Statistics in criminal justice: Third edition

Statistics in Criminal Justice: Third Edition

Volumn , Issue , 2007, Pages 1-674

  Weisburd, David ^a,b   Britt, Chester ^c  

^a HEBREW UNIVERSITY OF JERUSALEM   (Israel)

^b UNIVERSITY OF MARYLAND   (United States)

^c NORTHEASTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

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EID: 84892280185     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-0-387-34113-2     Document Type: Book

References (131)

1. A. J. Reiss, Jr., "Systematic Social Observation of Social Phenomenon," in Herbert Costner (ed.), Sociological Methodology (San Francisco: Jossey Bass, 1971), pp. 3-33.
2. See D. Black, "The Boundaries of Legal Sociology," in D. Black and M. Mileski (eds.), The Social Organization of Law (New York: Seminar Press, 1973), pp. 41-47.
3. See R. Sparks, H. Genn, and D. Dodd, Surveying Victims: A Study of the Measurement of Criminal Victimization (New York: Wiley, 1977).
4. The College Board, 2001 College-Bound Seniors: A Profile of SAT Program Test Takers, accessed at http://www.collegeboard.com/sat/cbsenior/yr2001/pdf/ NATL.pdf.
5. Most statistical software and spreadsheet packages allow for manipulation of many characteristics of a bar chart, including color, shading, patterning, and dimensions (two vs. three). While this allows for the construction of unique charts, the investigator should be wary of adding so much detail to a chart that the reader loses the point the investigator is trying to make.
6. For a good discussion of the benefits and drawbacks of various approaches, see Gary T. Henry, Graphing Data: Techniques for Display and Analysis (Thousand Oaks, CA: Sage, 1995).
7. Capital Punishment in the United States, ICPSR Study #6956, available through the National Archive of Criminal Justice Data (NACJD) at http://www.icpsr.umich.edu/NACJD.
8. The entire GSS database is publicly available and can be accessed at http://www.icpsr.umich.edu/GSS. Data presented here are drawn from a 1991 study.
9. Federal Bureau of Investigation, Crime in the United States, available at http://www.fbi.gov/ucr/ucr.htm.
10. With this method, it is possible that the defined median value will fall between two categories of an ordinally measured variable. In that case, you simply note that the median falls between these two categories.
11. Sometimes the median for ordinal-level variables is also calculated using this method. In such cases, the researcher should realize that he or she is treating the variable under consideration as an interval-level measure. Only for an interval-level measure can we assume that the units of measurement are constant across observations.
12. L. Sherman, J. D. Schmidt, D. Rogan, P. Gartin, E. G. Cohn, D. J. Collins, and A. R. Bacich, "From Initial Deterrence to Long-Term Escalation: Short-Custody Arrest for Poverty Ghetto Domestic Violence," Criminology 29 (1991): 821-850.
13. 3-In words, this equation tells us to take the deviation between a value and the mean and cube it, then sum these values over all observations; the sum of the cubed deviations is then divided by the sample size (N) multiplied by the standard deviation cubed. The measure of skewness will have a value of 0 if the distribution is symmetrical, a negative value if the distribution is negatively skewed, and a positive value if the distribution is positively skewed. The greater the value of the measure, the greater the degree of positive or negative skewness.
14. J. Horney and I. H. Marshall, "An Experimental Comparison of Two Self-Report Methods for Measuring Lambda," Journal of Research in Crime and Delinquency 29 (1992): 102-121.
15. If you are working with SPSS or another computer package, you will notice that the result you get computing the variance by hand using this formula and the result provided by the computer package are slightly different. For example, SPSS computes a variance of 167.18 for the distribution provided in Table 5.6. The difference develops from the computer's use of a correction for the bias of sample variances: 1 is subtracted from the N in the denominator of Equation 5.4. The correction is used primarily as a tool in inferential statistics and is discussed in Chapter 10. Though it is our view that the uncorrected variance should be used in describing sample statistics, many researchers report variances with the correction factor for sample estimates. When samples are larger, the estimates obtained with and without the correction are very similar, and thus it generally makes very little substantive difference which approach is used.
16. As discussed in footnote 1, SPSS and many other computer packages would provide a slightly different result, based on the use of a correction of -1 in the denominator.
17. See Albert J. Reiss, Occupations and Social Status (New York: Free Press, 1961).
18. M. Maltz, "Deviating from the Mean: The Declining Significance of Significance," Journal of Research in Crime and Delinquency 31 (1994): 434-463.
19. Some statisticians prefer to call the research hypothesis the "alternative" hypothesis, because we can, in theory, choose any value as the null hypothesis, and not just the value of zero or no difference. The alternative hypothesis, in this case, can be defined as all other possible outcomes or values. For example, you could state in your null hypothesis that the professor's grades are, on average, five points higher than those of other professors in the college. The alternative hypothesis would be that the professor's grades are not, on average, five points higher than those of other professors.
20. See Lawrence Sherman and David Weisburd, "General Deterrent Effects of Police Patrol in Crime 'Hot Spots': A Randomized Study," Justice Quarterly 12:4 (1995): 625-648.
21. Because of rounding error, the total for our example is actually slightly larger than 1 (see Table 7.5).
22. See Anthony Braga, "Solving Violent Crime Problems: An Evaluation of the Jersey City Police Department's Pilot Program to Control Violent Crime Places," unpublished dissertation, Rutgers University, Newark, NJ, 1996.
23. Problem-oriented policing is an important new approach to police work formulated by Herman Goldstein of the University of Wisconsin Law School. See H. Goldstein, Problem-Oriented Policing (New York: McGraw-Hill, 1990).
24. The correction factor adjusts your test to account for the fact that you have not allowed individuals to be selected from the population more than once. Not including a correction factor makes it more difficult to reject the null hypothesis. That is, the inclusion of a correction factor will make it easier for you to reject the null hypothesis. One problem criminal justice scholars face in using a correction factor is that they often want to infer to populations that are beyond their sampling frame. For example, a study of police patrol at hot spots in a particular city may sample 50 of 200 hot spots in the city during a certain month. However, researchers may be interested in making inferences to hot spots generally in the city (not just those that exist in a particular month) or even to hot spots in other places. For those inferences, it would be misleading to adjust the test statistic based on the small size of the sampling frame. For a discussion of how to correct for sampling without replacement, see Paul S. Levy and Stanley Lemeshow, Sampling of Populations: Methods and Applications (New York: Wiley, 1991).
25. See Chester L. Britt, "Health Consequences of Criminal Victimization," International Review of Victimology 8 (2001): 63-73 for a description of the study.
26. There are certain specific situations in which the chi-square test does not require sampling with replacement; see B. S. Everitt, The Analysis of Contingency Tables (London: Chapman and Hall, 1997).
27. In this case, a stratified random sample was selected in order to ensure a broad sampling of white-collar offenders. For our example here, we treat the sample as a simple random sample. See David Weisburd, Elin Waring, and Ellen Chayet, "Specific Deterrence in a Sample of Offenders Convicted of White Collar Crimes," Criminology 33 (1995): 587-607.
28. What if we had defined a directional research hypothesis? In this case, we look to the column of the table for twice the value of the desired significance level, since we now have placed all risk of falsely rejecting the null hypothesis in only one direction. For example, for a 0.05 significance level, we turn to the test statistic for a 0.10 level.
29. e
30. See Douglas McDonald and David Weisburd, "Segregation and Hidden Discrimination in Prisons: Reflections on a Small Study of Cell Assignments," in C. Hartchen (ed.), Correctional Theory and Practice (Chicago: Nelson Hall, 1991).
31. David F. Greenberg, "The Weak Strength of Social Control Theory," Crime and Delinquency 45:1 (1999): 66-81.
32. Another alternative solution is to use another group of non-parametric tests, defined as 'e xact' tests, to estimate the observed significance level (see Alan Agresti, Categorical Data Analysis, New York, John Wiley, 1990). Such tests (e.g., Fisher's Exact Test) which develop a sampling distribution for each problem examined, have been made more practical with the advent of powerful Computers. SPSS provides exact tests for two by two tables as Computational options for cross tabulations. A few statistical programs (e.g., SAS) have begun to provide exact test options for larger tables.
33. F. H. Allport, "The J-Curve Hypothesis of Conforming Behavior," Journal of Social Psychology 5 (1934): 141-183.
34. Our hypothesized results mirror those found in prior studies; see R. J. Hernstein, "Some Criminogenic Traits of Offenders," in J. Q. Wilson (ed.), Crime and Public Policy (San Francisco: Institute for Contemporary Studies, 1983). Whether these differences mean that offenders are, on average, less intelligent than nonoffenders is an issue of some controversy in criminology, in part because of the relationship of IQ to other factors, such as education and social status.
35. By implication, we are asking whether it is reasonable to believe that our sample of prisoners was drawn from the general population. For this reason, the z-test can also be used to test for random sampling. If you have reason to doubt the sampling methods of a study, you can conduct this test, comparing the observed characteristics of your sample with the known parameters of the population from which your sample was drawn.
36. In principle, any distribution may be arranged in such a way that it conforms to a normal shape. This can be done simply by ranking scores and then placing the appropriate number within standard deviation units appropriate for constructing a standard normal distribution.
37. It would not make sense, however, to use a normal distribution test for nominal-scale measures with more than two categories. The normal distribution assumes scores above and below a mean. The sampling distribution of a proportion follows this pattern because it includes only two potential outcomes, which then are associated with each tail of the distribution. In a multicategory nominal-scale measure, we have more than two outcomes and thus cannot fit each outcome to a tail of the normal curve. Because the order of these outcomes is not defined, we also cannot place them on a continuum within the normal distribution. This latter possibility would suggest that the normal distribution could be applied to ordinal-level measures. However, because we do not assume a constant unit of measurement between ordinal categories, the normal distribution is often considered inappropriate for hypothesis testing with ordinal scales. In the case of a proportion, there is a constant unit of measurement between scores simply because there are only two possible outcomes (e.g., success and failure).
38. As noted on page 97 (footnote 1), computerized statistical analysis packages, such as SPSS, use this corrected estimate in calculating the variance and standard deviation for sample estimates.
39. Our statistical problem is that we assume that μ and σ are independent in developing the t distribution. When a distribution is normal, this is indeed the case. However, for other types of distributions, we cannot make this assumption, and when N is small, a violation of this assumption is likely to lead to misleading approximations of the observed significance level of a test.
40. Michael Pendleton, Ezra Stotland, Philip Spiers, and Edward Kirsch, "Stress and Strain among Police, Firefighters, and Government Workers: A Comparative Analysis," Criminal Justice and Behavior 16 (1989): 196-210.
41. The logic here follows simple common sense. If you select each case independently and randomly from a population, on each selection you have an equal probability of choosing any individual, whether male or female, college-educated or not, and so on. From the perspective of a particular group-for example, males-each time you choose a man, the method can be seen as independent and random. That is, the likelihood of choosing any male from the sample is the same each time you draw a case. Of course, sometimes you will draw a female. However, within the population of males, each male has an equal chance of selection on each draw. And if the sampling method is independent, then each male has an equal chance of being selected every time a case is selected.
42. See H. M. Blalock, Social Statistics (New York: McGraw-Hill, 1979), p. 231.
43. A test of statistical significance may be performed to assess differences in variances. It is based on the F distribution, which is discussed in detail in Chapter 12. The test takes a ratio of the two variances being examined: F=σ̂2 larger variance σ̂2 smaller variance
44. These data are available through the National Archive of Criminal Justice Data and can be accessed at http://www.icpsr.umich.edu/NACJD.
45. In practice, many statistics texts use the z-test for examples involving proportions. Generally this is done because a difference of proportions test is appropriate only for larger samples, and with larger samples, there is substantively little difference between the outcomes of these two normal distribution tests. We illustrate a difference of proportions problem using a t-test because it follows the logic outlined in Chapter 10. That is, in the case where σ is unknown, a t-test should be used. Moreover, most packaged statistical programs provide outcomes only in terms of t-tests.
46. See Chester Britt III, Michael Gottfredson, and John S. Goldkamp, "Drug Testing and Pretrial Misconduct: An Experiment on the Specific Deterrent Effects of Drug Monitoring Defendants on Pretrial Release," Journal of Research in Crime and Delinquency 29 (1992): 62-78.
47. In fact, although we do not examine their findings here, Britt and colleagues conducted their study in two Arizona counties.
48. Here we examine the t-test for dependent samples only in reference to mean differences for interval-level data. However, this test may also be used for dichotomous nominal-level data. Suppose you were assessing the absence or presence of some characteristic or behavior at two points in time. If each observation were coded as 0 or 1, then you would calculate the mean difference(Xd) and the standard deviation of the difference (sd) using the same equations as in this section. The only difference from the example discussed in the text is that you would work only with zeros and ones.
49. See Chapter 12 for an example of a rank-order test (the Kruskal-Wallis one-way analysis of variance).
50. The data are drawn from S. Wheeler, D. Weisburd, and N. Bode, Sanctioning of White Collar Crime, 1976-1978: Federal District Courts (Ann Arbor, MI: Inter-University Consortium for Political and Social Research, 1988).
51. For example, see G. Hornsnell, "The Effect of Unequal Group Variances on the F Test for Homogeneity of Group Means," Biometrika 40 (1954): 128-136
52. G. E. P. Box, "Some Theorems on Quadratic Forms Applied in the Study of Analysis of Variance Problems. I. Effect of Inequality of Variance in the One Way Classification," Annals of Mathematical Statistics 25 (1954): 290-302.
53. Most packaged statistical programs provide a test for equivalence of variances as an option with their ANOVA program. However, be careful not to automatically reject use of analysis of variance on the basis of a statistically significant result. In smaller studies, with samples of less than 50 per group, a finding of a statistically significant difference should make you cautious about using analysis of variance. In such a study, you may want to adjust the significance level, as suggested here, or consider alternative nonparametric tests (discussed later in the chapter). With larger samples, a statistically significant result at conventional significance levels should not necessarily lead to any adjustments in your test. For such adjustments to be made, the difference should be highly significant and reflect large actual differences among variance estimates.
54. However, in the special case of analysis of variance with only two samples, the researcher can use a directional research hypothesis. This will sometimes be done in experimental studies when the researcher seeks to examine differences across experimental and control groups, taking into account additional factors [e.g., see L. W. Sherman and D. Weisburd, "General Deterrent Effects of Police Patrol in Crime 'Hot Spots.' A Randomized Study," Justice Quarterly 12 (1995): 625-648.]
55. Indeed, note that the values of F with 1 degree of freedom for the between sum of squares are simply the values of t squared.
56. The data are taken from State Court Processing Statistics: 1990, 1992, 1994, 1996 and can be accessed through the National Archive of Criminal Justice Data web site at http://www.icpsr.umich.edu/NACJD.
57. See Jacob Cohen, Statistical Power Analysis for the Behavorial Sciences (Hillsdale, NJ: Lawrence Erlbaum, 1988), pp. 285-287.
58. Once again, there is no universally accepted definition of what is "small." There will be little question regarding the validity of your estimate of eta if your samples meet the 30 cases minimum defined for invoking the central limit theorem. Some statisticians suggest that you will gain relatively reliable estimates even for samples as small as 10 cases.
59. For a discussion of pairwise comparison tests, see A. J. Klockars and G. Sax, Multiple Comparisons (Quantitative Applications in the Social Science, Vol. 61) (London: Sage, 1986).
60. Most pairwise comparison tests, including Tukey's HSD test, require that the sample sizes of the groups examined be equal. While most statistical software packages provide adjustments of these tests to account for unequal sample sizes, there is still debate over whether the estimates gained can be relied upon [e.g., see Robert R. J. Sokal and F. J. Rohlf, Biometry: The Principles and Practice of Statistics in Biological Research, 3rd ed. (New York: W. H. Freeman, 1995), Chap. 9]. Irrespective of this debate, when unequal sample sizes are examined, the adjusted estimates are to be preferred over the unadjusted estimates.
61. Most statistical computing packages provide an alternative calculation that adjusts for ties. In practice, the differences between using this correction procedure and performing the unadjusted test are generally small. For our example, where there are a large number of ties relative to the sample size (14/30), the difference in the observed significance level is only 0.0001.
62. See Jacob Cohen, Statistical Power Analysis for the Behavioral Sciences (Hillsdale, NJ: Lawrence Erlbaum, 1988), pp. 215-271.
63. For a description of the study, see Chester L. Britt, "Health Consequences of Criminal Victimization," International Review of Victimology, 8 (2001): 63-73.
64. For all calculations of prediction errors, we have rounded the result to the nearest integer.
65. All the measures of association for ordinal variables that we discuss here are for grouped data that can be represented in the form of a table. In Chapter 14, we discuss another measure of association for ordinal variables-Spearman's r (rs)-that is most useful in working with ungrouped data, such as information on individuals. The difficulty we confront when using Spearman's r on grouped data is that the large number of tied pairs of observations complicates the calculation of this measure of association. Spearman's r is a more appropriate measure of association when we have ordinal variables with a large number of ranked categories for individual cases or when we take an interval-level variable and rank order the observations (see Chapter 14).
66. These two tau measures are different from Goodman and Kruskal's tau, which measures the strength of association between two nominal variables.
67. David F. Greenberg, "The Weak Strength of Social Control Theory," Crime and Delinquency 45:1 (1999): 66-81.
68. For a more detailed discussion of these issues, see Jean Dickson Gibbons, Nonparametric Measures of Association (Newbury Park, CA: Sage, 1993).
69. See Jacob Cohen, Statistical Power Analysis for the Behavioral Sciences (Hillsdale, NJ: Lawrence Erlbaum, 1988), pp. 79-80. In Chapter 21, we discuss in greater detail how statisticians develop standardized estimates of "effect size."
70. For a discussion of this issue, see. J. Fox, Linear Statistical Models and Related Methods (New York: Wiley, 1994).
71. It is good practice to examine the sample scatterplot of scores to assess whether this assumption is likely to be violated. We find no reason to suspect a violation of the assumption when we examine this scatterplot (see Chapter 15, Figure 15.2).
72. The table does not list a t-value for df = 56. We therefore interpolate from the values of df = 55 (2.004) and df = 60 (2.000).
73. Jihong Zhao and Quint Thurman, "A National Evaluation of the Effect of COPS Grants on Crime from 1994-1999," unpublished manuscript, University of Nebraska, December, 2001.
74. Note that there is no single accepted convention for representing the Y-intercept. Some researchers use the symbol (alpha), while others prefer to use a.
75. For a discussion of the relationship between age and crime, see David F. Farrington, "Age and Crime," in Crime and Justice: An Annual Review of Research, Vol. 7 (Chicago: University of Chicago Press, 1986), pp. 189-250.
76. It is important to note that this equation for the standard error of b is appropriate only if we have a bivariate regression model. If we have two or more independent variables, then a modified equation is necessary to calculate the standard error of the regression coefficients.
77. Except for rounding error, this result is the same as the one we obtained in testing the significance of the correlation coefficient for this relationship (4.4195 vs. 4.4188). In practice, you could use the correlation coefficient significance test result for defining the statistical significance of the regression coefficient. Indeed, in many texts only one formula is provided for both coefficients
78. discussed in the prior section. We assume an interval scale, a normal distribution (relaxed While we do not state the assumption of the tests below formally, they follow those when N is large), homoscedasticity, linearity, and independent random sampling.
79. We have likely violated the normality assumption of our test because we do not have knowledge about the shape of the joint distribution of age and number of arrests in the population and N = 15 cases is not enough to safely invoke the central limit theorem.
80. For a discussion of experimental methods in criminal justice, see E. Babbie and M. Maxfield, The Practice of Social Research in Criminal Justice (Belmont, CA: Wadsworth, 1995).
81. For a comparison of experimental and nonexperimental methods, see D. Weisburd, C. Lum, and A. Petrosino, "Does Research Design Affect Study Outcomes in Criminal Justice?" The Annals 578 (2001): 50-70.
82. It is important to note that bias can be caused by a nonlinear relationship between the excluded and the included variable. The assumption is that there is no systematic relationship of any form.
83. Mark Moore of Harvard University has argued, for example, that legal and ethical dilemmas make it difficult to base criminal justice policies about crime control on models that still include a substantial degree of statistical error. See M. Moore, "Purblind Justice: Normative Issues in the Use of Prediction in the Criminal Justice System," in A. Blumstein, J. Cohen, A. Roth, and C. A. Visher (eds.), Criminal Careers and "Career Criminals," Vol. 2 (Washington, DC: National Academy Press, 1986).
84. David Weisburd et al., The Abuse of Authority: A National Study of Police Officers' Attitudes (Washington, DC: The Police Foundation, 2001).
85. There may be times when you want to choose a category that does not include the largest number of cases as the reference. For example, if you wanted to compare a series of treatments to a no-treatment, or control, condition, it would make sense to have the control condition as the excluded category, even if it did not include the largest N. However, if the excluded category has a small number of cases, it may lead to instability in the regression estimates.
86. Michael H. Tonry, Sentencing Matters (New York, Oxford University Press, 1996).
87. These data are available through the National Archive of Criminal Justice Data and can be accessed at http://www.icpsr.umich.edu/NACJD
88. R. Agnew, 1992, Foundation for a general strain theory of crime and delinquency, Criminology, 30, 47-87.
89. D. Steffensmeier, J. Kramer, and J. Ulmer, 1995, Age differences in sentencing, Justice Quarterly, 12, 583-602.
90. Donald J Black, The Behavior of Law (New York; Academic Press, 1976).
91. A method called generalized least squares might also be used to deal with violations of our assumptions, though logistic regression analysis is generally the preferred method. See E. A. Hanushek and J. E. Jackson, Statistical Methods for Social Scientists (New York: Academic Press, 1977) for a comparison of these approaches.
92. See also David W. Hosmer and Stanley Lemeshow, Applied Regression Analysis, 2nd ed. (New York: Wiley, 2000). Another method, probit regression analysis, is very similar to that presented here, though it is based on the standard normal distribution rather than the logistic model curve. The estimates gained from probit regression are likely to be very similar to those gained from logistic regression. Because logistic regression analysis has become much more widely used and is available in most statistical software packages, we focus on logistic regression in this chapter.
93. Your calculator likely has a button labeled "e x," which performs this operation. If there is no ex button, then you should be able to locate a button labeled "INV" and another for the natural logarithm, ln. By pushing "INV" and then "ln" (the inverse or antilog), you will be able to perform this operation.
94. It should be noted, however, that maximum likelihood techniques do not always require an iterative process.
95. See John Tukey, Report to the Special Master, p. 5
96. Report to the New Jersey Supreme Court 27 (1997).
97. For a description of this study, see David Weisburd, Stephen Mastrofski, Ann Marie McNally, and Rosann Greenspan, Compstat and Organizational Change (Washington, DC: The Police Foundation, 2001).
98. Departments with 1,300 or more officers were coded in our example as 1,300 officers. This transformation was used in order to take into account the fact that only 5% of the departments surveyed had more than this number of officers and their totals varied very widely relative to the overall distribution. Another solution that could be used to address the problem of outliers is to define the measure as the logarithm of the number of sworn officers, rather than the raw scores. We relied on the former solution for our example because interpretation of the coefficients is more straightforward. In an analysis of this problem, a researcher would ordinarily want to compare different transformations of the dependent variable in order to define the one that best fit the data being examined.
99. See Stanton Wheeler, David Weisburd, and Nancy Bode, "Sentencing the White Collar Offender: Rhetoric and Reality," American Sociological Review 47 (1982): 641-659.
100. Some researchers have proposed alternative ways of calculating standardized logistic regression coefficients that allow for interpretations related to changes in probabilities. See, for example, Robert L. Kaufman, "Comparing Effects in Dichotomous Logistic Regression: A Variety of Standardized Coefficients," Social Science Quarterly 77 (1996): 90-109.
101. For example, see Andy Field, Discovering Statistics Using SPSS for Windows (London: Sage Publications, 2000).
102. As with standardized regression coefficients in OLS regression, you should not compare standardized logistic regression coefficients across models. Moreover, while we report standardized regression coefficients for the dummy variables included in the model, you should use caution in interpreting standardized coefficients for dummy variables. See Chapter 16, pages 474-475, for a discussion of this problem
103. D. R. Cox and E. J. Snell, The Analysis of Binary Data, 2nd ed. (London: Chapman and Hall, 1989).
104. See N. J. D. Nagelkerke, "A Note on a General Definition of the Coefficient of Determination, Biometrika 78 (1991): 691-692.
105. We discuss the Wald statistic in detail here, because it is the most common test of statistical significance reported in many statistical software applications. it should be noted, however, that some researchers have noted that the Wald statistic is sensitive to small sample sizes (e.g., less than 100 cases). The likelihood-ratio test discussed later in this chapter and in more detail in Chapter 19 offers an alternative test for statistical significance that is appropriate to both small and large samples (see J. Scott Long, Regression Models for Categorical and Limited Dependent Variables (Thousand Oaks, CA, Sage, 1997).
106. The difference between our result and that shown in Table 18.12 is due to rounding error.
107. This printout is identical to that in Table 18.5. It is reproduced here for easy reference as you work through the computations presented in this section
108. See, for example, D. Nagin, Group-based Modeling of Development (Cambridge, MA: Harvard University Press, 2005).
109. You can verify this identity by using the fact that the logarithm of a fraction is equal to the logarithm of the numerator minus the logarithm of the denominator: ln(x/y)=ln(P(Y=C1)P(Y=C2)/) =ln(P(Y=C1))-In(P(Y=C2)) and ln(P(Y=C2)P(Y=C3)/)=ln(P(Y=C2))-In(P(Y=C3)) When we put these two pieces together in a single equation, we have [In(P(Y=C1))-ln(P(Y=C2)]+[In(P(Y=C2))- ln(P(Y=C3)) =In(P(Y=C1))-In(P(Y=C2))+In(P(Y=C2))-In(P(Y=C3)) =In(P(Y=C1))-In(P(Y=C3))
110. =In(In(P(Y=C1))/In(P(Y=C3))) Which establishes the equality. We explain the practical implication of this equality below in our discussion of the interpretation of the coefficients from a multinomial logistic regression model.
111. While it may appear odd at first glance that we have not included those individuals who were acquitted at a trial, there were very few individuals who fell into this category. Like most jurisdictions, courts in California acquit relatively few individuals through a trial - it was about 1% of all cases in the 1990s. What this means is that once the prosecutor has filed charges against a defendant, rather than dismiss the case, it will likely result in the conviction of the defendant through either a guilty plea or a trial conviction. This also implies that a dismissal of the case functions much like an acquittal, but one made by the prosecuting attorney rather than a judge or jury.
112. It is worth pointing out that the binary logistic regression model presented in Chapter 18 is a special case of the multinomial logistic regression model, where m = 2. If you work through both Equations 19.2 and 19.3 above assuming that m = 2, you will be able to replicate the equations in the previous chapter.
113. Recall from footnote # 13 in Chapter 18 that the Wald statistic is sensitive to small samples (e.g., less than 100), while the LR test is not.
114. You should verify the statistical significance of each coefficient presented in Table 19.6 using a Wald test statistic with df = 1.
115. Min Sik Lee and Jeffery T. Ulmer, "Fear of Crime Among Korean Americans in Chicago Communities," Criminology 38:4 (2000): 1173-1206.
116. The sampling distribution for r will generally be normal and symmetric only for the case where r = 0, which is what allowed us to use the t distribution to test whether rp = 0 (i.e., the null hypothesis) in Chapter 14. When r ≠ 0, the sampling distribution is not symmetric around r, so we cannot calculate a confidence interval for r in the same way we did for sample means or the difference of sample means.
117. Ronald A. Fisher, Statistical Methods for Research Workers, 14th ed. (New York: Hafner, 1970).
118. Eric Silver, "Mental Disorder and Violent Victimization: The Mediating Role of Involvement in Conflicted Social Relationships," Criminology 40 (2002): 191-212.
119. See S. E. Brown "Statistical Power and Criminal Justice Research," Journal of Criminal Justice 17 (1989): 115-122. However, criminal justice researchers are not very different from researchers in other areas of social science
120. see also D. Weisburd "Design Sensitivity in Criminal Justice Experiments," Crime and Justice 17 (1991): 337-379.
121. Effect size can also be calculated for observed differences in a study. This is a common approach in meta-analysis, where a large group of studies are summarized for a single analysis. For example, in calculating effect size for a randomized experiment with one treatment and one control group, the researcher would substitute the outcome scores for both groups in the numerator of the ES equation, and the pooled standard deviation for the two outcome measures in the denominator. For a more detailed discussion of effect size and its use generally for comparing effects across different studies, see Mark Lipsey and David Wilson, Practical Meta-Analysis, Applied Social Research Methods Series 49 (Thousand Oaks, CA: Sage, 2001).
122. See M. W. Lipsey, Design Sensitivity: Statistical Power for Experimental Research (Newbury Park, CA: Sage, 1990).
123. For example, see J. Petersilia "Randomized Experiments: Lessons from BJA's Intensive Supervision Project," Evaluation Review 13 (1989): 435-458
124. and D. Weisburd, "Design Sensitivity in Criminal Justice Experiments,"Crime and Justice 17 (1991): 337-379.
125. For a review of this issue in criminal justice experiments, see D. Weisburd, "Design Sensitivity in Criminal Justice Experiments," Crime and Justice 17 (1991): 337-379.
126. Increasing the size of a sample may also affect the variability of study estimates in other ways. For example, it may become more difficult to monitor implementation of treatments as a study grows. It is one thing to make sure that 100 people or places receive a certain intervention, but quite another to ensure consistency of interventions across hundreds or thousands of subjects. Also, studies are likely to include more heterogeneous groups of subjects as sample size increases. For example, in one intensive probation study, eligibility requirements were continually relaxed in order to meet project goals regarding the number of participants; see J. Petersilia, "Randomized Experiments: Lessons from BJA's Intensive Supervision Project," Evaluation Review 13 (1989): 435-458. As noted earlier, as the heterogeneity of treatments
127. For example, see Jacob Cohen, Statistical Power Analysis for the Behavioral Sciences (Hillsdale, NJ: Lawrence Erlbaum, 1988)
128. M. W. Lipsey, Design Sensitivity: Statistical Power for Experimental Research (Newbury Park, CA: Sage, 1990)
129. and H. C. Kraemer and S. Thiemann, How Many Subjects: Statistical Power Analysis in Research (Newbury Park, CA: Sage, 1987). There also are software packages for computing statistical power
130. for example, see M. Borenstein, Power and Precision, Release 2.1 (2001).
131. Jacob Cohen, Statistical Power Analysis for the Behavioral Sciences (Hillsdale, NJ: Lawrence Erlbaum, 1988).