## Spatial Models of Parliamentary VotingThis book presents a simple geometric model of voting as a tool to analyze parliamentary roll call data. Each legislator is represented by one point and each roll call is represented by two points that correspond to the policy consequences of voting Yea or Nay. On every roll call each legislator votes for the closer outcome point, at least probabilistically. These points form a spatial map that summarizes the roll calls. In this sense a spatial map is much like a road map because it visually depicts the political world of a legislature. The closeness of two legislators on the map shows how similar their voting records are, and the distribution of legislators shows what the dimensions are. These maps can be used to study a wide variety of topics including how political parties evolve over time, the existence of sophisticated voting and how an executive influences legislative outcomes. |

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### Contents

Introduction | 1 |

Theory and Meaning | 2 |

A Theory of Spatial Maps | 4 |

The 1964 Civil Rights Act | 14 |

A Road Map to the Rest of This Book | 15 |

The Geometry of Parliamentary Roll Call Voting | 18 |

The Geometry in One Dimension | 19 |

The Geometry in More than One Dimension | 30 |

Statistical Issues | 113 |

Conclusion | 126 |

Practical Issues in Computing Spatial Models of Parliamentary Voting | 128 |

Standardized Measures of Fit | 129 |

How to Get Reasonable Starting Values for the Legislator Ideal Points | 130 |

How Many Dimensions Should I Estimate? | 141 |

The Problem of Constraints | 155 |

Computing Made Easy Some Simple Tricks to Make Estimation Tractable | 159 |

The Relationship to the Geometry of Probit and Logit | 37 |

Conclusion | 41 |

The Optimal Classification Method | 46 |

The OneDimensional Maximum Classification Scaling Problem The Janice Algorithm | 49 |

The Multidimensional Maximum Classification Scaling Problem | 60 |

Overall OC Algorithm | 82 |

Conclusion | 85 |

Appendix | 86 |

Probabilistic Spatial Models of Parliamentary Voting | 88 |

The Deterministic Portion of the Utility Function | 89 |

The Stochastic Portion of the Utility Function | 97 |

Estimation of Probabilistic Spatial Voting Models | 101 |

Conclusion | 160 |

Conducting Natural Experiments with Roll Calls | 162 |

MultipleIndividuals Experiments | 163 |

LargeScale Experiments Using DWNOMINATE | 172 |

Estimating a Common Spatial Map for Two Different Legislatures | 187 |

Conclusion | 195 |

Conclusion | 197 |

Unsolved Problems | 202 |

Conclusion | 209 |

211 | |

225 | |

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90th U.S. Senate agreement score matrix analyze APRE Bayesian chambers Chapter choice classification error columns configuration constraint correctly classified corresponding cutting lines cutting plane cutting plane procedure diagonal discuss DW-NOMINATE eigenvalues eigenvector equation estimated example geometry held fixed Hinich hypersphere Janice algorithm legislator ideal points legislator points legislator votes legislature Liberal-Conservative likelihood function logit maximum normal distribution normal vector normal vector line number of cutting number of legislators one-dimensional optimal classification percent perfect voting polytopes Poole and Rosenthal probabilistic problem produces quadratic utility random rank ordering roll call cutting roll call data roll call matrices roll call voting rows sample second dimension shown in Figure shows simple Southern Democrats space spatial map spatial model spatial theory spatial voting squared distances two-dimensional U.S. Congress U.S. House U.S. Senate unit hypersphere utility function utility model vote Nay vote Yea voters Yea and Nay