Wilcoxon Signed Rank Testing for a difference R+ RR

What does it do? Tests for a difference in averages (medians – the middle value – to be exact) Compares two cases (eg soil moisture content north & south of a hedge) Data must occur in natural pairs  Measurements at matched points on either side of a hedge  Measurements at corresponding points on two shores

Planning to use it? Your data occur in natural pairs You want to test for difference You have just two cases to compare You have five or more pairs of values If your data are likely to be normally distributed, it may be easier to get a significant result using the t-test Make sure that…

How does it work? You assume (null hypothesis) there is no difference between the two cases The test involves finding the differences between the pairs of data, and ranking the differences. If, for example, the values in the first sample were all much bigger, then all the differences would be positive

Doing the test These are the stages in doing the test: 1.Write down your hypotheseshypotheses 2.Finding the differencesdifferences 3.Doing the rankingranking 4.Look at the tablestables 5.Make a decisiondecision Click here Click here for an example

Hypotheses H 0: There is no difference between population 1 and population 2 For H 1, you have a choice, depending on what alternative you were looking for. H 1: Population 1 is larger than population 2 eg: Soil moisture content is higher on the north of the hedge than the south of the hedge orH 1: Population 1 is different to population 2 eg: Soil moisture content is different on the north and south sides of the hedge. Unless you have a good scientific reason for expecting one to be larger, you should choose “different” for H 1

Differences Work out the differences between the pairs of values, including signs Always do sample 1 – sample 2 not the other way round, where sample 1 is the one you expect to have larger values

Ranking We rank the differences, ignoring their signs. Rank from 1 for largest difference and ignore any zero differences Then work out  R+ = sum of ranks of positive differences  R- = sum of ranks of negative differences Eg: To rank the data –4, 3, 5, -6, -1, we’d put them in descending numerical order, ignoring signs: Ranks:12345 SoR+ = = 6R  = = 9

Tables This is a Wilcoxon table Sample size. Significance levels - note different values for 1 and 2-tailed

Make a decision If you used: H 1 : Population 1 is larger than population 2:  You are doing a 1-tailed test (1 alternative only considered)  Use W = R   If your W value is smaller than the tables value, you reject your null hypothesis If you used H 1 : Population 1 is different to population 2:  You are doing a 2-tailed test (both alternatives considered)  Choose W = the smaller of R+ and R-  If your W value is smaller than the tables value, you reject your null hypothesis

Soil Moisture North & South of Hedge Data were obtained for soil moisture content at seven matched points north and south of a hedge. Hypotheses: H 0: There is no difference in the soil moisture content on the north & south of the hedge H 1 The soil moisture is higher north of the hedge

The data Site North South

Differences We are expecting the North side to have a higher moisture content. So we do North - South N S N – S

Ranking We now rank the data ignoring signs (Rank 1 for largest difference – ignore 0 differences) N – S Rank Now we find R+ and R  R+ = = 20 R- = = 8

The test We are doing a 1-tailed test, so W = R- = 8 Critical value (5%) = 4 Since our W-value is larger than the tables value, we must accept H 0 – the moisture content is not significantly higher north of the hedge.