AP Statistics Curriculum 2007 NonParam 2MedianPair

General Advance-Placement (AP) Statistics Curriculum - Differences of Medians of Two Paired Samples

Distribution-free (or non-parametric) statistical methods provide alternative to the (standard) parametric tests that we saw earlier, and are applicable when the distribution of the data is unknown.

Motivational Clinical Example

Whitley and Ball reported on the relative risk of mortality from 16 studies of septic patients. The outcome measure of interest was whether the patients developed complications of acute renal failure. The relative risk calculated in each study compared the risk of dying between patients with and without renal failure. A relative risk of 1.0 means no effect, and relative risk \(\not= 1\) suggests beneficial or detrimental effect of developing acute renal failure in sepsis. The main goal of the study was to determine whether developing acute renal failure as a complication of sepsis impacts patient mortality from the cumulative evidence in these 16 studies. The data of this study is included below.

Study	Relative Risk	Sign (Relative Risk - 1)
1	0.75	-
2	2.03	+
3	2.29	+
4	2.11	+
5	0.80	-
6	1.50	+
7	0.79	-
8	1.01	+
9	1.23	+
10	1.48	+
11	2.45	+
12	1.02	+
13	1.03	+
14	1.30	+
15	1.54	+
16	1.27	+

We see the clear analogy of this study design to the Paired or One-Sample studies we saw before. However, if we were to plot these data (Relative Risk), we can see that their distribution is hardly symmetric, unimodal and bell-shaped (i.e., not Normal). Therefore, we cannot use the Paired T-test to test a Null-Hypothesis that the mean Relative Risk is 1 using this parametric test.

The Sign-Test

The Sign Test is a non-parametric alternative to the One-Sample and Paired T-Test. The Sign Test has no requirements for the data to be Normally distributed. It assigns a positive (+) or negative (-) sign to each observation according to whether it is greater or less than some hypothesized value. It measures the difference between the \(\pm\) signs and how distinct the difference is from our expectations to observe by chance alone. For example, if there were no effects of developing acute renal failure on the outcome from sepsis, about half of the 16 studies above would be expected to have a relative risk less than 1.0 (a "-" sign) and the remaining 8 would be expected to have a relative risk greater than 1.0 (a "+" sign). In the actual data, 3 studies had "-" signs and the remaining 13 studies had "+" signs. Intuitively, this difference of 10 appears large to be simply due to random variation. If so, the effect of developing acute renal failure would be significant on the outcome from sepsis.

Calculations

Suppose N+ is the number of "+" signs and we fix a significance level of \(\alpha= 0.05\). And consider the following two hypotheses:

\[H_o: N_+=8\] (equivalent to \(N_-=8\)): The effect of developing acute renal failure is not significant on the outcome from sepsis. \[H_1: N_+ \not=8\]: The effect of developing acute renal failure is significant on the outcome from sepsis.

Define the following test-statistics \[B_s = \max{(N_+ , N_-)}\], where \(N_+\) and \(N_-\) are the number of positive and negative signs, respectively.

Then the distribution of \(B_s \sim Binomial(n=16, p=8/16=0.5)\).

For our data, \(B_s = \max{(N_+ , N_-)}=\max{13,3}=13\) and the probability that such binomial variable exceeds 13 is \(P(Bin(16,0.5,13))=0.010635\). Therefore, we can reject the null hypothesis \(H_o\) and regard as the significant effect of developing acute renal failure on the outcome from sepsis.

The Sign Test Using SOCR Analyses

It is much quicker to use SOCR Analyses to compute the statistical significance of the sign test. This SOCR Sign Test Activity may also be helpful in understanding how to use the sign test method in SOCR.

Example

A set of 12 identical twins is given psychological tests to determine whether the first born of the set tends to be more aggressive than the second born. Each twin is scored according to aggressiveness; a higher score indicates greater aggressiveness. Because of the natural pairing in a set of twins these data can be considered paired.

Twin-Index	1st Born	2nd Born	Sign
1	86	88	-
2	71	77	-
3	77	76	+
4	68	64	+
5	91	96	-
6	72	72	0 (Drop)
7	77	65	+
8	91	90	+
9	70	65	+
10	71	80	-
11	88	81	+
12	87	72	+

We first plot the data using the SOCR Line Chart. Visually there does not seem to be a strong effect of the order of birth on baby's aggression.

Next, we can use the SOCR Sign Test Analysis to quantitatively evaluate the evidence to reject the null hypothesis that there is no birth-order effect on baby's aggressiveness.

Clearly the p-value reported is 0.274, and our data can not reject the null hypothesis.

The Wilcoxon Signed Rank Test

Like the Sign Test and the T-Test, the Wilcoxon Signed Rank Test involves comparisons of differences between measurements. It requires that the data are measured at an interval level of measurement, but does not require assumptions about the form of the distribution of the measurements. It should therefore be used whenever the distributional assumptions of the T-Test are not satisfied.

Example

Whitley and Ball reported data on the central venous oxygen saturation (SvO₂ (%)) from 10 consecutive patients at 2 time points; at admission and 6 hours after admission to the intensive care unit (ICU). The null hypothesis is that there is no effect of 6 hours of ICU treatment on SvO₂. Under the null hypothesis, the mean of the differences between SvO₂ at admission and that at 6 hours after admission should be zero.

Patient	On Admission	At 6 Hours	Difference	Rank
2	59.1	56.7	-2.4	1
7	58.2	60.7	2.5	2
9	56.0	59.5	3.5	3
10	65.3	59.8	-5.5	4
3	56.1	61.9	5.8	5
5	60.6	67.7	7.1	6
6	37.8	50.0	12.2	7
1	39.7	52.9	13.2	8
4	57.7	71.4	13.7	9
8	33.6	51.3	17.7	10

Clearly, we can reject the null-hypothesis at \(\alpha=0.05\), as the one- and two-sided alternative hypotheses p-values for the Wilcoxon Signed Rank Test reported by the SOCR Analysis are respectively

One-Sided p-value = 0.011

Two-Sided p-value = 0.022

Result interpretation

The Results tab of the SOCR Wilcoxon signed-rank test applet contains the following output (this is for the data data above, and will change for other datasets, accordingly):

Variable 1 = At_Admission

Variable 2 = 6_Hrs_Later

Results of Two Paired Sample Wilcoxon Signed Rank Test:

Wilcoxon Signed-Rank Statistic = 5.000

E(W+), Wilcoxon Signed-Rank Score = 27.500

Var(W+), Variance of Score = 96.250

Wilcoxon Signed-Rank Z-Score = -2.293

One-Sided P-Value = .011

Two-Sided P-Value = .022

Where:

• WStat, Wilcoxon Signed-Rank Statistic = data-driven estimate of the Wilcoxon Signed-Rank Statistic

DataCase[] combo = QSortAlgorithm.rankList(diffListAbs);

double wStat = 0;

int lenList = combo.length;

for (int i = 0; i < lenList; i++)

if ( combo[i].getSign() ) wStat = wStat + combo[i].getRank();

• E(W+), Wilcoxon Signed-Rank Score = expectation of the Wilcoxon Signed-Rank Statistic, see page 414 Rice book (Mathematical Statistics and Data Analysis, 2dn Edition, by John Rice, Duxbury 1995.)
• Var(W+), Variance of Score = variance of the Wilcoxon Signed-Rank Statistic, see page 414 Rice book (Mathematical Statistics and Data Analysis, 2dn Edition, by John Rice, Duxbury 1995.)
• Wilcoxon Signed-Rank Z-Score = \({WStat - E(W+) \over \sqrt{Var(W+)}}\)
• One-Sided P-Value = the one-sided (uni-directional) probability value expressing the strength of the evidence in the data to reject the null hypothesis that the two populations have the same medians (based on Gaussian, standard Normal, distribution).
• Two-Sided P-Value = the double-sided (non-directional) probability value expressing the strength of the evidence in the data to reject the null hypothesis that the two populations have the same medians (based on Gaussian, standard Normal, distribution).
• See more details on the SMHS EBook section.

Practice Problems

Treatment in Rats

Suppose 10 randomly selected rats were chosen to see if they could be trained to escape a maze. The rats were released and timed (seconds) before and after 2 weeks of training (N means the rat did not complete the maze-test). Do the data provide evidence to suggest that the escape time of rats is different after 2 weeks of training? Test using \(\alpha= 0.05\).

Rat	Before	After	Sign
1	100	50	+
2	38	12	+
3	N	45	+
4	122	62	+
5	95	90	+
6	116	100	+
7	56	75	-
8	135	52	+
9	104	44	+
10	N	50	+

Comparison and Validation of Automated Brain Volume Segmentation Methods

Brain Parsing

Automated brain volume segmentation is an important step in many modern computational brain mapping studies. Suppose we have two separate and competing versions of automated brain parsing (segmentation) algorithms that automatically tessellate (partition) the brain into 57 separate regions of interest (ROI's). An important question then is how consistent are these 2 different techniques, across the 57 ROIs. We can use the ROI volume as a measure of the resulting automated brain parcellation and compare the paired differences between the 2 methods across all ROIs. The image shows an example of a brain parcellated into these 57 regions and the table below contains the volumes of the 57 ROIs for the 2 different brain tessellation techniques. Use appropriate SOCR Analyses and relevant SOCR Charts to argue whether or not the 2 different methods are consistent and agree on their ROI labels.

Index	Volume_Intensity	ROI_Name	Method1_Volume	Method2_Volume
1	0	Background	9236455	9241667
2	21	L_superior_frontal_gyrus	78874	78693
3	22	R_superior_frontal_gyrus	69575	74391
4	23	L_middle_frontal_gyrus	67336	68872
5	24	R_middle_frontal_gyrus	68344	67024
6	25	L_inferior_frontal_gyrus	31912	21479
7	26	R_inferior_frontal_gyrus	26264	29035
8	27	L_precentral_gyrus	28942	33584
9	28	R_precentral_gyrus	35192	30537
10	29	L_middle_orbitofrontal_gyrus	10141	11608
11	30	R_middle_orbitofrontal_gyrus	9142	11850
12	31	L_lateral_orbitofrontal_gyrus	7164	5382
13	32	R_lateral_orbitofrontal_gyrus	5964	4947
14	33	L_gyrus_rectus	3840	1995
15	34	R_gyrus_rectus	2672	2994
16	41	L_postcentral_gyrus	24586	27672
17	42	R_postcentral_gyrus	21736	28159
18	43	L_superior_parietal_gyrus	25791	27500
19	44	R_superior_parietal_gyrus	28850	32674
20	45	L_supramarginal_gyrus	16445	22373
21	46	R_supramarginal_gyrus	11893	11018
22	47	L_angular_gyrus	20740	22245
23	48	R_angular_gyrus	20247	17793
24	49	L_precuneus	14491	12983
25	50	R_precuneus	15589	16323
26	61	L_superior_occipital_gyrus	6842	6106
27	62	R_superior_occipital_gyrus	5673	6539
28	63	L_middle_occipital_gyrus	15011	19085
29	64	R_middle_occipital_gyrus	19063	25747
30	65	L_inferior_occipital_gyrus	10411	8675
31	66	R_inferior_occipital_gyrus	12142	12277
32	67	L_cuneus	6935	9700
33	68	R_cuneus	7491	11765
34	81	L_superior_temporal_gyrus	29962	34934
35	82	R_superior_temporal_gyrus	30630	28788
36	83	L_middle_temporal_gyrus	27558	19633
37	84	R_middle_temporal_gyrus	26314	25301
38	85	L_inferior_temporal_gyrus	24817	24885
39	86	R_inferior_temporal_gyrus	25088	20661
40	87	L_parahippocampal_gyrus	6761	6977
41	88	R_parahippocampal_gyrus	6529	7964
42	89	L_lingual_gyrus	16752	14748
43	90	R_lingual_gyrus	20914	18500
44	91	L_fusiform_gyrus	16565	15020
45	92	R_fusiform_gyrus	14409	17311
46	101	L_insular_cortex	10779	9814
47	102	R_insular_cortex	8222	5599
48	121	L_cingulate_gyrus	14662	12490
49	122	R_cingulate_gyrus	16595	14489
50	161	L_caudate	1906	1608
51	162	R_caudate	2353	1997
52	163	L_putamen	3015	2622
53	164	R_putamen	2177	3758
54	165	L_hippocampus	3791	4454
55	166	R_hippocampus	3596	4673
56	181	cerebellum	174045	158617
57	182	brainstem	32567	28225

Moths Memory acquired at the Caterpillar Stage

See this example and analysis of data of moth odor memory demonstrated at post metamorphosis, which was acquired at the larvae stage (caterpillar).

References

• Whitley, E. and Ball, J. (2002) Statistics review 6: Nonparametric methods. Critical Care, 6(6): 509–513.

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• SOCR Home page: http://www.socr.ucla.edu

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