“Monday Journal Activity”

A Parametric test A Non-parametric test

“Many-sample location parameter tests” bout

Tanja Van HeckeFaculty of Applied Engineering Sciences,Ghent University,Ghent ,BelgiumTanja.VanHecke@hogent.be

Presented by: Mr. Benito Jr. B. CuananScientific Publishing and Statistics DepartmentStrategic Center for Diabetes ResearchKing Saud University

Suppose levels of Patotine (a non-existing hormone) of different groups are being investigated. Suppose the groups are: “the underweight group” , “the normal group” and “the overweight group”. Once data are collected, how should this three groups be compared?

-----The most common approach would be to compare the mean or median levels of the different groups.

But how?

Take note:It would be very awkward if in your research paper, you present your data like this:“ … figures in table 1.1 are the Tukey’sBiweight m-estimator…”

•Consider the table on the right.

•If we are to compare the 3 groups, what test statistics should we use?

Looking at the type of data ( in this case: continuous) and the number of groups to be compared( in this case: 3 groups), our “textbook” will surely suggest: …

USE: one-way ANOVA

Are you sure about that?

Have you checked the normality?

How about its scedasticity?

This paper described the comparison of the ANOVA and the Kruskal-Wallis test by means of the power when violating the assumption about normally distributed populations. The permutation method is used as a simulation method to determine the power of the test. It appears that in the case of asymmetric populations, the non-parametric Kruskal-Wallis test performs better than the parametricequivalent ANOVA method.

A parametric test The most commonly used test for location. Used to analyze the differences between group means and

their associated procedures. It models the data as:

yij= µi + εij ,

µi is the mean or expected response of data in the i-th treatment.

εij are independent, identicallydistributed normal random errors.

The one-way ANOVA is used to test the equality of k (k > 2) population means, so the null hypothesis is:

H0 : μ1 = μ2 = . . . = μk.Assumptions:1. The dependent variable is

normally distributed in each group that is being compared.

2. There is homogeneity of variances. This means that the population variances in each group are equal.

one-way ANOVA may yield inaccurate estimates of the p-value when the data are not normally distributed at all.

If the sample sizes are equal or nearly equal, ANOVA is very robust. If not, then the true p-value is greater than the computed p-value.

What if the assumptions are

violated?

This paper focused on the violation of the first(in our list) assumption of ANOVA,

that is: violation on NORMALITY of each group’s distribution.

A Non-parametric test The “analogue” of ANOVA in testing for location. Used to compare the medians of 3 or more independent

groups. It models the data as:

yij = ηi + φij ,

ηi is the median response of data in the i-thtreatment.

φij are independent, identicallydistributed continuous random errors

Unlike the ANOVA, this test does not make assumptions about normality. However, it assumes that each group have approximately the same shape.

Like most non-parametric tests, it is performed on the ranks of the measurement observations.

The null hypothesis of the Kruskal-Wallis test states that the samples are from identical populations.

When rejecting the null hypothesis of the Kruskal-Wallis test, then at least one of sample stochastically dominates at least one other sample.

POWER: Statistical power is defined as the probability that the test

correctly rejects the null hypothesis when the null hypothesis is false.

It is the “sensitivity” of the test. In this paper, empirical power were calculated.

PERMUTATION TEST:The steps for a multiple-treatment permutation test:

• Compute the F-value of the given samples, called Fobs.• Re-arrange the k n observations in k samples of size n.• For each permutation of the data, compare the F-value with the Fobs

For the upper tailed test, compute the p-value asp =#(F >Fobs)/ntot

If the p-value is less than or equal to the predetermined level of significance α, then we reject H0.

if we have 3 groups with 20 observations each, there are (60!)(20!)-3

possible different permutations of those observations into three groups.

That’s equivalent to 577,831,814,478,475,823,831,865,900

The Monte Carlo simulation was used.

random data from a specified distribution with given parameters were generated.

ANOVA and Kruskal-Wallis test were then conducted.

2500 samples from chosen distributions were tested at α = 0.05.

The following hypotheses were considered:Ho : µ1 = µ2 = µ3

and H1 : µ1 + d = µ2 and µ1 + 2d = µ3

Ex. @d=0.3:if µ1 = 1 , then µ2 =1.3 and µ3 =1.6

The two tests were compared in 3 distribution types, namely: Normal, Lognormal, and Chi-squared distribution.

Normal Distribution

Chi-squared distribution with 3 degrees of freedom

Lognormal Distribution( in red)

For non-symmetrical distributions, the non-parametrical Kruskal-Wallis test results in a higher power compared to the classical one-way anova.

The results of the simulations show that an analysis of the data is needed before a test on differences in central tendencies is conducted.

Although the literature and textbooks state that the F-test is robust under the violations of assumptions, these results show that the power suffers a significant decrease.