values of  which satisfy −2lp () = 12 (1 − ) − 2lp () ˆ

(6)

where 12 (1 − ) is the 1 −  quantile of the 12 distribution. Numerical aspects of these computations are discussed by DiCiccio and Tibshirani [5], Venzon and Moolgavkar [9] and SAS ([10], pp. 416–417). 2.5. Remarks on the delta, Fieller and likelihood procedures These three procedures are the most frequently used and recommended for obtaining approximate con
dence intervals for  [8, 11–14]. The delta procedure can be obtained from the Fieller by setting C = 0 in (4). It provides a solution symmetric about . ˆ As pointed out by one of the referees, the distribution of ˆ will typically be skewed and thus use of the symmetric interval will tend to lead to bias in the individual tail coverage probabilities. In addition, the delta and likelihood procedures will give similar results where the pro
le likelihood can be well approximated by a second-order Taylor series (Kalbeisch and Prentice, reference [15], p. 47). The Fieller approach was originally [7] developed to give an exact con
dence interval, having precisely the required coverage, for the ratio of normal means assuming the availability of observations from a bivariate normal distribution. In our situation the normality is approximate depending on the large sample theory of maximum likelihood estimates. Consequently the Fieller con
dence intervals will be only approximate and their eectiveness will depend on how well bivariate normality approximates the distribution of (ˆ0 ; ˆ1 ). The likelihood procedure, which is based on the asymptotic distribution of the likelihood ratio test, depends on similar large sample theory. However the invariance of the likelihood ratio Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

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D. FARAGGI, P. IZIKSON AND B. REISER

test to one-to-one parameter transformations (Kalbeisch and Prentice [15], p. 47) suggests that the likelihood-based method will depend less on the asymptotic normality of (ˆ0 ; ˆ1 ) and consequently may perform somewhat better than the Fieller method. 2.6. Adjusted likelihood ratio con
dence interval for  Recently a great deal of eort has gone into
nding small sample corrections to the asymptotic distribution of the likelihood ratio statistic, see for example DiCiccio et al. [16] and Severini [3]. These corrections are generally based on the signed square root of the likelihood ratio test statistic. The DiCiccio and Tibshirani [5] adjustment is particularly easy to implement for the logistic model. Under standard asymptotic theory ˆ − lp ())]1=2 R() = sgn(ˆ − )[2(lp () tends to the standard normal distribution. Denote the expectation of R by −z 0 . Then the 1 −  adjusted likelihood ratio con
dence bounds for  say ( L ;  U ) can be found as the respective solutions of the equations R() + z 0 = ± z1−=2 DiCiccio and Tibshirani show how z 0 can be estimated for likelihoods in the general exponential family and provide S+ algorithms to carry out the computations. 2.7. Approximate bootstrap con
dence (ABC) interval for  DiCiccio and Efron [2] obtain the ABC interval as a useful and readily computable approximation to the well known bias corrected accelerated con
dence interval procedure (BCa ) [17] by replacing bootstrap sampling with numerical derivatives. In the framework of the general exponential family they develop algorithms to carry out the computations. These are readily applicable to logistic regression. S+ programs to implement the calculations appear in the appendix of Efron and Tibshirani [18]. For all our simulations we also calculated con
dence intervals using the ABCq method [2, 18]. Since its performance was consistently worse than the ABC method we do not provide further details.

3. SIMULATION STUDY AND NUMERICAL EXAMPLE We carry out two sets of simulations. The
rst is based on the Sitter and Wu [8] simulations while the second is based on the Hoekstra [11] simulations. 3.1. Simulation set 1 We follow Sitter and Wu [8] and also Abdelbasit and Plackett [12] in basing our
rst set on the Hewlett data presented in Table I. Fitting the logistic model to this data results in ˆ = −0:01732, ˆ1 = 28:2422. The estimated response probabilities for each dose are presented in column 4 of Table I. Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

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CONFIDENCE INTERVALS FOR THE 50 PER CENT RESPONSE DOSE

Table I. Hewlett data. Log10 dose (xi ) 0.2810 0.2304 0.1523 0.0864 −0:0362 −0:0809 −0:1487 −0:2147 −0:3098

Number of subjects (ni )

Number aected (ri )

Estimated response probability

47 50 50 50 50 50 50 50 50

47 50 50 46 25 0 2 1 0

1.000 0.999 0.992 0.949 0.370 0.142 0.024 0.004 0.000

The simulation of new data sets proceeds as follows. For each dose in Table I generate a binomial count for ri using the estimated response probability and the sample size given in Table I for the particular dose under consideration (many statistical packages have built-in binomial variate generators). This set of counts along with their associated doses and sample sizes provides one data set. The process is repeated 10 000 times. All the simulations discussed in the remainder of this paper proceed similarly. Using the
tted model, 10 000 new data sets with the doses and sample sizes of Table I were generated, nominal 95 per cent con
dence intervals for the methods discussed in Section 2 were calculated and the observed coverage of these methods were obtained. This procedure was repeated using smaller (and equal) sample sizes at each dose. The results are summarized in Table II. The columns L and U give the lower and upper tail error rates while L + U represents the two-tailed error rates. The average length of the intervals is also presented. For smaller sample sizes some of the numerical maximum likelihood computations did not converge, as noted in Table II, and were discarded in the computations. For the Fieller method the C-values for all the cases were much smaller than one. The results in Table II for the delta and Fieller methods were similar to those found by Sitter and Wu. The case of sample sizes being the same as the original data set (≈ 50) is labelled as Full. For this case all methods provide very similar coverages with the delta method being somewhat worse than the others in terms of overall coverage and showing serious unbalance between the upper and lower tail error rates. The delta method is ‘liberal’, giving error rates larger than the nominal, while the Fieller method is ‘conservative’ with error rates less than the nominal. These tendencies are more pronounced with smaller sample sizes. In addition the Delta method is unbalanced between lower and upper error rates and this lack of balance increases with decreasing sample sizes. The Fieller procedure provides more balanced results. Although the delta method consistently gives the smallest average length of all the methods this is not very meaningful due to its liberalness in coverage. The observed error rate of the ABC method is much too low for ni = 10 but performs satisfactorily for larger sample sizes and provides balanced results. The likelihood ratio procedure gives results that, although liberal, are much closer to the nominal and have average length shorter than both the ABC and Fieller methods. The adjusted likelihood ratio method gave results similar to the likelihood ratio for large sample sizes and performed worse for small sample sizes. The
tted probabilities shown in the last column of Table I indicate a very steep rise in probabilities from zero to one over the given doses. In order to examine other patterns we carried out additional simulations. All combinations of  = 0:1; 0:2; 0:3 and 1 = 7; 14 were considered. These were also used by Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

Copyright ? 2003 John Wiley & Sons, Ltd.

1.5 1.4 1.5 1.5 1.1

7.4 1.6 3.8 2.7 1.1

U 8.9 3.0 5.3 4.2 2.2

L+U 0.065 0.089 0.068 0.069 0.073

length 1.8 1.8 2.4 3.1 2.3

L 6.5 2.3 3.5 3.1 2.3

U 8.3 4.1 5.9 6.2 4.6

L+U

ni = 20(2)

Cases which did not converge (1) 905 runs, (2) 72 runs, (3) 10 runs.

Delta Fieller LR Adj-LR ABC

L

ni = 10(1)

0.048 0.053 0.048 0.049 0.062

length 1.9 2.2 2.3 2.7 2.4

L 5.6 2.2 3.5 2.8 2.4

U 7.5 4.4 5.8 5.5 4.8

L+U

ni = 30(3)

0.040 0.042 0.040 0.040 0.050

length

2.2 2.5 2.5 2.8 2.7

L

4.2 2.4 3.0 2.6 2.7

U

Table II. Simulation comparison of 95 per cent con
dence intervals based on the Hewlett data.

6.4 4.9 5.5 5.4 5.4

L+U

Full

0.030 0.032 0.031 0.031 0.031

length

1982 D. FARAGGI, P. IZIKSON AND B. REISER

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CONFIDENCE INTERVALS FOR THE 50 PER CENT RESPONSE DOSE

1983

Sitter and Wu [8]. As can be seen from their Table I the corresponding response probabilities vary from close to zero to moderately and very high values at various rates of increase. In addition we consider  = −0:01732 and 1 = 2. This gives the response probabilities at the dose levels of Table I to be (0.644, 0.621, 0.584, 0.522, 0.491, 0.468, 0.434, 0.403, 0.358) which are much more centred about 0.5 and have a smaller range than the cases considered by Sitter and Wu. For each case considered, 10 000 simulations were performed and the observed coverages are presented in Table III. Runs that did not converge are footnoted in Table III and were discarded from the calculations. In addition there are a number of simulation runs in which the Fieller intervals are of in
nite length (that is, C¿1). The number of such cases is given in the superscript for the column L + U in the Fieller method. C¿1 generally occurs for smaller sample sizes and when the range of probabilities over doses is relatively small. For the Fieller method when C¿1 one-tailed error rates and average interval lengths are not meaningful. Sitter and Wu [8] do not include runs with C¿1 in their Fieller coverage. For the upper and lower tail Fieller error rates in Table III, only cases with C¡1 are included while the two-tailed error rates using all runs regardless of C are provided in parentheses. The Delta method’s performance is quite variable. Usually it gives quite dierent upper and lower tail error rates. For large sample sizes its two-tailed error rates are generally close to the nominal value. However there are situations where it can be quite liberal (for example,  = 0:3, 1 = 14) or conservative (for example,  = −0:01732, 1 = 2) and this can persist even for ni = 50. The Fieller procedure tends to be conservative although this property weakens with larger samples. For small sample sizes (ni = 10) this conservativeness can be substantial, giving observed two-tailed error rates as small as 3 per cent instead of the nominal 5 per cent. Although the Fieller lower and upper tail error rates tend to be approximately the same, substantial unbalance can still occur for smaller sample sizes. The likelihood ratio method’s coverage is usually quite close to its nominal value although there are cases where it performed conservatively. In our simulations the two-tailed nominal 5 per cent error rate ranged from 4.4 to 5.8 per cent. It also exhibits balance between the one-tailed error rates. The adjusted likelihood ratio method gave results quite similar to the likelihood ratio method. When there were improvements they were not substantial and sometimes for smaller sample sizes the adjustment gave error rates further from their nominal value. The ABC method generally resulted in two-tailed error rates much larger than their nominal value. We calculated average lengths of the con
dence intervals but do not report them, as they were not very informative. For moderate and large sample sizes they were quite similar over the various methods. 3.2. Simulation set 2 Hoekstra [11] carries out a limited simulation study with ni = 10 for dose values (1; 2; 3; : : : ; 10) and 1 = 1; 3; 6. Hoekstra permits  to vary uniformly between 5 and 6 and reports results of the two-tailed error rates based on 1000 simulations per case. The response probabilities for these models increase very steeply from 0 to 1. For 1 = 3; 6 this resulted in a response rate of either 0 per cent or 100 per cent for most doses. It also resulted in data in which, for 1 = 6, Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

Copyright ? 2003 John Wiley & Sons, Ltd.

7 7 7 7 7 7 7 7 7 7 7 7 14 14 14 14 14 14 14 14 14 14 14 14 2 2 2 2

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 0.3 0.3 0.3 0.3 0.1 0.1 0.1 0.1 0:2(1) 0.2 0.2 0.2 0:3(2) 0:3(3) 0.3 0.3 −0:01732 −0:01732 −0:01732 −0:01732

∗ Cases that did not † The superscript in

1

∗

0.9 1.1 1.3 2.0 0.0 0.3 0.6 1.0 0.0 0.0 0.0 0.3 3.2 2.9 2.6 2.8 1.3 1.5 1.9 1.8 0.0 0.0 0.0 0.3 0.4 0.7 1.1 1.4

L 3.3 3.1 2.7 2.8 4.7 4.3 4.0 4.1 8.5 6.6 5.3 5.1 2.9 2.4 2.5 2.5 3.4 3.3 2.9 3.2 11.7 8.2 6.7 6.0 0.2 0.4 0.5 1.0

U

Delta

4.2 4.2 4.0 4.8 4.7 4.6 4.6 5.1 8.5 6.6 5.3 5.4 6.1 5.3 5.1 5.3 4.7 4.8 4.8 5.0 11.7 8.2 6.7 6.3 0.6 1.1 1.6 2.4

L+U 2.4 2.0 2.1 2.7 2.3 2.6 2.5 2.2 0.3 2.0 2.5 2.7 2.2 2.5 2.4 2.7 2.7 2.8 2.8 2.5 0.2 1.8 1.9 2.3 2.7 2.4 2.4 2.4

L 1.9 2.2 2.0 2.2 2.1 2.5 2.3 2.7 2.9 2.5 2.5 2.8 1.9 2.0 2.2 2.4 2.0 2.3 2.1 2.6 2.8 2.8 2.7 3.0 2.5 2.5 2.5 2.6

U

(2:2)(5749) (3:5)(2804) (4:2)(1250) (4:9)(196)

(7:1)(500)

(3:6)(170) (4:5)(1)

(4:5)(13)

2.4 2.0 2.0 2.6 2.6 2.6 2.5 2.2 1.8 2.3 2.6 2.9 2.8 2.8 2.6 2.9 2.9 2.8 2.8 2.5 1.8 2.1 2.3 2.3 2.2 2.2 2.3 2.4

(4:3)(2)

4.3 4.2 4.1 4.9 4.4 5.1 4.8 4.9 3.1 4.5 5.0 5.5 4.1 4.5 4.6 5.1 4.7 5.1 4.9 5.1 3.0 4.6 4.6 5.3 5.2 4.9 4.9 5.0

L

L + U†

Fieller

converge: (1) 1 run; (2) 70 runs; (3) 1 run. the Fieller L + U column denotes the number of runs with C¿1.

10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50

n

2.4 2.5 2.2 2.3 2.5 2.8 2.5 2.9 3.4 3.0 2.6 2.8 2.6 2.2 2.4 2.5 2.7 2.7 2.4 2.7 4.0 3.4 3.2 3.2 2.2 2.6 2.6 2.6

U

LR

4.8 4.5 4.2 4.9 5.1 5.4 5.0 5.1 5.2 5.3 5.2 5.7 5.4 5.0 5.0 5.4 5.6 5.5 5.2 5.2 5.8 5.5 5.5 5.5 4.4 4.8 4.9 5.0

L+U 2.5 2.0 2.2 2.6 2.8 2.7 2.7 2.4 1.9 2.6 2.7 3.1 2.6 2.7 2.4 2.7 3.2 2.9 2.9 2.4 2.7 2.9 2.5 2.7 2.2 2.2 2.3 2.4

L 2.5 2.5 2.3 2.3 2.6 2.7 2.5 2.9 3.4 2.8 2.6 2.7 2.7 2.3 2.5 2.5 2.8 2.9 2.4 2.8 4.1 3.2 3.1 3.0 2.2 2.5 2.6 2.5

U 5.0 4.5 4.5 4.9 5.4 5.4 5.2 5.3 4.7 5.4 5.3 5.8 5.3 5.0 4.9 5.2 6.0 5.8 5.3 5.2 6.8 6.1 5.6 5.7 4.4 4.7 4.9 4.9

L+U

Adj-LR

3.6 2.4 2.2 2.7 3.5 3.2 2.9 2.5 1.1 2.8 3.1 3.2 3.0 2.9 2.6 2.8 4.6 3.1 3.3 2.7 1.5 3.1 2.7 2.8 3.9 4.5 4.0 3.8

L

2.8 2.9 2.5 2.5 3.4 3.2 2.9 3.1 4.3 3.2 2.7 2.8 2.9 2.4 2.6 2.6 3.6 3.2 2.9 3.0 6.4 3.9 3.2 3.2 3.0 4.3 4.2 3.9

U

ABC

6.4 5.3 4.7 5.2 6.9 6.4 5.8 5.6 5.4 6.0 5.8 6.0 5.9 5.3 5.2 5.4 8.2 6.3 6.1 5.7 7.9 7.0 5.9 6.0 6.9 8.8 8.2 7.7

L+U

Table III. Comparison of 95 per cent con
dence intervals for various values of , 1 , and sample sizes (doses from Table I).

1984 D. FARAGGI, P. IZIKSON AND B. REISER

Statist. Med. 2003; 22:1977–1988

CONFIDENCE INTERVALS FOR THE 50 PER CENT RESPONSE DOSE

1985

75 per cent of the simulation runs did not result in convergence of the maximum likelihood calculations. Clearly in such a situation the observed coverage is not of great interest. Consequently we repeated these computations using 10 000 simulations,
xing  = 5:5 (the midrange of the values considered by Hoekstra), and using 1 = 1; 2; 3; 4. The results are reported in Table IV. For all cases the C-values were substantially smaller than one. Cases for which the maximum likelihood calculations did not converge are indicated in a footnote to Table IV and were excluded from the error rate calculations. Table IV indicates that the delta method usually gives two-tailed error rates higher than the nominal 5 per cent value. There are only two cases in which they are lower than their nominal value and these correspond to the cases having the highest proportion of convergence problems. The Fieller method acts conservatively but generally has two-tailed error rates close to the nominal value. Usually these are closer to the nominal value than the corresponding delta method with the case when 1 = 4 being the exception. It is noteworthy that this case is also the most problematic in terms of convergence of the maximum likelihood calculations. The likelihood ratio method tends to be somewhat liberal but less so than the delta method and often has observed error rates close to their nominal value. Here also the case with 1 = 4 provides an exception for which the likelihood ratio gives quite conservative results. The adjusted likelihood ratio is usually very close to or the same as the likelihood ratio method and rarely improves on it. The ABC method tends to give error rates similar but somewhat higher than the likelihood-ratio procedure. For all methods the upper and lower tail error rates are quite similar.

3.3. Numerical example As an example of the dierences obtained by these methods consider the mortality of T. castaneum beetle data at various concentrations of insecticide (Table V) discussed by Zelterman [19] and taken from Hewlett and Plackett [20]. The resulting 95 per cent con
dence intervals for the delta, Fieller, likelihood ratio, adjusted likelihood ratio and ABC methods are presented in Table VI. For this particular case the delta and ABC intervals are quite similar, diering from the Fieller, likelihood ratio and adjusted likelihood ratio intervals which are also quite similar. In order to estimate, for this example, the coverage of the
ve methods of obtaining con
dence intervals we follow Loh [21] and Efron and Tibshirani (reference [18], pp. 263–266) who discuss a bootstrap method for calibrating an approximate con
dence interval procedure in order to improve its coverage properties. As part of this development they provide an estimator of coverage based on a particular data set which we use. To estimate these coverages, model (1) is
tted to the data of Table V by maximum ˆ Using these parameter estimates 10 000 data likelihood giving ˆ0 , ˆ1 and consequently . sets are generated. For each data set the con
dence intervals are calculated and then the observed coverage of ˆ for each of the
ve methods is obtained. In addition to the 95 per cent con
dence intervals based on the dierent methods Table VI gives upper (U ), lower (L) and two-tailed (L + U ) error rates for the 95 per cent intervals. In this case at least, it is the delta method and not the Fieller procedure that produces overly conservative results. The Fieller, likelihood ratio and adjusted likelihood ratio coverages are very similar to each other and to the nominal values while the ABC method gives too large error rates. Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

Copyright ? 2003 John Wiley & Sons, Ltd.

10 20 30 50 10 20 30 50 10 20 30 50 10 20 30 50

1 1 1 1 2(1) 2(2) 2 2 3(3) 3(4) 3(5) 3(6) 4(7) 4(8) 4(9) 4(10)

3.2 2.6 2.6 2.8 3.4 2.9 2.8 2.9 1.6 3.4 3.6 3.2 0.3 2.6 2.7 3.3

L 3.3 2.9 2.9 2.7 3.2 3.1 3.0 2.7 1.4 3.3 3.7 3.1 0.5 2.5 2.7 3.4

U

Delta

6.5 5.5 5.5 5.5 6.6 6.0 5.8 5.6 3.0 6.7 7.3 6.3 0.8 5.1 5.4 6.7

L+U 2.2 2.3 2.4 2.6 1.6 2.1 2.3 2.6 0.4 1.3 2.1 2.3 0.1 0.3 1.1 1.6

L 2.4 2.6 2.2 2.6 1.6 2.3 2.5 2.4 0.4 1.3 2.1 2.1 0.1 0.3 1.1 1.5

U

Fieller

4.6 4.9 4.6 5.2 3.2 4.4 4.8 5.0 0.8 2.6 4.2 4.4 0.2 0.6 2.2 3.1

L+U 2.9 2.6 2.6 2.8 2.6 2.5 2.6 2.7 1.3 2.6 2.8 2.7 0.3 0.9 1.2 1.6

L 3.1 2.9 2.4 2.7 2.4 2.7 2.8 2.5 1.0 2.4 2.9 2.5 0.4 0.6 1.2 1.5

U

LR

6.0 5.5 5.0 5.5 5.0 5.2 5.4 5.2 2.3 5.0 5.7 5.2 0.7 1.5 2.4 3.1

L+U 2.9 2.6 2.6 2.8 2.6 2.5 2.5 2.7 1.3 2.6 2.8 2.7 0.3 1.2 1.2 2.6

L 3.1 2.9 2.4 2.7 2.4 2.7 2.9 2.5 1.0 2.4 2.9 2.5 0.5 0.9 1.2 2.5

U

Adj-LR

that did not converge: (1) 276 runs; (2) 8 runs; (3) 1978 runs; (4) 209 runs; (5) 19 runs; (6) 1 run; (7) 4597 runs; (8) 1421 runs; (9) 383 runs; (10) 33 runs.

∗ Cases

n

1

6.0 5.5 5.0 5.5 5.0 5.2 5.4 5.2 2.3 5.0 5.7 5.2 0.8 2.1 2.4 5.1

L+U

3.2 2.7 2.9 2.8 2.7 2.5 2.7 2.9 1.3 2.0 2.6 2.5 0.3 1.2 2.0 2.7

L

3.3 3.0 2.7 2.7 2.5 2.6 3.0 2.6 1.0 1.8 2.8 2.3 0.4 0.9 1.9 2.5

U

ABC

Table IV. Comparison of 95 per cent con
dence intervals for  = 5:5, various 1 and sample sizes (doses 1; 2; : : : ; 10).

6.5 5.7 5.6 5.5 5.2 5.1 5.7 5.5 2.3 3.8 5.4 4.8 0.7 2.1 3.9 5.2

L+U

1986 D. FARAGGI, P. IZIKSON AND B. REISER

Statist. Med. 2003; 22:1977–1988

1987

CONFIDENCE INTERVALS FOR THE 50 PER CENT RESPONSE DOSE

Table V. Beetle mortality data. Log10 dose (xi )

Number of subjects (ni )

Number killed (ri )

50 49 50 50 50 49

15 24 26 24 29 29

1.08 1.16 1.21 1.26 1.31 1.35

Table VI. Ninety-
ve percent con
dence intervals using the dierent methods and their tail coverage probabilities (2.5 per cent nominal on each tail) for the beetle data. 95 per cent con
dence interval Delta Fieller Likelihood ratio Adjusted likelihood ratio ABC

1.176, 1.161, 1.162, 1.162, 1.179,

1.295 1.320 1.319 1.318 1.297

L

U

L+U

0.5 2.5 2.4 2.5 4.4

1.0 2.6 2.3 2.4 4.2

1.5 5.1 4.7 4.9 8.6

4. DISCUSSION When discussing the closely related probit model, Finney [13] recommended the use of the Fieller procedure over the delta method. Abdelbasit and Plackett [12], based on a small simulation study, disagreed with Finney and found the delta method to provide satisfactory results. Sitter and Wu [8] criticized the Abdelbasit and Plackett work as being based on such a large sample size as to override any dierences in the two procedures. In the context of the logistic model they carried out a much more extensive simulation study with various sample sizes and found that Fieller intervals outperformed those based on the delta method, especially for small sample sizes. Even for large sample sizes, where the two-tailed coverage of the delta intervals is often satisfactory, they found that the Fieller interval does ‘a much better job of matching the one-tailed error rates’. Williams [14] compared the Fieller and likelihood ratio con
dence intervals and recommended the use of the likelihood ratio intervals,
nding the Fieller procedure to produce overly conservative results. Hoekstra [11], in a very limited simulation study, compared the Fieller, delta and likelihood ratio intervals. He found that ‘Fieller’s method seems to yield no improvement over the simpler delta method’ and in some cases found the likelihood ratio interval to be conservative. The disagreement in the literature over these three methods motivated this paper. Based on our extensive simulations for a wide variety of shapes of response probabilities over dose we conclude that for con
dence intervals for the 50 per cent response dose: (i) The delta method cannot be recommended. Even for large sample sizes its coverage can be either strongly conservative or strongly liberal and for small sample sizes it can provide one-tailed error rates quite far from their nominal value. The only situation where the delta method might be considered is that of Hoekstra [11] where the response probabilities are generally very close to either 0 or 1 with only one or two doses giving Copyright ? 2003 John Wiley & Sons, Ltd.

Statist. Med. 2003; 22:1977–1988

1988

(ii) (iii)

(iv) (v)

D. FARAGGI, P. IZIKSON AND B. REISER

intermediate values. This is generally a poor design for which the maximum likelihood estimation procedure will frequently not converge. The Fieller procedure is usually somewhat conservative with error rates close to their nominal value but in some cases can be quite conservative. The likelihood ratio method gives error rates somewhat higher than Fieller. Over all the combinations of sample size and parameter values considered, the likelihood ratio method gave two-sided error rates closer to the nominal 5 per cent than the Fieller method in about 60 per cent of the cases. Adjusting the likelihood ratio does not lead to any improvements. The ABC method is often overly conservative.

Consequently the likelihood ratio method should generally be used but if one prefers to have conservative con
dence intervals, the Fieller method should be chosen. Many packaged computer programs such as SAS now routinely give likelihood ratio con
dence intervals for the parameters of logistic regression. It would be useful if this could be extended to deal with functions of these parameters such as the 50 per cent response dose considered in this paper. ACKNOWLEDGEMENTS

We would like to thank the referees whose comments led to improvements in this paper. This paper was partly written while B. Reiser was visiting the Department of Biostatistics, University of Michigan. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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Statist. Med. 2003; 22:1977–1988