Nonparametric tests are chosen depending on the type of variable and on number of groups which are compared. Let us again discuss our Second example with values of minimal inhibitory concentrations (MICs) of essential oils with and without antibiotic gatifloxacin against two bacterial species.
First let us discuss criteria for dependent samples. They are grouped by the type of data:
For order and continuous data we use different criteria depending on the number of groups. When we have two groups, the most commonly used criteria are Wilxocon test and sign test, also Hodges-Lehman estimator sometimes is used. Unlike first two criteria which compare medians in groups, the Hodges-Lehman estimator test estimates confidence intervals which depend on the mean and error of the mean.
The sign test uses only directional information (assesses at which column a value is bigger for every case), while the Wilcoxon test uses both direction and magnitude information. Thus, the Wilcoxon test is more powerful statistically than the sign test. However, the Wilcoxon test assumes that the difference between pairs of scores is ordinally scaled, and this assumption is usually difficult to test.
When we have to compare several groups, analogue of parametric ANOVA, the Friedman’s ANOVA, is used or the Kendall’s coefficient of concordance.
Criteria for independent samples differ by mechanism of comparison: they may compare distributions, ranges, medians, or estimate confidence intervals. Also selection of criterion depends on a number of groups – different criteria are used when we compare two groups or several groups.
The most commonly used are Mann-Whitney U test and Kolmogorov-Smirnov test for two groups, and Kruskall-Wallis ANOVA for several groups.
Performing non-parametric statistics analysis in SPSS
We continue our second examples about essential oils in combinations with gatifloxacin against two bacterial species (see Second example). First, we will compare MICs of tea tree oil alone and in the presence of gatifloxacin. This will be the analysis of dependent groups:
1. Click the Analyze menu, point to Nonparametric Tests, and select Related Samples… :
The Nonparametric Tests: Two or More Related Samples dialog box opens:
2. Leave selected by default Automatically compare observed data to hypothesized in Objective tab.
3. Click the Fields tab, where select fields (variables) “Tea tree” and “Tea tree + Gati”:
Click the transfer arrow button . The selected variables are moved to the Test Fields: list box.
4. Click the Settings tab and select Automatically choose the tests based on the data:
In this way statistical tests will be chosen automatically.
5. Click the Run button.
6. An Output Viewer window opens and displays the statistics for nonparametric tests.
There is one more variant to start nonparametric tests:
1) Click the Analyze menu, point to Nonparametric Tests, point to Legacy Dialogs and select 2 Related Samples… :
The Two-Related-Samples Tests dialog box opens:
2) Select the “Tea Tree” variables; click the transfer arrow button . The selected variable is moved to the left cell of Test Pairs: list box (Variable 1).
3) Select the “Tea Tree + Gati” variable; click the transfer arrow button . The selected variable is moved to the right cell of Test Pairs: list box (Variable 2).
4) In the Test Type section select statistical test which should be used. By default Wilcoxon test is already selected.
5) Click the OK button. The Output Viewerwindow opens with statistical results:
The main thing to look for in the Output Viewer window is the significance value. This is the probability that the null hypothesis is correct. The Null hypothesis is also written here – it states that there are no differences between groups, that is medians are equal.
Since we normally work with a significance value of 0.05, i.e. a 95% certainty of getting the right answer:
If the significance value is less than 0.05, we reject the null hypothesis.
If the significance value is greater than or equal to 0.05, we accept the null hypothesis.
Here significance value is 0.007. We reject the null hypothesis about absence of differences and we can conclude that differences between groups are statistically significant: MICs of tea tree oil in the presence of gatifloxacin is significantly different from MIC of tea tree oil without gatifloxacin.
The next research question: Are there differences between MIC of tea tree oil with and without gatifloxacin either against E. coli or against E. faecalis?
To answer this question we have to split file. Now we want to compare MICs in the media with and without gatifloxacin separately for two species, for E. faecalis and for E. coli. We choose split criterion – the species of microorganism.
Procedure of specifying analysis is totally the same as for the whole dataset (see above). If we want to select tests ourselves instead of automatically, in the Settings tab we should first select Customize tests and then we can select appropriate criteria, for example, Sign test (2 samples) and Wilcoxon matched-pair signed-rank (2 samples):
After clicking the Run button an Output Viewer window appears with statistical results.
For E. coli differences between MICs of tea tree oil without gatifloxacin and MICs of tea tree oil with gatifloxacin by using Sign test are significant (p = 0.039) but by using Wilcoxon test are not significant (p = 0.091):
For E. faecalis – the same results. Differences between MICs of tea tree oil without gatifloxacin and MICs of tea tree oil with gatifloxacin by using Sign test are significant (p = 0.021) but by using Wilcoxon test are not significant (p = 0.074):
Now we will repeat the same procedure for thyme oil without gatifloxacin and in the presence of gatifloxacin. For this purpose we select vriables “Thyme” and “Thyme + Gati”.
For thyme oil with E. coli differences are significant using both criteria: p = 0.006 for Sign test and p = 0.004 for Wilcoxon test:
The same is for E. faecalis – differences in MICs of thyme oil without and with gatifloxacin in the medium are significant using both criteria:
The next research question: Do the MICs of studied essential oils significantly differ between two bacterial species – E. coli and E. faecalis?
In other words now we want to compare MICs of tea tree oil against E. faecalis and against E. coli as well as to compare other MICs between these microorganisms. We will use criteria for independent samples (see above).
Before analyze of independent groups we should reset splitting the file. In the Split File dialog box select Analyze all cases, do not create groups (see Splitting the file).
Then we should specify variables and tests for the analysis:
1. Click the Analyze menu, point to Nonparametric Tests, and select Independent Samples… :
The Nonparametric Tests: Two or More Independent Samples dialog box opens:
2. Leave the selected by default Automatically compare distributions across groups in the Objective tab.
3. Click the Fields tab, where select variables “Tea tree”, “Tea tree + Gati”, “Thyme” and “Thyme + Gati”:
click the upper transfer arrow button . The selected variables are moved to the Test Fields: list box.
4. Select the “Species” variable, click the lower transfer arrow button . The selected variable is moved to the Groups: list box.
5. In the Settings tab leave selected by default Automatically choose the tests based on the data:
6. Click the Run button.
7. An Output Viewer window opens and displays the statistics for nonparametric tests:
The results of analysis demonstrate that statistically significant differences between E. faecalis and E. coli are characteristic only of tea tree oil without gatifloxacin. The analysis of means revealed that MIC of tea tree oil against E. coli was lower than against E. faecalis (1.09 and 2.75 % vol./vol., respectively).
Therefore, summarizing above written, during comparison of MICs of two essential oils alone and in the presence of antibiotic gatifloxacin we performed the following steps:
We calculated descriptive statistics for MICs of 2 essential oils in the absence and in the presence of sub-inhibitory concentration of gatifloxacin against two groups of bacteria – E. coli and E. faecalis.
We assessed distribution of these variables – for all MICs it was non-normal.
We compared different MICs between themselves using nonparametric criteria for dependent samples.
We compared MICs for different groups of microorganisms using nonparametric criteria for independent samples.
Performing these calculations allowed us to formulate following conclusions:
The MICs of both studied oils in the presence of gatifloxacin significantly differ from MICs of oils alone, that is gatifloxacin enhances activity of both oils.
Activity of tea tree oil alone was different against studied strains with higher activity against E. coli (because MIC against E. coli was lower, as activity is opposite to MIC).
Activity of tea tree oil in the presence of gatifloxacin and activity of thyme oil either alone or with gatifloxacin was equal against E. coli and E. faecalis.