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Dr. Peddada is a PLOS ONE Editorial Board member and also currently serving as an Academic Editor of PLOS ONE. Dr. Peddada confirms that this does not alter the authors' adherence to all the PLOS ONE policies on sharing data and materials.

Conceived and designed the experiments: SDP. Performed the experiments: LF. Analyzed the data: LF. Wrote the paper: SDP LF AI.

Constraints arise naturally in many scientific experiments/studies such as in, epidemiology, biology, toxicology, etc. and often researchers ignore such information when analyzing their data and use standard methods such as the analysis of variance (ANOVA). Such methods may not only result in a loss of power and efficiency in costs of experimentation but also may result poor interpretation of the data. In this paper we discuss constrained statistical inference in the context of linear mixed effects models that arise naturally in many applications, such as in repeated measurements designs, familial studies and others. We introduce a novel methodology that is broadly applicable for a variety of constraints on the parameters. Since in many applications sample sizes are small and/or the data are not necessarily normally distributed and furthermore error variances need not be homoscedastic (i.e. heterogeneity in the data) we use an empirical best linear unbiased predictor (EBLUP) type residual based bootstrap methodology for deriving critical values of the proposed test. Our simulation studies suggest that the proposed procedure maintains the desired nominal Type I error while competing well with other tests in terms of power. We illustrate the proposed methodology by re-analyzing a clinical trial data on blood mercury level. The methodology introduced in this paper can be easily extended to other settings such as nonlinear and generalized regression models.

In many applications researchers are typically interested in testing for trends or patterns in mean response among two or more experimental or study groups rather than just testing if the groups are different or not. For instance, toxicologists are typically interested in detecting dose-related trends in mean response such as trends in tumor incidence as the dose of a toxin increases

The power of the Williams test was estimated by averaging 1000 simulated where the critical values are estimated using 10,000 bootstrap samples. The power for F-test was determined using PROC POWER in SAS (9.0). The null hypothesis was that the means of the four dose groups were equal (and zero) and the alternative hypothesis was that the means of the four dose groups have an increasing trend with dose. Data representing the four dose groups were simulated from normal populations with dose means taken to be 0, 0.1, 0.5 and 1, respectively. The actual values of the doses are irrelevant for the two methods described here. The population standard deviation for the four populations was taken to be 1. Corresponding to the 14 different patterns of total sample sizes, namely, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 76, 80, 100, 116, the powers of the two methods are plotted. The Type I error was set to 0.05.

The field of statistics that deals with statistical methods designed to test ordered or constrained hypotheses is commonly called order restricted inference or constrained statistical inference. There exists a very large body of literature on order restricted (constrained inference) spanning nearly sixty years with four books written on the subject, including a recent book by Silvapulle and Sen

In many applications it is common for researchers to be interested in comparing the population means of two or more experimental conditions or groups after adjusting for covariates. Depending upon the study design, as in repeated measurement designs, it is common to use linear mixed effects models to account for the underlying dependence structure as well as the covariates. There exists several decades of literature on statistical inference in linear mixed effects models and numerous books have been written on the subject

In the absence of any covariates, especially continuous covariates, Mukerjee

It was not until

We present the methodology in Section 2. Results of our simulation study are reported in Section 3. The proposed methodology is illustrated in Section 4 using data from a clinical trial comparing succimer treatment to placebo in children exposed to mercury

Let

denote a linear mixed effects models where

Let

where the inequalities are component-wise, with at least one strict inequality. For example, if one is interested in testing a simple order

The matrix

Suppose

The weights

There are three differences between our approach and the above LRT approach. Firstly, rather than using the RMLE which, at each step of the iteration, projects the unconstrained estimator

Let

where

where

where

The bootstrap methodology to obtain the p-values is described in the flow chart in

We evaluated the Type I error and power of the proposed EBLUP bootstrap test for the case of simple order using the proposed statistic (4). We compared our method with the asymptotic likelihood ratio test

The data were simulated using model (1) with the number of subjects per each treatment

Here different values of

All results are based on 500 simulation runs. We used 500 bootstrap runs to generate the null distribution for the proposed bootstrap test. In all simulations the nominal value for Type I error was taken to be 0.05.

Complete simulation scenarios and results are presented in

Asymp-LRT | Proposed method | |||

3 | 10 | 1 | 0.03 | 0.05 |

3 | 50 | 1 | 0.01 | 0.03 |

3 | 10 | 0.2 | 0.05 | 0.04 |

3 | 10 | 2 | 0.04 | 0.05 |

3 | 50 | 2 | 0.01 | 0.03 |

5 | 10 | 1 | 0.02 | 0.03 |

5 | 10 | 0.2 | 0.02 | 0.04 |

5 | 10 | 2 | 0.02 | 0.04 |

Asymp-LRT | Proposed method | |||||||||||||

3 | 10 | 0 | 0.00 | 1.25 | 1 | 0.82 | 0.84 | |||||||

3 | 10 | 0 | 1.26 | 1.26 | 1 | 0.82 | 0.84 | |||||||

3 | 50 | 0 | 0.55 | 0.55 | 1 | 0.85 | 0.89 | |||||||

3 | 10 | 0 | 0.73 | 1.45 | 1 | 0.86 | 0.89 | |||||||

5 | 10 | 0 | 0.00 | 0.00 | 0.00 | 1.27 | 1 | 0.65 | 0.86 | |||||

5 | 10 | 0 | 1.24 | 1.24 | 1.24 | 1.24 | 1 | 0.58 | 0.86 | |||||

5 | 10 | 0 | 0.37 | 0.74 | 1.11 | 1.48 | 1 | 0.80 | 0.90 | |||||

5 | 10 | 0 | 0.81 | 0.81 | 0.81 | 1.62 | 1 | 0.74 | 0.93 |

Asymp-LRT | Proposed method | ||||||||

3 | 10 | 0.1 | 0.10 | 2.37 | 1 | 0.04 | 0.03 | ||

3 | 10 | 0.1 | 0.20 | 0.20 | 1 | 0.03 | 0.04 | ||

3 | 10 | 0.1 | 0.09 | 0.36 | 1 | 0.03 | 0.03 | ||

3 | 50 | 0.1 | 0.10 | 0.01 | 1 | 0.01 | 0.03 | ||

3 | 50 | 0.1 | 0.02 | 0.02 | 1 | 0.02 | 0.04 | ||

5 | 10 | 0.1 | 0.10 | 0.10 | 0.10 | 0.16 | 1 | 0.01 | 0.04 |

5 | 10 | 0.1 | 0.20 | 0.20 | 0.20 | 0.20 | 1 | 0.01 | 0.04 |

5 | 10 | 0.1 | 0.11 | 0.44 | 0.99 | 1.76 | 1 | 0.01 | 0.04 |

5 | 10 | 0.1 | 0.11 | 0.11 | 0.11 | 0.45 | 1 | 0.02 | 0.05 |

Asymp-LRT | Proposed method | ||||||||

3 | 10 | 0 | 0.00 | 1.54 | 1 | 0.82 | 0.88 | ||

3 | 10 | 0 | 0.45 | 0.45 | 1 | 0.81 | 0.82 | ||

3 | 10 | 0 | 0.30 | 0.60 | 1 | 0.82 | 0.80 | ||

3 | 50 | 0 | 0.00 | 0.10 | 1 | 0.70 | 0.74 | ||

3 | 50 | 0 | 0.15 | 0.15 | 1 | 0.86 | 0.92 | ||

3 | 50 | 0 | 0.08 | 0.16 | 1 | 0.95 | 0.93 | ||

5 | 10 | 0 | 0.00 | 0.00 | 0.00 | 0.40 | 1 | 0.42 | 0.78 |

5 | 10 | 0 | 0.45 | 0.45 | 0.45 | 0.45 | 1 | 0.68 | 0.81 |

5 | 10 | 0 | 0.33 | 0.66 | 1.00 | 1.33 | 1 | 0.96 | 0.88 |

5 | 10 | 0 | 0.34 | 0.34 | 0.34 | 0.67 | 1 | 0.71 | 0.82 |

Asymp-LRT | Proposed method | ||||||||

3 | 10 | 0.10 | 0.10 | 0.04 | 1 | 0.02 | 0.03 | ||

3 | 10 | 0.10 | 0.20 | 0.20 | 1 | 0.03 | 0.05 | ||

3 | 10 | 0.10 | 0.01 | 0.04 | 1 | 0.03 | 0.03 | ||

3 | 50 | 0.10 | 0.02 | 0.02 | 1 | 0.01 | 0.01 | ||

3 | 50 | 0.10 | 0.01 | 0.03 | 1 | 0.01 | 0.01 | ||

5 | 10 | 0.10 | 0.10 | 0.10 | 0.10 | 0.16 | 1 | 0.02 | 0.04 |

5 | 10 | 0.10 | 0.04 | 0.04 | 0.04 | 0.04 | 1 | 0.02 | 0.03 |

5 | 10 | 0.10 | 0.01 | 0.04 | 0.09 | 0.16 | 1 | 0.02 | 0.05 |

5 | 10 | 0.10 | 0.01 | 0.01 | 0.01 | 0.04 | 1 | 0.03 | 0.03 |

Asymp-LRT | Proposed method | ||||||||

3 | 10 | 0 | 0.00 | 1.54 | 1 | 0.90 | 0.98 | ||

3 | 10 | 0 | 0.45 | 0.45 | 1 | 0.81 | 0.85 | ||

3 | 10 | 0 | 0.30 | 0.60 | 1 | 0.85 | 0.90 | ||

3 | 50 | 0 | 0.00 | 0.20 | 1 | 0.92 | 0.88 | ||

3 | 50 | 0 | 0.15 | 0.15 | 1 | 0.57 | 0.68 | ||

3 | 50 | 0 | 0.08 | 0.16 | 1 | 0.89 | 0.69 | ||

5 | 10 | 0 | 0.00 | 0.00 | 0.00 | 0.40 | 1 | 0.44 | 0.77 |

5 | 10 | 0 | 0.45 | 0.45 | 0.45 | 0.45 | 1 | 0.66 | 0.83 |

5 | 10 | 0 | 0.10 | 0.20 | 0.30 | 0.40 | 1 | 0.79 | 0.75 |

5 | 10 | 0 | 0.34 | 0.34 | 0.34 | 0.67 | 1 | 0.72 | 0.94 |

5 | 50 | 0 | 0.00 | 0.00 | 0.00 | 0.20 | 1 | 0.97 | 0.96 |

5 | 50 | 0 | 0.20 | 0.20 | 0.20 | 0.20 | 1 | 0.66 | 0.92 |

5 | 50 | 0 | 0.08 | 0.08 | 0.08 | 0.16 | 1 | 0.87 | 0.74 |

We illustrate the proposed methodology using data from patients in a randomized placebo-controlled, double-blind clinical trial the Treatment of Lead-exposed Children trial, or TLC

Denoting the difference in the mean log mercury levels at the

with at least one strict inequality. The three UMLEs were 0.59, 0.92, 0.73 and the MINQUE-based constrained estimates were 0.59, 0.82, 0.82. Using the proposed test we obtained a bootstrap p-value of 0.109. Thus, we are not able to reject the null hypothesis in favor of a trend in the difference in means. These results are consistent with conclusions of

Inequality constraints arise naturally in many applications, such as toxicology, where researchers are interested in studying dose-response of a chemical, or gene expression studies in oncology, where a researcher may be interested in understanding the changes in gene expression according to cancer stage. We proposed a new method to test for inequality constraints. Since the method uses Rao’s MINQUE theory (cf.

In the manuscript we considered one-dimensional order constraints. In toxicology, researchers are often interested in dose

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Authors thank Grace Kissling and Bahjat Qaqish for their careful reading of the manuscript and for their comments that helped improve the presentation of the paper. Authors also thank Drs. Rogan and Cao for providing us the data that were analyzed in this paper.