## Small Sample Methods for the Analysis of Clustered Binary DataThere are several solutions for analysis of clustered binary data. However, the two most common tools in use today, generalized estimating equations and random effects or mixed models, rely heavily on asymptotic theory. However, in many situations, such as small or sparse samples, asymptotic assumptions may not be met. For this reason we explore the utility of the quadratic exponential model and conditional analysis to estimate the effect size of a trend parameter in small sample and sparse data settings. Further we explore the computational efficiency of two methods for conducting conditional analysis, the network algorithm and Markov chain Monte Carlo. Our findings indicate that conditional estimates do indeed outperform their unconditional maximum likelihood counterparts. The network algorithm remains the fastest tool for generating the required conditional distribution. However, for large samples, the Markov chain Monte Carlo approach accurately estimates the conditional distribution and is more efficient than the network algorithm. |

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### Contents

ESTIMATION APPROACHES FOR SMALL OR SPARSE SAMPLES | 25 |

COMPUTATIONAL METHODS FOR CONDITIONAL ESTIMATION | 35 |

SMALL SAMPLE PROPERTIES OF PARAMETER ESTIMATES | 59 |

ACCURACY AND COMPUTATIONAL EFFICIENCY OF MCMC | 74 |

SUMMARY | 84 |

APPENDIX A PROGRAMS FOR PARAMETER ESTIMATION FROM | 94 |

clustexampmod c C program for gaining basic information prior | 110 |

simc txt R program for generating random samples from a quadratic | 143 |

CIFalse txt R program for calculating 1 α conﬁdence intervals | 157 |

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&& numsteps Airbag analysis approach Arcs asymptotic baseline probability binomial calculated chain Monte Carlo change[1 changeposs CMLE MLE CMLE computational conditional distribution conditional likelihood conditional models confidence interval corneal grafts correlated binary covariate Data in Table deﬁned dehne diﬀerent double double)(*(current eﬃciency Equation Estimate SD exact distribution example experiencing a crash exponential family ﬁrst ﬁxed Fofa Fofb Fofb*a Fofck Gofa Gofb Gofck increase log odds logistic regression long nh low birth weight Low Weight Birth malformations Markov chain Markov chain Monte maximum likelihood MCMC method median Mehta MLE CMLE MLE MLE estimate ncol,int network algorithm NewNode newtable nnodes,int nodes npos NULL number of clusters number of successes numclust observed odds ratio one-sided outcomes p-value Parameter Estimates Patel quadratic exponential model random eﬀects model reference set rejection sampling small sample speciﬁc stop Sturmfels subnodes ref sufficient statistics void Zeger