Abstract
Chi-squared is a useful and convenient statistic for testing hypotheses about multinomial distributions (see Section 4.2 at p. 103). This is important because a wide range of applied problems can be formulated as hypotheses about “cell frequencies” and their underlying expectations. For example, in Section 4.6.2, Silver “butterfly” straddles, the question arises whether the price change data are distributed normally; this question can be reduced to a multinomial problem by dividing the range of price changes into subintervals and counting how many data points fall into each interval. The cell probabilities are given by the normal probabilities attaching to each of the intervals under the null hypothesis, and these form the basis of our expected cell frequencies. Chi-squared here, like chi-squared for the fourfold table, is the sum of the squares of the differences between observed and expected cell frequencies, each divided by its expected cell frequency. While slightly less powerful than the Kolmogorov-Smirnov test—in part because some information is lost by grouping the data—chi-squared is easier to apply, can be used in cases where the data form natural discrete groups, and is more widely tabled.
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Notes
- 1.
The combined significance of the six studies was very high. See Section 8.1.
- 2.
The possibility of errors in three or more tests also exists, but is considered negligible.
- 3.
These numbers are not reflective of the rate at which bench warrants were issued, but rather of the sampling design of the retrospective study. The τ B measure (like the odds ratio) yields the same result for retrospective and prospective data in 2 × 2 tables, but (unlike the odds ratio) not in larger tables.
- 4.
Several courts suggested that strict adherence to the Parole Commission guidelines would violate a constitutional or statutory requirement of individualized determination. See, e.g., Geraghty v. United States Parole Comm’n, 579 F.2d 238, 259–63 (3rd Cir. 1978), rev’d on other grounds, 445 U.S. 388 (1980); United States v. Cruz, 544 F.2d 1162, 1164 (2d Cir. 1976); United States v. Norcome, 375 F. Supp. 270, 274, n.3 (D.C.), aff’d, 497 F.2d 686 (D.C. Cir. 1974). These courts rejected only exclusive reliance on the guidelines; as one court put it, the guidelines could be used “as a tool but not as a rule.” Page v. United States, 428 F. Supp. 1007, 1009 (S.D. Fla. 1977). But other courts held that there was no inconsistency between individualized sentencing and strict adherence to parole guidelines, because each defendant was individually evaluated to obtain the guideline score. See, e.g., Daniels v. United States Parole Comm’n, 415 F. Supp. 990 (W.D. Okla. 1976). But see United States v. Booker, 543 U.S. 220 (2005) in which the Court held that the federal statute making the Federal Sentencing Guidelines mandatory violated the Sixth Amendment guarantee of trial by jury.
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Finkelstein, M.O., Levin, B. (2015). Comparing Multiple Proportions. In: Statistics for Lawyers. Statistics for Social and Behavioral Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5985-0_6
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