# Wilcoxon one-tailed signed rank test

It's regarding a one-tailed Wilcoxon sign test.

I am comparing (fictional) ratings of liking before and after an apple is peeled.

The design is complicated. I will do my best to explain.

There are 9 apples. Participants are asked to rate how much they like each apple. After that they are asked if the apple should be peeled. Those who answered yes, are asked to rate how much they like the apple after it was peeled while those who answer no move to rating the next apple. (please remember this is fictional, my actual research is diff. but I don't want to post it here so using a fictional research which is nearly similar).

So basically, I have pre and post peeling ratings.

Participant one may have 5 pre and post ratings while part. 2 has 7 pre and post ratings.

I expect pre peeling ratings to be more than post peeling ratings, hence one-tailed.

This is the output from SPSS:

Descriptive Statistics
N Mean Std. Deviation Minimum Maximum Percentiles
25th 50th (Median) 75th rating1 379 4.000 1.08588 1.00 5.00 3.0000 4.0000 5.0000 rating2 379 3.500 1.70245 1.00 5.00 3.0000 4.0000 5.0000

Ranks
N Mean Rank Sum of Ranks rating2 - rating1 Negative Ranks 135a 63.00 7975.00 Positive Ranks 0b .00 .00 Ties 254c
Total 389
a rating2 < rating1
b rating2 > rating1
c rating2 = rating1

Test Statisticsa

                           rating2 - rating1
Z              -10.331b
Asymp. Sig. (2-tailed)     .000


a Wilcoxon Signed Ranks Test
b Based on positive ranks.

As far as I can see, the difference is signficant p<0.001 (two-tailed).

How about one-tailed? How do I know if it is signficant at one-tailed as my hypothesis is one-tailed...I have predicted that post peeling ratings will be less than pre-peeling.

• If it's significant two-tailed, it's certainly significant one tailed. (More formally, if you wanted the p-value, for most symmetric tests you would simply halve the p-value, since the area in the desired tail is half the area in both tails.) There's probably a way to make SPSS give you the test one tailed. Commented Dec 3, 2013 at 17:39
• Hi Glen. Can you help me further? Is wilcoxon appropriate since each participant is contributing several pairs of data? see above. Thank you! Commented Dec 3, 2013 at 17:41
• So only the ones that vote yes to shaving get to vote the shaved womans? Commented Dec 3, 2013 at 19:44

You can get the one-tailed p-value just by dividing in half the two-tailed p-value. But keep in mind that it's generally not advisable to use one-tailed tests.

• You're right Michael, the sign-rank test applied on the differences is fine for paired data. I think I read the question too fast, I'll edit my answer. Commented Dec 3, 2013 at 16:37
• Thank you for your answers. I really appreciate it. OK, so my p value being less than 0.001, significant 2-tailed. For one-tailed, I halve 0.05 , gives 0.025...my p value is hence still significant at one-tailed. Now, a look at the median for pre and post shows that the medians are exactly SAME...median for pre shaving: 3 (25th percentile), 4 (50th p) 5 (75th p)...it is exactly the same as the post shaving medians...3 , 4,5. Commented Dec 3, 2013 at 16:37
• sorry, ..continuing above..medians are same for pre and post..how can the test be significant then? Commented Dec 3, 2013 at 16:47
• I have another question please: The same participant contributed several pairs of data...the part. had to rate several women(9 women) pre and post shaving ...so participants all contribute 9 pairs of data...I have 70 participants..so basically (70 x 9)..630 pairs of data ...is wilcoxon still appropriate? Thank you so much for replies!!! shabnam Commented Dec 3, 2013 at 16:48
• @shabnam: In your original question, you are talking about a single woman. Could you please edit the question accordingly? Using Wilcoxon test is clearly not appropriate for clustered data. Btw: The Wilcoxon signed-rank test is not directly linked to median differences. Commented Dec 3, 2013 at 16:53

Well, I see several issues with the study, any which one of which could be fatal.

1. Your data is NOT i.i.d. A single subject will contribute several results, hence they are not independent. So you can not lump all together in a single sample. A subject who does not like apples will not want to peel any and will not contribute any data point, and a subject who loves apples will want to peel all of them, and will contribute a disprtoportionate of the data... Not a random sample...
2. Wilcoxon sign rank test (WSRt) is NOT a test of the median, it is a test of the pseudo median. Very hard to intuiitively interpret.
3. WSRt relies heavily on an assumption of symmetry to be interpretable as a test of the median. I am pretty sure your data is NOT symmetrical (you indicate as much: the medians are identical, but you get a significant result). All that result shows is that your data is very assymetrical, not that there is any interpretable difference in medians, or means, or ...
4. Single tailed test is just fine: there must be a way to get SPSS to give you a single trailed result. But all that test says is that the data is (very) assymetric...