Standard Article

# Isotonic Inference

Published Online: 15 JUL 2005

DOI: 10.1002/0470011815.b2a15061

Copyright © 2005 John Wiley & Sons, Ltd

Book Title

## Encyclopedia of Biostatistics

Additional Information

#### How to Cite

Hirotsu, C. 2005. Isotonic Inference. Encyclopedia of Biostatistics. 4.

#### Publication History

- Published Online: 15 JUL 2005

- Abstract
- Article
- Tables
- References

Isotonic inference concerns situations in which a set of parameters is assumed, a priori, to satisfy certain order restrictions. In the most common case, where data are arranged in ordered groups, the mean value of a random variable is assumed to change monotonically with the ordering of the groups. It is then reasonable to take account of the order restrictions in making inferences about the group means, such as point or interval estimations or significance tests. Isotonic inference extends more generally to situations where there are various shape constraints on response curves, such as convexity, concavity, or sigmoidicity.

One approach to such problems is to assume a parametric model that incorporates those order or shape constraints such as a linear regression equation or a particular dose–response function. The inference based on the parametric model can, however, be considerably biased and variable when the specified model is incorrect. It has been pointed out in environmental toxicology applications, for example, that no parametric dose–response model can be assumed to hold generally at very low doses of interest, and yet a monotone and convex relationship might reasonably, and more reliably, be assumed. We are therefore concerned in this article mainly with methods of inference that avoid the need to specify a rigid parametric model, but nevertheless allow for those order restrictions.

There is a large literature on estimation and testing (*see* Hypothesis Testing) in the areas of isotonic and order-restricted inferences, and comprehensive surveys of these areas include [3] and [43].

One general approach to the isotonic inference is maximum likelihood estimation. The problem of finding order-restricted maximum likelihood estimates is often solved by using isotonic regression. In its simplest case an explicit solution is obtained by the pool-adjacent-violators method, but in more general cases it is solved only by some nonlinear programming, see [46] and [10], for example, or by the aid of a formal Bayesian approach, as in [42]. For a restricted likelihood ratio test the usual asymptotic chi-square distribution theory does not apply. In some cases, the resulting distributions are known to be a mixture of χ^{2} distributions, but in other cases some computer-intensive methods such as parametric bootstrap tests [10], or an asymptotic conservative approximation method [46] may be used. The maximum likelihood approach is outlined in another article (*see* Isotonic Regression). Here we are concerned mainly with other approaches to isotonic inference. As a natural method of incorporating prior knowledge in particular applications, a Bayesian approach is also briefly mentioned.

### 1 The Case for Isotonic Inference

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

The data in Table 1 are measurements of the half-life of an antibiotic drug in relation to the dose administered. The usual analysis of variance (ANOVA) is obviously inappropriate, because of the ordering of the doses, and one possible approach is to assume a parametric model. The simplest model for the monotone relationship is linear regression. However, it is generally difficult to assume that a linear relationship holds over a wide range of an explanatory variable. For the dose–response relationship there are of course more natural response curves, such as a sigmoid function, but it is still often difficult to assume a particular model for the given set of data. Furthermore, it also sometimes suffices to show an overall upward trend or to detect a steep change-point in the responses. It is then unnecessary to assume a rigid parametric model, and a nonparametric trend test or some multiple comparisons procedure is more appropriate (*see* Simultaneous Inference). We need assume only a monotone relationship in the mean half-life,

- (1)

where at least one inequality is strong, so that the null model H_{0}: μ_{1} = ⋯ = μ_{a} is excluded.

Dose (mg/kg) | Data (h) | Average | ||||
---|---|---|---|---|---|---|

5 | 1.17 | 1.12 | 1.07 | 0.98 | 1.04 | 1.076 |

10 | 1.00 | 1.21 | 1.24 | 1.14 | 1.34 | 1.186 |

25 | 1.55 | 1.63 | 1.49 | 1.53 | 1.550 | |

50 | 1.21 | 1.63 | 1.37 | 1.50 | 1.81 | 1.504 |

200 | 1.78 | 1.93 | 1.80 | 2.07 | 1.70 | 1.856 |

The data in Table 2 show ordinal (i.e. ordered) categorical data typical of a Phase III comparative clinical trial. Assuming a multinomial model with cell probabilities *p*_{i j}, the null hypothesis that the two treatments are equal can be expressed as *p*_{1j} = *p*_{2j}, *j* = 1, …, 4, or equivalently as

- (2)

where a dot denotes the summation with respect to the suffix replaced by the dot. Eq. 2 is the familiar independence hypothesis for a two-way contingency table. However, the usual goodness-of-fit chi-square test is inappropriate, since we are interested in a more restricted alternative

where at least one inequality is strong, implying that treatment 1 is superior to treatment 2 in efficacy or vice versa. Ordered categorical data are a special case of rank data with many ties, and any method for rank data can be applied to ordered categorical data, and vice versa.

Drug | Not effective | Slightly effective | Effective | Excellent |
---|---|---|---|---|

AMPC | 3 | 8 | 30 | 22 |

S6472 | 8 | 9 | 29 | 11 |

If ordered categorical data are obtained at several doses, as in Table 3, then we are interested in testing the two-way ordered alternative:

which implies that higher doses are superior to lower doses in efficacy.

Drug | Undesirable | Slightly undesirable | Not useful | Slightly useful | Useful | Excellent |
---|---|---|---|---|---|---|

Placebo | 3 | 6 | 37 | 9 | 15 | 1 |

AF 3 (mg/kg) | 7 | 5 | 33 | 21 | 10 | 1 |

AF 6 (mg/kg) | 5 | 6 | 21 | 16 | 23 | 6 |

A similar hypothesis

has been considered for normal means from a two-way layout experiment, which implies that the differences, μ_{i j} − μ_{i′j}, tend upwards as the level *j* increases for any *i* > *i*′; see [14].

### 2 Various Extensions of the Monotone Relationship

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

A monotone dose–response relationship may be disturbed by toxicity at higher doses, and a nonparametric testing procedure for the downturn (or “umbrella”) hypothesis,

has been proposed in [50]; here τ is an unknown turning point.

Some other extensions arise when responses show monotone relationships with the passage of time. Frequently encountered examples include the monotonic change of occurrence probabilities of some events, increasing treatment effects, and increasing hazard rates with time. For instance, the hypothesis

arises from the analysis of the age–period–cohort effects model (*see* Age–Period–Cohort Analysis) where only the second-order differences are estimable in each effect along with the time axis. Hypothesis H_{6} is equivalent to H′_{6}:μ_{i} − 2 µ_{i + 1} + μ_{i + 2} ≥ 0, and may be called the “convexity hypothesis”. Convexity, concavity, and sigmoidicity constraints are commonly employed also in the field of bioassay as reasonable shape constraints on a dose–response relationship (*see* Biological Assay, Overview; Quantal Response Models).

As seen from the above examples, isotonic inference is closely related to change-point analysis. Actually, a one-sided change-point model may be formulated as a set of particular monotone relationships,

- (3)

with τ an unknown change-point parameter, so that a useful statistic for change-point analysis is useful also for isotonic inference. Interestingly, 3 defines *a* − 1 edges of the convex cone defined by the simple ordered alternative 1; see [24].

For other extensions, including tree-structured, star-shaped, unimodality, and symmetry models, the reader is referred to [3] and [43].

### 3 Testing a Simple Ordered Alternative in Normal Means

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

We wish to test a simple ordered alternative H_{1} in the one-way layout model

where the ɛ_{i j} are assumed to be independently distributed as normal N(0, σ^{2}) with known variance σ^{2}. Then there are two major streams of overall trend tests and multiple contrast type tests. Most cases of unknown variance can be dealt with similarly, if an unbiased variance estimator distributed as a multiple of χ^{2} is available.

#### 3.1 Overall Trend Tests

One possible approach is the restricted likelihood ratio test developed extensively in [3] (*see* Isotonic Regression). The approach does not, however, possess any obvious optimal property for such restricted alternatives, and is rather difficult to extend to higher-way problems.

Abelson & Tukey [1] proposed a linear score statistic which maximizes the minimum power in the region defined by H_{1} within the class of linear tests. This has been extended to the most stringent and somewhere most powerful (MSSP) test for a more general restricted alternative by Schaafsma [44, 45]. In the balanced case, Abelson & Tukey's score is determined by equalizing powers at all the *a* − 1 edges of H_{1} and is given by

Extending Taguchi's idea [52], the cumulative χ^{2} test was introduced in [15], and its power has been compared with that of the previous two approaches. The test statistic χ*^{2} is the sum of squares of the standardized accumulated statistics

- (4)

where *N*_{i} = *n*_{1} + ⋯ + *n*_{i}, , and with , *i* = 1, …, *a*. The χ*^{2} statistic is characterized by the strong positive correlations between the serial components , and in particular by the expansion for the balanced case in a series of independent χ^{2} variables,

where is the 1 df χ^{2} statistic for detecting the departure from the null model in the direction of Chebyshev's *l*th order orthogonal polynomial. Hence χ*^{2} tests mainly, but not exclusively, a linear trend; see [19] and [36] for details.

#### 3.2 Multiple Contrast Type Tests

Several multiple comparison procedures have been proposed for ordered parameters. Williams [55] proposed a closed testing procedure based on the maximum likelihood estimator for defining the maximal noneffective dose level. Marcus [30] modified the method by changing the estimator at the control level from *y*_{1}. to , the maximum likelihood estimator of μ_{1}, so that his statistic is the maximal component of Bartholomew's . The limiting distribution of the latter statistic is obtained in [56] and more recently, an exact recursive integration procedure for calculating its distribution function is obtained in [29]. The maximal component of χ*^{2} has been proposed also for this purpose, and is called the “max *t*” method, where *t* stands for the of 4. The statistic is characterized as the likelihood ratio test for the change-point hypothesis H_{7}, and an exact and very efficient algorithm for calculating the *P* value has been obtained by Hawkins [12] (*see* Change-point Problem). The power functions of these closed multiple testing procedures have been compared in [30, 49], and [26]. More general multiple tests for ordered parameters are obtained in [32].

#### 3.3 Confidence Interval

A confidence interval taking advantage of order restrictions can be obtained by inverting an appropriate test for order restricted alternatives. For example, Marcus & Peritz [31], Schoenfeld [47] and Hirotsu & Srivastava [25] obtain confidence intervals for normal means by inverting a multiple contrast type test, the restricted likelihood ratio test, and the max *t* test, respectively. Wynn [59] gives a general methodology for obtaining one-sided confidence intervals, and Hayter [13] obtains confidence intervals based on the one-sided studentized range test. Miwa & Hayter [35] obtain confidence intervals for the differences of the ordered normal means in the one-way layout setting taking the advantages of the one- and two-sided procedures. In particular, in the bioassay problem, Schmoyer [46] obtains improved upper confidence bounds for the responses at very low doses by assuming sigmoidicity in the dose–response curve.

Hwang & Peddada [28] develop a methodology under a general order restriction, which has been extended recently to a test procedure by Peddada, Prescott and Conaway [37].

There is no extensive work on the design of experiments on the ordered parameters, although an optimal allocation has been discussed in [23] (*see* Optimal Design).

#### 3.4 Applications

A test of Abelson & Tukey, the cumulative χ^{2} test, and some of the multiple comparison procedures, are now applied to the data in Table 1. Since the variance σ^{2} is unknown, it is replaced by the usual unbiased estimate of variance, .

The linear score statistic of Abelson & Tukey is calculated as

with scores , and *c*_{3} = 0, giving a *P* value of 2.2 × 10^{−8} as evaluated by the *t* distribution with 19 df.

The null distribution of cumulative χ^{2} statistic is well approximated by , a multiple of the χ^{2} variable with df *f*, where the constants *d* and *f* are given by

- (5)

with . An even better approximation based on the expansions by Laguerres' orthogonal polynomials, and also the approximation under the alternative hypothesis, are given in [16]. Then the *P* value of the statistic

can be evaluated as 1.1 × 10^{−8} by the *F* distribution with df (*f*, ∑*n*_{i} − *a*), where *f* = 2.067 from 5.

The maximal component of the is obtained at the partition between levels 2 and 3:

the one-sided *P* value of which is evaluated as 1.1 × 10^{−7} by the recurrence formula based on the Markov property of . According to the closed testing procedure of [32], the process proceeds to the final step where the *t* statistic between levels 1 and 2 shows a nonsignificant result at the one-sided significance level 0.10, thus suggesting finally the difference between the dose levels (1,2) and (3,4,5).

In applying the Williams [55] procedure and the modified Williams procedure of [30], we need the maximum likelihood estimaters of the μ_{i}, which are

by the pool-adjacent-violators method. Since , both statistics coincide and equal

where we take the repetition number *m* as the harmonic mean of the *n*_{i}s for referring approximately to the tables for upper percentiles in the balanced case by [55] and [56], respectively. In any case, the statistic *w* is highly significant and the closed testing procedure stops with the nonsignificant result between levels 1 and 2, thus again suggesting a difference between the dose levels (1, 2) and (3, 4, 5).

For a more general likelihood *L*(**y**, **θ**, **ν**) with the ordered parameter **θ**, and possibly with the nuisance parameter **ν**, arguments similar to those used above apply if the asymptotic normality of the likelihood estimators is assured. In particular, the cumulative χ^{2} and the max *t* statistics can be based on the cumulative efficient scores evaluated at the null hypothesis and extended easily to two-way problems; see [7, 17, 18] for details.

### 4 Testing Ordered Alternatives in Binomial Probabilities

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

The data in Table 4 are from a dose–response clinical trial. Assuming that the *y*_{i} are independently distributed as binomial *B*_{in}(*n*_{i}, *p*_{i}) we are interested in testing the simple ordered alternative

Dose (mg/day) | Improved | Not improved |
---|---|---|

100 | 20 | 16 |

150 | 23 | 18 |

200 | 27 | 9 |

225 | 26 | 9 |

300 | 9 | 5 |

If the quantitative measures *d*_{1} < ⋯ < *d*_{a} are attached to the *y*_{i}, then the locally most powerful test against a wide range of monotone relationships of *p*_{i} to *d*_{i} is obtained by Cochran [6] and Armitage [2] (*see* Trend Test for Counts and Proportions). For the case where there is no information on *d*_{i}, the likelihood ratio test has been developed by Chacko [5]. The tests based on the cumulative χ^{2} and its maximal component have also been extended as follows:

- (6)

where is given by 4 with σ replaced by [*Y*(1 − *Y*)]^{1/2}, *Y* = ∑*y*_{i}/∑*n*_{i}, and *y*_{i·} replaced by *y*_{i} in defining *Y*_{i}. Formula 5 is also valid for the χ*^{2} to give a two-sided *P* value of 0.113 when applied to Table 4. For max *t*, another exact algorithm is available based on the Markov property of the to give a one-sided *P* value of 0.044 for Table 2 at the partition between levels (1, 2) and (3, 4, 5); see [26, 57, 58] for the algorithm. The Cochran–Armitage test gives a slightly larger one-sided *P* value of 0.049 since there is a slight downturn tendency in this example.

### 5 Analyzing the Two-Way Contingency Table with Ordered Column Categories

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

#### 5.1 Two-Sample Problem

First consider the two-sample problem presented by Table 2. A popular approach to the analysis is to use a nonparametric test based on a linear score statistic such as Wilcoxon's (*see* Wilcoxon–Mann–Whitney Test). Now, for the two-sided alternative H_{2} the two statistics,

- (7)

- (8)

can be defined in terms of the accumulated efficient scores, where is the goodness-of-fit χ^{2} statistic for the 2 × 2 table formed by accumulating the first *j* and the remaining *b* − *j* columns. The is, however, identical to the of 6 if the binomial data are arranged in a 2 × *b* table in an obvious way and exactly the same distribution theory applies also to this case. The two-sided *P* values are 0.039 for χ*^{2} and 0.154 for max χ^{2}, whereas it is 0.025 for the Wilcoxon test. In this case the Wilcoxon test shows the smallest *P* value, since approximately a linear trend is observed in *p*_{1j}/*p*_{2j}, *j* = 1, …, 4. If these tests are applied to the last two rows of Table 3 for comparing AF 3 mg and AF 6 mg, then the two-sided *P* values are 0.0128, 0.0096, and 0.0033 for the Wilcoxon, the χ*^{2}, and max χ^{2} methods, respectively. It has been verified by simulation that, when evaluated as two-sample nonparametric tests, the Wilcoxon method is useful for the location shift of the underlying symmetrical and light-tailed distributions such as the logistic or normal, the max χ^{2} method is useful for skewed or heavy-tailed distributions, and the cumulative χ^{2} method is characterized by its robustness, having relatively high power over a wide range of underlying distributions—normal, heavy-tailed, or skewed.

Another important approach to the problem is to assume an underlying continuous distribution for each treatment and to compare the parameters describing those distributions. The proportional-odds and proportional-hazards models are important examples; see [34] for details.

#### 5.2 General a-Sample Problem

For a general *a*-sample problem the Wilcoxon test is extended to the Kruskal–Wallis test. The same type extensions are available for the χ*^{2} and its maximal component by defining the in 7 and 8 as the goodness-of-fit χ^{2} statistic for the accumulated *a* × 2 table for the partition between columns *j* and *j* + 1. The constants for the χ^{2} approximation of χ*^{2} are obtained by

with , *C*_{j} = *y*_{·1} + ⋯ + *y*_{·j}, and . The max χ^{2} can be evaluated by the calculation algorithm based on the Markov property of the subsequent s [26].

For the row-wise multiple comparisons based on the cumulative χ^{2}, the statistic

is defined where ⊗ is a Kronecker product, **z** a vector of arranged in dictionary order, **C***′ a *b* − 1 × *b* matrix defined so that the (*j*, *j*′) th element of **C***′**C*** is (γ_{j}/γ_{j′})^{1/2} for *j* ≤ *j*′ and the maximum is taken over all **a** that satisfy **a′a** = 1, and . When *a* ≥ *b* and under the null model, the statistic *S* is asymptotically distributed as the largest root of the Wishart matrix *W*(**C***′**C***, *a* − 1), which is well approximated by γ_{(1)}χ^{2}(*a* − 1) with γ_{(1)} the largest root of **C***′**C***. The statistic *S* gives the Scheffé-type multiple comparison test, and has been applied to taste-testing data of five foods in five ordered categorical responses of [4] to obtain the significant classification of rows (foods) (1, 2), (3, 4) and (5). The max χ^{2} is also applied to the data for multiple comparisons of the columns to obtain a highly significant classification (1, 2, 3) and (4, 5). The resulting block interaction model is expressed as

if *i* belongs to the μth subgroup of rows and *j* to the νth subgroup of columns. The goodness-of-fit χ^{2} has been compared with the fitting of the proportional-odds model [51] and its extension [33]; see [20] for details. The Scheffé-type multiple comparison method is applied to the normal distribution model in [21], for classifying subjects based on the upward, flat, and downward tendencies defined by repeated measurements.

### 6 Two-Way Contingency Table with Natural Orderings in Both Rows and Columns

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

Assuming a multinomial model *M*(*y*_{··}, *p*_{i j}) for the data *y*_{i j} in Table 3 the cumulative χ^{2} statistic and its maximal component are defined for testing H_{3} using the cumulative efficient scores evaluated at the null hypothesis. These are

with the being the goodness-of-fit χ^{2} for the 2 × 2 tables obtained from partitioning and accumulating rows and columns at *i* = 1, …, *a* − 1, and *j* = 1, …, *b* − 1, respectively. The χ**^{2} is for the two-sided version of H_{3}, and max max χ^{2} is applicable to both one- and two-sided problems. When applied to Table 3 the two-sided *P* values are approximately 0.0065 for the χ**^{2} and exactly 0.0142 for max max χ^{2}. The details of the *P* value calculations are given in [22]. As a semiparametric model for the ordered two-way table the constant-odds ratio model has been proposed by Wahrendolf [54] based on Plackett's [38] coefficient of association for bivariate distributions (*see* Association, Measures of).

As an example of the higher-way layouts, a 2 × J × *K* table comparing two treatments based on bivariate allele frequencies, is analyzed in [27]. An example of highly fractional factorial experiments with ordered categorical responses is given in [11]; see also the discussion following that article (*see* Factorial Experiments).

### 7 Bayesian Approach to Isotonic Inference

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

Since the purpose of an isotonic inference is to make use of the prior knowledge to enhance the efficiency of test and estimation, it is natural to consider a Bayesian approach. For example, an essentially complete class of tests for orderly constrained hypothesis is obtained as the whole set of Bayes tests with a prior distribution defined on those constrained supports. The cumulative χ^{2} and max *t* methods are derived from this idea; see [7, 17, 53]. More specifically, in bioassay problems, the Dirichlet prior has been introduced for the successive differences of the responses for doses *d*_{i}, *p*(*d*_{i}) − *p*(*d*_{i − 1}), *i* = 1, …, *a* + 1;*p*(*d*_{0}) = 0, *p*(*d*_{a + 1}) = 1, reflecting the nondecreasing nature of the dose–response relationship, see [40, 41], for example. Shaked & Singpurwalla [48] discuss the defect of the Dirichlet prior, and introduce concavity constraints on the shape of a dose–response curve, reflecting a situation encountered in practice. Because of computational difficulties, however, they are unable to compute posteriors beyond modal estimates. The computational problem was overcome later by Gelfand & Kuo [8], who showed how a sampling-based approach could be used to develop the desired marginal posterior distributions and their features, for Dirichlet and product–beta priors; see also [9]. Ramgopal et al. [39] consider convex, concave, and ogive constraints to specify the shape of dose–response curves, and extend the sampling-based approach to calculating any posterior feature of interest in these generalized constrained problems.

### References

- Top of page
- The Case for Isotonic Inference
- Various Extensions of the Monotone Relationship
- Testing a Simple Ordered Alternative in Normal Means
- Testing Ordered Alternatives in Binomial Probabilities
- Analyzing the Two-Way Contingency Table with Ordered Column Categories
- Two-Way Contingency Table with Natural Orderings in Both Rows and Columns
- Bayesian Approach to Isotonic Inference
- References

- 11963). Efficient utilization of non-numerical information in quantitative analysis: general theory and the case of the simple order, Annals of Mathematical Statistics 34, 1347–1369.& (
- 21955). Test for linear trends in proportions and frequencies, Biometrics 11, 375–386.(
- 31972). Statistical Inference Under Order Restrictions. Wiley, Chichester., , & (
- 41962). Optimal scaling for ordered categories, Psychometrika 27, 355–374., & (
- 51966). Modified chi-square test for ordered alternatives, Sankhy$\bar{\hbox{a}}$, Series B 28, 185–190.(
- 61954). Some methods for strengthening the common χ(
^{2}test, Biometrics 10, 417–451. - 71991). Tests for independence in contingency tables with ordered categories, Journal of Multivariate Analysis 36, 56–67.& (
- 81991). Nonparametric Bayesian bioassay including ordered polytomous response, Biometrika 78, 657–666.& (
- 91992). Bayesian analysis of constrained parameter and truncated data problems using Gibbs sampling, Journal of the American Statistical Association 87, 523–532., & (
- 101991). Constrained maximum likelihood exemplified by isotonic convex logistic regression, Journal of the American Statistical Association 86, 717–724.(
- 111990). A critical look at accumulation analysis and related methods (with discussion), Technometrics 32, 119–130.& (
- 121977). Testing a sequence of observations for a shift in location, Journal of the American Statistical Association 72, 180–186.(
- 131990). A one-sided Studentized range test for testing against a simple ordered alternative, Journal of the American Statistical Association 85, 778–785.(
- 141978). Ordered alternatives for interaction effects, Biometrika 65, 561–570.(
- 151979). The cumulative chi-squares method and Studentized maximal contrast method for testing an ordered alternative in a one-way analysis of variance model, Reports of Statistical Application Research, Union of Japanese Scientists and Engineers 26, 12–21.(
- 161979). An(
*F*-approximation and its application, Biometrika 66, 577–584. - 171982). Use of cumulative efficient scores for testing ordered alternatives in discrete models, Biometrika 69, 567–577.(
- 181983). Defining the pattern of association in two-way contingency tables, Biometrika 70, 579–589.(
- 191986). Cumulative chi-squared statistic as a tool for testing goodness of fit, Biometrika 73, 165–173.(
- 201990). Discussion on Hamada and Wu's paper, Technometrics 32, 133–136.(
- 211991). An approach to comparing treatments based on repeated measures, Biometrika 78, 583–594.(
- 221997). Two-way change-point model and its application, Australian Journal of Statistics 39, 205–218.(
- 231987). Optimal allocation of observations for inference on& (
*k*ordered normal population means, Australian Journal of Statistics 29, 151–165.Direct Link: - 242002). Changepoint analysis as a method for isotonic inference, Scandinavian Journal of Statistics 29, 125–138.& (
- 252000). Simultaneous confidence intervals based on one-sided max& (
*t*test, Statistics & Probability Letters 49, 25–37. - 261992). Multiple comparison procedure based on the maximal component of the cumulative chi-squared statistic, Biometrika 79, 381–392., & (
- 272001). An exact test for the association between the disease and alleles at highly polymorphic loci with particular interest in the haplotype analysis, Biometrics 57, 769–778., , & (
- 281994). Confidence interval estimation subject to order restrictions. Annals of Statistics 22, 67–93.& (
- 292002). On the isotonic range statistic for testing against an ordered alternative, Journal of Statistical Planning and Inference 105, 347–362., & (
- 301976). The powers of some tests of the equality of normal means against an ordered alternative, Biometrika 63, 177–183.(
- 311976). Some simultaneous confidence bounds in normal models with restricted alternatives, Journal of the Royal Statistical Society, Series B 38, 157–165.& (
- 321976). On closed testing procedure with special reference to ordered analysis of variance, Biometrika 63, 655–660., & (
- 331980). Regression models for ordinal data, Journal of the Royal Statistical Society, Series B 42, 109–142.(
- 34 & (
- 351999). Combining the advantages of one-sided and two-sided test procedures for comparing several treatment effects, Journal of the American Statistical Association 94, 302–307.& (
- 361986). On testing against ordered alternatives in analysis of variance models, Biometrika 73, 493–499.(
- 372001). Biometric 57, 1219–1227., & (
- 381965). A class of bivariate distributions, Journal of the American Statistical Association 60, 516–522.(
- 391993). Nonparametric Bayesian bioassay with prior constraints on the shape of the potency curve, Biometrika 80, 489–498., & (
- 401972). A Bayesian approach to bioassay, Biometrics 28, 841–858.(
- 41 (
- 421996). Maximum likelihood estimation under order restrictions by the prior feedback method, Journal of the American Statistical Association 91, 167–172.& (
- 431988). Order Restricted Statistical Inference. Wiley, New York., & (
- 441966). Hypothesis testing problems with the alternative restricted by a number of inequalities,(
*Doctoral dissertation.*University of Groningen, Noordhoff, Groningen. - 451968). A comparison of the most stringent and the most stringent somewhere most powerful tests for certain problems with restricted alternatives, Annals of Mathematical Statistics 39, 531–546.(
- 461984). Sigmoidally constrained maximum likelihood estimation in quantal bioassay, Journal of the American Statistical Association 79, 448–453.(
- 471986). Confidence bounds for normal means under order restrictions, with application to dose-response curves, toxicology experiments, and low-dose extrapolation, Journal of the American Statistical Association 81, 186–195.(
- 481990). A Bayesian approach for quantile and response probability estimation with applications to reliability, Annals of the Institute of Statistical Mathematics 42, 1–19.& (
- 491979). The comparison of treatment with control group means in toxicological studies, Applied Statistics 28, 144–151.(
- 501986). Recursive nonparametric testing for dose response relationships subject to downturn at high doses, Biometrika 73, 589–596.& (
- 511964). A scaling procedure for ordered categorical data, Biometrics 20, 592–607.(
- 521966). Statistical Analysis (in Japanese). Maruzen, Tokyo.(
- 531979). Test and estimation problems under restricted null and alternative hypotheses (in Japanese), Journal of Economics 45, 2–10.(
- 541980). Inference in contingency tables with ordered categories using Plackett's coefficient of association for bivariate distributions, Biometrika 76, 15–21.(
- 551971). A test for differences between treatment means when several dose levels are compared with a zero dose control, Biometrics 27, 103–117.(
- 561977). Some inference procedures for monotonically ordered normal means, Biometrika 64, 9–14.(
- 571983). The power of likelihood ratio and cumulative sum tests of a change in a bionomial probability, Biometrika 70, 455–464.(
- 581986). Confidence regions and tests for a change point in a sequence of exponential family of random variables, Biometrika 73, 91–104.(
- 591975). Integrals for one-sided confidence bounds: A general result, Biometrika 62, 393–396.(