1. Biostatistics course Part 17 Non-parametric methods
Dr. C. Nicolas Padilla Raygoza
Department of Nursing and Obstetrics
Division of Health Sciences and Engineers
Campus Celaya-Salvatierra
University of Guanajuato

2. Biosketch Médico Cirujano por la Universidad Autónoma de Guadalajara.
Pediatra por el Consejo Mexicano de Certificación en Pediatría.
Diplomado en Epidemiología, Escuela de Higiene y Medicina Tropical de Londres, Universidad de Londres.
Master en Ciencias con enfoque en Epidemiología, Atlantic International University.
Doctorado en Ciencias con enfoque en Epidemiología, Atlantic International University.
Profesor Asociado C, Department of Nursing and Obstetrics, Division of Health Sciences and Engineerings, Campus Celaya Salvatierra, University of Guanjuato.

3. Competencies
The reader will know the non-parametric methods and when he(she) can use them.
He (she) will apply non-parametric methods in an appropriate form.
He (she) can obtain a confidence interval in non-paramethric analysis
He (she) will apply Wilcoxon sum rank test
He (she) will apply Wilcoxon
He (she) will apply r Spearman.

4. Introduction Parametric methods
They are base in means, standard deviations or probabilities.
The Normal distribution is not always appropriate
To study variables with a few observations,
Non-symmetrical distributions, or
Variables that can have more than two values
Until now, we have using methods that assume that variable has a distribution with some characteristics: a) the distribution is Normal, in quantitative data, and b) in binary data, the distribution is binomial, with a probability, p.
Means, standard deviations and probabilities, are calling parameters and the methods to make inferences about these parameters, are calling parametric methods.
There are other distributions; with them, we do not assume that they are Normal or binomials; as when the sample size is small, quantitative data do not have Normal distribution or when categorical data have more than two values.

5. Introduction (contd…)
When this happens, we use other anaylisis methods
Non-parametric methods
They are not based in the same assumptions that parametric methods, but also have some assumptions.

6. Categories (ranking), means, medians
Some non-parametric methods use rankings en lugar de los real values.
Categories are use to compare data, more for their ranking that for their size.
Patient
Glucose in blood (mg/dl) 1
135 2
225 3 70 4
100 5
110 6
150 7 90 8 9
170 10 60 11 80
From these data, we van see that patient 1 have a lesser value of glucose in blood that patient 2. When we can to study the order, we can summarize data and apply statistic tests, without waiting that data have a particular distribution. Median is the center of the ordinal classification (ranking).

7. Categories (ranking), means, median
Ranked in ascending order
Patient
Glucose in blood (mg/dl)
Ranking 10 60 1 3 70 2 11 80 7 90 4
100 5 8 6
110
135
150 9
170
225
Median is the center of categories distribution.

8. Are mean and median equals?
To use mean and confidence interval is adequate, the distribution of values should be symmetric.
To the median and confidence intervals are adequate, no need for assumptions.
To calculate the confidence interval of a median, we need only calculate the probability that the sample data are grouped symmetrically around the true median. There are tables published for the confidence interval of a median. Having already located the middle, the table lists the values for lower limit and upper confidence interval.

9. Are mean and median equals?
Using the order (ranking) instead of original values, reduces the need for assumptions about the distribution, the calculations are simpler and faster.
The disadvantage is that the original values are lost.
Thus, non-parametric methods are used only to test hypotheses, not for estimation purposes.

10. Non-parametric methods
Situation
Non-paramethric method
Paramethric methods
One sample
Wilcoxon signed rank test
Z statistic ( t test)
Two indpendent samples
Wilcoxon sum rank test
Z statistic for two independent samples (t test)
Two paired samples
Z-paired statistic (t-paired test)
One sample, two quantitative variables
Correlation coefficient of Spearrman
Correlation coefficient of Pearson

11. Data of one sample
The table show data of glucose levels in blood from 11 patients.
We want to know if the mean is 100 mg/dl.
Patient
Glucose in blood (mg/dl)
Ranking 10 60 1 3 70 2 11 80 7 90 4
100 5 8 6
110
135
150 9
170
225
If we use a parametric method to know the answer to this question, we should assume that distribution of glucose levels in blood in this sample are approximately Normal and then, probe:
Null hypothesis
Ho: μ = 100
Alternative hypothesis
H1: μ ≠ 100
Then we calculate a t-statistic test ( no Z) because the little sample size. _
Mean X =
s = 49.01
SE = 15.52
t = 1.11
P > 0.05

12. Data of one sample
Alternative no parametric test is Wilcoxon signed rank test.
It can be used to evaluate if the values in the sample are centered in 100 mg/dl.
This test does not require Normality of the distribution of data, but requires that the distribution is symmetrical, but not necessarily take the form of "bell" as Normal.

13. Data of one sample
Wilcoxon signed rank test is calculate by six steps:
1. To calculate the difference between each observation and the interest value, 100 mg/dl.
2. You should exclude any difference = 0.
3. To classify and order (ranking) differences by magnitude , not taken into accoun the sign.
4. Sum the rankings of positive differences.
5. Sum the rankings of negative differences.
6. Select the more little sums and call it T.
This T value is search in the critical values table from Wilcoxon signed rank test. We can read the table in the row corresponding to differences non-zero. Each row has ranges different of values corresponding to different p-values.
If the T-values is out of range of the column or exactly equal at one of range- values, the p-value is less that corresponding to the column.
If the T-value is between the values of range, p-value is higher that corresponding to the column.

14. Data of one sample Patient Glucose in blood (mg/dl)
Differences with 100 mg/dl
Rnking 10 60
-40 6 3 70
-30 4 11 80
-20 7 90
-10 2
100 8 5
110 1
135 35
150 50 9
170
225
125
Positive differences = 30
Negative differences = 15
Two differences = 0
T=15
From the table of critical values for Wilcoxon signed rank test for one sample, with sample size of 11 – 2, 0 differences, n = 9, the first column show the range from 10 – 35 (T=15 is between the range) and correspond to p=0.2.
To obtain confidence interval, n=11, in the table, we search 11 sample size and the 95% confidence interval is between ranking 2 and 10.

15. Two independent groups
30 teenagers with acute apendicitis, were distributed 15 to underwent traditional apendicectomia and 15 with laparoscopic apedicectomia.
For both groups, we evaluate post-surgical pain.
Post-surgical pain
Traditional
Laparoscopy
None 1 3
Slight 5 7
Moderate 4
Severe
Total 15
The median in the group of traditional appendectomy is moderate pain in the laparoscopic group is slight.

16. Two independent groups
To compate post-surgical pain in both groups, we can use Wilcoxon rank sum test.
We define the null hypothesis Ho: the two distributions overlap.
We define alternative hypothesis Hi: the two distributions are not overlap.

17. Two independent groups
Wilcoxon rank sum test has three steps:
We order the values in both groups in ascendant order.
To calculate T as the sum of rankings of more short sample or one of two if the sample size is equal.
To compare T-value in the critical values of Wilcoxon rank sum test.

18. Two independent groups
Post-surgical pain
Traditional
Laparoscopy
Rankings
None 1 1+ 3
Slight 5 9+ 7 15
Moderate
21+ 4 25
Severe
29+ 30
Total
T = = 60
In the critical values of Wilcoxon table, n1,n2 (15,15), we search the 60 value and it correspond to p<0.05

19. Two paired groups
The table show hours of improvement given by two analgesics in 12 patients with rheumatoid arthritis.
To test that the improvement is the same with both analgesics, we can use paired-t test or Wilcoxon signed ranking test.
With both methods, we calculate the difference of improvement in hours for each patient.
Patient
A Analgesic
B Analgesic 1
3.5 2
3.6
5.7 3
2.6
2.9 4
2.4 5
7.3
9.9 6
3.4
3.3 7
14.9
16.7 8
6.6
6.0 9
2.3
3.8 10
2.0
4.0 11
6.8
9.1 12
8.5
26.9

20. Two paired groups
With Wilcoxon signed rank test, it is no requirement the Normality, but the data should be symmetrical to both sides of the median.
Ho: difference in medians = 0 Hi= difference in medians ≠ 0
Patient
A Analgesic
B Analgesic
Difference
Rankings 1
3.5 2
3.6
5.7
-2.1 8 3
2.6
2.9
-0.3 4
2.4
0.2 5
7.3
9.9
-2.6 10 6
3.4
3.3
0.1 7
14.9
16.7
-1.8
6.6
6.0
0.6 9
2.3
3.8
-1.5
2.0
4.0
-2.0 11
6.8
9.1
-2.3 12
8.5
26.9
-18.4
If we are using t-test, assume that the distribution of differences is Normal and note Ho: δ = 0 Hi:δ ≠ 0 δ is the mean of differences.
δ = SE = 1.48 t= p>0.10

21. Two paired groups
We calculate the Wilcoxon signed rank test for differences, making the following:
1.- Count how many differences non-zero.
2.- Order the differences by their magnitude, without take into account the sign.
3.- Sum rankings of positive differences.
4.- Sum rankings of negative differences.
5.- Select the more shor of the two sums and call it T. (Sum of negative differences = 59, sum of positive differences = 7, T=7).
6.- Compare the T-value in the critical values tables for Wilcoxon signed rank test. T=7, p<0.05.
With these result, we rejected the null hypothesis (differences of medians is not 0).

22. Spearman’s correlation of ranks
Table and graphic show incidence of colon cancer and average of meat intake per capita in 13 countries.
Country
Incidence colon ca
Mean of intake of meat 1 10 2 8 9 3 11 5 4 12 22 33 6 67 37 7 73 32 48 41 31 21 29 17 13
We can measure correlation between two quantitative variables, using the r Pearson correlation. For the relationship between meat intake and incidence of colon cancer is r=0.65; but the both variables should have a Normal distribution. When they are not Normal, we can apply two strategies:
1.- To transform the variables ( logarithmic or squared) to come more Normal, or
2.- To use an equivalent non- parametric

23. Spearman ranks correlation
It is appropiate for monotonic relationships, non-lineal.
It is calculate at the same time that r’s Pearson, only using the rankings.
To calculate it, we need three steps:
To order the values of first variable,
To order the values of second variable,
To apply the formulae of r’s Pearson, using the rankings instead of original values.

24. Spearman ranks correlation
Country
Incidence colon ca
Mean of meat intake
Ranking of cancer
Ranking of meat intake 1 10 3 2 8 9 7 11 5 4 12 22 33 6 67 37 73 32 13 48 41 31 21 29 17
r= Σ(x – median of x)(y-median of y) / √Σ(x – median of X)2 Σ(y-median of y)2 = 0.74

25. Comparison of methods Example Parametric method Non-parametric method
Glucose in blood
t test for one sample p>0.05
Wilcoxon signed rank test, p>0.2
Intensity of surgical pain
t test for two independent samples p<0.05
Wilcoxon sun rank test p<0.05
Improvement of pain
t paired test p>0.1
Wilcoxon signed rank test, p<0.05
Corrlation between colon cancer and meat intake
R Pearson, r= 0.65
R Spearman, r=0.74

26. Bibliografy
1.- Last JM. A dictionary of epidemiology. New York, 4ª ed. Oxford University Press, 2001:173.
2.- Kirkwood BR. Essentials of medical ststistics. Oxford, Blackwell Science, 1988: 1-4.
3.- Altman DG. Practical statistics for medical research. Boca Ratón, Chapman & Hall/ CRC; 1991: 1-9.