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**Sensitivity analysis in linear regression.**
*(English)*
Zbl 0648.62066

Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons. xiv, 315 p. £25.95 (1988).

Linear regression analysis is one of the most used - and misused - statistical techniques. Important techniques for improving an analysis based on regression are those known as regression diagnostics. Fortunately, there has been a rapid and remarkable development of regression diagnostics over the past ten to fifteen years. This has been manifested in several text books and articles in scientific journals. For a further development it is important to extend the diagnostics to a sensitivity analysis, including not only a study of effects of individual observations and groups of observations on a fitted model, but also systematic examinations of the impacts of the different model assumptions made, and of the different principles of statistical inference adapted. The characteristic feature of the present book is that it takes a major step towards a development of sensitivity analysis in linear regression.

The first two chapters contain a presentation of the model and the inference procedures used, including a summary of the standard least squares regression results, a review of the assumptions on which these results are based, and a discussion of the properties of the projection matrix. The impact of individual and multiple observations on the fitted model, including a detailed discussion of the nature of outliers, leverage points, and influential points, is presented in Chapter 4 (impact of individual observations), Chapter 5 (impact of multiple observations), and Chapter 6 (joint impact of a variable and an observation). The impact of some of the model assumptions made are examined in Chapter 3 (impact of different selections of explanatory variables), Chapter 7 (impact of measurement errors in variables), and in Chapter 8 (impact of distributional assumptions about the error terms). Some computational methods are presented in Chapter 9. An Appendix contains a brief presentation of matrix norms and proofs of some theorems. The ideas presented in the book are illustrated by several examples.

In conclusion, the book gives an important contribution to the understanding of linear regression analysis and presents tools for improving an analysis. The book is therefore highly recommended to all who work with regression analysis and as a supplement to a course in regression analysis. The book is a pioneer in sensitivity analysis, although it fails to cover all the relevant aspects of sensitivity analysis.

In particular, a discussion of the impact of the principles of statistical inference adapted is absent, and the approach to study the effect of distributional assumptions (through the concept of generalized linear models) gives, in the present form, more a comparison of a continuous and a discrete distribution for the response variable than a comparison of realistic alternatives to the normal distribution, and is therefore less interesting. Finally, an error in a proof of a theorem is found (Theorem 4.4). The statement seems, however, to be correct, possibly with an additional condition.

The first two chapters contain a presentation of the model and the inference procedures used, including a summary of the standard least squares regression results, a review of the assumptions on which these results are based, and a discussion of the properties of the projection matrix. The impact of individual and multiple observations on the fitted model, including a detailed discussion of the nature of outliers, leverage points, and influential points, is presented in Chapter 4 (impact of individual observations), Chapter 5 (impact of multiple observations), and Chapter 6 (joint impact of a variable and an observation). The impact of some of the model assumptions made are examined in Chapter 3 (impact of different selections of explanatory variables), Chapter 7 (impact of measurement errors in variables), and in Chapter 8 (impact of distributional assumptions about the error terms). Some computational methods are presented in Chapter 9. An Appendix contains a brief presentation of matrix norms and proofs of some theorems. The ideas presented in the book are illustrated by several examples.

In conclusion, the book gives an important contribution to the understanding of linear regression analysis and presents tools for improving an analysis. The book is therefore highly recommended to all who work with regression analysis and as a supplement to a course in regression analysis. The book is a pioneer in sensitivity analysis, although it fails to cover all the relevant aspects of sensitivity analysis.

In particular, a discussion of the impact of the principles of statistical inference adapted is absent, and the approach to study the effect of distributional assumptions (through the concept of generalized linear models) gives, in the present form, more a comparison of a continuous and a discrete distribution for the response variable than a comparison of realistic alternatives to the normal distribution, and is therefore less interesting. Finally, an error in a proof of a theorem is found (Theorem 4.4). The statement seems, however, to be correct, possibly with an additional condition.

Reviewer: Hans Nyquist (Umeå)

### MSC:

62J05 | Linear regression; mixed models |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |