The Mann-Whitney U Test provides a non-parametric alternative to The Two Sample Student’s T-Test. While I would recommend the latter simply due to its own innate robustness, the Mann-Whitney U Test will appear from time to time in research papers. Therefore, for this reason, and for a greater understanding as it pertains to the inner workings of the underlying methodology, the Mann-Whitney U Test should at the very least, be momentarily contemplated.

__Example:__A scientist creates a chemical which he believes changes the temperature of water. He applies this chemical to water and takes the following measurements:

70, 74, 76, 72, 75, 74, 71, 71

He then measures temperature in samples which the chemical was not applied.

74, 75, 73, 76, 74, 77, 78, 75

Can the scientist conclude, with a 95% confidence interval, that his chemical is in some way altering the temperature of the water?

For this, we will utilize the code:

N1 <- c(70, 74, 76, 72, 75, 74, 71, 71)

N2 <- c(74, 75, 73, 76, 74, 77, 78, 75)

wilcox.test(N2, N1, alternative = "two.sided", paired = FALSE, conf.level = 0.95)

N1 <- c(70, 74, 76, 72, 75, 74, 71, 71)

N2 <- c(74, 75, 73, 76, 74, 77, 78, 75)

wilcox.test(N2, N1, alternative = "two.sided", paired = FALSE, conf.level = 0.95)

Which produces the output:

*Wilcoxon rank sum test with continuity correction*

data: N2 and N1

W = 50.5, p-value = 0.05575

alternative hypothesis: true location shift is not equal to 0

data: N2 and N1

W = 50.5, p-value = 0.05575

alternative hypothesis: true location shift is not equal to 0

From this output we can conclude:

With a p-value of 0.05575 (0.05575 > .05), we can state that, at a 95% confidence interval, that the scientist's chemical is not altering the temperature of the water.

The t-test equivalent of this analysis would resemble:

(If we were measuring mean values)

With a p-value of 0.05575 (0.05575 > .05), we can state that, at a 95% confidence interval, that the scientist's chemical is not altering the temperature of the water.

The t-test equivalent of this analysis would resemble:

(If we were measuring mean values)

**t.test(N2, N1, alternative = "two.sided", var.equal = TRUE, paired=FALSE, conf.level = 0.95)**

Which produces the output:

Two Sample t-test

data: N2 and N1

t = 2.4558, df = 14, p-value = 0.02773

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

0.3007929 4.4492071

sample estimates:

mean of x mean of y

75.250 72.875

Two Sample t-test

data: N2 and N1

t = 2.4558, df = 14, p-value = 0.02773

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

0.3007929 4.4492071

sample estimates:

mean of x mean of y

75.250 72.875

Below are the steps necessary to perform the above analysis within the SPSS platform.

__Mann-Whitney U Test Example:__For this particular test, data must be structured in an un-conventional manner. The cases are combined into one single variable, with their group identity providing their initial designation.

Below is our example data set:

From the

**“Analyze”**menu, select

**“Nonparametric Tests”**, then select

**“Legacy Dialogues”**, followed by

**“2 Independent Samples”**.

This should populate the menu below:

Select

This will generate the output below:

Select

**“N1N2”**, and utilize the top center arrow to designate these values as**“Test Variable(s)”**. Once this has been completed, utilize the bottom center arrow to designate**“Group”**as our**“Grouping Variable”**. Two groups exist, which we must specifically define. To achieve this, click**“Define Groups”**, then enter the value**“1”**into the input adjacent to**“Group 1”**. Next, enter the value**“2”**into the input adjacent to**“Group 2”**. Once this step has been completed, click**“Continue”**, and then click**“OK”**.This will generate the output below:

The two values from the output that are relevant for our purposes are those labeled

**“Asymp Sig.”**and

**“Exact Sig”**. There is some debate amongst researchers as to which value should be utilized for reaching a statistical conclusion. Some recommend utilizing

**“Exact Sig”**when conducting analysis that contains only a few data points, and relying on

**“Asymp Sig”**when working with larger data sets.

Remember, SPSS and R calculate output values differently for both the Mann-Whitney U Test, and the Wilcox Ranked Signed Rank Test. This differentiation arises from the methodology utilized to resolve rank order.

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