SMHS NonParamInference - Revision history

Dinov at 21:24, 10 February 2026

2026-02-10T21:24:58Z

Dinov at 21:24, 10 February 2026

2026-02-10T21:24:06Z

Dinov: /* Applications & Problems */

2025-12-08T18:42:29Z

‎Applications & Problems

Dinov: /* Wilcoxon-Mann-Whitney (WMW) Test */

2025-12-08T18:40:19Z

‎Wilcoxon-Mann-Whitney (WMW) Test

Dinov: /* The Wilcoxon Signed Rank Test */

2025-12-08T18:36:27Z

‎The Wilcoxon Signed Rank Test

Dinov: /* The Sign Test */

2025-12-08T18:35:34Z

‎The Sign Test

Dinov at 22:35, 7 December 2025

2025-12-07T22:35:04Z

Dinov at 22:34, 7 December 2025

2025-12-07T22:34:30Z

Show changes

Dinov at 20:35, 17 March 2023

2023-03-17T20:35:30Z

Dinov at 20:31, 17 March 2023

2023-03-17T20:31:42Z

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← Older revision		Revision as of 21:24, 10 February 2026
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	*[https://sda.statisticalcomputing.org/learning use the SOCR SDA app to complete the Non-parametric test Learning Module]		*[https://sda.statisticalcomputing.org/learning use the SOCR SDA app to complete the Non-parametric test Learning Module]
		+
		+	<hr>
		+	* SOCR Home page: https://socr.umich.edu
		+
		+	{{translate\|pageName=https://wiki.socr.umich.edu/index.php?title=SMHS_NonParamInference}}

← Older revision		Revision as of 21:24, 10 February 2026
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	===References===		===References===
−	*[~~http~~://~~wiki~~.~~stat~~.~~ucla.edu~~/~~socr/index.php/Probability_and_statistics_EBook#Chapter_XII:_Non~~-~~Parametric_Inference SOCR~~]	+
		+	*[https://sda.statisticalcomputing.org/learning use the SOCR SDA app to complete the Non-parametric test Learning Module]

@@ Line 244: / Line 244: @@
   Total "+" signs: 9. Total "-" signs: 1. Total valid <math>n=10</math>.
   Check significance using Sign Test table or Binomial calculation for 9/10 successes.
 '''Problem 6.2: Brain Volume Segmentation'''
 Comparing two algorithms for brain segmentation across 57 regions (ROI).

@@ Line 156: / Line 156: @@
 '''Motivational Example:''' Soil pH levels at two different locations (Location 1 vs. Location 2).
 * Assumption Check: Histograms show data is not Normal; small sample size (<math>n=9</math>) makes T-test risky.
-* Hypothesis: <math>H_0</math>: The distributions of Location 1 and Location 2 are identical. <math>H_a</math>: The distributions differ (shift in location.
+* Hypothesis: \(H_0\): The distributions of Location 1 and Location 2 are identical. <math>H_a</math>: The distributions differ (shift in location.
 '''Calculation (U Statistic):'''
 The logic relies on ranking all observations together (pooled).
@@ Line 174: / Line 174: @@
 # Output: p-value = 0.0009172 (Significant Difference)
 </pre>
 ====[[AP_Statistics_Curriculum_2007_NonParam_2PropIndep#General_McNemar_test_of_marginal_homogeneity_for_a_single_category|McNemar Test]]====
 This test analyzes paired nominal data (e.g., Before/After binary outcomes). It specifically tests for marginal homogeneity.

@@ Line 89: / Line 89: @@
 ====[[AP_Statistics_Curriculum_2007_NonParam_2MedianPair#The_Wilcoxon_Signed_Rank_Test|The Wilcoxon Signed Rank Test]]====
-The Sign Test ignores the *magnitude* of differences, utilizing only their direction. The [[AP_Statistics_Curriculum_2007_NonParam_2MedianPair#The_Wilcoxon_Signed_Rank_Test|Wilcoxon Signed Rank Test]] improves power by incorporating the ranks of the absolute differences.
+The Sign Test ignores the magnitude of differences, utilizing only their direction. The [[AP_Statistics_Curriculum_2007_NonParam_2MedianPair#The_Wilcoxon_Signed_Rank_Test|Wilcoxon Signed Rank Test]] improves power by incorporating the ranks of the absolute differences.
 '''Assumptions:'''
 * Data are paired and drawn independently.
@@ Line 151: / Line 151: @@
              paired=TRUE, alternative = "two.sided")
 </pre>
 ====[[AP_Statistics_Curriculum_2007_NonParam_2MedianIndep#The_Wilcoxon-Mann-Whitney_Test|Wilcoxon-Mann-Whitney (WMW) Test]]====
 Also known as the Mann-Whitney U test, this is the nonparametric analogue to the Independent Samples T-test. It assesses whether two independent samples come from the same distribution.

@@ Line 64: / Line 64: @@
 We calculate the probability of observing 13 or more successes in 16 trials under the null hypothesis (<math>p=0.5</math>):
 <math>P(B_{S} \ge 13 | B_{S} \sim Bin(16,0.5)) = 0.0106</math>
 '''Critical Note on P-Values:''' The value <math>0.0106</math> is the one-sided probability. Since our hypothesis is two-sided (<math>\ne</math>), the p-value is <math>2 \times 0.0106 = 0.0212</math>. Since <math>0.0212 < 0.05</math>, we reject <math>H_0</math> and conclude that acute renal failure has a significant effect on mortality.
 '''Example 2: Twin Aggressiveness'''
@@ Line 84: / Line 87: @@
 * P-value = 0.274.
 * Conclusion: We cannot reject the null hypothesis at the 5% level. There is no strong evidence of a birth-order effect.
 ====[[AP_Statistics_Curriculum_2007_NonParam_2MedianPair#The_Wilcoxon_Signed_Rank_Test|The Wilcoxon Signed Rank Test]]====
 The Sign Test ignores the *magnitude* of differences, utilizing only their direction. The [[AP_Statistics_Curriculum_2007_NonParam_2MedianPair#The_Wilcoxon_Signed_Rank_Test|Wilcoxon Signed Rank Test]] improves power by incorporating the ranks of the absolute differences.

← Older revision		Revision as of 22:35, 7 December 2025
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	===References===		===References===
−	* SOCR ~~Non-Parametric Inference EBook.~~	+	*[http://wiki.stat.ucla.edu/socr/index.php/Probability_and_statistics_EBook#Chapter_XII:_Non-Parametric_Inference SOCR]

@@ Line 143: / Line 143: @@
 : \(E(W_{+})\), Wilcoxon Signed-Rank Score = 27.500: [Wilcoxon Signed-Rank Score = expectation of the Wilcoxon Signed-Rank Statistic]
 : \(Var(W_{+})\), Variance of Score = 96.250: [Variance of Score = variance of the Wilcoxon Signed-Rank Statistic]
-: Wilcoxon Signed-Rank Z-Score = -2.293 \(\bigg \{Z_{score}=\frac{W_{stat}-E(W_{+})}{\sqrt{Var(W_{+})}} \bigg \}\(
+: Wilcoxon Signed-Rank Z-Score = -2.293 \(\bigg \{Z_{score}=\frac{W_{stat}-E(W_{+})}{\sqrt{Var(W_{+})}} \bigg \}\)
+:
 : One-Sided P-Value = .011: [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)]
@@ Line 180: / Line 180: @@
 The null hypothesis yields an expected value of \(W \sim 0\).  The standard deviation \(W\) is given by:
-\(\(\sigma_W = \sqrt{ \frac{n(n+1)(2n+1)}{6}}.\(\(
+\(\sigma_W = \sqrt{ \frac{n(n+1)(2n+1)}{6}}.\)
 The test statistics is \(Z= \frac{W \pm 0.5}{\sigma_W}\). The \(\pm 0.5\) is added as continuity correction, as \(W\) takes decimal values due to the artifact of the process of assigning tied ranks.
@@ Line 318: / Line 318: @@
 *Marginal homogeneity occurs when the row totals equal to the column totals, a and d in each equation can be cancelled; leaving b equal to c: a+b=a+c, c+d=b+d. In this example, marginal homogeneity would mean there was no effect of the treatment.
-* [[AP_Statistics_Curriculum_2007_NonParam_2PropIndep#General_McNemar_test_of_marginal_homogeneity_for_a_single_category|The McNemar statistic]] is shown: \(χ_{0}^{2}=\frac{(b-c)^{2}}{b+c} \simχ_{df=1}^{2}\(
+* [[AP_Statistics_Curriculum_2007_NonParam_2PropIndep#General_McNemar_test_of_marginal_homogeneity_for_a_single_category|The McNemar statistic]] is shown: \(χ_{0}^{2}=\frac{(b-c)^{2}}{b+c} \simχ_{df=1}^{2}\).
 *The marginal frequencies are not homogeneous if the \(χ_{0}^{2}\) result is significant (say at \(p < 0.05\)). If b and/or c are small \((b + c < 20)\) then \(χ_{0}^{2}\) is not approximated by the [[AP_Statistics_Curriculum_2007_Chi-Square#Chi-Square_Distribution|Chi-Square Distribution]] and a [[SMHS_NonParamInference#The_Sign_Test|Sign Test]] should be used instead.
 *An interesting observation when interpreting McNemar's test is that the elements of the main diagonal contribute no information whatsoever to the decision, i.e., in the above example, pre- or post-treatment condition is more favorable.
@@ Line 376: / Line 376: @@
 :To test marginal homogeneity for one single category (in this case poor evaluation) means to test row/column marginal homogeneity for the first category (poor). This is achieved by collapsing all rows and columns corresponding to the other categories.
-:: \(χ_{o}^{2}=\frac{(b-c)^{2}}{b+c}=\frac{(-8)^{2}}{46}=1.39\sim χ_{df=1}^{2}\(
+:: \(χ_{o}^{2}=\frac{(b-c)^{2}}{b+c}=\frac{(-8)^{2}}{46}=1.39\sim χ_{df=1}^{2}\)
-:: \(P( χ_{df=1}^{2}>1.39)=0.238405\(
+:: \(P( χ_{df=1}^{2}>1.39)=0.238405\)
 : Therefore, we don’t have sufficient evidence to reject the null hypothesis of two evaluators were consistent in their ratings of students.
@@ Line 428: / Line 428: @@
 : \(H_{a}\): At least one of the populations tends to yield larger observations than at least one of the other populations.
-: Suppose \({X_{i,1},X_{i,2},…,X_{i,n_i}}\) represents the values of the \(i^{th}\) sample, where \(i\le i\le k.\(
+: Suppose \({X_{i,1},X_{i,2},…,X_{i,n_i}}\) represents the values of the \(i^{th}\) sample, where \(i\le i\le k.\)
-:Test statistics: \(T=(\frac{1}{S^{2}})\sum_{i=1}^{k}\frac{R_{i}^{2}}{n_{i}}-\frac{N(N+1)^{2}}{4}\), where \(S^{2}=\frac{\Bigg(\frac{1}{N-1}\sum R(X_{ij})^{2}-N(N+1)\Bigg)^{2}}{4}.\(
+:Test statistics: \(T=(\frac{1}{S^{2}})\sum_{i=1}^{k}\frac{R_{i}^{2}}{n_{i}}-\frac{N(N+1)^{2}}{4}\), where \(S^{2}=\frac{\Bigg(\frac{1}{N-1}\sum R(X_{ij})^{2}-N(N+1)\Bigg)^{2}}{4}.\)
 :Note: If there are no ties, then the test statistic is reduced to:
-:\(T=\frac{12}{N(N+1)} \sum_{i=1}^{k}\frac{R_{i}^{2}}{n_{I}} -3(N+1).\(
+:\(T=\frac{12}{N(N+1)} \sum_{i=1}^{k}\frac{R_{i}^{2}}{n_{I}} -3(N+1).\)
 :However, the SOCR implementation allows for the possibility of having ties; so it uses the non-simplified, exact method of computation. Multiple comparisons have to be done here. For each pair of groups, the following is computed and printed at the Result Panel.
-: \(|\frac{R_{i}}{n_{i}}-\frac{R_{j}}{n_{j}}|\) > \(t_{1-\frac{\alpha}{2}}\frac{\Bigg(\frac{S^{2}(N-1-T)}{N-K}\Bigg)^{\frac{1}{2}}}{\Bigg(\frac{1}{n_{i}} +\frac{1}{n_{j}}\Bigg)^{\frac{1}{2}}}\(
+: \(|\frac{R_{i}}{n_{i}}-\frac{R_{j}}{n_{j}}|\) > \(t_{1-\frac{\alpha}{2}}\frac{\Bigg(\frac{S^{2}(N-1-T)}{N-K}\Bigg)^{\frac{1}{2}}}{\Bigg(\frac{1}{n_{i}} +\frac{1}{n_{j}}\Bigg)^{\frac{1}{2}}}\)
 : The SOCR computation employs the exact method instead of the approximate one (Conover 1980), since computation is easy and fast to implement and the exact method is somewhat more accurate.
@@ Line 484: / Line 484: @@
 ====[[AP_Statistics_Curriculum_2007_NonParam_VarIndep|Fligner-Killeen test]]: Variance Homogeneity (Differences of Variances of Independent Samples)====
 Different tests for variance equality can be applied. It is frequently necessary to test if \(k\) samples have equal variances. [[SMHS_HypothesisTesting#Testing_a_claim_about_variance_.28or_standard_deviation.29|Homogeneity of variances]] is often a reference to equal variances across samples. Some statistical tests, for example the analysis of variance, assume that variances are equal across groups or samples.
-*Calculation: The [[AP_Statistics_Curriculum_2007_NonParam_VarIndep|(modified) Fligner-Killeen test (the median-centering Fligner-Killeen test)]] provides the means for studying the homogeneity of variances of k populations \(\big\{X_{ij}\), for \(1\le i \le n_{j}\) and \(1 \le j \le k\big\}.\) The test jointly ranks the absolute values of \(|X_{ij}-\tilde X_{j}|\) and assigns increasing scores \(a_{N,i}=\Phi^{-1} \left( \frac{1+\frac{i}{N+1}}{2}\right)\), based on the ranks of all observations. In this test, \(\tilde X_{j}\) is the sample ''median'' of the \(j^{th}\) group, and \(\Phi\((.) is the cumulative distribution function for [[SMHS_ProbabilityDistributions#Normal_distribution|Normal distribution]].
+*Calculation: The [[AP_Statistics_Curriculum_2007_NonParam_VarIndep|(modified) Fligner-Killeen test (the median-centering Fligner-Killeen test)]] provides the means for studying the homogeneity of variances of k populations \(\big\{X_{ij}\), for \(1\le i \le n_{j}\) and \(1 \le j \le k\big\}.\) The test jointly ranks the absolute values of \(|X_{ij}-\tilde X_{j}|\) and assigns increasing scores \(a_{N,i}=\Phi^{-1} \left( \frac{1+\frac{i}{N+1}}{2}\right)\), based on the ranks of all observations. In this test, \(\tilde X_{j}\) is the sample ''median'' of the \(j^{th}\) group, and \(\Phi\)(.) is the cumulative distribution function for [[SMHS_ProbabilityDistributions#Normal_distribution|Normal distribution]].
 *Fligner-Killeen test statistics: \(χ_{o}^{2}=\frac{\sum_{j=1}^{k}n_{j}(\overline A_{j}-\bar a)^{2}} {V^{2}}\) , where \(\overline A_{j}\) is the mean score for the \(j^{th}\) sample, \(\bar a\) is the overall mean score of all \(a_{Nj}\) and \(V^{2}\) is the sample variance of all scores. We have \(N=\sum_{j=1}^{k}n_{j},\overline A_{j}=\frac{1}{n_{j}}\sum_{j=1}^{n_j}a_{N,m_i}\), where \(a_{N,m_i}\) is the increasing rank score for the \(i^{th}\) observation in the \(j^{th}\) sample, \(\bar a=\frac {1}{N} \sum_{i=1}^{N}a_{N,i},V^{2}=\frac{1}{N-1}\sum_{i=1}^{N}(a_{N,i}-\bar a)^{2}\) .