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    The Wilcoxon Sign Rank Test: A Geological Example

    Geological data is often not normally distributed, and with data that is not normally distributed parametric methods should not be used.  In this case, I did a paired experiment, which is a simplified example of blocking, where comparisons are made between similar experimental units. This blocking needs to be done prior to performing the experiment.

    Wilcoxon sign rank test can be used when we can’t assume a normal distribution in a paired experiment.

    We will use the fictional data below to determine if Arsenic (As) is increasing in soil samples over a 1-year period.

    Location IDFeb 2018 (As ppm)Feb 2019 (As ppm)
    11811
    2127
    31716
    41010
    51417
    6810
    7915
    81317
    91220
    10935
    Figure 1: Fictional As soil sample data.

    In this sample data set, I used a Shapiro-Wilk test to test for normality (∝=0.05) (“Shapiro-Wilk Test Calculator”, 2020). For February 2018, we calculated a p-value of 0.457 that our sample set is normal, so we assume that it is normal. For February 2019, our calculated p-value is 0.047, this is statistically significant and therefore we assume it is not normally distributed.

    As usual we state our hypothesis:

    We state our level of significance at ∝=0.05, and our number of observations n=9(we ignore values with no differences such as location 4). First, we need to calculate some new information for our test: the absolute difference, and the rank.

    Location IDFeb 2018 (As ppm)Feb 2019 (As ppm)DifferenceAbsolute DifferenceRank
    11811-777
    2127-555
    31716-111
    4101000
    51417333
    6810222
    7915666
    81317444
    91220888
    1093526269
    Figure 2: Calculation for the Wilcoxon signed rank test.

    As this is a Wilcoxon signed rank test, we need to know the rank sum of the negative and positive differences. We ignore sample pairs with no differences.

    Our Wilcox test statistic is the smallest of these two calculations, therefore wstat=13. We use a table to find that the critical value for n=10 and ∝=0.05, is wcrit=8, is  (“Wilcoxon Signed-Ranks Table”, 2020). Therefore,  wstat>wcrit, and we cannot reject the null hypothesis. It is possible there has been no change in As levels between the two years. If our critical value was greater than our test statistic, we would be able to reject the null and confirm statistically significant change in As values.

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