缺失数据的统计功效分析：结构方程建模方法

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"《Statistical Power Analysis with Missing Data》是一本使用结构方程建模方法进行缺失数据统计功效分析的专业书籍。作者是Adam Davey和Jyoti Savla，分别来自天普大学和弗吉尼亚理工学院。这本书由Routledge出版社发行，探讨了在存在缺失数据的情况下如何进行统计功效分析的挑战和方法。" 统计功效分析是统计学中的一个重要概念，它衡量了在给定的显著性水平下，一项研究能否有足够的能力检测到一个实际存在的效应。在实际研究中，尤其是在社会科学、医学研究等领域，数据的缺失情况是普遍存在的，这为统计分析带来了复杂性。本书通过结构方程建模（Structural Equation Modeling, SEM）这一强大工具，提供了解决这个问题的策略和方法。结构方程模型是一种统计建模技术，可以同时处理多个变量之间的因果关系，并允许处理复杂的依赖结构。在缺失数据的背景下，SEM可以用来估计模型参数，同时考虑到数据不完整的影响。书中可能涵盖了以下关键知识点： 1. **缺失数据类型**：书中可能会区分不同的缺失数据类型，如完全随机缺失（MCAR）、马尔可夫缺失（MAR）和非随机缺失（NMAR），并讨论每种类型对分析的影响。 2. **处理缺失数据的方法**：可能包括完整的案例分析、多重填补方法（如多重插补）、删失模型等。这些方法的原理和适用条件会有所阐述。 3. **SEM在缺失数据中的应用**：书中可能会详细介绍如何构建和估计SEM模型，以及如何在有缺失数据的情况下进行模型诊断和验证。 4. **统计功效计算**：介绍如何在考虑缺失数据的情况下计算统计功效，可能包括基于模拟的方法和其他专门针对缺失数据的功效计算技术。 5. **实例分析**：通过真实或模拟的数据集，展示如何应用所介绍的方法进行统计分析，帮助读者理解和掌握实际操作。 6. **软件实现**：可能会提到一些常用的统计软件，如Mplus、R、SPSS等，以及如何在这些软件中实现缺失数据的统计功效分析。 7. **伦理和实践问题**：讨论在处理缺失数据时的伦理考量，以及在设计研究时如何减少数据缺失以提高统计功效。通过对这些知识点的深入学习，读者将能够更有效地处理和分析含有缺失数据的复杂研究，提高研究结果的可靠性和有效性。本书对于从事统计分析、社会科学和医学研究的学者及学生来说，是一本宝贵的参考资料。

Introduction

Overview and Aims

Missing data are a nearly ubiquitous aspect of social science research. Data

can be missing for a wide variety of reasons, some of which are at least

partially controllable by the researcher and others that are not. Likewise,

the ways in which missing data occur can vary in their implications for

reaching valid inferences. This book is devoted to helping researchers

consider the role of missing data in their research and to plan appro‑

priately for the implications of missing data. Recent years have seen an

extremely rapid rise in the availability of methods for dealing with miss‑

ing data that are becoming increasingly accessible to non‑methodologists.

As a result, their application and acceptance by the research community

has expanded exponentially.

It was not so very long ago that even highly sophisticated researchers

would, at best, acknowledge the extent of missing data and then proceed

to present analyses based only on the subset of participants who provided

complete data for the variables of interest. This “list‑wise deletion” treat‑

ment of missing values remains the default option in nearly every statisti‑

cal package available to social scientists.

Researchers who attempted to address issues of missing data in more

sophisticated ways risked opening themselves to harsh criticism from

reviewers and journal editors, often being accused of making up data or

being treated as though their methods were nothing more than statisti‑

cal sleight of hand. In reality, it usually requires stronger assumptions to

ignore missing data than to address them. For example, the assumptions

required to reach valid conclusions based on list‑wise deletion actually

require a much greater leap of faith than the use of more sophisticated

approaches.

Fortunately, the times have changed quickly as statistical software

developers have gone to greater lengths to incorporate appropriate tech‑

niques in their software. At the same time, many social scientists with

sophisticated methodological skills have helped to facilitate a better con‑

ceptual understanding of missing data issues among non‑methodologists

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2 Statistical Power Analysis with Missing Data

(e.g., Acock, 2005; Allison, 2001; Schafer & Graham, 2002). There is now

the general expectation within the scientic community that researchers

will provide more sophisticated treatment of missing data. However, the

implications of missing data for social science research have not received

widespread treatment to date, nor have they made their way into the plan‑

ning of sound research, being something to anticipate and perhaps even

incorporate deliberately.

Statistical power is the probability that one will nd an effect of a given

magnitude if in fact one actually exists. Although statistical power has a

long history in the social sciences (e.g., Cohen, 1969; Neyman & Pearson,

1928a, 1928b), many studies remain underpowered to this day (Maxwell,

2004; Maxwell, Kelly, & Rausch, 2008). Given that the success of publish‑

ing one’s results and obtaining funding typically rest upon reliably iden‑

tifying statistically signicant associations (although there is a growing

movement away from null hypothesis signicance testing; see Harlow,

Mulaik, & Steiger, 1997), there is considerable importance to learning how

to design and conduct appropriate power analyses, in order to increase

the chances that one’s research will be informative and in order to work

toward building a cumulative body of knowledge (Lipsey, 1990). In addi‑

tion to determining whether means differ, assumptions (e.g., normality,

homoscedasticity, etc.) are met, or a more parsimonious model performs

as well as one that is more complex, there is greater recognition today that

statistically signicant results are not always meaningful, which places

increased emphasis on the choice of alternative hypotheses.

We have several aims in this volume. First, we hope to provide social

scientists with the skills to conduct a power analysis that can incorpo‑

rate the effects of missing data as they are expected to occur. A second

aim is to help researchers move missing data considerations forward in

their research process. At present, most researchers do not truly begin

to consider the inuence of missing data until the analysis stage (e.g.,

Molenberghs & Kenward, 2007). This volume can help researchers to

carry the role of missing data forward to the planning stage, before any

data are even collected or a grant proposal is even submitted.

Power analyses that consider missing data can provide more accurate

estimates of the likelihood of success in a study. Several of the consider‑

ations we address in this book also have implications for ways to reduce

the potential effects of missing data on loss of statistical power, many

under the researcher’s control. Finally, we hope that this volume will pro‑

vide an initial framework in which issues of missing data and their asso‑

ciations with statistical power can become better explored, understood,

and even exploited.

As resources to conduct research become more difcult to obtain, the

importance of statistical power grows as well, along with demands on

researchers to plan studies as efciently as possible. At the same time,

Y100315.indb 2 7/15/09 2:58:43 PM

Introduction 3

the increase in application of complex multivariate statistics and latent

variable models afforded by greater computing power also places heavier

demands on data and samples. In the context of structural equation mod‑

eling, there is generally also a greater theoretical burden on researchers

prior to the conduct of their analyses. It is important for researchers to

know quite a bit in advance about their constructs, measures, and models.

The process of hypothesis testing has become multivariate and, particu‑

larly in nonexperimental contexts, subsequent stages of analysis are often

predicated on the outcomes of previous stages. Fortunately, it is precisely

these situations that lend themselves most directly to power analysis in

the context of structural equation modeling.

History is also on our side. Designs that incorporate missing data

(so‑called incomplete designs) have been a part of the social sciences for

a very long time, although we do not often think of them in this way.

Some of the classic examples include the Solomon four‑group design

for evaluating testing by treatment interactions (Campbell & Stanley,

1963; Solomon, 1949) and the Latin squares design. More recent exam‑

ples include cohort‑sequential, cross‑sequential, and accelerated longi‑

tudinal designs (cf. Bell, 1953; McArdle & Hamagami, 1991; Raudenbush

& Chan, 1992; Schaie, 1965). However, with the exception of the latter

example, these designs have not typically been analyzed as missing

data designs but rather analyzed piecewise according to complete data

principles.

In Solomon’s four‑group design (Table 1.1), for example, researchers

are typically directed to test the pretest by intervention interaction using

(only) posttest scores. Finding this to be nonsignicant, they are then

advised either to pool across pretest and non‑pretest conditions for a

more powerful posttest comparison or to consider the analysis of gain

scores (posttest values controlling for pretest values). Each approach dis‑

cards potentially important information. In the former, pretest scores

are discarded; in the latter, data from groups without pretest scores are

discarded. It is very interesting to note that Solomon himself initially

recommended replacing the two missing pretest scores with the aver‑

age scores obtained from O1 and O3 in Table 1.1, leading Campbell

and Stanley (1963) to indicate that “Solomon’s suggestions concerning

Table 1.1

Solomon’s Four‑Group Design

Pretest Intervention Posttest

R O1 X O2

R O3 O4

R ? X O5

R ? O6

Y100315.indb 3 7/15/09 2:58:43 PM

4 Statistical Power Analysis with Missing Data

these are judged unacceptable” (p. 25) and to suggest that pretest scores

essentially be discarded from analysis. If you think about it, the random

assignment to groups in this design suggests that the mean of O1 and

O3 is likely to be the best estimate, on average, of the pretest means for

the groups for which pretest scores were deliberately unobserved. How

modern approaches differ from Solomon’s suggestion is that they cap‑

ture not just these point estimates, but they also factor in an appropriate

degree of uncertainty regarding the unobserved pretest scores. In this

design, randomization allows the researcher to make certain assump‑

tions about the data that are deliberately not observed.

In contrast, compare this approach with the accelerated longitudinal

design, illustrated in Table 1.2. Here, one can deliberately sample three (or

more) different cohorts on three (or more) different occasions and subse‑

quently reconstruct a trajectory of development over a substantially lon‑

ger period of development by simultaneously analyzing data from these

incomplete trajectories. In this minimal example, just 2 years (with three

waves of data collection) of longitudinal research yields information about

4 years of development with, of course, longer periods possible through

the addition of more waves and cohorts. Different assumptions, such as

the absence of cohort differences, are required in order for this design to

be valid. However, careful design can also allow for appropriate evalu‑

ation of these assumptions and remediation in cases where they are not

met. In fact, testing of some hypotheses would not even be possible using

a complete data design, as is the case with Solomon’s four‑group design.

On the other hand, an incomplete design means that it may not be pos‑

sible to estimate all parameters of a model. Because it is never observed,

the correlation between variables at ages 12 and 16 cannot be estimated in

the example above. Often, however, this has little bearing on the questions

we wish to address, or else we can design the study in a way that allows

us to capture the desired information.

This book has been designed with several goals in mind. First and fore‑

most, we hope that it will help researchers and students from a variety of

disciplines, including psychology, sociology, education, communication,

Table 1.2

Accelerated Longitudinal Design

Age

12 13 14 15 16

Cohort 1 O O O ? ?

Cohort 2 ? O O O ?

Cohort 3 ? ? O O O

Y100315.indb 4 7/15/09 2:58:43 PM

Introduction 5

management, nursing, and social work, to plan better, more informative

studies by considering the effects that missing data are likely to have on

their ability to reach valid and replicable inferences.

It seems rather obvious that whenever we are missing data, we are miss‑

ing information about the population parameters for which we wish to

reach inferences, but learning more about the extent to which this is true

and, more importantly, steps that researchers can take to reduce these

effects, forms another important objective of this book. As we will see, all

missing data were not created equal, and it is very often possible to con‑

duct a more effective study by purposefully incorporating missing data

into its design (e.g., Graham, Hofer, & MacKinnon, 1996).

Even the statistical literature has devoted considerably more attention

to ways in which researchers can improve statistical power over and

above list‑wise deletion methods, rather than to consider how appropri‑

ate application of these techniques compares with availability of com‑

plete data. Under at least some circumstances, researchers may be able to

achieve greater statistical power by incorporating missing data into their

designs.

A third goal of this volume is to help researchers to anticipate and eval‑

uate contingencies before committing to a specic course of action and to

be in a better position to evaluate one’s ndings. In this sense, conducting

rigorous power analyses appropriate to the range of missing data situa‑

tions and statistical analyses faced in a typical study is analogous to the

role played by pilot research when one is developing measures, manipula‑

tions, and designs appropriate to testing hypotheses. As we shall see, the

techniques presented in this book are appropriate to both experimental

and nonexperimental contexts and to situations where data are missing

either by default or by design. They can be used as well, with only minor

modications, in either an a priori or a posteriori fashion and with a single

parameter of interest or in order to evaluate an entire model just as in the

complete data case (e.g., Hancock, 2006; Kaplan, 1995).

Statistical Power

Because the practical aspects of statistical power do not always receive a

great deal of attention in many statistics and research methods courses,

before launching into consideration of statistical power, we begin by rst

reviewing some of the components and underlying logic associated with

statistical power. The rest of this book elaborates on the importance of

each of these elements in the examples presented.

Y100315.indb 5 7/15/09 2:58:43 PM

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