Definition and properties of the non-parametric methods

Strengths and weaknesses of non-parametric methods

Parametric and non-parametric methods: correspondence and comparison

Situations in which non-parametric tests are used

*Key Words and Terms:*

Friedman

Kendal

Kruskall Wallis

Non-parametric

Rank correlation coefficient

Wilcoxon Rank sum test

Sign test

Wilcoxon Signed rank test

Spearman rank order correlation coefficient

* *

1.0INTRODUCTION

1.1 DEFINITION AND NATURE

Non-parametric methods do not conform to normality assumptions. They were first introduced as rough, quick and dirty
methods and became popular because of their ease and not being constrained by normality assumptions. They were later found
to be powerful and valid even in conditions of normally distributed data. They are about 95% as efficient as the more complicated
and involved parametric methods. Their popularity is likely to wane with availability of easy computing that negates the major
advantage of simplicity of non-parametric tests.

1.2 ADVANTAGES

These methods are simplicity itself. They are easy to understand and employ. They do not need complicated mathematical
operations thus leading to rapid computation. They have few assumptions about the distribution of data. All they require is
to array the data in ranks. They can be used for non-Gaussian data. They can also be used for data whose distribution is not
known because there is no need for normality assumptions. They have the further advantage that they can be used for data that
is expressed only as ranks. These methods are more robust; we can gain robustness at the expense of power.

1.3 DISADVANTAGES:

Non-parametric methods can be used efficiently for small data sets. With data sets that have many observations,
ranking becomes cumbersome and the methods can not be applied with ease. It is
difficult to estimate precision because computation of confidence intervals is cumbersome. These methods are also not easy
to use with complicated experimental designs. Non-parametric are less efficient than parametric methods for normally-distributed
data. They will require a larger sample size than comparable parametric methods to be able to reject a false null hypothesis.
Hypothesis testing with non-parametric methods is less specific than hypothesis testig with parametric methods.

1.4 CHOICE BETWEEN PARAMETRIC AND NON-PARAMETRIC

Parametric methods are most powerful (ie lower type 2 error) when normality assumptions hold. They are also more
efficient in using all available data. Non-parametric are less powerful and less efficient for normal data. They are concerned
with the direction and not the size of the difference between the groups being compared. They are most powerful for non-normal
data. Non-parametic methods should never be used where parametric methods are possible. Non-parametric should therefore be
used only if the test for normality is negative. In general non-parametric methods are used where the assumptions of the central
limit theorem do not apply. Non-parametric methods are also used in situations in which the distribution of the parent population
is not known. If the unknown distribution is not normal the non-parametric tests are the right choice. If the distribution
is normal then at least 95% efficiency is achieved compared to parametric tests. In case of ordinal or ranked data there is
no other analytic choice other than non-parametric tests.

When faced with a data set it is worth testing it for normality in order to decide the analysis to be used. The
Lilliefor test is a simple test of normality without assuming any particular mean or standard deviation. The variables are
standardized and the standardized variables are tested for normality.

1.5 CORRESPONDENCE OF PARAMETRIC & NON PARAMETRIC

Situation |
Parametric
test |
Non-parametric
test |

1
sample |
z-test,
t-test |
Sign
test |

2
independent sample means |
t-test |
Rank
Sum test |

2
paired sample means |
t-test |
Signed
Rank Test |

3
or more independent sample means |
ANOVA
(1-way) |
Kruskall
Wallis |

Multiple
comparisons of means |
ANOVA
(2-way) |
Friedman |

Correlation |
Pearson |
Spearman |

Comparing
survival curves |
Proportional
hazards regression |
Log
rank test |

Virtually each parametric test has an equivalent non-parametric one as shown in the table above. Note that the
Mann-Whitney test gives results equivalent to those of the signed rank test. The Kendall gives results equivalent to those of the Spearman coefficient. The signed rank and rank sum
tests are based on the median.

2.0 SPEARMAN RANK CORRELATION COEFFICIENT

The 2 samples being compared are ordered from high to low and a rank is assigned to each observation. The differences
between the ranks of corresponding pairs of observations are computed. The differences are then squared. The sum of the squared
differences is computed and are summed up. The Spearman rank correlation coefficient is then determined as 6 x {sum of squared
differences}/n(nxn-1) where n is the sample size. This can be presented in symbols as r_{s} = {1 - 6åd^{2}}
/ {n(n^{2} – 1)}where n = number of pairs and d = (rank of x) – (rank of y). The significance of the correlation
coefficient is determined by using the t test where t = [{r_{s}(n-2)^{1/2}}] / [{1 - r_{s}^{2}}^{1/2}]^{
}with n-2 degrees of freedom.^{ }There are tables that can be used to look up the significance of the rank correlation
coefficient. The advantage of rank correlation is that comparisons can be carried out even if actual values of the observations
are not known. It suffices to know the ranks.

3.0 THE SIGN TEST FOR ONE SAMPLE TESTS

The concept of the sign test is very simple to grasp. The test is based on the binomial proportion. If a value
is picked at random from any distribution, the probability that it will be less than the median is 0.5. The probability that
it is more than the median is also 0.5. The sign test thus makes assumptions based only on the median and does not refer at
all to any population parameters. In a 1 sample test we want to test the null hypothesis H_{0} : sample median = population
median. The sign + is assigned to observations above the population median. The sign – is assigned to observations below
the population median. If H_{0} is true, the number of pluses is equal to the number of minuses. Zero is assigned
to any tie; ties are not counted. The p-value can be read off the appropriate tables depending on the number of pluses and
minuses.

4.0 TESTS FOR 2 SAMPLES

SIGNED RANK TEST FOR 2 PAIRED SAMPLES

This is a test of the hypothesis H_{0}: sample median1 = sample median2. The differences between the first
and second measurement are computed by simple subtraction. The sign of the difference is ignored for the moment and ranks
are assigned to the difference from high to low. A positive or negative sign is assigned to each rank according to the sign
of the original difference. The appropriate tables are then used to look up the p value for the given sample size, sum of
positive ranks, and sum of negative ranks. Alternatively the z test statistic can be used being defined as z = (w_{o}
- w_{e})/s_{w }= (w_{o} - w_{e}) / {2n + 1) w_{e}/6}^{1/2}
where w_{e} = (1/2) n(n + 1) and w_{o} = the sum of positive ranks.

RANK SUM TEST FOR TWO INDEPENDENT SAMPLES

Both samples are combined and the observations are ordered from low to high. A rank is assigned to each observation
while making sure that the original group of each observation is not mixed up. The ranks of the group with the smaller size
are added up. The rank is referred to the appropriate tables and the p-value is looked up. Alternatively the z test statistic
based on the ranks of the smaller sample can be used defined as follows: z = (w_{o} - w_{e})/s_{w} = (w_{o} - w_{e})/ [n_{1}n_{2 }(n_{1} + n_{2}
+1)/12}]^{1/2} where w_{o }= sum of ranks in othe smaller sample, w_{o} = expected sum of ranks =
{n_{1}(n_{1} + n_{2} +1)}/2.

5.0 TESTING 3 or MORE SAMPLES

KRUSKALL WALLIS

This a 1-way test for 3 or more independent sample means. It is the non-parametric equivalent of 1-way ANOVA. The
observations from various groups are combined and are arranged in order of magnitude from the lowest to the highest. Ranks
are assigned before the observations are restored to their original groups. The ranks are summed in each group and the test
statistic is constructed as follows: H = 12/N(N+1) ^{k}å_{j=1} R_{j}^{2}
/ n_{j }– N(N+1) where n_{j} = number of observations in a group, N= number of observations combined
for all groups, R_{j} = sum of ranks in the jth group, k = number of groups.

FRIEDMAN

This is a (2-way
test for 3 or more independent sample means.