4-191125033339.pptx kolmogorov smirnov test

Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA
Kolmogorov-Smirnov test
Demystifying statistics series: Meta-analysis course Dr. S. A. Rizwan, M.D. Nov 2019 1
Dr. S. A. Rizwan, M.D.,
Public Health Specialist,
Saudi Board of Community Medicine,
Riyadh, Kingdom of Saudi Arabia.

Outline
• Introduction
• Uses
• Advantages and disadvantages
• Example
• Take home messages

Introduction
• KS test belongs a group of tests called goodness of fit tests
• Other members are chi square test and Anderson-Darling test
• Goodness of fit tests check whether there is a significant
difference between an observed frequency distribution and a
given theoretical (expected) frequency distribution
• Similar to what the Chi-Square test does, but the K-S test has
several advantages:
– More powerful test.
– Easier to compute and use, as no grouping of data is required.
– The test statistic is independent of the expected frequency distribution.
– It only depends on the sample size n.

Andrey Kolmogorov, Nikolai Smirnov

Introduction
• The hypotheses:
– H0: The observed frequency distribution is consistent with the theoretical
frequency distribution (Good fit).
– H1: The observed frequency distribution is not consistent with the theoretical
frequency distribution (Bad fit).
– α = Level of significance of the test.
• Here we use the cumulative probability distribution (CDF) of
observed and theoretical frequencies.

Introduction
• The basis of the test is that it relates the distance between the
cumulative fraction functions of the two samples as a number,
D, which is then compared to the critical-D value for that data
distribution.

K-S test statistic
• Here,
– Fe = the expected relative cumulative frequencies (CDF).
– Fo = the observed relative cumulative frequencies (CDF).
• If the gap between Fe and Fo is large then Ho should be rejected.
• A K-S test is a one tailed test.
• The critical values of Dn have been tabulated and can be found from the K-
S table for corresponding levels of significance and sample size n.
• The calculated value of Dn is compared with the critical value of Dn.
• If the calculated value > critical value, then reject H0.

K-S tables

Uses of K-S test
• We usually use this test to check the normality assumption.
• This test is used to test goodness of fit or shape and can be
used instead of the chi square test (one-sample K-S test).
• It simply needs the maximum difference between two
cumulative distribution curves.
• It can either compare a sample with a reference probability
distribution or it can directly compare two sample datasets
(two-sample K-S test).

The KS test

Steps in K-S test
• Sort data from smallest to largest.
• Convert the data distribution to a cumulative distribution S(x).
• Plot the cumulative distribution along with the comparison
distribution.
• Find the maximum absolute difference (D value).
• If D is greater than critical-D, then it can be concluded that the
distributions are indeed different, otherwise there is not
enough evidence to prove difference between the two
datasets.
• A P-value can also be calculated from the D-value and the
sample size of the two data sets

Strengths of the K-S test
• It is nonparametric.
• D-value result will not change if X values are transformed to
logs or reciprocals or any other transformation.
• No restriction on sample size.
• The D-value is easy to compute and the graph can be
understood easily.
• One sample K-S test can serve as a goodness-of-fit test and
can link data and theory.

Drawbacks
• The K-S test is less sensitive when the differences between
curves is greatest at the beginning or the end of the
distributions.
• It works best only when EDFs deviate the most near the
center of the distribution.
• The situation in which normality tests are needed -small
sample sizes- is also the situation in which they perform
poorly

Example 1
• H0: The distn. is normal with μ= 6.80, σ= 1.24.
• H1: Above not true.
• We obtain Fe values from the normal table, z= (X- μ)/ σ.
• The calculated value of Dn is the maximum value in the | Fe - Fo | column.
Thus, 0.0373.
• For the predecided level of significance, Dcritical = 0.0416.
• Dn < Dcritical , so accept H0 and conclude that it is a good fit.

Example 2

Example 3

Take home messages
• KS test belongs a group of tests called goodness of fit tests
• Used to check normality assumption
• The basis of the test is that it relates the distance between the
cumulative fraction functions of the two samples as “D”
• More powerful test than chi square test
• Less sensitive when the differences between curves is
greatest at the beginning or the end of the distributions
• SPSS tutorial https://www.spss-tutorials.com/spss-
kolmogorov-smirnov-test-for-normality/

Kindly email your queries to sarizwan1986@outlook.com
THANK YOU

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