The aim of this repository is to show the statistical workflow when comparing groups of data in order to prove if there is significant difference between them. As data I provide some borrowed from my master thesis (in spanish), which consisted in a first approach to establishing background tissue concentration in macroinvertebrates of rivers from mining areas of northern Spain. This time we will only perform a statistical analysis to compare three different taxa to set if their metal levels really differ between them. The three taxa are: Heptageniidae (scraper), Hydropsychidae (collector-filterer) and Rhyacophilidae (predator).
The first step is always to take a quick view of the basic features of the data, perfoming some descriptive analysis.
> summary(mydata) # quick first look at global data
Taxa As Se Cd Hg
Heptageniidae :27 Min. :0.1100 Min. : 0.500 Min. :0.010 Min. :0.0300
Hydropsychidae:23 1st Qu.:0.6325 1st Qu.: 1.930 1st Qu.:0.080 1st Qu.:0.0700
Rhyacophilidae:20 Median :1.0900 Median : 3.605 Median :0.255 Median :0.1000
Mean :1.9741 Mean : 4.961 Mean :0.622 Mean :0.1359
3rd Qu.:2.9775 3rd Qu.: 6.478 3rd Qu.:0.865 3rd Qu.:0.1500
Max. :7.2300 Max. :20.280 Max. :2.820 Max. :0.5200 
At first look we could think that there may be signifficant differences in the next cases:
- As -> the three taxas
- Se -> Hepta vs Hydro and Rhya
- Cd -> Hepta vs Hydro and Rhya
- Hg -> none
In the next step we will prove if our hypothesis is true or not.
To know which statistical test we need to use we have to know if our data follows a normal distribution or not. For this case we'll use three approaches:
- Skewness must be in range of -2 and 2
- p value os Shapiro-Wilk must be > 0.01
- p value of Levene's homogeneity of variance test must be > 0.01
This are the results:
--------------------------------------------------
Taxa Metal Shapiro.Wilk Skewness
---------------- ------- -------------- ----------
Heptageniidae As 0.292 0.23
Heptageniidae Se 0.216 0.77
Heptageniidae Cd 0.456 0.3
Heptageniidae Hg *0.009* 1.54
Hydropsychidae As *0.004* 0.68
Hydropsychidae Se 0.334 0.76
Hydropsychidae Cd 0.106 **2.6**
Hydropsychidae Hg *0* 1.74
Rhyacophilidae As *0.001* 0.28
Rhyacophilidae Se *0* 0.33
Rhyacophilidae Cd *0* 1.81
Rhyacophilidae Hg *0.001* 1.53
--------------------------------------------------
Table: *p<0.01; **-2<p>2
---------------------
Metal Levene.Test
------- -------------
As *0*
Se *0*
Cd *0*
Hg 0.154
---------------------
Table: *p<0.01
As we can see, some of the varaibles adjust to normal distribution and others don't. To better understand the difference we'll create some visualization for each one of the tests. For Shapiro-Wilk we can perform a QQ plot, which is commonly used to detect deviations from the normal distribution.
Cd in Heptageniidae fits the QQplot, Cd in Rhyacophilidae doesn't.
We can visualize the skweness with histograms.
In the case of Hydropsychidae the density line is skewed to the left.
Finally to better understand the Levene's test we can build some boxplots and pay attention to the whiskers.
Heptageniidae shows a lot of variance in all metals except in Hg, that's why Levene fails for the first three metals.
NOTE: One important thing to consider is the amount of samples that we are working with, less than 30 for each metal and taxa. The smallest the population the harder the probability of having normal ditribution. Outliers can have a big impact.
None of the metals pass the three test we stablished on all taxa. Transforming the data we may achieve normal distribution, so that's the next step. Take note though, that in Cd in Hydropsychidae showed skewness, a parameter that does not change even if the data is transformed,so we already know that the data referring to Cd must be treated with non-parametric test.
After a logarithm transformation this are the results for SW and Levene:
---------------------------------------
Taxa Metal Shapiro.Wilk
---------------- ------- --------------
Heptageniidae As *0*
Heptageniidae Se 0.912
Heptageniidae Cd 0.449
Heptageniidae Hg 0.18
Hydropsychidae As 0.243
Hydropsychidae Se 0.383
Hydropsychidae Cd *0.002*
Hydropsychidae Hg 0.162
Rhyacophilidae As 0.959
Rhyacophilidae Se 0.315
Rhyacophilidae Cd 0.815
Rhyacophilidae Hg 0.365
---------------------------------------
Table: *p<0.01
---------------------
Metal Levene.Test
------- -------------
As 0.609
Se 0.597
Cd 0.551
Hg 0.919
---------------------
Table: *p<0.01So now, all cases pass the Levene's Test, and only two cases do not pass the SW -> As and Cd. This means that we can perform parametric test with the transformed data for Se and Hg, and non-parametric test for As and Cd with original data.
First we run a global test to see if there are signfficant differences between groups, in this case an ANOVA. If the the p value is <0.05 there are differences, and we have to perform a paired comparison as a post-hoc analysis, in this case Bonferroni.
Results:
---------------
Metal Anova
------- -------
Se *0*
Hg 0.057
---------------
Table: *p<0.05
--------------------------------------------------
Comparisons Se Hg
-------------------------------- --------- -------
Heptageniidae - Hydropsychidae *0* 0.068
Heptageniidae - Rhyacophilidae *0.004* 0.28
Hydropsychidae - Rhyacophilidae *0.001* 1
--------------------------------------------------
Table: *p<0.05As non-parametric tests we'll use Kruskal-Wallis as global comparation and a Dunn's Test as post hoc paired comparison.
Results:
------------------------
Metal Kruskal.Wallis
------- ----------------
As *0*
Cd *0*
------------------------
Table: *p<0.05
--------------------------------------------------
Comparisons As Cd
-------------------------------- --------- -------
Heptageniidae - Hydropsychidae *0* *0*
Heptageniidae - Rhyacophilidae *0* *0*
Hydropsychidae - Rhyacophilidae *0.001* 0.079
--------------------------------------------------
Table: *p<0.05
So now we know which cases have signifficant differences we can visualize the boxplots from before with annotations of this cases
As a final step we can check the correlation between the metals and the taxa with a PCA
So most influencial metals in taxa differentiation are As and Se; Cd has a bit of differentiation power towards Heptageniidae; Hg shows same accumulation levels in three taxa.