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Biostatistics coursePart 17
Non-parametric methods
Dr. C. Nicolas Padilla RaygozaDepartment of Nursing and Obstetrics
Division of Health Sciences and EngineersCampus Celaya-Salvatierra
University of Guanajuato
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.
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.
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
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.
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
8 100
9 170
10 60
11 80
Categories (ranking), means, median
Ranked in ascending order
Patient Glucose in blood (mg/dl) Ranking
10 60 1
3 70 2
11 80 3
7 90 4
4 100 5
8 100 6
5 110 7
1 135 8
6 150 9
9 170 10
2 225 11
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.
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.
Are mean and median equals?
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
Wilcoxon signed rank test
Z-paired statistic (t-paired test)
One sample, two quantitative variables
Correlation coefficient of Spearrman
Correlation coefficient of Pearson
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 3
7 90 4
4 100 5
8 100 6
5 110 7
1 135 8
6 150 9
9 170 10
2 225 11
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.
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.
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 3
7 90 -10 2
4 100 0
8 100 0
5 110 10 1
1 135 35 5
6 150 50 7
9 170 70 8
2 225 125 9
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 5 4
Severe 4 1
Total 15 15
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.
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.
Two independent groups
Post-surgical pain Traditional Laparoscopy
Rankings
None 1 1+
None 3 3
Slight 5 9+
Slight 7 15
Moderate 5 21+
Moderate 4 25
Severe 4 29+
Severe 1 30
Total 15 15
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 3.5
2 3.6 5.7
3 2.6 2.9
4 2.6 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
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 ≠ 0Patient A Analgesic B Analgesic Difference Rankings
1 3.5 3.5 0
2 3.6 5.7 -2.1 8
3 2.6 2.9 -0.3 3
4 2.6 2.4 0.2 2
5 7.3 9.9 -2.6 10
6 3.4 3.3 0.1 1
7 14.9 16.7 -1.8 6
8 6.6 6.0 0.6 5
9 2.3 3.8 -1.5 4
10 2.0 4.0 -2.0 5
11 6.8 9.1 -2.3 7
12 8.5 26.9 -18.4 11
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.
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 1
2 8 9
3 11 5
4 12 5
5 22 33
6 67 37
7 73 32
8 48 8
9 37 41
10 31 12
11 21 29
12 17 3
13 3 1
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.
Spearman ranks correlation
Country
Incidence colon ca
Mean of meat intake
Ranking of cancer
Ranking of meat intake
1 10 1 3 1
2 8 9 2 7
3 11 5 4 5
4 12 5 5 4
5 22 33 8 11
6 67 37 12 12
7 73 32 13 10
8 48 8 11 6
9 37 41 10 13
10 31 12 9 8
11 21 29 7 9
12 17 3 6 3
13 3 1 1 2
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
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.
