Statistics for Forensic Toxicology

Abstract

Forensic toxicologists routinely deal with the analysis of numerical data. Statistical methods have a wide variety of applications, from initial method development and data interpretation to method validation and measurement uncertainty. In this chapter, basic concepts including data distributions, types of error, and hypothesis testing are explored. Using illustrative examples from forensic toxicology, the use of statistical tests is discussed, including the Student’s t-test and some of its variants, F-tests, and analysis of variance (ANOVA). The treatment of outliers, regression analysis, and tools for the evaluation of calibration models are also discussed.

Appendices

Appendix 1: Table of the critical values for the Student’s t-distribution

	α – upper tail probability p
d.f.	0.10	0.05	0.025	0.01	0.005	0.001	0.0005
1	3.078	6.314	12.076	31.821	63.657	318.310	636.620
2	1.886	2.920	4.303	6.965	9.925	22.326	31.598
3	1.638	2.353	3.182	4.541	5.841	10.213	12.924
4	1.533	2.132	2.776	3.747	4.604	7.173	8.610
5	1.476	2.015	2.571	3.365	4.032	5.893	6.869
6	1.440	1.943	2.447	3.143	3.707	5.208	5.959
7	1.415	1.895	2.365	2.998	3.499	4.785	5.408
8	1.397	1.860	2.306	2.896	3.355	4.501	5.041
9	1.383	1.833	2.262	2.821	3.250	4.297	4.781
10	1.372	1.812	2.228	2.764	3.169	4.144	4.587
11	1.363	1.796	2.201	2.718	3.106	4.025	4.437
12	1.356	1.782	2.179	2.681	3.055	3.930	4.318
13	1.350	1.771	2.160	2.650	3.012	3.852	4.221
14	1.345	1.761	2.145	2.624	2.977	3.787	4.140
15	1.341	1.753	2.131	2.602	2.947	3.733	4.073
16	1.337	1.746	2.120	2.583	2.921	3.686	4.015
17	1.333	1.740	2.110	2.567	2.898	3.646	3.965
18	1.330	1.734	2.101	2.552	2.878	3.610	3.922
19	1.328	1.729	2.093	2.539	2.861	3.579	3.883
20	1.325	1.725	2.086	2.528	2.845	3.552	3.850
21	1.323	1.721	2.080	2.518	2.831	3.527	3.819
22	1.321	1.717	2.074	2.508	2.819	3.505	3.792
23	1.319	1.714	2.069	2.500	2.807	3.485	3.767
24	1.318	1.711	2.064	2.492	2.797	3.467	3.745
25	1.316	1.708	2.060	2.485	2.787	3.450	3.725
26	1.315	1.706	2.056	2.479	2.779	3.435	3.707
27	1.314	1.703	2.052	2.473	2.771	3.421	3.690
28	1.313	1.701	2.048	2.467	2.763	3.408	3.674
29	1.311	1.699	2.045	2.462	2.756	3.396	3.659
30	1.310	1.697	2.042	2.457	2.750	3.385	3.646
40	1.303	1.684	2.021	2.423	2.704	3.307	3.551
60	1.296	1.671	2.000	2.390	2.660	3.232	3.460
120	1.289	1.658	1.980	2.358	2.617	3.160	3.373
1000	1.282	1.646	1.962	2.330	2.581	3.098	3.300
z	1.282	1.645	1.960	2.326	2.576	3.091	3.291
	80%	90%	95%	98%	99%	99.8%	99.9%
	Confidence level C

Appendix 2: Table of selected critical values for the F distribution for sample sets of 10 objects or 9 degrees of freedom and a significance level at α = 0.05. Refer to rows α = 0.05 for a one-tailed test and rows α = 0.025 for a two-tailed test

			d.f. in the numerator
		α	1	2	3	4	5	6	7	8	9
d.f. in the denominator	1	0.05	161.45	199.50	215.71	224.58	230.16	233.99	236.77	238.88	240.54
		0.025	647.79	799.50	864.16	899.58	921.85	937.11	948.22	956.66	963.28
	2	0.05	18.51	19.00	19.16	19.25	19.30	19.33	19.35	19.37	19.38
		0.025	38.51	39.00	39.17	39.25	39.30	39.33	39.36	39.37	39.39
	3	0.05	10.13	9.55	9.28	9.12	9.01	8.94	8.89	8.85	8.81
		0.025	17.44	16.04	15.44	15.10	14.88	14.73	14.62	14.54	14.47
	4	0.05	7.71	6.94	6.59	6.39	6.26	6.16	6.09	6.04	6.00
		0.025	12.22	10.65	9.98	9.60	9.36	9.20	9.07	8.98	8.90
	5	0.05	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77
		0.025	10.01	8.43	7.76	7.39	7.15	6.98	6.85	6.76	6.68
	6	0.05	5.99	5.14	4.76	4.53	4.39	4.28	4.21	4.15	4.10
		0.025	8.81	7.26	6.60	6.23	5.99	5.82	5.70	5.60	5.52
	7	0.05	5.59	4.74	4.35	4.12	3.97	3.87	3.79	3.73	3.68
		0.025	8.07	6.54	5.89	5.52	5.29	5.12	4.99	4.90	4.82
	8	0.05	5.32	4.46	4.07	3.84	3.69	3.58	3.50	3.44	3.39
		0.025	7.57	6.06	5.42	5.05	4.82	4.65	4.53	4.43	4.36
	9	0.05	5.12	4.26	3.86	3.63	3.48	3.37	3.29	3.23	3.18
		0.025	7.21	5.71	5.08	4.72	4.48	4.32	4.20	4.10	4.03

Appendix 3: Table of critical values of the Q test statistic (P = 0.05)

Q test statistic		Significance level α
	n	0.10	0.05	0.01
$Q=\frac{x_2-{x}_1}{x_n-{x}_1}$	3	0.886	0.941	0.988
	4	0.679	0.765	0.889
	5	0.557	0.642	0.780
	6	0.482	0.560	0.698
	7	0.434	0.507	0.637
$Q=\frac{x_2-{x}_1}{x_{n-1}-{x}_1}$	8	0.479	0.554	0.683
	9	0.441	0.512	0.635
	10	0.409	0.477	0.597
$Q=\frac{x_3-{x}_1}{x_{n-1}-{x}_1}$	11	0.517	0.576	0.679
	12	0.490	0.546	0.642
	13	0.467	0.521	0.615
$Q=\frac{x_3-{x}_1}{x_{n-2}-{x}_1}$	14	0.492	0.546	0.641
	15	0.472	0.525	0.616
	16	0.454	0.507	0.595
	17	0.438	0.490	0.577
	18	0.424	0.475	0.561
	19	0.412	0.462	0.547
	20	0.401	0.450	0.535
	21	0.391	0.440	0.524
	22	0.382	0.430	0.514
	23	0.374	0.421	0.505
	24	0.367	0.413	0.497
	25	0.360	0.406	0.489