AP Statistics Curriculum 2007 NonParam 2MedianIndep - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T17:24:10Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Calculations */

2014-12-01T22:21:24Z

‎Calculations

Jenny: /* WMW Test vs. Independent T-Test */

2010-06-28T21:05:47Z

‎WMW Test vs. Independent T-Test

Jenny: /* WMW Test vs. Independent T-Test */

2010-06-28T21:05:30Z

‎WMW Test vs. Independent T-Test

Jenny at 21:04, 28 June 2010

2010-06-28T21:04:14Z

Jenny at 21:01, 28 June 2010

2010-06-28T21:01:48Z

Jenny at 17:41, 28 June 2010

2010-06-28T17:41:24Z

PriscillaChui: /* The Wilcoxon-Mann-Whitney Test */

2008-03-10T23:12:25Z

‎The Wilcoxon-Mann-Whitney Test

PriscillaChui: /* Motivational Example */

2008-03-10T23:10:52Z

‎Motivational Example

PriscillaChui: /* Motivational Example */

2008-03-01T23:10:03Z

‎Motivational Example

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

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← Older revision		Revision as of 22:21, 1 December 2014
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	The maximum value of ''U'' is the product of the sample sizes for the two samples. In such a case, the "other" ''U'' would be 0.		The maximum value of ''U'' is the product of the sample sizes for the two samples. In such a case, the "other" ''U'' would be 0.
		+
		+	* [http://socr.umich.edu/Applets/WilcoxonRankSumTable.html SOCR WMW Critical scores Table]. These critical values for the WMW test-statistics are computed using the following R-script
		+
		+	cat("n1", " n2 ")
		+	for (a in c(1,5,10,100,200)) cat(a/2000, " ")
		+	for (x in 4:20) {
		+	for (y in 1:20) {
		+	cat(x,y, "")
		+	for (a in c(1,5,10,100,500)) {
		+	cat(if (qwilcox(a/2000,x,y,lower.tail = TRUE,
		+	log.p = FALSE)-1>=0) qwilcox(a/2000,x,y,lower.tail = TRUE,
		+	log.p = FALSE)-1 else 0, " ")
		+	}
		+	cat("\n")
		+	}
		+	}
		+
		+	Note that if F(x) denotes the CDF of the Wilcoxon-Mann-Whitney $U$ statistic, the R-function ''qwilcox'' computed the quantile function $Q(α)=\inf\{x∈N:F(x)≥α\}, α∈(0,1)$, for $U$. Since U is a discrete variable, for a given probability $\alpha$, we can't always find a critical value $x$ corresponding to $\alpha$, i.e., $F(x)=α$. However, there will be a minimum $x$, such that $F(x)= F(Q(α))>α$. If $C(α)$ denotes the critical value for the WMW test, $F(C(α))≤α$, to preserve the false-positive error rate of the test. Thus, $C(α)=\sup\{x∈N:F(x)≤α\},α∈(0,1)$ and if there exists a value $x$ such that $F(x)=α$, then, $C(α)=Q(α)$, otherwise, $C(α)=Q(α)−1$.

	===The Wilcoxon-Mann-Whitney Test Using SOCR Analyses===		===The Wilcoxon-Mann-Whitney Test Using SOCR Analyses===

@@ Line 97: / Line 97: @@
 * The T-test uses the raw measurements
-: Positive : Incorporates all of the data into calculations
+: Positive: Incorporates all of the data into calculations
-: Negative : Must meet normality assumption
+: Negative: Must meet normality assumption
 * Neither test is uniformly superior. If the data are normally distributed we use the T-test. If the data are not normal use the WMW test.

@@ Line 93: / Line 93: @@
 Both types of tests answer the same question, but treat data differently.
 *The WMW test uses rank ordering
-: Positive: doesn’t depend on normality or population parameters
+: Positive: Doesn’t depend on normality or population parameters
-: Negative: distribution free lacks power because it doesn't use all the info in the data
+: Negative: Distribution free lacks power because it doesn't use all the info in the data
 * The T-test uses the raw measurements

@@ Line 36: / Line 36: @@
 [[Image:SOCR_EBook_Dinov_NonParam_Wilcoxon_022408_Fig2.jpg|600px]]</center>
-The first of these two figures shows the index plot of the pH levels for both samples. The second figure shows the sample histograms of these samples, which are clearly not Normal-like. Therefore, the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] would not be appropriate to analyze these data.
+The first figure shows the index plot of the pH levels for both samples. The second figure shows the sample histograms of these samples, which are clearly not Normal-like. Therefore, the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] would not be appropriate to analyze these data.
 Intuitively, we may consider these group differences significantly large, especially if we look at the [[SOCR_EduMaterials_Activities_BoxPlot | Box-and-Whisker Plots]], but this is a qualitative inference that demands a more quantitative statistical analysis that can back up our intuition.
@@ Line 85: / Line 85: @@
 <center>[[Image:SOCR_EBook_Dinov_NonParam_Wilcoxon_022408_Fig4.jpg|600px]]</center>
-Clearly the p-value < 0.05, and therefore our data provides sufficient evidence to reject the null hypothesis and therefore assumes that there were significant pH differences between the two soil lots tested in this experiment..
+Clearly the p-value < 0.05, and therefore our data provides sufficient evidence to reject the null hypothesis. So we assumes that there were significant pH differences between the two soil lots tested in this experiment.
 : One-Sided P-Value for Sample2 < Sample1: 0.00040

@@ Line 31: / Line 31: @@
 </center>
-We see the clear analogy of this study design to the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent 2-sample designs]] we saw before. However, if we were to plot these data we can see that their distributions may be different or not even symmetric, unimodal and bell-shaped (i.e., not Normal). Therefore, we can not use the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] to test a Null-hypothesis that the centers of the two distributions (that the 2 samples came from) are identical, using this parametric test.
+We see the clear analogy of this study design to the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent 2-sample designs]] we saw before. However, if we were to plot these data we can see that their distributions may be different or not even symmetric, unimodal and bell-shaped (i.e., not Normal). Therefore, we cannot use the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |Independent T-Test]] to test a Null-hypothesis that the centers of the two distributions (that the 2 samples came from) are identical, using this parametric test.
 <center>[[Image:SOCR_EBook_Dinov_NonParam_Wilcoxon_022408_Fig1.jpg|600px]]

@@ Line 43: / Line 43: @@
 ==The Wilcoxon-Mann-Whitney Test==
-The Wilcoxon-Mann-Whitney (WMW) Test (also known as Mann-Whitney U test, Mann-Whitney-Wilcoxon test, or Wilcoxon rank-sum test) is a non-parametric test for assessing whether two samples come from the same distribution. The null hypothesis is that the two samples are drawn from a single population, and therefore that their probability distributions are equal. It requires that the two samples are independent, and that the observations are [[AP_Statistics_Curriculum_2007#Types_of_Data |ordinal]] or continuous measurements.
+The Wilcoxon-Mann-Whitney (WMW) Test (also known as Mann-Whitney U Test, Mann-Whitney-Wilcoxon Test, or Wilcoxon Rank-Sum Test) is a non-parametric test for assessing whether two samples come from the same distribution. The null hypothesis is that the two samples are drawn from a single population, and therefore that their probability distributions are equal. It requires that the two samples are independent, and that the observations are [[AP_Statistics_Curriculum_2007#Types_of_Data |ordinal]] or continuous measurements.
 ===Calculations===
-The ''U'' statistic for the WMW test may be approximated for sample sizes above about 20 using the [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distribution]].
+The ''U'' statistic for the WMW test may be approximated for sample sizes above about 20 using the [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal Distribution]].
 The ''U'' test is provided as part of [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] - [[SOCR_EduMaterials_AnalysisActivities_Wilcoxon |see this activity]].
@@ Line 80: / Line 80: @@
 The maximum value of ''U'' is the product of the sample sizes for the two samples.  In such a case, the "other" ''U'' would be 0.
-===The Wilcoxon-Mann-Whitney Test using SOCR Analyses===
+===The Wilcoxon-Mann-Whitney Test Using SOCR Analyses===
-It is much quicker to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the statistical significance of the sign test. This [[SOCR_EduMaterials_AnalysisActivities_Wilcoxon | SOCR Wilcoxon-Mann-Whitney Test activity]] may also be helpful in understanding how to use this test in SOCR. We have the following results for the pH data above:
+It is much quicker to use [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to compute the statistical significance of the sign test. This [[SOCR_EduMaterials_AnalysisActivities_Wilcoxon | SOCR Wilcoxon-Mann-Whitney Test Activity]] may also be helpful in understanding how to use this test in SOCR. We have the following results for the pH data above:
 <center>[[Image:SOCR_EBook_Dinov_NonParam_Wilcoxon_022408_Fig4.jpg|600px]]</center>

@@ Line 3: / Line 3: @@
 As we discussed in the [[AP_Statistics_Curriculum_2007_NonParam_2MedianPair | paired case]], non-parametric statistical methods provide alternatives to the (standard) [[EBook#Chapter_VIII:_Hypothesis_Testing | parametric tests that we saw earlier]], and they are applicable when the distribution of the data is unknown.
-==Motivational Example==
+===Motivational Example===
 Nine observations of surface soil [http://en.wikipedia.org/wiki/PH pH] were made at two different (independent) locations.  Does the data suggest that the true mean soil [http://en.wikipedia.org/wiki/PH pH] values differ for the two locations?  Note that there is no pairing in this design, even though this is a balanced design with 9 observations in each (independent) group. Test using <math>\alpha = 0.05</math>, and be sure to check any necessary assumptions for the validity of your test.