实用多元统计与数学建模手册：解决跨学科问题的权威指南

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《应用多元统计与数学建模手册》是一本由霍华德·E·A·廷斯利（Howard E.A. Tinsley）和史蒂文·D·布朗（Steven D. Brown）共同编辑的专业书籍，其出版商是elsevier科学与技术书籍，于2000年4月发行。该书针对的是那些在各个学科领域中广泛应用的多元统计学和数学模型，旨在帮助实践中的研究人员有效地运用这些复杂工具，而不必深入探讨其数学原理。本书的核心价值在于提供实际操作指导，帮助读者理解何时以及如何选择合适的多变量方法和数学建模技术来解决特定问题。它强调了将模型和统计作为解决问题的工具，而非仅仅关注理论层面。每一章都首先讨论了特定技术的应用范围，明确指出它可以解答的问题类型以及局限性，以便用户能够有针对性地使用。作者阵容强大，包括贝茜·J·贝克尔（Betsy J. Becker）等多位来自教育、心理学、社会科学和医学领域的专家，他们在各自的专业背景下分享了实践经验，提供了丰富的案例研究和实用建议。例如，迈克尔·T·布朗（Michael T. Brown）可能介绍了在教育心理学中如何利用多元统计分析；罗伯特·Cudeek（Robert Cudeek）则可能讨论了心理测量中的模型构建方法；而马克·L·戴维森（Mark L. Davison）和莱内·V·达维斯（Rene V. Dawis）可能会涉及教育评估或社会科学研究中的多元数据分析。《应用多元统计与数学建模手册》不仅涵盖了生物科学、社会科学，还扩展到人文学科等领域，反映了这些技术的广泛适用性和跨学科需求。通过阅读这本书，读者不仅可以学习到最新的多元统计学和建模技术，还能了解到如何将它们融入实际研究项目中，提高研究效率和结果的可靠性。这是一本实用性极强的指南，适合无论是初学者还是经验丰富的研究人员，想要更新知识、提升技能并优化研究方法的人们。书中详尽的内容和实例使得读者能够更好地理解和掌握多元统计与数学建模在现代科研中的关键作用。

8 HOWARD E. A. TINSLEY AND STEVEN D. BROWN

of critical issues to consider in interpreting the results of your statistical

analyses.

Often throughout the remainder of this chapter we will refer in a

generally inclusive sense to all of the multivariate statistical techniques and

mathematical modeling testing procedures covered in the

Handbook.

will use terms such as "statistical techniques" or "multivariate procedures"

for ease of communication when making general references to the proce-

dures covered in the

Handbook.

I. DATA PREPARATION

"Garbage in, garbage out," has been used as a cautionary metaphor for

so long that it is now clich6. Nevertheless, the underlying message that it

conveys is even more important when relying on computers to perform

multivariate procedures than it was at midcentury when researchers per-

formed univariate analyses by hand using mechanical desk calculators. The

answers that will emerge from a statistical analysis are dependent on the

quality of the data submitted to analysis, so it is critical that the data be

as accurate as possible. For this reason, the first step in the analysis process

is a close inspection of the data to correct any errors and to deal with

missing data and outliers. It is ironic that the painstaking preparation of

your data for analysis may take days or even weeks, while the primary

analysis often takes only a few hours.

A. Accuracy

It is likely that all data files of the size submitted to multivariate statistical

analysis or used in testing mathematical models contains some errors. These

can occur as a result of respondent error (e.g., responding in the wrong

place on a scan sheet or using the wrong set of response alternatives), an

error by data-entry personnel, or even an error in scanning the answer sheet.

Although it is not widely known, scanning errors occur as a consequence of

stray marks on the form, dust on the glass, and other idiosyncrasies of the

scanning process.

With small data files it is prudent to proofread the data by comparing

the computerized data file against the original answers. However, that will

seldom be possible with the large data files used in multivariate procedures.

Instead, begin by preparing a frequency distribution for each item in the

data file and inspect these meticulously for out-of-range, unusual, and

missing responses. Check possible errors by comparing original answers

given by the respondent against the number appearing in the data file. For

example, a missing response may have been recorded on the answer sheet

too lightly for the scanner to record it. An out-of-range response may be

I. MULTIVARIATE STATISTICS AND MATHEMATICAL MODELING 9

the result of a stray mark on the answer sheet, despite the presence of a

valid answer for that question. The necessary remedial action is clear when

these types of errors are identified, and making the necessary corrections

is unlikely to introduce significant error into the data. Make all corrections

in the data file and on the original (answer sheet or questionnaire) simulta-

neously to avoid the possibility of future confusion about which is the

"correct" response.

Other errors that are due to more subtle causes may still be correctable.

For example, the respondent might have answered each question faithfully

but skipped one circle on a scan sheet. That would be apparent if the

response to one item appears to be missing and the highest item number

answered is one greater than the number of questions. Another possibility

is the presence of out-of-range responses (e.g., an answer of "4" when only

three response alternatives were available). Depending on the pattern of

responses, you might decide to change such a response to "3" under the

assumption that the respondent intended to use the highest response alter-

native and made a clerical error.

Also examine combinations of responses to determine whether the

pattern of responses is logically consistent. For example, respondents might

describe themselves as an only child with two older siblings, as a 15-year-

old having a doctorate, as a school teacher having only a high school

education, or as having been in a relationship longer than they have been

alive (see Behrens, 1997). In instances such as these the likelihood of

an error is high, but the way to correct the error is not obvious. You

may decide to leave the data unchanged, delete the response that seems

most likely to be wrong, or even to delete the respondent from the data

file.

Other errors involving combinations of items can occur when item

responses are summed to create a composite score. Dawis (chapter 3, this

volume) provides a thorough review of scaling methods and the potential

problems that can occur. When all of the responses that are aggregated to

produce the composite are within the acceptable range, then the summary

score of the individual must be within the acceptable range. Nevertheless,

the composite score may be quite atypical for the sample of respondents.

(We will return to this issue in the section on outliers.) Also examine the

distribution of composite scores for the entire sample and the mean and

standard deviation to determine whether they are plausible. Both the shape

of the distribution and the mean and standard deviation of the sample are

important. The proper use of multivariate procedures requires more than

a knowledge of the techniques themselves; it also requires a thorough

knowledge of the theory and research pertinent to the substantive issues

under investigation (Comrey & Lee, 1992). Hetherington (chapter 2) dis-

cusses the critical role of theory as a guide to the informed use of multivari-

ate statistics and model-testing procedures.

| 0 HOWARD E. A. TINSLEY AND STEVEN D. BROWN

Given the likelihood that errors will occur in most large data files, it

is best to think carefully about these possibilities before beginning the data

preparation and to develop an a priori strategy for taking remedial action

when such errors are detected. Develop a consistent strategy for dealing

with such errors and record all editing decisions for future reference. It is

important to avoid the introduction of alternative errors into the data file

under the guise of correcting errors.

Two particularly troublesome problems when performing multivariate

statistics and testing mathematical models are missing data and outliers.

B. Missing Data

Venter and Maxwell (chapter 6, this volume), Huberty and Petoskey (chap-

ter 7, this volume), and DiLalla (chapter 15, this volume) discuss alternative

procedures for dealing with the problem of missing data. They explain the

advantages and disadvantages of such procedures as pairwise and listwise

deletion of missing data, briefly describe and evaluate alternative proce-

dures for substituting estimated values in the place of the missing data, and

provide references to more extensive treatments of the missing data options.

Although we touch upon some of the procedures they review, we have

structured this discussion to minimize redundancy. We recommend you

turn to the sections of those chapters dealing with missing data after reading

this chapter to gain a more complete understanding of the advantages and

disadvantages of the alternative strategies for dealing with missing data.

The presence of a small amount of randomly distributed missing data

in a large data file is not likely to cause serious problems, and most of the

standard procedures for dealing with missing data will be adequate. On

the other hand, the presence of systematic missing data can be serious.

Systematic missing data occurs when the variable for which information is

missing is related to other variables included in the data file (e.g., informa-

tion about political affiliation is missing when you believe that political

affiliation is related to other important variables, such as income level and

attitudes towards social issues).

Although it is tempting to assume that any missing data is randomly

distributed, we recommend that you test to determine whether the missing

data is randomly distributed vis-a-vis the other variables of interest. Create a

dummy variable to divide the sample into those who provided the requested

information (coded 0) and those who did not (coded 1) and compare the

two subgroups statistically on those variables you believe are related to

the missing data. You can conclude that the missing information is not

systematically related to the other variables if the means of the two groups

do not differ significantly. Dropping the cases having missing data will most

likely address the problem adequately when the missing data is random

and only a few respondents have missing data. However, the missing data

I. MULTIVARIATE STATISTICS AND MATHEMATICAL MODELING I I

is not random if there is a significant mean difference, and the problem is

especially troublesome when the significant mean difference accounts for

a substantial amount of variance. Under those circumstances dropping the

cases having missing data would introduce a distortion into the data.

Another possibility is to drop variables from the data file. This option

is most practical when the missing values occur primarily for one or a

relatively few variables and those variables are not critical to the research,

or the data file contains another variable for which complete data is available

that correlates highly with the variable having missing data.

Numerous strategies have been used for estimating (imputing) missing

values. Tabachnick and Fidell (1996) argue that it is reasonable to make

a well-educated guess if the sample is large, the number of missing values

is small, and the researcher has extensive knowledge of the research area.

This recommendation again underscores the importance of possessing a

thorough knowledge of the substantive area under investigation. The effec-

tive use of multivariate procedures requires both a familiarity with the

techniques themselves and with the substantive area under investigation.

More typically, missing values are replaced with a mean value calculated

on the basis of the entire sample or the subgroup to which the person

belongs. Using the mean for the entire sample is a conservative procedure

that does not change the overall mean, but it does reduce the variance in

scores on that variable, and therefore the correlation of that variable with

other variables. Using the mean for the subgroup to which the respondent

belongs does not change the mean of the group, but it does reduce the

within-group heterogeneity on the variable. Because some tests of signifi-

cance examine the relation of the variance between groups to the variance

within groups, this procedure introduces a slight bias in favor of finding a

statistically significant difference between groups and increases the likeli-

hood of a Type I error.

We recommend a variation of that strategy when one of the values in

a composite is missing. For example, suppose a respondent failed to answer

one of the items on the 10-item Self-Esteem Scale (SES; Rosenberg, 1965).

Calculate that respondent's mean score on the nine SES items that were

answered and substitute that mean in place of the missing value. This

strategy is conservative in that it does not change the respondent's overall

mean on the scale, but it inflates the internal consistency of the scale slightly.

Another procedure is to use regression to calculate predicted values for

the missing data. Use the other variables in the data file as the independent

variables and calculate a regression equation based on the data of all those

respondents who do not have missing data for the (dependent) variable.

Impute the resulting predicted values in place of the missing values. This

process can be iterated; the predicted value can be inserted in place of the

missing value and the data of all the respondents used to generate a second

predicted missing data value for each respondent having missing data. Then

12 HOWARD E. A. TINSLEY AND STEVEN D. BROWN

compare the second missing data value to the first missing data value and

repeat this process until the predicted values stabilize.

Despite it's seeming sophistication, this procedure is feasible only when

the data file contains at least one variable having no missing data that is a

reasonable predictor of the variable with missing data. Furthermore, this

procedure reduces the variance in the variable having missing data and

introduces greater consistency among the variables. Moreover, the regres-

sion equation will sometimes yield out-of-range estimated scores that must

be discarded or adjusted to fall within the acceptable range.

A point often overlooked by researchers is that the presence of missing

data may itself be useful as a predictor in your research. The respondent's

decision not to answer certain items represents a sample of behavior that

may have a meaningful relation to the issues under investigation. You can

test this possibility by creating a dummy variable that distinguishes cases

having complete data from cases having missing data. Including this infor-

mation about the respondent in your analysis might convert the liability of

missing data to an asset.

Finally, whenever missing data are imputed, it is good practice to rerun

your analyses using only those cases having complete data. Compare the

results obtained using those cases having complete data results to those

based on the entire sample when missing data is imputed. You can have

confidence in the results based on the entire sample if they are essentially

the same as those based on the subgroup having complete data. If the

results of the two analyses are different, however, you need to investigate

the reasons for the difference.

C. Outliers

Outliers are values on one or a combination of variables that are so extreme

they exert a disproportionate influence on the analysis. As such, they distort

the statistics produced by the analysis and lead to results that do not

generalize. Outliers can be univariate (i.e., cases with an extreme value on

a single variable) or multivariate (i.e., cases with an unusual combination

of scores on two or more variables). The proper handling of outliers involves

a three-step process of detecting the outliers, determining the cause of the

outliers, and reducing the influence of the outliers. Venter and Maxwell

(chapter 6, this volume) and Huberty and Petoskey (chapter 7, this volume)

describe diagnostic procedures for detecting the presence of outliers and

remedial procedures for nullifying their negative effects. As with our treat-

ment of missing data, we have structured our discussion of outliers to

minimize redundancy. For example, Venter and Maxwell distinguishes

among outliers, leverage, and influence, a distinction we do not draw

in our discussion. After reading this chapter we recommend you read

the sections of chapters 6 and 7 dealing with outliers to gain a more com-

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