Using non-parametric statistics to carry out Kolmogorov-Simonov tests and Kruskal-Wallis tests
Here we will carry out both Kolmogorov-Simonov tests and Kruskal-Wallis tests when looking for the information of salaries of graduates of the courses listed.
Question
(i) A number of different patterns were used by subjects climbing a portable straight ladder according to specified instructions. The ascent times for 5 subjects who used a lateral gait and 6 who used a diagonal gait were.
Lateral: 0.86, 1.31, 1.64, 1.51, 1.53
Diagonal: 1.27, 1.82, 1.66, 0.85, 1.45, 1.35
(a)Carry out the Kolmogorov-Smirnov test manually for these data. State your conclusions
(b)Ascent times for a further 4 subjects who used a mixed gait were also available. Describe the Kolmogorov-Smirnov test to compare the three groups
(ii) To gain information on salaries for its graduates, a school of mathematics and statistics selected a random sample of students from each of the subjects (1) mathematics (2) statistics (3) applied mathematics, and (4) actuarial science. The salaries (in 1000’s of euro) three years after graduating were:
| Mathematics | 35.4 | 40.5 | 40.3 | 42.3 |
| Statistics | 44.7 | 38.9 | 39.5 | 40.1 |
| App.Mathematics | 46.7 | 37.6 | 38.2 | 40.7 |
| Actuarial Science | 40.0 | 41.0 | 40.5 | 42.1 |
Compare the four groups manually using the Kruskal-Wallis test. State the assumptions made and state your conclusions.
Solution:
The problem objective is to describe the difference of the climbing times of the ladder in two populations, one which is later gait and the other which is Diagonal gait. On the basis of the available ample data now, we need to test whether a significant difference in the mean climbing times for the two gaits
let μ_1-μ_1=0 is the difference between mean climbing times for the lateral gait and diagonal giat.
The null hypothesis is given as:
The mean climbing for lateral gait is identical to that of the diagonal gait
Ho: μ_1-μ_2=0
The alternative hypothesis is given as
The mean climbing time for lateral gait differs significantly from that of diagonal gait
Ha: μ_1-μ_2≠0
Level of significance α=0.05
Assumptions:
The samples are independent random samples and the two population distributions have the same shape and spread. Therefore, we can use the Kolmogorov Smirnov test for the significance of the above hypothesis.
Test static for Kolmogorov-smirnov statistics (D) =Maximum|Fo(X)-Ft(X)|=0.333
P-values of test = 0.818
Since the p-value is greater than 0.05, we fail to reject the null hypothesis. Therefore, we conclude that the mean climbing for lateral gait is identical to that of the diagonal gait
When we have a third group of mixed gait, we have
| lateral | diagonal | Mixed |
| 0.86 | 1.27 | 2.13 |
| 1.31 | 1.82 | 3.13 |
| 1.64 | 1.66 | 3.3 |
| 1.51 | 0.85 | 2.36 |
| 1.53 | 1.45 | 2.98 |
| 1.35 | 1.35 |
The Kolmogorov test can only be used for two samples; it cannot be used for three groups.
ii.
Assumption of Kruskalwalis test
- The samples must be random
- The samples are mutually independent
- The measurement scale is at least ordinal and the variable is continuous
Null hypothesis: the population median is equal
Alternative hypothesis: the population median is not all equal
| mathematics(1) | statistics (2) | applied maths (3) | actuarial (4) |
| 35.4 | 44.7 | 46.7 | 40 |
| 40.5 | 38.9 | 37.6 | 41 |
| 40.3 | 39.5 | 38.2 | 40.5 |
| 42.3 | 40.1 | 40.7 | 42.1 |
Calculations
| Rank 1 | Rank 2 | Rank 3 | Rank 4 |
| 5 | 19 | 20 | 10 |
| 13.5 | 8 | 6 | 16 |
| 12 | 9 | 7 | 13.5 |
| 18 | 11 | 15 | 17 |
| 2.5 | 2.5 | 2.5 | 2.5 |
| n=5 | n=5 | n=5 | n=5 |
H=0.029×2216.5-63
H=0.3286
P-value = 0.95457
Critical value χ^2 (3)=7.81
Since Kruskalwalis test statistic (D) < critical value or since p > 0.05, we fail to reject the null hypothesis and conclude that the test is not statistically significant. Therefore, we can say that the distribution of the salaries after three years of graduating is the same.