Trend tests for the evaluation of exposureresponse relationships in epidemiological exposure studies
 Ludwig A Hothorn^{1_53}Email author,
 Michael Vaeth^{2_53} and
 Torsten Hothorn^{3_53}
DOI: 10.1186/1742557361
© Hothorn et al. 2009
Received: 29 May 2006
Accepted: 06 March 2009
Published: 06 March 2009
Abstract
One possibility for the statistical evaluation of trends in epidemiological exposure studies is the use of a trend test for data organized in a 2 × k contingency table. Commonly, the exposure data are naturally grouped or continuous exposure data are appropriately categorized. The trend test should be sensitive to any shape of the exposureresponse relationship. Commonly, a global trend test only determines whether there is a trend or not. Once a trend is seen it is important to identify the likely shape of the exposureresponse relationship. This paper introduces a best contrast approach and an alternative approach based on orderrestricted information criteria for the model selection of a particular exposureresponse relationship. For the simple change point alternative H _{1} : π _{1} = ...= π _{ q } <π _{ q+1} = ... = π _{ k } an appropriate approach for the identification of a global trend as well as for the most likely shape of that exposureresponse relationship is characterized by simulation and demonstrated for real data examples. Power and simultaneous confidence intervals can be estimated as well. If the conditions are fulfilled to transform the exposureresponse data into a 2 × k table, a simple approach for identification of a global trend and its elementary shape is available for epidemiologists.
Introduction
Child cancer and magnetic fields
Exposure/μ Tesla  j  n_{cancer}  n_{no cancer}  n_{j}  p _{ j }  RR _{ j1} 

0–0.05  1  1698  4759  6457  0.263   
0.051–0.101  2  0  9  9  0  0.000 
0.1010.15  3  2  3  5  0.4  1.525 
0.151.20  4  1  3  4  0.25  0.953 
0.2010.25  5  1  3  4  0.25  0.953 
0.2510.30  6  0  4  4  0  0.000 
0.3010.35  7  0  2  2  0  0.000 
0.3510.85  8  1  0  2  0.5  1.906 
0.8511.6  9  2  0  2  1  3.812 
>1.61  10  2  0  2  1  3.812 
Although this example is seriously unbalanced, real epidemiological exposure studies with many unexposed or lowexposure cases but few highexposure cases can be found. The appropriate evaluation of such epidemiological exposure studies is a statistical challenge. Many similar examples can be found in the literature, e.g. a casecontrol study for respiratory cancer possibly caused by longterm exposure to coke oven emissions [3].
In exposure studies, an unexposed group, E_{1}, is commonly compared with several exposure groups, E_{2},..., E_{k}. The outcome of the study is the number of cases suffering from the disease being investigated, such as a specific tumor, and the number of observations without the disease (controls), i.e. the risk of disease in each category of exposure. One important objective in exposure epidemiology is causation; the demonstration of a global exposureresponse relationship represents one of the causation criteria, according to Hill [4]. A global trend test leads to identification of a trend, whereas model selection allows inference of the likelihood of a particular elementary model.
The sampling strategy of epidemiological exposure studies is either a cohort study, in which a 2 × k contingency table represents the data, or a casecontrol study, in which two multinomial distributions are compared. However, the likelihood ratio test of identical multinomials against the elementary odds ratios alternative, for a sufficient total number of observations, is equivalent to the comparison of the k independent binomial proportions against a simple ordered alternative (Agresti and Coull, [5]; Hothorn et al., [6]). Therefore, it is appropriate to evaluate both designs by means of an asymptotic trend test for a 2 × k contingency table.
Numerous methods, including modelbased (e.g. Royston et al., [7]) and testbased approaches (e.g. Dosemeci and Benichou, [8]), are used to analyze exposureresponse relationships. A basic problem is that the shape of the exposureresponse is unknown a priori and is an outcome of the study. However, the choice of model or test greatly depends on the shape of the exposureresponse. Therefore, a broad class of models or tests should be used, but that, in turn, leads to a model selection dilemma. Model selection is an intricate component of statistical problems. Model selection in this case is not the objective, but is only a tool for identifying the correct trend from several possible elementary alternatives. An alternative hypothesis can be decomposed into its underlying elementary alternatives, e.g. the simple order alternative H _{1}: π _{1} ≤ π _{2} ≤ π _{3} can be decomposed into the three elementary hypotheses : π _{1} = π _{2} <π _{3}, : π _{1} <π _{2} = π _{3}, : π _{1} <π _{2} <π _{3}.
The pvalue, a commonly used outcome of a trend test, is frequently insufficient for epidemiological studies. Information concerning the shape of the exposureresponse and/or a measure of the magnitude of the effect, such as relative risks or odds ratios, is desirable for a significant trend. Thus, the level of the false positive decision rate (α) should be controlled. In addition, an approach with a minimum false negative decision rate (β) (respective maximum power π = 1  β) for the global test decision and a maximum correct decision rate for the selected model should be identified. The correct classification rate, the proportion of correctly identified elementary alternatives, is used as a major performance measure later on.
The exposure in casecontrol studies is frequently measured on a continuous scale. Categorization at preselected cutoff points of a small number of ordered categories is common; for example, four categories of trihalomethane exposure (Jones et al., [9]), or three categories of lifetime dose of hair dye (Benavente et al., [10]). Inappropriately chosen cutoff points dramatically reduce the power of the trend test (Greenland, [11]). Some exposures are naturally grouped, for example 2–3 cups of coffee per day, by the impreciseness of the definitions, such as "cup" and "coffee" (Ascherio et al., [12]). An example of ordinal definition of the exposure is given in a casecontrol study of Norwegian nickel refinery workers (Grimsrud et al., [13]). The exposurerelated associations between smokingadjusted lung cancer rates and cumulative exposure to different forms of nickel used the categories "low," "medium," and "high."
The best approach, in terms of both power and interpretation, occurs when a single cutoff point exists and is known a priori, resulting in a twosample test "above" vs. "below" the cutoff point. This is because an odds ratio and its onesided confidence interval can be estimated. The trend test approach discussed here is designed for naturally grouped exposure with a single change point. For continuous exposure models a continuous covariate can be used. However, the choice of an appropriate model – such as linear, logistic, or other – remains open and model selection influences the inference.
In this paper, a trend test for the comparison of k ordered binomial proportions using a change point alternative is presented. Either a single change point is directly of interest or the change point alternative is pivotal, i.e. many other elementary monotone alternatives can be generated from it. The concept of multiple contrasts is used because of the simplicity and the availability of the distribution under the alternative. After a significant trend test, information is provided that determines which contrast was the "best," and therefore, which exposureresponse shape describes the data most accurately. Alternatively, an information criterionbased approach for the likelihood ratio test under monotone orderrestriction according to Anraku [14] is examined.
Therefore, the primary objective of this paper is not just describing the exposureresponse relationship but also identifying the most likely elementary exposureresponse model with a control of the false model classification rate.
Analysis
Global tests on exposureresponse relationships
Principle of 2 by k tables for epidemiological exposure studies
E_{1}  ....  E_{k}  Total  

Disease  n_{11}  ...  n_{k1}  n_{.1} 
No disease  n_{10}  ...  n_{k0}  n_{.0} 
Sample size  n_{1.}  ...  n_{k.}  n_{..} 
The estimator for the proportions per exposure group is p _{ j } = n _{ j1}/n _{ j.} j = 1,..., k, the total is p = n _{.1}/n.., and the expected values for the proportions are denoted as π _{ j }. The hypotheses system for a monotone order is:
H _{0}: π _{1} = π _{2} = ... = π _{ k } against
H _{1}: π _{1} ≤ π _{2} ≤ ... ≤ π _{ k } with at least one strict inequality.
For simplicity, assume increasing effects with increased exposure; analogously, a directional decision for a decrease is possible.
There are an extensive number of publications concerning orderrestricted tests, including the analysis of 2 × k contingency tables (e.g. Agresti and Coull, [5]; Leuraud and Benichou, [15]). However, no uniformly powerful trend test exists for all possible alternative shapes. The possible shapes can be seen as different equalityinequality patterns of H_{1}. This can be seen for an extreme convex shape {0, 0, 0, π}. Clearly, the "Helmert's contrast" is most powerful because of the optimal pooling of all the lower exposures and the comparison with the high exposure: p _{4}  (p _{1} + p _{2} + p _{3})/3. However, power for Helmert's contrast is greatly reduced for the extreme concave shape {0, π, π, π}. The shape of the exposureresponse relationship is unknown a priori. Irrespective of numerous recent alternative proposals, the likelihood ratio test represents an appropriate solution for this situation. This test is numerically complicated, particularly concerning its distribution under the alternative, which is needed for power/sample size calculations (Robertson et al., [16]). The multiple contrast test according to Bretz and Hothorn [17] approximates its power and is simpler. There are 2 ^{ k } 1 different shapes for k exposure groups, and for each shape a contrast with a minimum false negative rate (β) can be defined. The idea is to select the best contrast, which is sensitive for a certain shape. The best contrast is simply tested by a maximum test. Because the proportions p _{ j } are asymptotically normally distributed, their linear combination (denoted as contrast) is also normally distributed, and therefore, the single contrast test statistic is asymptotically normally distributed, where ∑_{ j } c _{ j } = 0 guarantees a level α test under the null hypothesis. Different variance estimators can be used, but to keep the problem simple, the commonly used pooled estimator p is used here. Asymptotic test versions are used throughout. The contrast coefficients, c _{ j }, are specific for each contrast test; for example the Helmert's contrast [c _{ i } = 1; j = 1,..., k  1 and c _{ k } = k]. A multiple contrast test is the maximum of s predefined single contrast tests , i = 1,..., s where c _{ i } = (c _{ i1},..., c _{ ik }) is a k vector of contrasts. Under the null hypotheses, the joint distribution of the linear contrast tests t _{ SingleC }(c _{ i }) i = 1,..., s is an svariate normal distribution with a zero vector of means and a nonproductmoment correlation matrix. The correlation between two arbitrary contrasts, a = (a _{1},..., a _{ k }) and b = (b _{1},..., b _{ k }), is .
Contrast coefficients for the balanced design with four exposures groups
Type of contrasts  No. of contrasts  Alternative  Contrast c _{ j } 

Isotonic  2^{k}1  π _{1} <π _{2} = π _{3} = π _{4}  {3 1 1 1} 
π _{1} = π _{2} <π _{3} = π _{4}  {1 1 1 1}  
π _{1} = π _{2} = π _{3} <π _{4}  {1 1 1 3}  
π _{1} <π _{2} <π _{3} <π _{4}  {3 1 1 3}  
π _{1} = π _{2} <π _{3} <π _{4}  {1 1 0 2}  
π _{1} <π _{2} = π _{3} <π _{4}  {1 0 0 1}  
π _{1} <π _{2} <π _{3} = π _{4}  {2 0 1 1}  
Change point  k1  π _{1} <π _{2} = π _{3} = π _{4}  {3 1 1 1} 
π _{1} = π _{2} <π _{3} = π _{4}  {1 1 1 1}  
π _{1} = π _{2} = π _{3} <π _{4}  {1 1 1 3}  
Up/down  2  π _{1} <π _{2} = π _{3} = π _{4}  {3 1 1 1} 
π _{1} = π _{2} = π _{3} <π _{4}  {1 1 1 3}  
Single (linear)  1  π _{1} <π _{2} <π _{3} <π _{4}  {3 1 1 3} 
However, the correct classification rates for the most likely elementary alternative (shape of the exposureresponse) were found to be unsatisfactory for isotonic contrasts (Hothorn et al., [6]). Therefore, a special case of orderrestricted inference is considered for step shapes only and denoted as a change point alternative (Hirotsu and Marumo, [18]). Two situations should be considered: i) threshold level studies assuming that an exposureresponse reveals a single change point, which can be characterized by a lower part, an upper part, and an abrupt change between both; and ii) exposureresponse studies with continuous exposure data where the change point alternative is a special and substantial component of the allpattern alternative, which can simplify the evaluation. In some epidemiological problems this question arises. An example of a threshold level study is a diabetes study (PastorBarriuso et al., [19]) with the relationship between 2hour plasma glucose and mortality, where the following questions were formulated: i) Does a certain glucose level exists that markedly increases the mortality risk? ii) Can this change point be estimated? Proposals in the literature are directed only at proof of the existence of such a change point. However, epidemiologists not only want to know that such a change exists, but also where this change is located. Here it is demonstrated that the estimation of the change point q is characterized by its correct classification rate by means of multiple contrast tests, that is, in a testing framework. The hypotheses system for a change from q to q+1 is:H _{0}: π _{1} = π _{2} = ... = π _{ k } H _{1}: π _{1} = ... = π _{ q } <π _{ q+1} = ... π _{ k } q ∈ (1,..., k  1)
These step contrasts reveal a nice ability to transform the ksample problem into an unbalanced twosample problem, which can be used later for estimation of the unadjusted relative risk (or odds ratio) "above/below" the change point. Moreover, the step contrasts belong to a broader class of multiple contrasts. Isotonic contrasts approximate the power of the likelihood ratio test for the monotone ordered hypothesis. The bivariate up/down proposals (Neuhaeuser and Hothorn, [21]; Stewart and Ruberg, [22]) only use the two extreme contrasts (Table 3). Therefore, the change point alternative represents a compromise for testing trends. It is much less dependent on the power of the shape compared with the frequently used single linear contrast test, although only k instead of 2^{ k }  1 isotonic contrasts were used. The multiple contrast test (above) is defined for differences of proportions, but can be reformulated for the relative risk, commonly used in epidemiology (see Appendix A).
It seems that a multiple contrast test may be a different approach to the commonly used logistic model. However, a strong relationship between the multiple contrast test and the score test in a logistic model exists, which allows the correction for additional confounders (Hothorn et al., [6]).
Identification of the exposureresponse shape
However, the global trend tests provide no answer as to which particular alternative exists. Two different approaches can be used to answer this question: i) the best contrast approach; and ii) a model selection approach based on the information criterion for order restriction. This paper explores the identification of one of the possible k  1 elementary alternatives; that is, a classification into . Consequently, the correct classification rate, or the proportion of correctly identified elementary alternatives, is used as a performance measure later on.
The global test decision for the multiple contrast approach is based on the maximum of all included single contrasts , i = 1,..., s, where each single contrast is power optimal for a particular type of alternative (Table 3). Therefore, this maximum contrast approach can be used as an estimator for the exposureresponse shape, where the classification is performed after a significant trend test for control α. For example, two alternatives are possible for a design with three exposure groups: π _{1} = π _{2} <π _{3} or : π _{1} <π _{2} = π _{3}. Assume that the number of diseased cases, n _{11},..., n _{ k1}, is drawn from k binomial random variables with parameters π _{ j } and n _{j.} A possible exposureresponse is described by a contrast vector, c = (c _{1},..., c _{ k }). The problem is to estimate the underlying exposureresponse relationship when s contrast vectors are given. A simple estimator is the function Ψ : (n _{11},..., n _{ k1}) → {1,..., s} which can be derived from the associated contrast test, i.e. . Then explore variability of the simple estimator, Ψ_{1}. How likely is each of the s possible values under the observed data? This question can be addressed via the parametric bootstrap. Repeated realizations from k binomial distributions with sample sizes n _{ j }. and the estimated success parameter p _{ j } = n _{ j1}/n _{ j.} for j = 1,..., k are drawn.

Draw B bootstrap samples

Compute

Compute the relative frequency of each possible value from 1,..., s
This is a measure for the variance of the estimator. Under special circumstances, an improved estimator can be computed by a majority voting according to Breiman [23] over , where I denotes the indicator function. This approach is designated the "parametric bootstrap best contrast" approach.
The model selection approach, based on the information criterion for orderrestriction of normally distributed variables according to Anraku [14], can be modified for proportions and the change point alternative. The AIC criterion for the unrestricted maximum likelihood estimator : (with l( ) = loglikelihood, p = dimension of θ) was modified for orderrestricted maximum likelihood estimators: . The penalty term is calculated for each model using the level of probabilities under an orderrestriction. The explicit formulas for a design with three exposure groups, such as the nullmodel M_{0} and the two change point models M_{1} and M_{2}, are given in Appendix B. The ORICapproach represents a model estimation approach, where model M _{0}{H _{0}: π _{1} = π _{2} = π _{3}}, model M _{1} { : π _{1} = π _{2} <π _{3}}, or model M _{2} { : π _{1} = π _{2} <π _{3}} will be estimated as a "best fitted" model.
Simulation study
The simulation study is structured in two parts: i) empirical comparison between the bestcontrast approach and the ORIC approach for a design with three groups; and ii) investigation of the best contrast approach for more general designs. Fifty thousand pseudorandom 2 × k tables (k ranging from 3 to 7) were generated and 10,000 bootstrap samples were drawn. Two criteria are used, the correct classification rate – the empirical decision rate for the correct model – and the power.
Part I
Correct classification rates for several spontaneous rates π _{0}
π _{ j }  True Change q  ORIC(M_{0}, M_{1}, M_{2})  Max(H^{1}, H^{2})  

M_{0}  M_{1}  M_{2}  H^{1}  H^{2}  
0.3/0.3/0.3  0  .758  .112  .129  .514  .486 
0.1/0.1/0.3  2  .001  .979  .021  .987  .004 
0.1/0.3/0.3  1  .001  .020  .980  .030  .961 
0.2/0.2/0.4  2  .002  .958  .041  .936  .023 
0.2/0.4/0.4  1  .005  .029  .967  .040  .926 
0.3/0.3/0.5  2  .006  .940  .054  .906  .034 
0.3/0.5/0.5  1  .004  .053  .943  .044  .882 
0.4/0.4/0.6  2  .009  .940  .052  .887  .036 
0.4/0.6/0.6  1  .009  .053  .940  .039  .885 
Part II
Correct classification rates and power for several dimensions, sample sizes, unexposed rates, and noncentralities
Dimension  k  3  4  5  6  7 

Correct classif. rate  .992  .987  .977  .971  .971  
Power  .828  .845  .861  .899  .889  
Sample size  n_{j.}  25  50  75  100  125 
Correct classif. rate  .809  .973  .978  .987  .989  
Power  .393  .618  .742  .845  .903  
Unexpos. rate  Π _{1}  .01  .06  .11  .16  .20 
Correct classif. rate  .987  .903  .817  .767  .766  
Power  .845  .488  .373  .312  .266  
Noncentrality  Δ  0.03  0.05  0.07  0.09  0.11 
Correct classif. rate  .953  .973  .985  .994  .998  
Power  .479  .773  .904  .972  .991 
Asymmetrical cumulative false classification rates
Alternative  True Change  H^{1}  H^{2}  H^{3}  H^{4}  H^{5}  Cum. over.  Cum. under. 

.01/.01/.01/.01/.01/.07  5  .000  .000  .001  .027  .972    0.028 
.01/.01/.01/.01/.07/.07  4  .000  .002  .012  .847  .139  0.139  0.014 
.01/.01/.01/.07/.07/.07  3  .000  .011  .819  .119  .051  0.17  0.011 
.01/.01/.07/.07/.07/.07  2  .004  .809  .117  .038  .032  0.187  0.004 
.01/.07/.07/.07/.07/.07  1  .711  .135  .052  .050  .053  0.29   
Extreme unbalanced exposure data
Correct classification rates for extreme unbalanced designs
Sample sizes  N  Alternative  Power  Correct classif. rate 

200/200/200/200  800  .05/.05/.05/.10  .682  .935 
540/200/40/20  800  .05/.05/.05/.10  .251  .758 
200/200/200/200  800  .05/.05/.10/.10  .792  .831 
540/200/40/20  800  .05/.05/.10/.10  .425  .687 
200/200/200/200  800  .05/.10/.10/.10  .603  .783 
540/200/40/20  800  .05/.10/.10/.10  .755  .854 
400/400/400/400  1600  .05/.05/.05/.10  .915  .971 
1340/200/40/20  1600  .05/.05/.05/.10  .266  .749 
400/400/400/400  1600  .05/.05/.10/.10  .968  .916 
1340/200/40/20  1600  .05/.05/.10/.10  .438  .667 
400/400/400/400  1600  .05/.10/.10/.10  .903  .904 
1340/200/40/20  1600  .05/.10/.10/.10  .832  .883 
9740/200/40/20  10000  .05/.05/.05/.10  .252  .702 
Unbalanced designs, where the smallest sample size occurs in the informative groups (large change point s), reveal a clearly reduced classification rate. However, that decrease, compared with the balanced design, is much weaker than the related power loss. A further reduction occurs for the "inbetween" change points as long as the sample size of the pooled informative groups is still smaller than the lower exposure groups. A further substantial increase of the sample size for the unexposed group had almost no influence on the classification rate.
Since a sample size of n _{ j } = 1 is possible, in principle, for this approach, the impact of the continuous exposure categorization can be demonstrated quantitatively with respect to power and classification rate. When a single change point exists, the best approach is the categorization below or above this change point. The true alternative is never known a priori when dealing with real data. Therefore, appropriate categorization may be helpful and inappropriate categorization can greatly reduce the sensitivity.
The asymptotic power for the change point alternative is available (Bretz and Hothorn, [17]). Based on an Rcode, the power can be calculated for an arbitrary sample size pattern, which shapes the exposure response and dimensions k. Power estimation for unbalanced designs can be found in [6] whereas a serious power loss can be observed when the sample size in the informative high exposure groups is very small compared with the sample size in the unexposed or low exposure groups.
Evaluation of the example
The pvalue for the global trend test (change point alternative) and the classification rate of the best contrast approach is determined using an implementation of the proposed procedures in R (R Development Core Team, [24]). The most likely change point, q, and simultaneous confidence intervals for the related change point contrasts can be calculated for the 2 × k contingency table data. A marginal confidence interval can be estimated for each elementary contrast because it represents a linear combination of the proportions p _{ j }. Simultaneous confidence intervals for the maximum of several contrasts can be estimated using a multivariate normal distribution. A detailed description for the estimation of simultaneous confidence intervals for several multiple contrast tests can be found in [25] where the particular problems for binomial data were described recently [26]. The software is available as the R library bindosres as additional file 2. This file can be installed in the private R program via "Install packages from local zip files",
Child cancer and magnetic fields
Exposure/μ Tesla  j  p _{ j }  Pattern  Lower confidence limit 

0–0.05  1  0.263  {10,9,8,7,6,5,4,3,2} vs.1  .716 
0.051–0.101  2  0  {10,9,8,7,6,5,4,3} vs.{1,2}  .410 
0.1010.15  3  0.4  {10,9,8,7,6,5,4} vs.{1,2,3}  .327 
0.151.20  4  0.25  {10,9,8,7,6,5} vs.{1,2,3,4}  .246 
0.2010.25  5  0.25  {10,9,8,7,6} vs.{1,2,3,4,5}  .139 
0.2510.30  6  0  {10,9,8,7} vs.{1,2,3,4,5,6}  .108 
0.3010.35  7  0  {10,9,8} vs.{1,2,3,4,5,6,7}  .343 
0.3510.85  8  0.5  {10,9} vs.{1,2,3,4,5,6,7,8}  .534 
0.8511.6  9  1  10 vs.{1,2,3,4,5,6,7,8,9}  .563 
>1.61  10  1 
Conclusion
Trend tests for the analysis of 2 × k tables using epidemiological exposure data are described to identify the change point alternatives. Not only is the identification of a trend of interest important, but also the information regarding the particular types of alternatives. The best contrast approach for the multiple contrast test is useful for identifying the type of alternative or the change point, whereas a parametric bootstrap is suitable for an assessment of the variability. Both the bootstrapped best contrast and the ORIC approach are appropriate for different dimensions, noncentralities, sample sizes, and the unexposed group rates (due to the asymmetry in binomial testing). The consequences of unbalanced designs – of a large number in the unexposed or low exposure groups and a small number in the high exposure groups – can be calculated depending on the expected shape. Simultaneous confidence intervals for the change point alternative are also available.
Approaches that test a global trend in epidemiological exposure data and also provide information on the pattern of the exposureresponse relationship are rare. The most competitive approach is the fractional polynomials model [7], which is a specific multivariable regression approach.
Most epidemiological studies are characterized not only by the primary exposure factor but also by several covariates, such as gender, age, occupational status, and competing risk characteristics. Therefore, the best contrast approach within the framework of the generalized linear model is recently available [27]. Using the related R library (multcomp), real data can be evaluated using the contrast option "Changepoint" [28].
The suitability of such a simple change point alternative in epidemiological exposure studies should be critically discussed and some real data examples tested. Clearly, such a change point test describes the exposureresponse of the population only. Further investigations are required to demonstrate that this simple approach can be utilized to estimate the center of the individuallevel change point distribution. Moreover, the above approach is not limited to change point alternatives: other trend alternatives, such as Williamstype trends [29], can be assumed as well.
Appendix A
Formulation contrast tests for the relative risk
For general contrasts hold true
Appendix B
The ORIC approach for three binomials and the change point alternative. The three models are:
M _{0}{H _{0}: π _{1} = π _{2} = π _{3}}, M _{1} { : π _{1} = π _{2} <π _{3}}, M _{2} { : π _{1} <π _{2} = π _{3}}. The likelihood is . With the expected values π _{ j } and their crude estimators: , ,
The are the maximum likelihood estimates under the simple order restriction: . The likelihood for the nullmodel M_{0} is:
where provided w _{ j } = n _{ j }
where , for
and
where for
and
The modelspecific ORIC are: ORIC(M _{ r }) = log L ( )  penalty(M _{ r }).
Where the penalty terms are
Withw(M _{0}) = n _{1.} + n _{2.} + n _{3.} w(M _{1}) = n _{1.} + n _{2.}, n _{3.} w(M _{2}) = n _{1.}, n _{2} + n _{3.}
Because P{1,1, w(M _{0})} = 1 ORIC(M _{0}) = L( )  1
Because
Because
Declarations
Acknowledgements
This paper was worked out during the first author's sabbatical at the Department of Biostatistics of the University of Aarhus, under the financial support of the Volkswagen Stiftung grand no. II/78 956.
Authors’ Affiliations
References
 Lausen B, Lerche R, Schumacher M: Maximally selected rank statistics for doseresponse problems. Biometrical Journal 2002, 44:131–147.View ArticleGoogle Scholar
 Olsen JH, Nielsen A, Schulgen G: Residence near high voltage facilities and risk of cancer in children. British Medical Journal 1993, 307:891–895.View ArticlePubMedGoogle Scholar
 Costantino JP, Redmond CK, Bearden A: Occupationally related cancer risk among coke oven workers: 30 years of followup. Journal of Occupational and Environmental Medicine 1995, 37:597–604.View ArticlePubMedGoogle Scholar
 Hill AB: Principles of medical statistics. 9 Edition Oxford University Press, New York 1971, 309–323.Google Scholar
 Agresti A, Coull BA: The analysis of contingency tables under inequality constraints. Journal of Statistical Planning and Inference 2002, 107:45–73.View ArticleGoogle Scholar
 Hothorn LA, Vaeth M, Hothorn T: Trend tests for the evaluation of doseresponse relationships in epidemiological exposure studies. [http://www.biostat.unihannover.de/report/ReportAarhus2003.pdf] Research Report 2003–3 Department of Biostatistics, University of Aarhus 2003.
 Royston P, Ambler G, Sauerbrei W: The use of fractional polynomials to model continuous risk variables in epidemiology. International Journal of Epidemiology 1999, 28:964–974.View ArticlePubMedGoogle Scholar
 Dosemeci M, Benichou J: An alternative test for trend in exposureresponse analysis. J Expo Anal Environ Epidemiol 1998,8(1):9–15.PubMedGoogle Scholar
 Jones AQ, Dewey CE, Dore K, Majowicz SE, McEwen SA, WaltnerToews D: Exposure assessment in investigations of waterborne illness: a quantitative estimate of measurement error. Epidemiologic Perspectives & Innovations 2006, 3:6.View ArticleGoogle Scholar
 Benavente Y, Garcia N, DomingoDomenech E: Regular use of hair dyes and risk of lymphoma in Spain. International Journal of Epidemiology 2005, 34:1118–1122.View ArticlePubMedGoogle Scholar
 Greenland S: Avoiding power loss associated with categorization and ordinal scores in doseresponse and trend analysis. Epidemiology 1995, 6:450–454.View ArticlePubMedGoogle Scholar
 Ascherio A, Weisskopf MG, O'Reilly EJ: Coffee consumption, gender, and Parkinson's disease mortality in the Cancer Prevention Study II cohort: The modifying effects of estrogen. American Journal of Epidemiology 2004, 160:977–84.View ArticlePubMedGoogle Scholar
 Grimsrud TK, Berge SR, Haldorsen T, Andersen A: Exposure to different forms of nickel and risk of lung cancer. American Journal of Epidemiology 2002, 156:1123–1132.View ArticlePubMedGoogle Scholar
 Anraku K: An information criterion for parameters under a simple order restriction. Biometrika 1999, 86:141–152.View ArticleGoogle Scholar
 Leuraud K, Benichou J: Tests for monotonic trend from casecontrol data: Cochranarmitagemantel trend test, isotonic regression and single and multiple contrast tests. Biometrical Journal 2004, 46:731–749.View ArticleGoogle Scholar
 Robertson T, Wright FT, Dykstra RL: Order restricted statistical inference. New York. Wiley 1988.Google Scholar
 Bretz F, Hothorn LA: Detecting doseresponse using contrasts: asymptotic power and sample size determination for binomial data. Statistics in Medicine 2002, 21:3325–3335.View ArticlePubMedGoogle Scholar
 Hirotsu C, Marumo K: Changepoint analysis as a method for isotonic inference. Scandinavian Journal of Statistics 2002, 29:125–138.View ArticleGoogle Scholar
 PastorBarriuso R, Guallar E, Coresh J: Transition models for changepoint estimation in logistic regression. Statistics in Medicine 2003, 22:1141–1162.View ArticlePubMedGoogle Scholar
 Hirotsu C, Kuriki S, Hayter AJ: Multiple comparison procedures based on the maximal component of the cumulative chisquare statistic. Biometrika 1992, 79:381–392.View ArticleGoogle Scholar
 Neuhaeuser M, Hothorn LA: Trend tests for dichotomous endpoints with application in carcinogenicity studies. Drug Information Journal 1997, 30:463–469.Google Scholar
 Stewart WH, Ruberg SJ: Detecting dose response with contrasts. Statistics in Medicine 2000, 19:913–921.View ArticlePubMedGoogle Scholar
 Breiman L: Bagging Predictors. Machine Learning 1996, 24:123–140.Google Scholar
 R Development Core Team. R: [http://www.Rproject.org] A language and environment for statistical computing R Foundation for Statistical Computing, Vienna, Austria 2005.Google Scholar
 Hothorn LA: Multiple comparisons and multiple contrasts in randomized doseresponse trials – confidence interval orient approaches. J Biopharm Stat 2006, 16:7111–731.View ArticleGoogle Scholar
 Schaarschmidt F, Sill M, Hothorn LA: Approximate simultaneous confidence intervals for multiple contrasts of binomial proportions. Biometrical Journal 2008, 50:782–792.View ArticlePubMedGoogle Scholar
 Hothorn T, Bretz F, Westfall P: Simultaneous inference in general parametric models. Biometrical Journal 2008, 50:346–363.View ArticlePubMedGoogle Scholar
 Bretz F, Hothorn T, Westfall P: On multiple comparisons in R. R News 2002, 3:14–17.Google Scholar
 Williams DA: Comparison of several dose levels with a zero dose control. Biometrics 1972, 28:519–531.View ArticlePubMedGoogle Scholar
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