# Analysis of Incomplete Multivariate Data J. L. Schafer Department of Statistics The Pennsylvania State University USA CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 1997 by Chapman & Hall/CRC First edition 1997 First CRC Press reprint 1999 Originally published by Chapman & Hall No claim to original U.S. Government works International Standard Book Number 0-412-04061-1 7 8 9 0 ## Contents Preface xiii 1 Introduction 1 1.1 Purpose 1 1.2 Background 2 1.2.1 The EM algorithm 3 1.2.2 Markov chain Monte Carlo 3 1.3 Why analysis by simulation? 4 1.4 Looking ahead 6 1.4.1 Scope of the rest of this book 6 1.4.2 Knowledge assumed on the part of the reader 7 1.4.3 Software and computational details 7 1.5 Bibliographic notes 8 2 Assumptions 9 2.1 The complete-data model 9 2.2 Ignorability 10 2.2.1 Missing# Analysis of Incomplete Multivariate Data J. L. Schafer Department of Statistics The Pennsylvania State University USA CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress. This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. Visit the CRC Press Web site at www.crcpress.com © 1997 by Chapman & Hall/CRC First edition 1997 First CRC Press reprint 1999 Originally published by Chapman & Hall No claim to original U.S. Government works International Standard Book Number 0-412-04061-1 7 8 9 0 ## Contents Preface xiii 1 Introduction 1 1.1 Purpose 1 1.2 Background 2 1.2.1 The EM algorithm 3 1.2.2 Markov chain Monte Carlo 3 1.3 Why analysis by simulation? 4 1.4 Looking ahead 6 1.4.1 Scope of the rest of this book 6 1.4.2 Knowledge assumed on the part of the reader 7 1.4.3 Software and computational details 7 1.5 Bibliographic notes 8 2 Assumptions 9 2.1 The complete-data model 9 2.2 Ignorability 10 2.2.1 Missing