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Ton J. Cleophas · Aeilko H. Zwinderman

Regression
Analysis
in Medical
Research
for Starters and 2nd Levelers
Regression Analysis in Medical Research
Ton J. Cleophas • Aeilko H. Zwinderman

Regression Analysis in
Medical Research
for Starters and 2nd Levelers
Ton J. Cleophas Aeilko H. Zwinderman
Department Medicine Department of Epidemiology and Biostatistics
Albert Schweitzer Hospital Academic Medical Center
Sliedrecht, The Netherlands Amsterdam, Noord-Holland, The Netherlands

Additional material to this book can be downloaded from http://extras.springer.com.

ISBN 978-3-319-71936-8 ISBN 978-3-319-71937-5 (eBook)

https://doi.org/10.1007/978-3-319-71937-5

Library of Congress Control Number: 2017963262

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Preface

The authors, as professors in statistics and machine learning at European universi-

ties, are worried that their students find regression analyses harder than any other
methodology in statistics. This is serious, because almost all of the novel method-
ologies in current data mining and data analysis include elements of regression
analysis. It is the main incentive for writing a 26-chapter edition, consisting of:
– Over 26 major fields of regression analysis
– Their condensed maths
– Their applications in medical and health research as published so far
– Step-by-step analyses for self-assessment
– Conclusion and reference sections
The edition is a pretty complete textbook and tutorial for medical and health-care
students, as well as a recollection/update bench and help desk for professionals.
Novel approaches to regression analyses already applied in published clinical
research will be addressed: matrix analyses, alpha spending, gatekeeping, kriging,
interval censored regressions, causality regressions, canonical regressions, quasi-
likelihood regressions and novel non-parametric regressions. Each chapter can be
studied as a stand-alone and covers one of the many fields in the fast-growing world
of regression analyses. Step-by-step analyses of over 40 data files, both hypothesized
and real data, stored at extras.springer.com are included for self-assessment
purposes.
Traditional regression analysis is adequate for epidemiology but lacks the preci-
sion required for clinical investigations. However, in the past two decades, modern
regression methods have proven to be much more precise. And so it is time that a
book described regression analyses for clinicians. The current edition is the first to
do so.

Sliedrecht, The Netherlands Ton J. Cleophas

Amsterdam, NH, The Netherlands Aeilko H. Zwinderman

v
Contents

1 Continuous Outcome Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Data Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Data Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Defining the Intercept “a” and the Regression Coefficient “b”
from the Regression Equation y ¼ a + bx . . . . . . . . . . . . . . . . . 4
1.5 Correlation Coefficient (R) Varies Between 1 and +1 . . . . . . . 5
1.6 Computing R, Intercept “a” and Regression Coefficient
“b”: Ordinary Least Squares and Matrix Algebra . . . . . . . . . . . . 5
1.7 SPSS Statistical Software for Windows for Regression
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 A Significantly Positive Correlation, X Significant Determinant
of Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Simple Linear Regression Uses the Equation y ¼ a + bx . . . . . . 13
1.10 Multiple Regression with Thee Variables Uses Another
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.11 Real Data Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.12 SPSS Statistical Software for Windows for Regression
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.13 Summary of Multiple Regression Analysis of 3 Variables . . . . . 17
1.14 Purposes of Multiple Linear Regression . . . . . . . . . . . . . . . . . . 18
1.15 Multiple Regression with an Exploratory Purpose,
First Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.16 Multiple Regression for the Purpose of Increasing Precision,
Second Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.17 Multiple Regression for Adjusting Confounding,
Third Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.18 Multiple Regression for Adjusting Interaction, Fourth Purpose . . . 34
1.19 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vii
viii Contents

2 Dichotomous Outcome Regressions . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Cox Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3 Confirmative Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.1 Confirmative Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2 High Performance Regression Analysis . . . . . . . . . . . . . . . . . . 62
3.3 Example of a Multiple Linear Regression Analysis as Primary
Analysis from a Controlled Trial . . . . . . . . . . . . . . . . . . . . . . . 63
3.4 Example of a Multiple Logistic Regression Analysis as Primary
Analysis from a Controlled Trial . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Example of a Multiple Cox Regression Analysis as Primary
Analysis from a Controlled Trial . . . . . . . . . . . . . . . . . . . . . . . 71
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Dichotomous Regressions Other Than Logistic and Cox . . . . . . . . . 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Binary Poisson Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 Negative Binomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Probit Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Tetrachoric Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.5.1 The Yule Approximation . . . . . . . . . . . . . . . . . . . . . . 90
4.5.2 The Ulrich Approximation . . . . . . . . . . . . . . . . . . . . . 90
4.6 Quasi-Likelihood Regressions . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Polytomous Outcome Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.2 Multinomial Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.3 Ordinal Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4 Negative Binomial and Poisson Regressions . . . . . . . . . . . . . . . 111
5.5 Random Intercepts Regression . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.6 Logit Loglinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.7 Hierarchical Loglinear Regression . . . . . . . . . . . . . . . . . . . . . . 124
5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6 Time to Event Regressions Other Than Traditional Cox . . . . . . . . . 131
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.2 Cox with Time Dependent Predictors . . . . . . . . . . . . . . . . . . . . 132
6.3 Segmented Cox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.4 Interval Censored Regressions . . . . . . . . . . . . . . . . . . . . . . . . . 137
6.5 Autocorrelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.6 Polynomial Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Contents ix

7 Analysis of Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.2 Little Difference Between Anova and Regression Analysis . . . . 147
7.3 Paired and Unpaired Anovas . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8 Repeated Outcomes Regression Methods . . . . . . . . . . . . . . . . . . . . . 155
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8.2 Repeated Measures Analysis of Variance (Anova) . . . . . . . . . . . 156
8.3 Repeated Measures Anova Versus Ancova . . . . . . . . . . . . . . . . 159
8.4 Repeated Measures Anova with Predictors . . . . . . . . . . . . . . . . 164
8.5 Mixed Linear Model Analysis Without Random Interaction . . . . 166
8.6 Mixed Linear Model with Random Interaction . . . . . . . . . . . . . 170
8.7 Doubly Repeated Measures Multivariate Anova . . . . . . . . . . . . 172
8.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9 Methodologies for Better Fit of Categorical Predictors . . . . . . . . . . 179
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
9.2 Restructuring Categories into Multiple Binary Variables . . . . . . 180
9.3 Variance Components Regressions . . . . . . . . . . . . . . . . . . . . . . 183
9.4 Contrast Coefficients Regressions . . . . . . . . . . . . . . . . . . . . . . . 186
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10 Laplace Regressions, Multi-instead of Mono-exponential
Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
10.2 Regression Analysis with Laplace Transformations with Due
Respect to Those Clinical Pharmacologists Who Routinely
Use It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.3 Laplace Transformations: How Does It Work . . . . . . . . . . . . . . 196
10.4 Laplace Transformations and Pharmacokinetics . . . . . . . . . . . . . 197
10.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
11 Regressions for Making Extrapolations . . . . . . . . . . . . . . . . . . . . . . 201
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
11.2 Kriging Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.2.1 Semi Variography . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
11.2.2 Correlation Levels between Observed Places and
Unobserved Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
11.2.3 The Correlation between the Known Places
and the Place “?” . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
11.3 Markov Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
12 Standardized Regression Coefficients . . . . . . . . . . . . . . . . . . . . . . . 213
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
12.2 Path Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
x Contents

12.3 Structural Equation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 216

12.4 Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
13 Multivariate Analysis of Variance and Canonical Regression . . . . . 227
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
13.2 Multivariate Analysis of Variance (Manova) . . . . . . . . . . . . . . . 229
13.3 Canonical Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
14 More on Poisson Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
14.2 Poisson Regression with Event Outcomes per Person per
Period of Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
14.3 Poisson Regression with Yes/No Event Outcomes
per Population per Period of Time . . . . . . . . . . . . . . . . . . . . . . 241
14.4 Poisson Regressions Routinely Adjusting Age and Sex
Dependence, Intercept-Only Models . . . . . . . . . . . . . . . . . . . . . 243
14.5 Loglinear Models for Assessing Incident Rates with Varying
Incident Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
14.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
15 Regression Trend Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
15.2 Linear Trend Testing of Continuous Data . . . . . . . . . . . . . . . . . 250
15.3 Linear Trend Testing of Discrete Data . . . . . . . . . . . . . . . . . . . 252
15.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
16 Optimal Scaling and Automatic Linear Regression . . . . . . . . . . . . . 255
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
16.2 Optimal Scaling with Discretization and Regularization Versus
Traditional Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . 257
16.3 Automatic Regression for Maximizing Relationships . . . . . . . . . 261
16.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
17 Spline Regression Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
17.2 Linear and the Simplest Nonlinear Models of the
Polynomial Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
17.3 Spline Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
17.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
18 More on Nonlinear Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
18.2 Testing for Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
18.3 Logit and Probit Transformations . . . . . . . . . . . . . . . . . . . . . . . 284
18.4 “Trial and Error” Method, Box Cox Transformation,
ACE/AVAS Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
Contents xi

18.5 Sinusoidal Data with Polynomial Regressions . . . . . . . . . . . . . . 288

18.6 Exponential Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
18.7 Spline Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
18.8 Loess Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
18.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
19 Special Forms of Continuous Outcomes Regressions . . . . . . . . . . . . 299
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
19.2 Kernel Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
19.3 Gamma and Tweedie Regressions . . . . . . . . . . . . . . . . . . . . . . 307
19.4 Robust Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
19.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
20 Regressions for Quantitative Diagnostic Testing . . . . . . . . . . . . . . . 319
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
20.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
20.3 Deming Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
20.4 Passing-Bablok Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
20.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
21 Regressions, a Panacee or at Least a Widespread Help for Data
Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
21.2 How Regressions Help You Make Sense of the Effects
of Small Changes in Experimental Settings . . . . . . . . . . . . . . . . 329
21.3 How Regressions Can Assess the Sensitivity of Your
Predictors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
21.4 How Regressions Can Be Used for Data with Multiple
Categorical Outcome and Predictor Variables . . . . . . . . . . . . . . 334
21.5 How Regressions Are Used for Assessing the Goodness
of Novel Qualitative Diagnostic Tests . . . . . . . . . . . . . . . . . . . . 340
21.6 How Regressions Can Help You Find Out About Data Subsets
with Unusually Large Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 343
21.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
22 Regression Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
22.2 Data Example, Principles of Regression Trees . . . . . . . . . . . . . . 360
22.3 Automated Entire Tree Regression from the LDL Cholesterol
Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
22.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
23 Regressions with Latent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 365
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
23.2 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
23.3 Partial Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
xii Contents

23.4 Discriminant Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

23.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
24 Partial Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
24.2 Data Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
24.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
25 Functional Data Analysis I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
25.2 Principles of Principal Components Analysis and Optimal
Scaling, a Brief Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
25.3 Functional Data Analysis, Data Example . . . . . . . . . . . . . . . . . 397
25.3.1 Principal Components Analysis . . . . . . . . . . . . . . . . . . 398
25.3.2 Optimal Scaling with Spline Smoothing . . . . . . . . . . . . 401
25.3.3 Optimal Scaling with Spline Smoothing Including
Regularized Regression Using either Ridge, Lasso,
or Elastic Net Shrinkages . . . . . . . . . . . . . . . . . . . . . . 402
25.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
26 Functional Data Analysis II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
26.2 Statistical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408
26.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
26.4 Applications in Medical and Health Research . . . . . . . . . . . . . . 413
26.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Chapter 1
Continuous Outcome Regressions
General Principles Regression Analysis I

Abstract The current chapter reviews the general principles of the most popular
regression models in a nonmathematical fashion. First, simple and multiple linear
regressions are explained as methods for making predictions about outcome
variables, otherwise called dependent variables, from exposure variables, otherwise
called independent variables. Second, additional purposes of regression analyses are
addressed, including
1. an exploratory purpose,
2. increasing precision,
3. adjusting confounding,
4. adjusting interaction.
Particular attention has been given to common sense rationing and more intuitive
explanations of the pretty complex statistical methodologies, rather than bloodless
algebraic proofs of the methods.

Keywords Linear regression · Multiple linear regression · Exploratory purpose

· Precision · Confounding · Interaction

1.1 Introduction

The authors as teachers in statistics at universities in France and the Netherlands, have
witnessed, that students find regression analysis harder than any other methodology in
statistics. Particularly, medical and health care students rapidly get lost, because of
dependent data and covariances, that must be accounted all the time. The problem is
that high school algebra is familiar with equations like y ¼ a + b x, but it never addresses
equations like y ¼ a + b1 x1 + b2 x2, let alone y ¼ a + b1 x1 + b2 x2 + b1 x1. b2 x2 .
In the past 30 years the theoretical basis of regression analysis has changed little,
but an important step, made by the European Medicine Agency (EMA) last year,
was, that the EMA has decided to include directives regarding baseline
characteristics in the statistical analysis of controlled clinical trials. And regression

© Springer International Publishing AG 2018 1

T. J. Cleophas, A. H. Zwinderman, Regression Analysis in Medical Research,
https://doi.org/10.1007/978-3-319-71937-5_1
2 1 Continuous Outcome Regressions

methods have started to obtain a reason for being here, while a short time ago their
use was limited to hypothesis-generating rather than hypothesis-testing research.
The current chapter reviews the general principles of the most popular regression
models in a nonmathematical fashion. First, simple and multiple linear regression is
explained. Second, the main purposes of regression analyses are addressed. Just like
with previous editions of the authors (References section), particular attention has
been given to common sense rationing, and more intuitive explanations of the pretty
complex statistical methodologies, rather than bloodless algebraic proofs of the
methods.

1.2 Data Example

An example is given. One of the authors is an internist. Constipation is a typical

internists’ diagnosis. This is a study of the effect of a new laxative as compared to
that of a standard treatment. Thirty five patients are treated in a randomized cross-
over study with a novel and a standard laxative, bisacodyl. The numbers of monthly
stools is the outcome.

The above table shows, that the new treatment performed much better than the
standard treatment did with stool frequencies rising from around 10 times per month
to over 30 times per month. No further statistical analysis is needed. Statistical tests
1.3 Data Plot 3

are only useful for small differences. In this example the effect is like with penicilline
for infectious disease. It is simply fantastic. And so, the improvement is already
obvious from the data table. No need for statistical testing. The analysis thus stops
here. But the data can be used for another purpose shown in the next section.

1.3 Data Plot

50

40

30

20

10
VAR00001

0
2 4 6 8 10 12 14 16 18 20
VAR00002

The above plot with the standard treatment results on the x-axis and the new
treatment results on the y-axis shows something very interesting. There is a nice
pattern in the data, the better the effect of the standard treatment, the better the effect
of the new treatment. We can draw a line according to the equation y ¼ a + bx. For
every x-value this line provides the best predictable y-value in the population. The
y-values measured are called the dependent variable, the x-values measured the
independent variable. The b value in the equation is called the regression coefficient,
in algebra, usually, called the direction coefficient, the a value is the intercept, the
place where the line crosses the y-axis. The line is called the best fit line, there is no
better line for describing the measured data. It has the shortest distance to all of the
measured values.
4 1 Continuous Outcome Regressions

1.4 Defining the Intercept “a” and the Regression

Coefficient “b” from the Regression Equation y ¼ a + bx

The a and b values from the equation y ¼ a + bx can be found by the use of the least
square method.

b = direction coefficient = Ʃ(x- xmean )(y-ymean) Ʃ(x-xmean)2

Sum of products x times y / sum of squares x
SPxy / SSx

SPxy / SSx is an anova-like term and can be analyzed with anova (analysis of
variance).

The enumerator of the above ratio, SPxy (sum of products x- and y-variable), is
the covariance of the x- and y-variable. It is found by computing (x-xmean)( y-ymean)
for each patient, and then adding up the results. The denominator SSx (sum of
squares x-variable) is equal to the squared standard deviation of the x-variable.
a ¼ intercept y
 b x

Another important term in regression analysis is the R-value or r-value.

R ¼ correlation coefficient ¼ SPxy=√ðSSx:SSyÞ

R looks a lot like b, and it is also an anova-like term that can be analyzed with
analysis of variance. Why is it, just like b an important term in regression analysis? R
is a measure for the strength of association between the measured x and y values. The
stronger the association, the better the x predicts y. R varies between 1 and +1, 1
and 1 mean, that the association is 100%, 0 means it is 0%.


R2 ¼ SP2 xy= SS2 x SS2 y ¼ covariance xy=ðvariance x times variance yÞ

R2 varies from 0 to 1, it is nonnegative, and, therefore, often more convenient

than R.
1.6 Computing R, Intercept “a” and Regression Coefficient. . . 5

1.5 Correlation Coefficient (R) Varies Between 1 and +1

R varies between 1 and +1. The strongest association is either 1 or +1. The
weakest association is zero. The underneath graph gives an example of three
crossover studies with different levels of association. Ten patients are treated with
two treatments. The left study shows a very strong negative correlation: if one
treatment performs well, the other does not at all. The study in the middle gives a
zero correlation: the x-outcome can not predict the y-outcome. The right study shows
a strong positive correlation: if one treatment performs well, the other does so too. In
the left and right studies the x-data can be used as very good predictors of the y-data.
treatment effects of vasodilator 1

50 50 50
r ≈ -1 r≈0 r ≈ +1
(Raynaud attacks/wk)

40 40 40

30 30 30

20 20 20

10 negative correlation 10 zero correlation 10 positive correlation

0 0 0
10 20 30 40 10 20 30 40 10 20 30 40
treatment effects of vasodilator 2 (Raynaud attacks/wk)

1.6 Computing R, Intercept “a” and Regression Coefficient

“b”: Ordinary Least Squares and Matrix Algebra

With big data files it is pretty laborious to compute R, “a”, and “b”. Regression
analysis is also possible with just four outcome values, and the computations are
similar but less laborious. A real data example of just 4 outcome values studies the
effect of age of a magnetic resonance imager on its yearly repair costs. The
relationship and best fit regression line between the age and the costs per year is in
the underneath graph. We will use this small study to explain how computations
must be done.
6 1 Continuous Outcome Regressions

7,00

6,50
costs per year x $5000

6,00

5,50

5,00

4,50

4,00

1,00 2,00 3,00 4,00 5,00

age (years)

First, an ordinary least square computation will be performed.

age (years) repair costs per year

(x $5000)

x y xy x2 n
_________________________________________________________________
5 7 35 25 1
3 7 21 9 1
3 6 18 9 1
1 4 4 1 1
_________________________________________________________________
Σx = 12 Σy = 24 Σxy = 78 Σx2 = 44 Σn = 4

Y = A + BX is the equation of a linear regression.

x, y, a, and b, and their upper case X, Y,....are often used pell-mell.

Ym = mean of observed Y-values.

Xm = mean of observed X-values.

1.6 Computing R, Intercept “a” and Regression Coefficient. . . 7

B= Σ (X - Xm )(Y-Ym ) Σ XY - n Xm Ym
=
Σ(X-Xm )2 Σ X2 - n Xm2

A = Ym - BXm

We will now calculate B, the slope of the best fit regression line otherwise called
regression or direction coefficient, and A, the intercept, the place where the best fit
regression line crosses the Y-axis.

B = 78 - 4.3.6 = 78-72 = 6 = 0.75

44-4.32 44-36 8

A = 6 - 0.75.3 = 6 - 2.25 = 3.75

Second, matrix algebra will be used for finding the parameters of the best fit
regression line
We will try and determine the intercept and regression coefficient of the linear
regression equation with y as dependent and x as independent determinant.
Y ¼ A + BX is the equation of a linear regression.
It can be written and analyzed in the form matrices.
X is an n matrix (otherwise called column consistent of n x-values).
Y is an n matrix (otherwise called column consistent of n x-values).
A and B are also column vectors consistent of single values.
X’ is the transpose of the vector X, if the column is replaced with the same
value row.
Matrices can not only be transposed but also inversed, multiplied, added up.

X'X = 2 x 2 matrix X'X

X'Y = 2 x 1 Column vector X'Y

(X'X)-1 = the inverse of the X'X matrix

The X’X and XY matrices can be easily calculated by hand.

8 1 Continuous Outcome Regressions

12 = Σx
4 = Σn
44 = Σx2
12 = Σx
24 = Σy
78 = Σxy

It is, however, pretty complex to compute the inverse of the X’X matrix by hand,
and, therefore, the online matrix calculator “matrixcalc.org” was used. First enter the
values of the X0 X matrix at the appropriate place, then press inverse. The result is
given underneath.

Now the underneath three matrices can be written, and used for identifying the a
and b values. The XY0 and (X0 X)1 must be multiplied as shown.

X'X = 12 4
44 12

X'Y = 24
78

(X'X)-1 = -3/8 1/8

11/8 -3/8

(X'X)-1 . X'Y = -3/8 x 24 +1/8 x 78

11/8 x 24 -3/8 x 78

= -9 + 9.75
33 -29.25

= 0.75 +3.75

b = 0.75
a = 3.75
1.7 SPSS Statistical Software for Windows for Regression Analysis 9

The above computations give an overview of the procedure which provides the a
and b values of the regression equation:
y ¼ a + bx.

1.7 SPSS Statistical Software for Windows for Regression

Analysis

SPSS statistical software for Windows is used for regression analysis. First enter the
data, for example, from an Excel-file. The command is in the Menu.
Command:
Analyze. . . .Regression. . . .Linear. . . .Dependent: stools after new laxative. . . .Inde-
pendent: stools after bisacodyl. . . .click OK.
In the output sheets the underneath tables are given.
10 1 Continuous Outcome Regressions

We were mainly interested in the magnitudes of the b-values and the R or R

square values. A lot of information is supplied, more than we asked for.
The upper table gives the R and R square values. If R square had been 0, then no
correlation at all would have been in the data. If it had been 1, then correlation would
have been 100%. We would be 100% certain about the y-value to be predicted from
any x-value. However, the correlation was only 0.630. The level of correlation is
1.7 SPSS Statistical Software for Windows for Regression Analysis 11

63%. We can be 63% certain about y knowing x. Generally R square values smaller
than 0.25, means a very poor correlation, R squares between 0.25 and 0.50 means a
reasonable level of correlation. A correlation level over 0.50 means a strong
correlation.
In the middle is the anova table. The correlation is dependent not only on the
magnitude of the R square value but also on the data sample size. Of course, with
only 3 values on a line less certainty is provided than it is with 35 values like in our
example. Anova (analysis of variance) is used to test, whether the r square value is
significantly different from an R square value of 0. If so, then the line is not a chance
finding, and it means, that it can be used for making predictions from future x-values
about future y-values with 63% certainty. A typical anova-table is given. If the sum
of squares regression, 2128.393, is divided by the sum of squares total, then you will
find 0.630, which equals the R square value from the upper table. The sums of
squares are adjusted for degrees of freedom, a measure for the sample size of the
study, and a test statistic, the F value, is produced. It is very large and corresponds
with a p-value <0.0001. This means, that the correlation found is much larger than a
correlation of 0.
At the bottom the coefficients table is given. In the B column the intercept and the
regression coefficient are given, 8.647 and 2.065. The equation of the best fit
regression line: y ¼ a + bx ¼ 8.647 + 2.065 times bisacodyl data. The b-term can
just like the r square term be used for testing the level of correlation between x and
y. If statistically significant, then x is a significant determinant, otherwise called
independent or orthogonal determinant of y.
The above three tables are somewhat redundant. If you square the test statistic
t ¼ 7.491, then you will find the value 56.110, which is equal to the test statistic of
the F-test of the middle table. Indeed, anova with two variables produces an F-value
equal to the square root of the t-value from a t-test. Here an anova is equal to a t-test
squared. Also redundant is the standardized beta coefficient of 0.794, because it is
given already in the upper table as the R value of the data.
12 1 Continuous Outcome Regressions

1.8 A Significantly Positive Correlation, X Significant

Determinant of Y
50

40

30

20

10
VAR00001

0
2 4 6 8 10 12 14 16 18 20

VAR00002

So far we have demonstrated a significantly positive correlation, x is a significant

determinant of y. Maybe, also a positive correlation exists between the new laxative
and the age of the patients in the study. For example, the new laxative may work
better, the better the bisacodyl worked and the older the patients are. In this situation
we have three observations in a single person:
(1) efficacy datum of new laxative
(2) efficacy datum of bisacodyl
(3) age.
How do we test?
We first have to define the variables:
y variable presents the new laxative data
x1 variable bisacodyl data
x2 variable age data.
The regression equation for 3 variables is given
y ¼ a + b 1x 1 + b 2x 2
1.10 Multiple Regression with Thee Variables Uses Another Equation 13

The kind of equations have never been taught to you at high school. We will use
here x1 and x2 to predict y.

1.9 Simple Linear Regression Uses the Equation y ¼ a + bx

With simple linear regression the equation y ¼ a + bx is used for making predictions
from x about y. For example, if we fill out

x-value = 0, then the equation turns into y = a

x = 1, y=a+b
x = 2, y = a + 2b

For each x-value the equation gives the best predictable y-value, and all y-values
constitute a regression line which ¼ the best fit regression line for the data, meaning
the line with the shortest distance from the y-values.

Y-axis

X-axis

1.10 Multiple Regression with Thee Variables Uses Another

Equation

Multiple regression with 3 variables uses the equation y ¼ a + b1x1 + b2x2.

The underneath orthogonal 3 axes graphical model can be used for visual
modeling.
14 1 Continuous Outcome Regressions

Y-axis

0
1
2
3
X2-axis

X1-axis

If we fill out:

x1 = 0, then the equation turns into y = a + b2x2

x1 = 1, a + b1 + b2x2
x1 = 2 a + 2b1 + b2x2
x1 = 3 a + 3b1 + b2x2.

Each x1 -value has its own regression line, and all of the regression lines
constitute a regression plane, which is the best fit plane, i.e., the plane with the
shortest distance from the x-values in a 3 dimensional space. Of course, this space
does have uncertainty caused by (1) x1- and x2-uncertainty (explained) and by
(2) residual uncertainty (unexplained).
1.11 Real Data Example 15

1.11 Real Data Example

The 35 patient crossover study is used once more, with age (years) added as x2
-variable. The regression equation is y ¼ a + b1x1 + b2x2. In order to find the
coefficients a, b1 and b2 , just like with the simple linear regression, the least square
method is used.
Essentially, three equations should be helpful to solve three unknown values a, b1
, and b2. The underneath so-called normal equations are adequate for the purpose
(n ¼ sample size).
Σy ¼ na þ b1 Σx1 þ b2 Σx2
Σx1 y ¼ aΣx1 þ b1 Σx21 þ b2 Σx1 x2
Σx2 y ¼ aΣx2 þ b1 Σx1 x2 þ b2 Σx22

However, a lot of calculus is required, and we will, therefore, gratefully apply

SPSS statistical software for computations.
16 1 Continuous Outcome Regressions

1.12 SPSS Statistical Software for Windows for Regression

Analysis

SPSS statistical software for Windows is used for regression analysis. First enter the
data, for example from an Excel-file. Then command in the Menu:
Statistics. . . .Regression. . . .Linear. . . .Dependent stools after new laxative. . . .Inde-
pendent stools after bisacodyl, age. . . .click OK.
In the output sheets the underneath tables are given.
1.13 Summary of Multiple Regression Analysis of 3 Variables 17

We were mainly interested in the magnitudes of the a- and b-values, and the R of
R square values. A lot of information is supplied, more than we asked for. First, the
above tables are the tables of a multiple regression rather than a simple linear
regression. SPSS has named the y-variable VAR00001, the x1 -variable
VAR00002, and the x2 -variable VAR00003.
The upper table gives the R and R square values. If R square had been 0, then no
correlation at all would have been in the data. x1 and x2 would have determined y in
no way. If it had been 1, then correlation would have been 100%. We would be
100% certain about the y-value to be predicted from any of the x1 – and x2 -values.
However, the correlation was only 0.719 ¼ 72%. The x1 – and x2 -values determine
the y-values by 72%. The level of correlation is 72%. We can be 72% certain about y
when knowing the x1 – and x2 -values. Generally R square values smaller than 0.25,
means a very poor correlation, R squares between 0.25 and 0.50 means a reasonable
level of correlation. A correlation level over 0.50 means a strong correlation. So,
72% is pretty good. As compared to the R square of the simple linear regression the
level has increased from 63% to 72%. This is as expected: if you have additional
predictors about y, then you will probably have greater certainty to make predictions.
In the middle is the anova table. The correlation is dependent not only on the
magnitude of the R square value, but also on the data sample size. Of course, with
3 values on a line less certainty is provided than it is with 35 values like in our
example. Anova calculates whether the r square value is significantly different from
an R square value of 0. If so, then the line is not a chance finding, and it means that it
can be used for making predictions from future x-values about future y-values with
72% certainty.
At the bottom the coefficients table is given. In the B column the intercept and the
regression coefficient a are given, and 2.065. The equation of the best fit regression
line: y ¼ a + bx ¼ 1.547 + 1.701 times bisacodyl data +0.426 times age data. The
b-terms can just like the r square term be used for testing the level of correlation
between x-variables and y-variable. If statistically significant, then the x-variables
are significant determinants, otherwise called independent or orthogonal determi-
nants of y.

1.13 Summary of Multiple Regression Analysis of

3 Variables

A summary of the above multiple regression analysis with 3 variables is given.

18 1 Continuous Outcome Regressions

Y-axis

0
1
2
3
X2-axis

X1-axis
Regression plane

The regression equation of the above example is y ¼ 1.547 + 1.701 x1 + 0.426 x2.
The overall correlation between the x-values and the y-values is R2 ¼ 0.719
(p < 0.0001).
The independent determinants of y are x1 and x2.
The numbers of stools will be close to 0, if x1 and x2 are 0.
With every stool on bisacodyl, y will rise by 1.7
With every year of age, y will rise by 0.4.
Note: in case of > 3 variables the multiple regression will get multi-dimensional,
and, visually, beyond imagination, but mathematically this is no problem at all.

1.14 Purposes of Multiple Linear Regression

The main purposes of multiple linear regression are four.

1. Exploratory purpose. When searching for significant predictors of a study out-
come, here otherwise called independent determinants of the outcome y, you may
include multiple x-variables, and, then, test, if they are statistically significant
predictors of y. An example of a linear regression model with 10 x-variables is
given below.

y ¼ a þ b1 x1 þ b2 x2 þ . . . . . . . . . :b10 x10
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charitable grant.
[333] Sura ix. 30.
[334] In some treaties given by Belâdzori and others, as
concluded at the first conquest, some of these disabilities are
mentioned; but I doubt their genuineness. Though the law was
such, the practice varied greatly. Under intolerant Caliphs, such
as the orthodox Abbassides, the poor Christians were always
liable to have a fresh order issued to demolish all but their ancient
churches, close the Christian schools, &c.
[335] According to Caussin de Perceval, the strongholds along
the Tigris, as well as the Euphrates—Tekrît, Hît, &c.—were only
now reduced by the Arabs; but, according to the best traditions,
these towns fell into the hands of the Moslems, shortly after the
battle of Câdesîya.
[336] The story of this inroad and widespread rising is told by
tradition with the extremest brevity; but it is very evident that the
position of Abu Obeida must, for some little time, have been very
critical. Lebeau conjectures that the naval attack was led from
Egypt by Constantine, the son of Heraclius; and M. Caussin de
Perceval thinks that this is probable (vol. iii. p. 512).
[337] It seems almost certain that Khâlid did so serve, though
there are other traditions to the effect that he never served under
any other general than Abu Obeida. He may have led an
independent expedition.
[338] Now Diâr Bekr.
[339] Byzantine historians tell us that the Roman governor of
Edessa (Roha) concluded a treaty with Iyâdh, by which he bound
himself to pay 100,000 pieces of gold, as black-mail, with the view
of preserving his province from Saracen inroad, but that Heraclius
disowned the humiliating condition, and deposed the governor.
There is no hint of this in our Arabian authorities.
[340] Four thousand of the Beni Iyâdh were sent back in a
body to Mesopotamia from Asia Minor, and resumed their
allegiance to the Caliph, though continuing to profess the
Christian faith. The remainder dispersed on the borders of the two
kingdoms.
[341] That is, their tax was called úshr (‘tenth’), the tithe paid
by the believer, instead of jazia. It may be doubted whether the
intolerant condition, forbidding the education of the children in
Christian doctrine, was meant otherwise than as a nominal
indication of the supremacy of Islam. It certainly was not enforced
(if at all) with any rigour, for we read of this great tribe continuing
in the profession of Christianity under the Omeyyad, and even
under the Abbasside, dynasties. And in still later times they had
their bishops at Ana, on the Euphrates. (See Caussin de
Perceval, v. iii. p. 324.)
We now part with that invaluable author, whose history closes
with this narrative.
[342] Nothing illustrates the vagueness of the Syrian narrative
so forcibly as the uncertainty of the year in which Cæsarea fell.
Byzantine historians make the siege last seven years, and place
the fall in the year a.h. 19, that is, a.d. 640. Various traditions
place it in every year between a.h. 14 and a.h. 20, and represent
the siege as having lasted, some three, some four, some seven
years.
A Jew is said to have betrayed the town by discovering to the
Arabs an undefended aqueduct, through which they effected an
entrance. The population was mixed; 70,000, we are told, were
Greeks, fed (murtazac) from the public stores; 30,000
Samaritans; and 200,000 (?) Jews. It was a sad fate that of the
captives. It is mentioned incidentally that two were made over to
the daughters of Asâd ibn Zorâra, one of the twelve leaders, in
place of two from Ain Tamar, who had died in their service.
Multitudes of Greeks—men and women—must have pined
miserably in a strange land and in hopeless servitude. And
amongst these there must have been many women of gentle birth
forced into menial office, or if young and fair to look upon,
reserved for a worse fate—liable, when their masters were tired of
them, to be sold into other hands. No wonder that Al Kindy, in his
Apology, inveighs, with scathing denunciation, against the slavery
practised in these Moslem crusades.
[343] Calansua, or helmet, worn by the captains of the Syrian
army.
[344] Khâlid is no great favourite of Abbasside tradition. He
belonged to a branch distant from that of the Prophet, which
attached itself to the Omeyyads, of whom, in the struggle with Aly,
Abdallah son of Khâlid was a staunch adherent.
The general outline of Khâlid’s case is clear, though there is
variety in the details. According to some accounts, Omar returned
to him all the property he had confiscated. Others say that, when
pressed to do so, he said, ‘Nay, that be far from me. I am but the
agent of the Moslems, and am bound to administer their property
faithfully. I will never give it back.’
Tabari gives yet another account. Omar wrote to Abu Obeida
commanding him to arraign Khâlid; but adding that if he would
confess his guilt in the affair of Mâlik ibn Noweira, he would
pardon him and restore him to his Government. Khâlid repaired
for counsel to his sister Fâtima, then with her husband in Syria.
She dissuaded him from confessing; for if he did so, it would only
give Omar—who was determined on his ruin—a handle to depose
him with disgrace. He bent down, and, kissing her forehead, said:
‘It is the truth, my sister.’ So he returned to Abu Obeida, and
refused to make any confession. Thereupon Bilâl, as in the text,
stripped off his kerchief, and so on, as in the text. At the
conclusion of the trial Abu Obeida, by order of the Caliph,
confiscated half of his property, even to his sandals—taking one
and leaving the other.
[345] For an account of the persecution and martyrdom,
avenged by the invasion of the Abyssinian Negus, see Life of
Mahomet, vol. i. p. clxii. For the treaty of Mahomet, vol. iii. p. 299
(second edition, p. 158).
[346] The expulsion of the Jews is ordinarily assigned to the
twentieth year of the Hegira; that of the Christians took place
earlier. For the conquest of Kheibar, see Life of Mahomet, p. 395;
and for the death-bed saying of the Prophet, ibid. p. 503. That the
Peninsula should be wholly and exclusively Moslem was a
sentiment so closely connected with the inspiration of Mahomet,
when he declared in the Corân that he was ‘sent a prophet to the
Arabs,’ and so forth, that it might well have recurred in the
feverish delusions of his last illness. But whether or no, the
utterance—whatever its purport—was evidently not taken at the
time as an obligatory command. Had it been so, we may be sure
that Abu Bekr would have made it his first concern to give effect
to it, and no other reason would have been required to justify the
act. As it is, various reasons are assigned for the expatriation of
the Christians. First, we are told that they took usury greedily;
next, that they fell to variance among themselves, and asked to
be removed; lastly, that they were growing so strong that Omar
became afraid of them. As regards the Jews, we are told that they
were guilty of murder, and also that they attacked the Caliph’s
son.
The governors of the districts to which they emigrated had it in
charge to treat them fairly. The Christians received special
consideration, and the tale of raiment (which the heads of the
community collected by yearly circuits among their people in Irâc
and Syria) was reduced by successive Caliphs as the numbers of
the tribe diminished by conversion to Islam or other cause.
Fadak, a dependency of Kheibar, was long a source of
discontent to the descendants of Fâtima, who, as we have seen,
claimed it for her patrimony; but Abu Bekr reserved it for the poor
and the kinsmen of the Prophet (Beni Hâshim). Certain of the
Omeyyad Caliphs took possession of it as their private property. It
was repeatedly released to the claimants as an act of justice or of
piety (notably by Omar II., the pietist of the Dynasty); but it was
always soon resumed again.
[347] For example, the grandsons of the Prophet got 5,000
pieces each, like the men of Bedr. As to Abbâs, his uncle, some
say he was rated at 5,000 pieces, others 7,000, and some again
as high as 12,000 or even 24,000; but these last figures are
evidently a pandering of tradition to glorify Abbâs and exalt the
Abbasside dynasty under courtly influence. Abbâs was of course
respected in the time of Omar as the Prophet’s uncle; but he
never took any leading part at the Caliph’s court; and indeed his
antecedents, during the life of Mahomet, were not much to his
credit. See Life of Mahomet, p. 417. Ayesha was allotted 2,000
pieces extra ‘for the love the Prophet bare to her;’ but according
to some, she declined to take it. The slave-concubines (Safia and
Juweiria) were at first rated at 6,000, but at the solicitation of the
other widows they were placed on an equality with them.
[348] For these see ibid. pp. 368, 371, chap. xix.
[349] Thus certain of the Dihcâns, or Persian Talookdars, who
threw in their lot with the invading army, had a high rank, with the
title to 1,000 pieces, conferred upon them.
[350] See Life of Mahomet, p. 486.
[351] The dole was fixed, after a trial of what was sufficient as
a monthly ration, for the support of sixty poor persons. Two jarîbs
of grain, accordingly, was the portion appointed, as a minimum, to
which every indigent believer of whatever race was entitled.
 [352] The jealous susceptibilities of the rival tribes were
continually breaking forth; as for example, in the election of a
Muedzzin in place of the one killed at Câdesîya to proclaim the
times of prayer to the army, on which a free fight arose that nearly
ended in bloodshed.
[353] Belâdzori, p. 458.
[354] Omar gave out that if the revenues sufficiently increased,
he intended to advance the stipend of every man in the upper
grades to 4,000 dirhems. It is said also that he contemplated the
issue of a sumptuary ordinance both for Syria and Irâc, by which
1,000 dirhems were to be considered the allowance for the
support of the stipendiary’s family, 1,000 for his personal
expenses, 1,000 for house and furnishings, and the remainder for
hospitable entertainment; but that he died before he could issue
the order. The object of such a rule, and the practicability of giving
effect to it, are however doubtful.
[355] See Life of Mahomet, p. 555; and The Corân: its
Composition and Teaching, Society for Promoting Christian
Knowledge.
[356] This is the received derivation of the era called the Year
of Ashes. Others call it so because the land was pulverised, dark
and dusty, without a blade of grass or of any green thing.
[357] The secretary of Wâckidy has several pages filled with
traditions about Omar’s treatment of the famine, and self-denying
solicitude for his people. He refused to ride a horse during the
famine because it consumed corn. He chided his son for eating a
cucumber, when men around were dying of hunger, and so forth.
There may be much of exaggeration; but at the bottom of it all lies
a fine trait in Omar’s character.
[358] Ayla, on the Gulf of Acaba, at the head of the Red Sea.
[359] Here again the Kâtib Wâckidy gives a great array of
traditions regarding Omar’s prayers and the service for rain.
Some of these which notice the part taken by Abbâs (but they are
comparatively few in number) have been eagerly seized by the
Abbasside annalists to glorify the patriarch, and through him the
dynasty descended from him. The tale is cast in the supernatural
type of the Prophet’s life. A man finding a sheep which he had
slaughtered to be nothing but mere skin and bone without a drop
of blood, in his distress invokes Mahomet, who thereupon
appears to him in a vision, assures him that he shares the
distress of his people, and bids him tell Omar ‘to call to mind that
which he had forgotten.’ A general assembly is summoned in the
Great Mosque, and after much heart-searching as to what the
Prophet meant by these words, they betake themselves to prayer.
Omar seizes the hand of Abbâs, and for the sake of the Prophet’s
aged kinsman, beseeches the mercy of Heaven. Then Abbâs
himself prays, and the people weep floods of tears. The heavens
are suddenly overcast, and the rain descends. Thereupon Abbâs
is saluted as ‘the Waterer of the two Holy Places,’ i.e. of Mecca
and Medîna.
[360] We are told that Amru, to meet the famine, established a
shipping service between Egypt and the ports of the Hejâz, that
the trade in grain thus begun was permanently established, and
that prices were thereafter little higher at Medîna than in Egypt.
But Egypt was not conquered till two years later; and in the hostile
state of the border preceding the conquest, it is impossible that a
peaceful trade in corn could have sprung up. We must therefore
conclude that tradition here anticipates that which occurred
shortly after, when Omar reopened the communication from the
Nile to Lake Timsa and Suez, and Egypt found a rich customer in
the markets of Medîna and the Hejâz.
[361] The council was held at Sargh, near Tebûk, on the
confines of Syria. During the discussion Abd al Rahmân quoted a
saying of Mahomet:—‘If pestilence break out in a land, go not
thither; if thou art there, flee not from it.’ Omar’s views were more
reasonable, and he justified them by this illustration: ‘Suppose
that ye alight in a valley, whereof one side is green with pasture,
and the other bare and barren, whichever side ye let loose your
camels upon, it would be by the decree of God; but ye would
choose the brow that was green.’ And so he judged that in
removing the people from the scene of danger into a healthier
locality, he was making no attempt to flee from the decree of God.
[362] He purposed to make a circuit of all the provinces
subject to his sway. Aly, we are told, even recommended a
second hijra, or transfer of the Caliph’s court to Kûfa (evidently a
proleptic tradition anticipatory of the move eventually made by Aly
himself to that capital). What induced Omar to give up the project
of visiting Irâc is not very clear. The ordinary story is that Káb the
Rabbin (a Jew from Himyar, converted about this time, who will
be noticed more hereafter) dissuaded him from it: ‘Of evil,’ he
said, ‘the East hath nine parts, and of good but one; while the
dwellings of Satan and every kind of plague are there. On the
contrary, the West hath nine parts good, and but one of evil.’
Thereupon, the tradition proceeds, Omar abandoned the idea of
visiting Irâc.
[363] Before, having the double meaning of ‘he is before you,’
that is, in your presence; or (as they took it) ‘in advance of you,’
and farther on the road.
[364] Shorahbîl, who had the command of the province of the
Jordan (Ordonna), was put aside as weak and unfitted for the
office; or rather his government was apparently placed under that
of Amru, who was in command of all the Holy Land. The
appointment of Muâvia as the brother of Yezîd, the late governor
of Damascus, was in every way natural and expected.
[365] For Bilâl and his office of Muedzzin, see Life of
Mahomet, p. 204.
[366] The male population alone, we are told, numbered
600,000. There were 70,000 (according to others 40,000) male
Jews of an age to pay the poll tax, and 200,000 Greeks, of whom
30,000 effected their escape by sea before the siege. The baths
were 4,000 in number, the theatres 400, and the harbour held
12,000 vessels of various size.
[367] The narrative is almost more fugitive, and the chronology
less certain, than in the case of Syria. The expedition is variously
placed at from a.h. XVI. to XXV. The earlier date is due probably
to the notion (before explained) that Amru assisted Medîna with
corn in the year of famine; the later date, to the attempt of the
Greeks to retake Alexandria, a.h. 25. The best authenticated date
is that which I have followed. The received account is this. Amru
obtained permission for the campaign from Omar at Jâbia,
probably on his last visit to Syria. When the Caliph returned to
Medîna and reflected on the seriousness of the enterprise, he
repented of having allowed Amru to go on with so small a force,
and sent orders that if he had not already entered Egypt, he was
to return. Warned probably of its purport, Amru did not open the
packet till he had crossed the boundary; and so he went forward.
When Omar was informed of this he sent Zobeir with 12,000 men
to reinforce him. Other accounts say that Amru’s entire force
consisted of 12,000 men, despatched from Palestine and Medîna,
in three bodies, one after another. Some stories are told, but they
look apocryphal, of Amru having visited Alexandria, before his
conversion, many years previously.
 [368] For the communications of this Mucoucus with Mahomet
see Life, pp. 385 and 440.
[369] Memphis, in the vicinity of modern Cairo. The advance
was probably made by Salahiya up the Pelusian branch of the
Nile, to the north of Ismailia and Wolseley’s recent line of march.
[370] Later historians (whose accounts, however, bear the
mark of being apocryphal) represent the Moslem army as at one
time in considerable peril, surrounded and hemmed in at
Heliopolis by the rising waters of the Nile. Mucoucus having
retired to an island on the farther side of the Nile, broke up the
bridge across it. Deputations were then sent by boat to and fro;
and the Mussulman envoys delivered speeches before
Mucoucus, exhorting and threatening the governor, much in the
style of those recited at the Persian Court before the battle of
Câdesîya. Mucoucus, who is represented as favourable to Islam,
at last entered into terms with the invaders.
[371] Heraclius died in February, a.d. 641.
[372] The tale of Amru being taken prisoner in an attack on the
outworks is not mentioned by any early authority, and seems to
possess no foundation. The story is, that when carried before the
authorities, his freedman, who had been captured with him,
slapped Amru on the face, and so deceived the Greeks into the
belief that he was a common soldier who might be set at liberty.
[373] Here again we see the same nervous fear on the part of
Omar, lest his soldiers, wandering too far, or beyond some great
river, should be surprised and cut off, as led him at the first to
forbid an advance on Persia. Ghîzeh, properly Jîzeh, j in Egypt
being pronounced as hard g.
[374] This name Câhira, or City of the Victory, is of later date.
[375] There is here, as in respect of other countries, a great
profusion and variety of tradition, having for its object to prove
that Egypt was taken by force of arms, and could therefore be
treated as a conquered country; rather than that it capitulated,
and was the subject of treaty and stipulations. There was always
a strong pressure to prove the former, as it gave the invaders a
better standing in courts of law as against the natives, in such
claims as that pressed by Zobeir.
[376] The ancient canal appears to have followed very closely
the line of the Fresh-water Canal of the present day. We are not
favoured with many particulars; but there is no doubt that during
Omar’s reign vessels did make the voyage from Cairo to the
coast of Arabia, establishing thus a regular traffic between the two
countries; and therefore the work must have been very quickly
finished by the forced labour of the teeming population.
The reader who is curious about the previous attempts to unite
the Nile with the Red Sea will find the subject discussed by Weil
(vol. i. pp. 120–122). The attempt was made so far back as the
time of Pharaoh Nechos, and subsequently by Darius, who is said
to have made communication practicable from Bubastis, on the
eastern or Tanitic estuary of the Nile, to the head of the Red Sea.
A second canal was opened, under the Ptolemies at Phacusa (Tel
Fakhûs), nearer to the Mediterranean. This (taking apparently the
line of the Salahiya canal) must have presented greater difficulties
in maintaining communication through the system of lagoons
leading to the Red Sea, and so it was too shallow to be of much
use, excepting in high flood. One of these lines (the former most
probably) was eventually deepened by Trajan, and remained
navigable to the end at least of the third century of our era. It was
this canal, no doubt, which was now cleared out and deepened by
Amru. Reference is made by Weil to the following authorities:
Bähr’s Herodotus, vol. ii. p. 158; Revue des Deux Mondes, vol.
xxvii. p. 215.
[377] This tale (which is not given by our earliest authorities)
is, no doubt, based upon a custom of the Egyptians, who, as we
learn from Lane, cast, year by year, the effigy of a maiden,
decked in bridal attire, into the river, calling it ‘the Bride of the
Nile.’ But whether the tale be real or fictitious, the sentiment
conveyed in it is indicative of that virtue in the Moslem faith which
carries the special providence of God into the life of every day.
[378] Amru is said to have been so pleased with Barca as to
declare that if he had not possessed a property and home in the
Hejâz, he would have settled there.
[379] The circumstances of the siege (a strange contrast to
the bombardment, which recently crowded the horrors of months
into so many hours) are narrated with the utmost brevity; and
indeed tradition very much confuses the second siege with the
first. Eutychius speaks of the investment of the city by the Arabs
lasting fourteen months. He also tells us that George the
Patriarch fled to Constantinople, and that for ninety-seven years
there was no Melchite patriarch for Egypt. A Maronite patriarch
seems to have succeeded.
I should mention that by later and less reliable authorities a
long correspondence is given as having passed between Amru
and Omar, in which the latter upbraids his lieutenant for not
remitting ‘as large a revenue as that which Egypt yielded to the
Pharaohs.’ Amru resented the imputation; whereupon Omar sent
his legate, Mohammed ibn Maslama, to set on foot an
investigation into the revenues of the country; and also
superseded Amru in the government of Upper Egypt by Abdallah
Ibn Abu Sarh. The correspondence (though accepted by Weil)
appears to me altogether apocryphal. It was contrary to Omar’s
character to write in the harsh and unreasonable tone of these
letters, or to press his governors for funds at the expense of the
provinces which they administered. Nor did he stand in any
urgent need of the additional revenue, as these letters would
imply; for the treasures of the world were flowing at this time in a
full tide into Medîna. As to Ibn Abu Sarh, he did not supersede
Amru till the reign of his foster-brother Othmân.
[380] The earlier operations of Otba have been narrated
above, p. 91.
[381] The ancient capital of Khuzistan, where extensive ruins
and colonnades still mark the extent and magnificence of this
once regal city. Weil doubts whether the expedition reached so far
as Persepolis. But I can only follow our authorities, who certainly
represent Alâ as advancing to its vicinity.—Weil, vol. ii. p. 87.
[382] Omar, as we shall see farther on, had an unconquerable
dread of committing his troops to the sea.
[383] Otba died the same year, a.h. 17; and Moghîra
succeeded him, as related above (somewhat prematurely), p. 91.
[384] One of the three brothers who defended Medîna in the
attack on Abu Bekr—supra, p. 14.
[385] Tostar, otherwise named Shuster.
[386] These conquests are variously placed by different
traditions in a.h. XVII., XIX. and even XX. They immediately
preceded the great campaign of Khorasan.
[387] Shushan, the ancient capital of Media, now called Sûs.
Loftus gives an interesting history and description of Sûs, with a
picture of the tomb of Daniel. (Travels in Chaldæa and Susiana,
1857, p. 322.) Our authorities say that Omar gave orders for the
body of Daniel, which (as the legend goes) was still exposed to
view, being honourably interred.
Mr. Baring, Secretary of the Teheran Legation, visited the spot
in 1881, and found it much altered. The conical steeple, shown in
Loftus’ picture, was removed, when three or four years ago the
tomb was rebuilt; and it was then surrounded by a gallery with a
railing of brass and woodwork overlooking the river.
[388] Two thousand dirhems, the same as was given to
warriors of Câdesîya and the Yermûk. And stipends of like
amount were granted to the Persian nobles who had recently
joined the Moslem army in Khuzistan.
[389] It is remarkable that one of the arguments said to have
been used, even on this late occasion, was that if the Caliph
quitted Medîna there would be a risk of the Arab tribes of the
Peninsula again rising up in apostasy and rebellion.
[390] The spies were the famous Amr ibn Mádekerib (the
warrior-poet met with before) and Toleiha. The latter (the
quondam prophet of the Beni Asad) was long in returning from his
scouting expedition—so much so that the army, becoming
anxious, began to speak among themselves: ‘What if Toleiha hath
apostatised the second time!’ When he made his appearance,
therefore, there was a shout of joy. Toleiha, hearing of it, was
much hurt at the imputation. ‘Even had it been the old Arab faith,’
he said, ‘which I once professed much more this blessed faith of
Islam, I should have disdained to change it for the jargon of these
barbarians.’
[391] The battle was fought at Bowaj Rûd. Nóeim demolished
the fortifications of Rei, and laid the foundations of a new city. The
ruins of Rei, some five or six miles south-east of Teheran, are still
to be seen of considerable extent. See Porter’s Travels in Georgia
and Persia.
[392] The Zoroastrians must still have been numerous,
especially in the outlying provinces, even in the Abbasside reigns.
The social and political inducements brought to bear on them,
and to induce a profession of Islam that was at first but
superficial, are well brought out in ‘The Apology of Al Kindy’
(Smith and Elder, 1882). See especially the speech of Al Mâmûn,
pp. xii. and 33.
 [393] It is difficult to account for the origin of so strange a tale.
It illustrates the heterogeneous materials of which our authorities
are still composed.
[394] Ascalon is stated to have fallen as late as a.h. XXIII., i.e.
a.d. 643. If so, it must have held out so long only in virtue of its
maritime position. But we have no details.
[395] Omar presided every year, excepting the first of his
Caliphate, when the struggle with the Byzantine and Persian
empires was at its height. He is also said to have thrice visited
Mecca for the Omra, or Lesser Pilgrimage. (Life of Mahomet, p.
xii.)
[396] The superstition attributing the cessation of the volcano
to an extraordinary dole of alms is not worse than that which
seeks to check the devastations of Vesuvius by the liquefaction of
the blood of St. Januarius in the cathedral of Naples.
[397] Omar consulted Amru on the subject, who was of the
same mind, and said—

Dûd ála ûd
Fa in yaksar al ’ûd
Halak al dûd.

‘An insect floating on a splinter; if the splinter break, the insect

perisheth;’ signifying thereby the risks of the mariner.
[398] Otba came on a pilgrimage to Mecca, and there
besought Omar to allow him to resign his government. Omar
refused, and as Otba died on his way back, the Caliph was much
distressed. He visited his tomb to pray over it, and said that he
would have reproached himself as the cause of his death—‘had it
not been already written in the decrees of the Lord.’
[399] We have met Moghîra in the lifetime of the Prophet. First
at Hodeibia, where the murder was cast in his teeth by his uncle,
and subsequently at the demolition of the great idol of Tâyif, &c.
(Life of Mahomet, pp. 370, 467.) He was red-haired, one-eyed,
obese and repulsive in appearance, but insinuating in manner
and speech. One of his eighty concubines, when his ill looks were
mentioned, said, ‘Yes, he is a sweet conserve but on a beggarly
dish.’
The aged princess whom he demanded in marriage on the fall
of Hîra, was Hind, daughter of Nómân V. Some threescore years
before she had been married to Adi, who, when tutor to her father,
had caught a glimpse of her in the church at Hîra. Adi was
executed for some offence by the Chosroes, and Hind then retired
into a convent near Hîra, called, after her, Dâira Hind. See the
strange story of Moghîra’s coarse conduct towards her as related
by M. Caussin de Perceval, vol. ii. p. 150; and Life of Mahomet
(1st edition), vol. i. pp. clxxix. et seq.
For the law of evidence on the charge of adultery, see Life of
Mahomet, p. 313. The whole story is significant as manifesting
the deterioration of Arab life from the ancient spirit and customs,
which, amongst the Bedouins, admitted of social intercourse
between the sexes without such scandals. The lady’s name was
Omm Jamîl, of the Beni Aámir ibn Sassâá, and is said by Tabari
to have been a widow. ‘This lady used openly to visit Moghîra and
other chief men of Bussora, a custom common amongst some of
the ladies of that time.’ But the old Arab chivalry towards the sex
was rapidly disappearing under the system which raised the
slave-girl giving issue to her lord to the position of Omm Walad, or
freed-wife, and her children to the same legitimacy as the children
of the noble-born. This, coupled with the laxity of divorce and re-
marriage, was speedily lowering the position of the sex, and
rendered the strict use of ‘the Veil’ an absolute necessity for the
decent observances of social life; and gradually, but surely,
bringing about the wretched condition of women, together with the
seclusion of the harem, as we now find it in Moslem lands.
[400] In the action of Autâs following the field of Honein, his
uncle, who commanded, was slain; and Abu Mûsa took up the
banner and routed the enemy. He had more physical than moral
courage, as we shall see at the great Arbitration.
[401] It is not said that he punished the calumniator. What was
the fault of the girl which led to her imprisonment is not clear.
Possibly there was some scandal of undue influence over Abu
Mûsa, to whom some say she was given as a bribe by his
predecessor Moghîra. As regards the gift to the poet, Weil
remarks that for a smaller offence of the same kind, Khâlid was
deposed with ignominy—which is true. This is the same Ziâd of
whom we have heard before, as the putative son of Abu Sofiân,
destined hereafter to assume a prominent position.
[402] Above, p. 166.
[403] See Life of Mahomet, p. 72. He was one of the friendless
converts whose freedom Abu Bekr purchased, and thus saved
him from the persecution of the Coreish.
[404] The manner in which Moghîra got hold of the secret is
characteristic of his artfulness. He perceived Jobeir in close
conference with the Caliph. Now Omar had apprised Jobeir of his
intention to appoint him Governor of Kûfa; but bade him, for the
present, to keep the matter secret. Moghîra, suspecting the truth,
sent one of his wives with a present of viands to Jobeir’s wife,
who, caught in the trap, accepted the congratulatory gift. Moghîra,
thus assured that his suspicions were well founded, hurried off to
Omar, and representing that he had got hold of a weak fellow,
who could not even keep the secret of his nomination for a day,
got the appointment (as in the text) for himself. Some say that
Omar afterwards intended to reappoint Sád (who seems to have
been removed on very inadequate grounds) to Kûfa, but that he
died before he could give effect to the intention.
[405] Sura xxviii. 4.
[406] See Life of Mahomet, p. 64. His height only equalled that
of an ordinary man seated.
[407] An extraordinary grant of one hundred dirhems was
made to each. The civil list and pensions were settled by Omar in
his Dewân; but the means of paying the allowances was by local
assignments; so that each city was dependent on its endowment,
from which all the expenditure of administration had to be met.
[408] According to some authorities, however, neither Abu
Bekr nor Omar appointed any Câdhy to Kûfa or Bussorah.
[409] The calculation was already by strictly lunar notation of
months, according to the Arab calendar; for that had been fixed
by a Divine ordinance at the Farewell Pilgrimage. (Life of
Mahomet, p. 486.) But the commencement of the era, and
numbering of the years, was introduced only now. Note that the i
is short in Hegira.
[410] See Life of Mahomet, p. 349.
[411] Take, for example, two lines with the play on the name
Leila, or night—

I thought of Leila, but the heavens are between us;

Neither is her night (Leila) mine; nor my night hers.

[412] Many stories are told of Omar’s stern punishment of

wine-drinkers. The house of one who surreptitiously trafficked in
spirits, he caused to be burned over his head. Another culprit,
expelled for drinking, escaped to the Byzantine territory and
apostatised.
[413] See The Corân: its Teaching and Precepts, p. 61.
[414] For a description of the shameless demoralisation that
prevailed, especially among the youth of Damascus and
Baghdad, I must refer to the learned and elaborate work of H. von
Kremer, Culturgeschichte des Orients unter dem Chalifen.
[415] One of the wives was a captive maiden from Yemen,
who, having, as his bond-maid, borne him a son and daughter,
became, ipso facto, free. No mention is made of other slave-girls
in his harem; but this affords no presumption that he did not
consort with such; for no account is made of servile concubines,
and they are rarely or never mentioned, unless they chanced to
bear offspring.
It was his daughter from whom the tradition is derived that he
had no special weakness for the sex, and married chiefly for the
sake of issue.
[416] In the tradition both the maidens are spoken of as Omm
Kolthûm; but that must have been by anticipation, since they were
so called as having sons of that name.
[417] On one occasion Hind repaired to Syria and warned
Muâvia against giving money to his father, Abu Sofiân, who was
in need, lest he should incur the reproach of Omar and the
people; and Muâvia accordingly sent him away with only one
hundred dinars. But tradition, through Abbasside channels,
begins now to take so strong and bitter a tinge of hatred against
the Omeyyad family, that tales regarding it must be received with
caution.
[418] By some authorities he was now sixty-three; but this was
a favourite age with traditionists, being that at which the Prophet
died (supra, p. 119). He was born before the ‘Sacrilegious War,’
which lasted ten years, a.d. 580–590 (Life of Mahomet, p. 14); but
his birth was probably at the end of the last great battle, which
terminated that war. This would make him twenty-six at his
conversion, and fifty-five at his death. If born at the
commencement of the war, he would now be ten years older. The
true date may lie between the two extremes; and it is not unlikely
that he was near sixty years of age at his death.
 [419] Moghîra, when recently appointed to Kûfa, may have left
him at Medîna; or, more likely, he may have accompanied his
master from Kûfa to the Hejâz, it being the season of pilgrimage
when the governors presented themselves.
[420] The following story is told even by the earliest
authorities:—Káb (the converted Jewish doctor, of whom mention
has been made already) came to the Caliph and said, ‘Omar, thou
hast but three days to live.’ ‘Strange,’ said Omar, ‘for I feel quite
well and strong.’ ‘Nevertheless,’ continued Káb, ‘thus and thus I
find it foretold in the Towrât.’ Next day he came again, and told
Omar he had but two days left. After he was struck down, Káb
came to visit him, and Omar said, on this occasion, to those about
him,—‘Káb spake the truth,’ adding this couplet—

Káb warned me that in three days I should die; in the

prophecy of Káb there is no doubt;
I fear not death; and verily I am dying; but the fear of the
wolf followeth in its wake.

For wolf (zeib) some read sin (zanb). It is difficult to say what
can have given rise to this strange tradition. Possibly Káb, seeing
the sullen and threatening attitude of Abu Lulû, may have warned
him accordingly.
[421] It is possible that Abd al Rahmân’s subsequent
renunciation of the Caliphate in the coming conclave may have
led to the tradition of this supposed conversation with Omar; but I
give the tradition as I find it; and the facts as stated in the text are
not in themselves improbable.
[422] The selection of Soheib was, no doubt, made advisedly.
It will be remembered that Mahomet is thought to have, in a
manner, pointed out Abu Bekr as his successor by nominating
him, when he was himself laid aside, to preside at the public
prayers. Soheib had, of course, no pretensions to the office. He
had been a slave at Mecca, but was much revered because of his
early conversion (Life of Mahomet, p. 72). So his appointment on
this occasion was very suitable.
[423] A stalwart warrior. Mahomet used to say that in the field,
the voice of Abu Talha was better than a thousand men. At
Honein he slew twenty of the enemy with his own hand.
[424] Some traditions omit the words ‘Jews and Christians,’
giving thus to the sentence a general bearing; but the mention of
covenant or treaties would seem to imply that tribes or people
were meant other than Mahometans; and the best supported
traditions are as in the text.
[425] Backî al Gharcad
[426] There is the tradition of a long conversation between Ibn
Abbâs and Omar, in which the former pressed the right of his
family to the Caliphate; and Omar answered, attributing the claim
to envy. The whole is a mere Abbasside invention; for neither Aly
nor Abbâs, nor any one of the house of Hâshim, seems even to
have dreamed of any such pretension till after the dissensions
which broke out after Omar’s death. Fâtima was the only
discontented person, and that, as we have seen, was about the
property left by the Prophet withheld from her by Abu Bekr, not
about any claim to the Caliphate.
[427] As in the Oriental style, the bed, or matting, was spread
upon the ground, Abdallah had but to raise his father’s head and
remove it outside the pillow; so placing it on the ground, and
afterwards raising it upon his lap.
[428] Some traditions give the date of his death three days
later, i.e. on the last day of Dzul Hijj. This, no doubt, arises from
that having been the date on which the new Caliph was chosen,
and Omar’s reign is conventionally spoken of as also lasting up to
that day—the last day of the year a.h. 23. There is another
tradition that he was wounded on Wednesday, 23rd of Dzul Hijj,
and buried on the Sunday following, i.e. on the 27th.
[429] Bilâl used to say that the only way to soothe Omar, when
in a rage, was to recite in his hearing passages from the Corân,
which invariably assuaged his wrath. This may, perhaps, have
reference to the period of his conversion, when having struck his
sister, and made blood to flow, he was moved to repentance by
the reading of a Sura. (See Life of Mahomet, p. 96.)
[430] Such were Abd al Rahmân, Zobeir, Othmân, Aly, and
Talha. The tradition as given by the Secretary of Wackîdy (fol.
235) may also mean that he was unwilling to sully their name by
subjecting them to the sordid surroundings and associations of
provincial government.
[431] Thus, for example, while journeying in Arabia in the year
of famine, he came upon a poor woman, seated, with her hungry
and weeping children, round a fire, whereon was an empty pot.
Omar ran on to the next village, procured bread and meat, filled
the pot, and cooked an ample meal; leaving the little ones
laughing and at play.
[432] When some one proposed his son Abdallah, Omar was
angry and declared that the government had been long enough in
his family. ‘Besides’ (alluding apparently to some scandal in his
domestic life) ‘how could I appoint a man who was so weak as not
to divorce his wife?’ They say, also, that Omar once praised
Sâlim, the freedman of Hodzeifa, slain at Yemâma, as one who
would have been fit for the Caliphate—‘a man beloved of the
Prophet, and a lover of the Lord.’ But this could only have been
as a mere figure of speech.
[433] Others say that the conclave was held in the house of
Miswar, a citizen of Medîna; and that there Abd al Rahmân spent
the last decisive night in separate conference with Aly and then
with Othmân. For Micdâd, see Life of Mahomet, p. 239. Moghîra
and Amru are characteristically said to have sat at the door of the
house to make it appear as if they, too, had had a hand in the
election. Amru had probably come to Medîna with the other
governors on pilgrimage.
[434] For the two rival families see Life of Mahomet, pp. xx.
and xxviii. The Electors were, in reality, selected very evenly.
Zobeir was cousin to Aly both on the father’s side and the
mother’s. Sád and Abd al Rahmân belonged to the Beni Zohra, a
distant branch of Coreishite descent. Sád, however, was likewise
the nephew of Mahomet’s mother, Amina. Some say that he voted
for Othmân; others that, being pressed by Aly, he went over to his
side. Talha was of the Beni Taym, the clan of Abu Bekr. The
impartiality of Abd al Rahmân is impugned by the partisans of Aly,
as being the brother-in-law of Othmân, whose uterine sister he
married; and this probably was the relationship hinted at by Aly in
his appeal to Abd al Rahmân.
We are getting now into the full flood of Abbasside tradition,
which becomes entirely partisan and untrustworthy, with the view
of exalting the claims of the Prophet’s family and defaming the
Omeyyads. Of this class of traditions is the following:—Aly
complained to Abbâs that he was sure to be outvoted in the
conclave because Sád would go with his kinsman Abd al
Rahmân, and vote for Othmân, brother-in-law of the latter; and
that then, the votes being equally divided, Abd al Rahmân would
have the casting-vote. On this Abbâs reproached Aly for having
neglected the advice, given by him now and on former occasions,
to claim the Caliphate as his right, and to have nothing to do with
electors or arbitration. He had told him years before to demand
the Caliphate from Mahomet, and he had neglected to do so. ‘And
now,’ said Abbâs, ‘the Caliphate will leave our family for ever.’ All
this is patent fabrication.
[435] The Beni Makhzûm was a powerful branch of the
Coreish, but far removed by descent from the clan of Hâshim, and
having little sympathy with it. It was Khâlid’s tribe. To understand
the taunts here bandied, it must be remembered that Abu Sarh
(his proper name is Abdallah Ibn Abu Sarh) was the foster-brother
of Othmân, and bore a bad repute (as we shall see below) as
having deceived Mahomet, and been proscribed at the capture of
Mecca. Ammâr (as has been stated before) was son of a bond-
woman called Sommeyya. See on the tradition of her martyrdom,
Life of Mahomet (1st ed.), vol. ii. p. 126.
[436] The inaugural address was delivered on the 3rd
Moharram or Nov. 10, the interval between the election and
speech at installation being presumably taken up in receiving the
oath of allegiance from all present at Medîna.
[437] Quoted from the Corân, Sura xii. v. 19.
[438] His attitude in discharging the invidious task was that of
a loyal and unselfish patriot. He disclaimed the Caliphate for
himself. Night and day engaged unceasingly in canvassing the
sentiments of the leading chiefs, he did his best to compose the
antagonistic claims of the selfish Electors. What was the
immediate cause of his action when in the Mosque he nominated
Othmân, it is not possible to say. Abbasside traditions assume
that the cause was the conscientious scruples of Aly in hesitating
to swear that he would follow strictly the precedents of Abu Bekr
and of Omar in his conduct of the Caliphate. The Corân and the
precedent of Mahomet he would implicitly obey, but the precedent
of the first Caliphs only so far as he agreed in the same. In the
tenor of the traditions relating how Abd al Rahmân first
questioned Aly and then Othmân, and in their replies, I hardly find
sufficient ground for this assumption; and it looks very much of a
piece with the Abbasside fabrications of the day. One tradition
ascribes the hesitancy of Aly to the cunning counsel of Amru,
who, beforehand, advised him not to give a direct reply, lest Abd
al Rahmân should think him too grasping; while he advised
Othmân to answer unconditionally—as if Aly were so simple as to
have been caught by such transparent guile.
 [439] Aly, however, maintained his view, and sought, when he
became Caliph, to give practical effect to it. He searched for
Obeidallah, and would, we are told, have put him to death. But
Obeidallah made his escape to Syria, where he was safe under
the rule of Muâvia.
[440] From this point begin the rough waters of the great
cataclysm. Tradition becomes deeply affected by faction,
especially the envenomed shafts of the party of Aly and the
Abbassides, under cover of which they built up their pretensions,
and, in the end, succeeded in supplanting the Omeyyad dynasty.
The evidence, therefore, must be received with caution as we go
along.
[441] Kabul is said to have been first attacked a.h. 24. The
early Moslems seem to have been as unfortunate (perhaps as
unwise) as ourselves in their expeditions against Afghanistan,
where they met with many sad reverses.
[442] Ascalon is said to have been reduced (apparently for the
first time) just before Omar’s death, A.H. 23; but the delay was
purely owing to its maritime position. This excepted, Syria had for
some years been under the firm yoke of Islam.
[443] For his full name (Abdallah ibn Sád ibn Abu Sarh), see
note at p. 290; but it may conveniently be abbreviated into Abu
Sarh.
[444] Party spirit has, no doubt, been freely used to magnify
the offence of Abu Sarh. He is supposed to be the person alluded
to in Sura vi. 94:—‘Who is more wicked than he who saith, I will
produce a Revelation, like unto that which the Lord hath sent
down?’ Vide Sale’s note in loco. The circumstances as quoted
there are altogether apocryphal. He must, however, have
deceived, if not betrayed, Mahomet, in some very marked way, to
have led to his proscription on the capture of Mecca—an
occasion on which the Prophet treated the inhabitants, with but
few exceptions, with mercy and even generosity. See Life, p. 425.
We have seen above (p. 248) that Omar is said by some to have
been dissatisfied with Amru’s administration in Egypt—so much
so, as to have superseded him partially by appointing Abu Sarh to
the command in Upper Egypt. The evidence of Omar’s
disapproval of Amru is imperfect, but there is no doubt that he
appointed Abu Sarh to Egypt, and that Othmân on his accession
found him already in power there.