Which statistical test to apply

Some names of the statistical tests:

Ref. Dtsch Arztebl Int 2010; 107(19): 343–8 DOI: 10.3238/arztebl.2010.0343

Edit:error in the picture

Mann Whitney U test is same as wilxon rank sum test 

Wilcoxon Signed rank test is done in paired data

Ref. Dtsch Arztebl Int 2010; 107(19): 343–8 DOI: 10.3238/arztebl.2010.0343

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Minimisation

The most important drawback of the randomization software is the problem of unmatched groups. In the process of randomization it is probable that the treatment groups develop significant differences in some prognostic factors, especially when the sample size is relatively small (<200). If these factors have important effects on the primary or secondary outcomes of the study, any important difference in the levels of these factors invalidate the trial results, and necessitate complicated statistical analysis with unreliable results. Various methods have been used to overcome the problem of unmatched trial groups including minimization and stratification, with minimization providing more acceptable results. With minimization the first subjects are enrolled randomly into one of groups. The subsequent subjects will be allocated to treatment groups after hypothetical allocation of each subject to every group, and then calculating an imbalance score. Using these imbalance scores, we can decide to which group the new subject must be allocated, to have the minimum amount of imbalance, in terms of prognostic factors. Pure minimization is indeed completely deterministic, that is, we can predict which group the next subject will be enrolled in, provided the factor levels of the new subject are known. This may invalidate the principle of trial blindness and introduce some bias into the trial. To overcome this shortcoming some elements of randomness are incorporated into the minimization algorithm, to make the prediction unlikely. Unfortunately the whole process of minimization is well beyond the skill of a typical clinical researcher, especially when the problem of unequal group allocations has to be taken into account. The difficulty in computation has resulted in a relatively less frequent use of minimization methods, in randomized clinical trials. The computer software can perform excellently in these situations, especially when the implementation has been logical. In the following sections, the aspects of two minimization programs are presented. Again the selection of these programs is based on the availability and ease of use.

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Hosmer Lemeshow test

Proposed grouping based on the values of estimated probabilities.


2 grouping strategies:

1. Based on percentiles of estimated probabilities
2. Based on fixed values of estimated probabilities

With the first method, use of g=10 groups result in the first group containing
n1=n/10 subjects having the smallest estimated probabilities and the last group
containing n10=n/10 subjects having the largest estimated probabilities.


With the second method, use of g=10 groups results in cut points defined at the
values k/10, k=1,2……9 & the groups contain all subjects with estimated
probabilities between adjacent cut points.

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Mann-Whitney U test

Mann-Whitney U test: It is the non-parametric alternative test to the unpaired t-test. It is a non-parametric test that is used to compare two sample means that come from the same population, and used to test whether two sample means are equal or not.  Usually, the Mann-Whitney U test is used when the data is ordinal or when the assumptions of the t-test are not met.

Assumptions:

1. Sample drawn from the population is random
2. Observations are independent
3. Ordinal measurement scale

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Wilcoxon sign test

Wilcoxon sign test: It is a type of test of significance done in paired data if the data is non-parametrically distributed or ordinal data. Compared to paired t-tests which analyzes if the average difference of two repeated measures is zero and require metric (interval or ratio) and normally distributed data.

The Wilcoxon signed rank test relies on the W-statistics.  For large samples with n>10 paired observations the W-statistics approximates a Normal Distribution.  The W statistics is a non-parametric test, thus it does not need multivariate normality in the data.

The first step of the Wilcoxon sign test is to calculate the differences of the repeated measurements and to calculate the absolute differences.

The next step of the Wilcoxon sign test is to order the cases by increasing absolute differences.

For the Wilcoxon signed rank test we can ignore cases where the difference is zero.  For all other cases we assign their relative rank. In case of tied ranks the average rank is calculated.  That is if rank 10 and 11 have the same observed differences both are assigned rank 10.5.

The next step of the Wilcoxon sign test is to sign each rank.  If the original difference < 0 then the rank is multiplied by -1; if the difference is positive the rank stays positive.

The W-statistic is simply the sum of the signed ranks.

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Kruskal-Wallis test

Kruskal-Wallis test: It is a nonparametric  test, and is used when the assumptions of one-way ANOVA are not met.
In the ANOVA, we assume that the dependent variable is normally distributed and there is approximately equal variance on the scores across groups. While we use kruskal Wallis test when these assumptions are not met. Therefore, the Kruskal-Wallis test can be used for both continuous and ordinal-level dependent variables.  However, like most non-parametric tests, the Kruskal-Wallis Test is not as powerful as the ANOVA.

Null hypothesis: samples (groups) are from identical populations.

Alternative hypothesis: at least one of the samples (groups) comes from a different population than the others.

The distribution of the Kruskal-Wallis test statistic approximates a chi-square distribution, with k-1 degrees of freedom, if the number of observations in each group is 5 or more.  If the calculated value of the Kruskal-Wallis test is less than the critical chi-square value, then the null hypothesis cannot be rejected.  If the calculated value of Kruskal-Wallis test is greater than the critical chi-square value, then we can reject the null hypothesis and say that at least one of the samples comes from a different population.

Assumptions:
1. Sample drawn is random
2. Observations are independent of each other
3. Measurement scale for the dependent variable is atleast ordinal

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Fisher Exact test

Fisher Exact test: It is a type of test of significance that is used in the place of chi square test in 2 by 2 tables, especially in cases of small samples. (frequency in one box is less than 5 or less than 20% of expected)

The Fisher Exact test tests the probability of getting a table that is as strong due to the chance of sampling. The word ‘strong’ is defined as the proportion of the cases that are diagonal with the most cases.

Generally used in one tailed tests. It can also be used as a two tailed test as well. It is sometimes called a Fisher Irwin test.

The Fisher Exact test uses the following formula:

p= ( ( a + b ) ! ( c + d ) ! ( a + c ) ! ( b + d ) ! ) / a ! b ! c ! d ! N !

In this formula, the ‘a,’ ‘b,’ ‘c’ and ‘d’ are the individual frequencies of the 2X2 contingency table, and ‘N’ is the total frequency.

This formula is used to obtain probability of the combination of the frequencies that are actually obtained. 

Assumptions:

Sample drawn by random sampling

Directional hypothesis is assumed. The directional hypothesis assumed (either a positive association or a negative association, but not both)

Data is not paired

Mutual exclusivity within the observations is assumed

Dichotomous level of measurement of the variables is assumed

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