AP Statistics Curriculum 2007 NonParam 2MedianIndep

General Advance-Placement (AP) Statistics Curriculum - Difference of Medians of Two Independent Samples

As we discussed in the paired case, non-parametric statistical methods provide alternatives to the (standard) parametric tests that we saw earlier, and they are applicable when the distribution of the data is unknown.

Motivational Example

Nine observations of surface soil pH were made at two different (independent) locations. Does the data suggest that the true mean soil pH values differs 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 \(\alpha = 0.05\), and be sure to check any necessary assumptions for the validity of your test.

Location 1	Location 2
8.10	7.85
7.89	7.30
8.00	7.73
7.85	7.27
8.01	7.58
7.82	7.27
7.99	7.50
7.80	7.23
7.93	7.41

We see the clear analogy of this study design to the 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 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.

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 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 Box-and-Whisker Plots, but this is a qualitative inference that demands a more quantitative statistical analysis that can back up our intuition.

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 ordinal or continuous measurements.

Calculations

The U statistic for the WMW test may be approximated for sample sizes above about 20 using the Normal Distribution.

The U test is provided as part of SOCR Analyses - see this activity.

For small samples, we can directly compute the WMW test-statistic as follows.

1. Choose the sample for which the ranks seem to be smaller. Call this Sample 1, and call the other sample Sample 2.

1. Taking each observation in Sample 2, count the number of observations in Sample 1 that are smaller than it (count 0.5 for any that are equal to it).

1. The total of these counts is U.

For larger samples, a formula can be used:

1. Arrange all the observations into a single ranked series. That is, rank all the observations without regard to which sample they come from.

1. Add up the ranks in Sample 1. The sum of ranks in Sample 2 follows by calculation, since the sum of all the ranks equals N (N + 1)/2, where N is the total number of observations.

1. "U" is then given by:

\[U_1=R_1 - {n_1(n_1+1) \over 2} \,\!,\]

where n₁ is the two sample size for Sample 1, and R₁ is the sum of the ranks in Sample 1.

Note that there is no specification as to which sample is considered Sample 1. An equally valid formula for U is

\[U_2=R_2 - {n_2(n_2+1) \over 2}. \,\!\]

The sum of the two values is then given by

\[U_1 + U_2 = R_1 - {n_1(n_1+1) \over 2} + R_2 - {n_2(n_2+1) \over 2}. \,\!\]

Knowing that R₁ + R₂ = N(N + 1)/2 and N = n₁ + n₂ , and doing some algebra, we find that the sum is

\[U_1 + U_2 = n_1 n_2. \,\!\]

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.

• 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

It is much quicker to use SOCR Analyses to compute the statistical significance of the sign test. This 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:

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

Two-Sided P-Value for Sample1 not equal to Sample2: 0.00079

WMW Test vs. Independent T-Test

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

Negative: Distribution free lacks power because it doesn't use all the info in the data

• The T-test uses the raw measurements

Positive: Incorporates all of the data into calculations

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.

Practice Examples

Urinary Fluoride Concentration in Cattle

The urinary fluoride concentration (ppm) was measured both for a sample of livestock grazing in an area previously exposed to fluoride pollution and also for a similar sample of livestock grazing in an unpolluted area.

Polluted	Unpolluted
21.3	10.1
18.7	18.3
21.4	17.2
17.1	18.4
11.1	20.0
20.9
19.7

References

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

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