Test statistics

There are several techniques that can be used for testing, if the sample is significantly different than their corresponding benchmark. The following section discusses the use of parametric or non-parametric tests for this purpose.

Page 27 of 78 7.6.1 Normality test or parametric tests

When testing data one of the most easy and used methods are the normal distributed tests. As the name implies these tests assume that the underlying distribution of the data is normal distributed. If the data, that is used, are not normal distributed the reliability of these tests are very low, and the findings from these tests cannot be trusted since the p-value may be

inaccurate. When the data distribution is not normal distributed the parametric tests are not well specified, and therefore non parametric tests are more reliable.

7.6.2 Jarque-Bera test for normality

The Jarque Bera test is a test that measure if a data sample is normal distributed. The test has the null hypothesis that the data are from a normal distribution. A normal distribution is expected to have a kurtosis of 3 and a skewness of 0 and if the data have large deviations from this the JB value increases. The JB test is defined as:

J

S

2

4

)

Where n is the number of observations, S is the sample skewness and K is the sample kurtosis.

When the sample is close to being normal distributed the K ≈ 3 and S ≈ 0 and then JB ≈ 0. The JB value is compared to the chi-square distribution with two degrees of freedom. Large values of JB will often reject normality.

In the empirical section, the analysis of the data is found not to be normal distributed and normal distributed tests cannot be used. Therefore these tests will not be described further.

7.6.3 Non-parametric tests

When the data sample is not normal distributed the non-parametric tests outperform the

parametric tests, both in terms of power and efficiency (Barber & Lyon 1996). Therefore when the distribution of the empirical data violate the assumptions of the parametric, (e.g., if the distributions are highly skewed) the non parametric tests are more reliable.

Nonparametric tests compare medians (where parametric compare means) and below is a description of the non parametric tests used in the analysis. There are 2 non parametric tests used: The Wilcoxon Signed Rank Test is used to compare a group to a value, and the Mann-Whitney test is used for comparing two unpaired groups.

The Mann-Whitney test

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Mann-Whitney U test is the non parametric alternative to the parametric t-test. Mann-Whitney U test is used to test whether the median of two not dependent samples are statistically equal or not. The test ranks each value of the 2 samples from lowest to highest; the ranks of the 2 samples are summed. If the 2 populations have the same median what is the chance that the sum of the rankings in each sample, are as different as found in the test. The sum of the ranks in the 2 samples are therefore compared to critical value tables to see if the observed

difference is a result of coincidences or due to statistical differences in the 2 samples.

The null hypothesis that is tested is: H0: No difference between samples.

And the alternative hypothesis is: H₁: The 2 samples are different.

The test is calculated by ranking all the samples from lowest to highest with no regard to which samples the observations is from. In both sample 1 and 2 the ranks of each observation is summed. The test value U from sample 1 is then calculated by the formula:

Where R1 is the sum of ranks in sample 1, n1 is the sample size for sample 1. The U value for sample 2 can be calculated by the same formula just by using R and n for sample 2.

The smallest of the U₁ and U₂ is the value used to test against significance tables. If the sample size is large U is approximately normal distributed and can be standardized by the formula:

Where mU is the mean of U and U is the standard deviation of U and they are calculated by:

The z value can then be compared to critical values for the chosen significant level of 0,05. If U or z exceeds the critical value the null hypothesis can be rejected.

Wilcoxon Signed Rank Test

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The Wilcoxon Signed Rank test is much similar to the Mann Whitney test but it is more useful when comparing a sample to a value. The null hypothesis of the Wilcoxon Signed Rank Test is:

H₀: The median of the sample is equal to zero And the alternative hypothesis: H₁: The median are not equal to zero.

The test is calculated by ranking the absolute values in the sample, with the lowest absolute value having the lowest rank. Then the ranks of the positive and the negative ranks are

calculated. Then tests are made of the mean of the positive and the mean of the negative ranks.

The hypothesis is that there is no significant difference between positive and negative ranks, and that the median therefore are zero.

The lowest value of the sum of ranks is chosen as U and then the z value is calculated as:

z U E t

t

Where is the expected value of the ranks and can be calculated as

and is the standard deviation and can be calculated as _t ^{n n 1}₂₄ ^{2n 1}

The z value can then be compared to critical values for the chosen significant level of 0,05. If z exceeds the critical value, the null hypothesis can be rejected.

Critique of the models

Non parametric test compare medians by ranking the values. If the data have large differences between 2 ranks, the model does not value these differences. If the data have extreme values the model does not take the extreme values into its calculations. Higher positive values than negative values will though be seen by differences in the mean of the 2 samples. Since our data is not normal distributed the parametric tests cannot be used. To make up for the weaknesses by the non parametric models the mean in each of the tests are also considered even though they are not statistically analyzed.