Statistics in Criminal Justice

Contents......Page 8
Preface......Page 14
A c k n o w l e d g m e n t s......Page 17
A b o u t t h e A u t h o r s......Page 18
 Introduction: Statistics as a Research Tool......Page 19
  T h e P u r p o s e o f S t a t i s t i c s I s t o C l a r i f y......Page 21
  S t a t i s t i c s A r e U s e d t o S o l v e P r o b l e m s......Page 22
  B a s i c P r i n c i p l e s A p p l y A c r o s s S t a t i s t i c a l T e c h n i q u e s......Page 23
  T h e U s e s o f S t a t i s t i c s......Page 25
   Inferential Statistics......Page 27
   Taking into Account Competing Explanations: Multivariate Statistics......Page 28
  C h a p t e r S u m m a r y......Page 29
  K e y T e r m s......Page 30
 Measurement: The Basic Building Block of Research......Page 31
 S c i e n c e a n d M e a s u r e m e n t : C l a s s i f i c a t i o n a s a F i r s t S t e p i n R e s e a r c h......Page 32
 L e v e l s o f M e a s u r e m e n t......Page 33
  Nominal Scales......Page 34
  Ordinal Scales......Page 36
  Interval and Ratio Scales......Page 37
 R e l a t i n g I n t e r v a l , O r d i n a l , a n d N o m i n a l S c a l e s : T h e I m p o r t a n c e o f C o l l e c t......Page 40
 W h a t I s a G o o d M e a s u r e ?......Page 41
 C h a p t e r S u m m a r y......Page 43
 K e y T e r m s......Page 44
 E x e r c i s e s......Page 45
 Computer Exercises......Page 48
  SPSS......Page 49
  Stata......Page 51
 Representing and Displaying Data......Page 54
  W h a t A r e F r e q u e n c y D i s t r i b u t i o n s a n d H i s t o g r a m s ?......Page 55
  E x t e n d i n g H i s t o g r a m s t o M u l t i p l e G r o u p s : U s i n g B a r C h a r t s......Page 61
  U s i n g B a r C h a r t s w i t h N o m i n a l o r O r d i n a l D a t a......Page 68
  P i e C h a r t s......Page 69
  T i m e S e r i e s D a t a......Page 70
  C h a p t e r S u m m a r y......Page 73
  S y m b o l s a n d F o r m u l a s......Page 74
  E x e r c i s e s......Page 75
   SPSS......Page 77
   Stata......Page 79
 Describing the Typical Case: Measures of Central Tendency......Page 83
 T h e M o d e : C e n t r a l T e n d e n c y i n N o m i n a l S c a l e s......Page 84
 T h e M e d i a n : T a k i n g i n t o A c c o u n t P o s i t i o n......Page 86
 T h e M e a n : A d d i n g V a l u e t o P o s i t i o n......Page 92
  Comparing Results Gained Using the Mean and Median......Page 95
  Other Characteristics of the Mean......Page 98
 S t a t i s t i c s i n P r a c t i c e : C o m p a r i n g t h e M e d i a n a n d t h e M e a n......Page 100
 C h a p t e r S u m m a r y......Page 103
 S y m b o l s a n d F o r m u l a s......Page 104
 E x e r c i s e s......Page 105
  SPSS......Page 108
  Stata......Page 109
  Recoding Variables......Page 110
 How Typical Is the Typical Case?: Measuring Dispersion......Page 112
 M e a s u r e s o f D i s p e r s i o n f o r N o m i n a l a n d O r d i n a l L e v e l D a t a......Page 113
  The Proportion in the Modal Category......Page 114
  The Percentage in the Modal Category......Page 115
  The Variation Ratio......Page 116
  Index of Qualitative Variation......Page 118
 M e a s u r i n g D i s p e r s i o n i n I n t e r v a l S c a l e s : T h e R a n g e , V a r i a n c e , a n d S t a n d a r......Page 120
  The Variance......Page 122
  The Standard Deviation......Page 125
  The Coefficient of Relative Variation......Page 129
  A Note on the Mean Deviation......Page 131
 C h a p t e r S u m m a r y......Page 133
 K e y T e r m s......Page 134
 S y m b o l s a n d F o r m u l a s......Page 135
 E x e r c i s e s......Page 136
  SPSS......Page 139
  Stata......Page 140
 The Logic of Statistical Inference: Making Statements About Populations from Sample Statistics......Page 143
 T h e D i l e m m a : M a k i n g S t a t e m e n t s A b o u t P o p u l a t i o n s f r o m S a m p l e S t a t i s t i c s......Page 144
 T h e R e s e a r c h H y p o t h e s i s......Page 147
 T h e N u l l H y p o t h e s i s......Page 149
 R i s k s o f E r r o r i n H y p o t h e s i s T e s t i n g......Page 151
 R i s k s o f E r r o r a n d S t a t i s t i c a l L e v e l s o f S i g n i f i c a n c e......Page 153
 D e p a r t i n g f r o m C o n v e n t i o n a l S i g n i f i c a n c e C r i t e r i a......Page 155
 C h a p t e r S u m m a r y......Page 156
 K e y T e r m s......Page 158
 E x e r c i s e s......Page 159
 Defining the Observed Significance Level of a Test: A Simple Example Using the Binomial Distribution......Page 163
  Sampling Distributions and Probability Distributions......Page 165
  The Multiplication Rule......Page 167
 D i f f e r e n t W a y s o f G e t t i n g S i m i l a r R e s u l t s......Page 169
 S o l v i n g M o r e C o m p l e x P r o b l e m s......Page 172
 T h e B i n o m i a l D i s t r i b u t i o n......Page 173
 U s i n g t h e B i n o m i a l D i s t r i b u t i o n t o E s t i m a t e t h e O b s e r v e d S i g n i f i c a n c e L e v......Page 177
 K e y T e r m s......Page 181
 S y m b o l s a n d F o r m u l a s......Page 182
 E x e r c i s e s......Page 183
  SPSS......Page 186
  Stata......Page 187
 Steps in a Statistical Test: Using the Binomial Distribution to Make Decisions About Hypotheses......Page 189
 T h e P r o b l e m : T h e I m p a c t o f P r o b l e m O r i e n t e d P o l i c i n g o n D i s o r d e r l y A c t i v i t......Page 190
  Shape of the Population Distribution......Page 192
  Sampling Method......Page 193
  The Hypotheses......Page 197
 S e l e c t i n g a S a m p l i n g D i s t r i b u t i o n......Page 198
 S i g n i f i c a n c e L e v e l a n d R e j e c t i o n R e g i o n......Page 200
  Choosing a One-Tailed or a Two-Tailed Rejection Region......Page 201
 M a k i n g a D e c i s i o n......Page 205
 C h a p t e r S u m m a r y......Page 206
 K e y T e r m s......Page 207
 E x e r c i s e s......Page 208
  SPSS......Page 212
  Stata......Page 213
 Chi-Square: A Test Commonly Used for Nominal-Level Measures......Page 215
 T e s t i n g H y p o t h e s e s C o n c e r n i n g t h e R o l l o f a D i e......Page 216
  The Chi-Square Distribution......Page 217
  Calculating the Chi-Square Statistic......Page 219
  Linking the Chi-Square Statistic to Probabilities: The Chi-Square Table......Page 220
  A Substantive Example: The Relationship Between Assault Victims and Offenders......Page 221
  The Test Statistic......Page 223
  A Substantive Example: Type of Sanction and Recidivism Among Convicted White-Collar Criminals......Page 224
  The Sampling Distribution......Page 226
  Significance Level and Rejection Region......Page 227
  The Test Statistic......Page 228
 E x t e n d i n g t h e C h i S q u a r e T e s t t o M u l t i c a t e g o r y V a r i a b l e s : T h e E x a m p l e o f C e......Page 230
  The Test Statistic......Page 233
 E x t e n d i n g t h e C h i S q u a r e T e s t t o a R e l a t i o n s h i p B e t w e e n T w o O r d i n a l V a r i a b l......Page 235
  Significance Level and Rejection Region......Page 237
  The Test Statistic......Page 238
  The Decision......Page 239
 C h a p t e r S u m m a r y......Page 240
 S y m b o l s a n d F o r m u l a s......Page 241
 E x e r c i s e s......Page 242
  SPSS......Page 247
  Stata......Page 249
 The Normal Distribution and Its Application to Tests of Statistical Significance......Page 252
 T h e N o r m a l F r e q u e n c y D i s t r i b u t i o n , o r N o r m a l C u r v e......Page 253
  Characteristics of the Normal Frequency Distribution......Page 255
  Scores......Page 257
  Test for Known Populations......Page 259
  The Sampling Distribution......Page 260
  Significance Level and Rejection Region......Page 262
  The Test Statistic......Page 263
 A p p l y i n g N o r m a l S a m p l i n g D i s t r i b u t i o n s t o N o n n o r m a l P o p u l a t i o n s......Page 265
  Computing the Mean and Standard Deviation for the Sampling Distribution of a Proportion......Page 270
  Testing Hypotheses with the Normal Distribution: The Case of a New Prison Program......Page 272
  The Sampling Distribution......Page 273
  Significance Level and Rejection Region......Page 274
 T e s t f o r M e a n s......Page 275
  Distribution......Page 277
  The Test Statistic......Page 278
 C h a p t e r S u m m a r y......Page 279
 K e y T e r m s......Page 280
 S y m b o l s a n d F o r m u l a s......Page 281
 E x e r c i s e s......Page 282
 Comparing Means and Proportions in Two Samples......Page 287
  The Case of Anxiety Among Police Officers and Firefighters......Page 288
  The Sampling Distribution......Page 291
  Significance Level and Rejection Region......Page 295
  The Test Statistic......Page 296
  The Decision......Page 297
  Test for Means......Page 298
  The Test Statistic......Page 299
 T e s t f o r D i f f e r e n c e s o f P r o p o r t i o n s......Page 300
  The Case of Drug Testing and Pretrial Misconduct......Page 302
  The Sampling Distribution......Page 304
  The Test Statistic......Page 305
 T e s t f o r D e p e n d e n t S a m p l e s......Page 306
  The Effect of Police Presence Near High-Crime Addresses......Page 307
  Significance Level and Rejection Region......Page 309
  The Test Statistic......Page 310
 T e s t f o r O r d i n a l S c a l e s......Page 311
 C h a p t e r S u m m a r y......Page 312
 S y m b o l s a n d F o r m u l a s......Page 313
 E x e r c i s e s......Page 314
  SPSS......Page 320
  Stata......Page 321
 Comparing Means Among More Than Two Samples: Analysis of Variance......Page 324
 A n a l y s i s o f V a r i a n c e......Page 325
  Developing Estimates of Variance Between and Within Groups......Page 326
  Partitioning Sums of Squares......Page 330
  Developing Estimates of Population Variances......Page 331
  A Substantive Example: Age and White-Collar Crimes......Page 333
  The Sampling Distribution......Page 336
  Significance Level and Rejection Region......Page 337
  The Test Statistic......Page 338
  Another ANOVA Example: Race and Bail Amounts Among Felony Drug Defendants......Page 342
  The Test Statistic......Page 344
 D e f i n i n g t h e S t r e n g t h o f t h e R e l a t i o n s h i p O b s e r v e d......Page 346
 M a k i n g P a i r w i s e C o m p a r i s o n s B e t w e e n t h e G r o u p s S t u d i e d......Page 349
 A N o n p a r a m e t r i c A l t e r n a t i v e : T h e K r u s k a l W a l l i s T e s t......Page 352
  The Test Statistic......Page 354
 C h a p t e r S u m m a r y......Page 355
 K e y T e r m s......Page 356
 S y m b o l s a n d F o r m u l a s......Page 357
 E x e r c i s e s......Page 359
  SPSS......Page 363
  Stata......Page 365
 Measures of Association for Nominal and Ordinal Variables......Page 369
 D i s t i n g u i s h i n g S t a t i s t i c a l S i g n i f i c a n c e a n d S t r e n g t h o f R e l a t i o n s h i p : T......Page 370
  Measures of Association Based on the Chi-Square Statistic......Page 373
  Proportional Reduction in Error Measures: Tau and Lambda......Page 376
  The Sampling Distribution......Page 383
 M e a s u r e s o f A s s o c i a t i o n f o r O r d i n a l L e v e l V a r i a b l e s......Page 385
  Gamma......Page 391
  and......Page 392
  A Substantive Example: Affectional Identification with Father and Level of Delinquency......Page 394
  Statistical Significance of Measures of Association for Ordinal Variables......Page 399
  The Test Statistic......Page 401
 C h o o s i n g t h e B e s t M e a s u r e o f A s s o c i a t i o n f o r N o m i n a l a n d O r d i n a l L e v e l V a r i......Page 403
 C h a p t e r S u m m a r y......Page 404
 K e y T e r m s......Page 406
 S y m b o l s a n d F o r m u l a s......Page 407
 E x e r c i s e s......Page 409
  SPSS......Page 413
  Stata......Page 414
 Measuring Association for Interval-Level Data: Pearson’s Correlation Coefficient......Page 416
 M e a s u r i n g A s s o c i a t i o n B e t w e e n T w o I n t e r v a l L e v e l V a r i a b l e s......Page 417
 P e a r s o n ’ s C o r r e l a t i o n C o e f f i c i e n t......Page 419
  The Calculation......Page 423
  A Substantive Example: Crime and Unemployment in California......Page 425
  Nonlinear Relationships and Pearson’s......Page 429
  Beware of Outliers......Page 433
 S p e a r m a n ’ s C o r r e l a t i o n C o e f f i c i e n t......Page 437
  : The Case of Age and Number of Arrests......Page 439
  The Decision......Page 444
  Unemployment and Crime in California......Page 445
 T e s t i n g t h e S t a t i s t i c a l S i g n i f i c a n c e o f S p e a r m a n ’ s......Page 446
  The Decision......Page 447
 K e y T e r m s......Page 448
 S y m b o l s a n d F o r m u l a s......Page 449
 E x e r c i s e s......Page 450
  SPSS......Page 453
  Stata......Page 454
 An Introduction to Bivariate Regression......Page 457
 E s t i m a t i n g t h e I n f l u e n c e o f O n e V a r i a b l e o n A n o t h e r : T h e R e g r e s s i o n C o e f f i......Page 458
  Calculating the Regression Coefficient......Page 460
  A Substantive Example: Unemployment and Burglary in California......Page 462
  Intercept......Page 463
  The Regression Line......Page 464
  Predictions Beyond the Distribution Observed in a Sample......Page 466
  Predicting Burglary Rates from Unemployment Rates in California......Page 467
  Choosing the Best Line of Prediction Based on Regression Error......Page 469
  Percent of Variance Explained......Page 471
  Percent of Variance Explained: Unemployment Rates and Burglary Rates in California......Page 474
  Statistical Significance of the Regression Coefficient: The Case of Age and Number of Arrests......Page 476
  The Test Statistic......Page 479
  The Test Statistic......Page 482
  Age and Number of Arrests......Page 485
  Unemployment Rates and Burglary Rates in California......Page 487
 C h a p t e r S u m m a r y......Page 488
 S y m b o l s a n d F o r m u l a s......Page 489
 E x e r c i s e s......Page 491
  SPSS......Page 495
  Stata......Page 496
 Multivariate Regression......Page 499
  Errors in Prediction......Page 500
  Correctly Estimating the Effect of......Page 502
  Comparing Regression Coefficients Within a Single Model: The Standardized Regression Coefficient......Page 510
 C o r r e c t l y S p e c i f y i n g t h e R e g r e s s i o n M o d e l......Page 512
  Taking into Account Ordinaland Nominal-Scale Measures in a Multivariate Regression......Page 513
 C h a p t e r S u m m a r y......Page 522
 K e y T e r m s......Page 523
 S y m b o l s a n d F o r m u l a s......Page 524
 E x e r c i s e s......Page 525
  SPSS......Page 528
  Stata......Page 530
 Multivariate Regression: Additional Topics......Page 532
  N o n l i n e a r R e l a t i o n s h i p s......Page 534
   Finding a Non-linear Relationship: Graphical Assessment......Page 536
   Incorporating Non-linear Relationships into an OLS Model......Page 537
    Interpreting Non-linear Coefficients......Page 538
    Note on Statistical Significance......Page 539
  I n t e r a c t i o n E f f e c t s......Page 540
   Interaction of a Dummy Variable and Interval-Level Variable......Page 541
  A n E x a m p l e : R a c e a n d P u n i s h m e n t S e v e r i t y......Page 543
   Interaction Effects between Two Interval-level Variables......Page 546
  A n E x a m p l e : P u n i s h m e n t S e v e r i t y......Page 551
  T h e P r o b l e m o f M u l t i c o l l i n e a r i t y......Page 552
  C h a p t e r S u m m a r y......Page 555
  E x e r c i s e s......Page 556
  Computer Exercises......Page 559
   SPSS......Page 560
   Stata......Page 561
 Logistic Regression......Page 566
  W h y i s I t I n a p p r o p r i a t e t o U s e O L S R e g r e s s i o n f o r a D i c h o t o m o u s D e p e n d e n t V a r i a b l e ?......Page 568
  L o g i s t i c R e g r e s s i o n......Page 573
   A Substantive Example: Adoption of Compstat in U.S. Police Agencies......Page 581
  I n t e r p r e t i n g L o g i s t i c R e g r e s s i o n C o e f f i c i e n t s......Page 585
   The Odds Ratio......Page 586
   The Derivative at Mean......Page 593
   Using Probability Estimates to Compare Coefficients......Page 595
   “Standardized” Logistic Regression Coefficients......Page 598
   Percent of Correct Predictions......Page 601
   Pseudo R2......Page 603
  S t a t i s t i c a l S i g n i f i c a n c e i n L o g i s t i c R e g r e s s i o n......Page 605
  C h a p t e r S u m m a r y......Page 609
  K e y T e r m s......Page 610
  S y m b o l s a n d F o r m u l a s......Page 611
  E x e r c i s e s......Page 613
   SPSS......Page 616
   Stata......Page 617
 Multivariate Regression with Multiple Category Nominal or Ordinal Measures: Extending the Basic Logistic Regression Model......Page 619
 M u l t i n o m i a l L o g i s t i c R e g r e s s i o n......Page 621
  A Substantive Example: Case Dispositions in California......Page 623
  Odds Ratio Equation......Page 624
  The Missing Set of Coefficients......Page 627
   Single Coefficients......Page 629
   Multiple Coefficients......Page 630
  O r d i n a l L o g i s t i c R e g r e s s i o n......Page 633
   Odds Ratio Using Cumulative Probabilities......Page 636
  S u b s t a n t i v e E x a m p l e : S e v e r i t y o f P u n i s h m e n t D e c i s i o n s......Page 637
   Interpreting the Coefficients......Page 638
   Statistical Significance......Page 639
   Score Test......Page 640
   Brant Test......Page 641
   Partial Proportional Odds......Page 642
   Severity of Punishment Example......Page 643
  C h a p t e r  S u m m a r y......Page 645
  S y m b o l s a n d F o r m u l a s......Page 646
  E x e r c i s e s......Page 647
   SPSS......Page 650
   Stata......Page 651
    Problems......Page 653
 Multilevel Regression Models......Page 655
  V a r i a n c e C o m p o n e n t s M o d e l......Page 658
   A Substantive Example: Bail Decision-Making Study......Page 659
   Fixed and Random Effects......Page 660
   Intraclass Correlation and Explained Variance......Page 662
   Bail Decision-Making Study......Page 663
  R a n d o m I n t e r c e p t M o d e l......Page 664
   Explained Variance......Page 665
   Centering Independent Variables......Page 666
   Bail Decision-Making Study......Page 668
   Between and Within Effects......Page 670
   Testing for Between and Within Effects......Page 671
  R a n d o m C o e f f i c i e n t M o d e l......Page 673
   Note on Explained Variance......Page 674
   Bail Decision-Making Study......Page 675
  A d d i n g C l u s t e r ( L e v e l 2 ) C h a r a c t e r i s t i c s......Page 678
   A Substantive Example: Race and Sentencing Across Pennsylvania Counties......Page 680
  C h a p t e r S u m m a r y......Page 682
  K e y Te r m s......Page 683
  S y m b o l s a n d F o r m u l a s......Page 684
  E x e r c i s e s......Page 685
   SPSS......Page 688
    Random Coefficient Models......Page 689
    Problems......Page 690
 Special Topics: Randomized Experiments......Page 692
  T h e S t r u c t u r e o f a R a n d o m i z e d E x p e r i m e n t......Page 694
  T h e M a i n A d v a n t a g e o f E x p e r i m e n t s : I s o l a t i n g C a u s a l E f f e c t s......Page 695
  n t e r n a l V a l i d i t y......Page 700
  S a m p l e S i z e , E q u i v a l e n c e , a n d S t a t i s t i c a l P o w e r......Page 701
   Statistical Foundations for Block Randomization......Page 703
   An Example: Jersey City Experiment......Page 706
   The Benefits of Block Randomized Trials......Page 707
  S t a t i s t i c a l P o w e r a n d B l o c k R a n d o m i z a t i o n......Page 709
  U s i n g C o v a r i a t e s t o I n c r e a s e S t a t i s t i c a l P o w e r i n E x p e r i m e n t a l S t u d i e s......Page 711
  E x a m i n i n g I n t e r a c t i o n T e r m s i n E x p e r i m e n t a l R e s e a r c h......Page 713
  C h a p t e r S u m m a r y......Page 715
  S y m b o l s a n d F o r m u l a s......Page 716
  E x e r c i s e s......Page 717
 Special Topics: Confidence Intervals......Page 720
  C o n f i d e n c e I n t e r v a l s......Page 722
  C o n s t r u c t i n g C o n f i d e n c e I n t e r v a l s......Page 726
   Confidence Intervals for Sample Means......Page 727
   Confidence Intervals for Sample Proportions......Page 728
   Confidence Intervals for a Difference of Sample Means......Page 729
   Confidence Intervals for Pearson’s Correlation Coefficient, r......Page 731
   Confidence Intervals for Regression Coefficients......Page 733
   Confidence Intervals for Logistic Regression Coefficients and Odds Ratios......Page 734
  C h a p t e r S u m m a r y......Page 735
  K e y T e r m s......Page 736
  E x e r c i s e s......Page 737
   SPSS......Page 740
   Stata......Page 741
    Problems......Page 742
 Special Topics: Statistical Power......Page 744
  S t a t i s t i c a l P o w e r......Page 746
   Setting the Level of Statistical Power......Page 748
   Statistical Significance and Statistical Power......Page 749
    Directional Hypotheses......Page 750
   Sample Size and Statistical Power......Page 751
   Effect Size and Statistical Power......Page 753
   E s t i m a t i n g S t a t i s t i c a l P o w e r a n d S a m p l e S i z e f o r a S t a t i s t i c a l l y P o w e r f u l S t u d y......Page 756
    Difference of Means Test......Page 759
    ANOVA......Page 760
    Correlation......Page 761
    Least-Squares Regression......Page 762
   S u m m i n g U p : A v o i d i n g S t u d i e s D e s i g n e d f o r F a i l u r e......Page 765
   K e y Te r m s......Page 766
   S y m b o l s a n d F o r m u l a s......Page 767
   Computer Exercises......Page 768
     Two-Sample Difference of Means Test......Page 769
     ANOVA......Page 770
    Correlation......Page 773
     OLS Regression......Page 774
     Problems......Page 775
 Factorials......Page 777
 Critical Values of X2 Distribution......Page 778
 Areas of the Standard Normal Distribution......Page 779
 Critical Values of Student’s t Distribution......Page 780
 Critical Values of the F-Statistic......Page 781
 Critical Value for P (Pcrit), Tukey’s HSD Test......Page 784
 Critical Values for Spearman’s Rank-Order Correlation Coefficient......Page 785
 Fisher r-to-Z* Transformation......Page 786
Glossary......Page 788
I n d e x......Page 796