Thanks but this is not really a normal question. Best wishes, David Booth

David Eugene Booth Thanks for your answer. As I am not from statistics background, may be I didn't ask the question properly. Check again, if it is understandable.

Dear Nushrat Chowdhury ,

These two tests are indicating whether you can assume that your data samples are normally distributed. As your sample size is small the recommended test is Shapiro-Wilk. Significance values less than 0.05 indicate evidence of non-normality. As many of your Shapir-Wilk sig values are below this threshold you should conclude that some of these data sets are not normally distributed, meaning that you should exercise caution in using a parametric test which might assume normally distributed data. It may still be possible to use a parametric test in a valid way. This is known as robust exceptions. However, as you haven't indicated what statistical model you are seeking to test, I cannot advise you any further. The issue of data transformation is separate.

I wouldn't expect PCB concentrations in sediments to follow a normal distribution or maybe a conditionally normal distribution. Your inclination to use a log transformation is a good one, as this kind of environmental contaminants data can often be approximated by a log-normal distribution.

You don't say what you are trying to do with this data, and it's difficult to give advice without knowing this.

If you are looking to present summary statistics, it may be reasonable to present the geometric means, even though the data don't follow a log-normal distribution. You can add to these an appropriate measure of dispersion or confidence intervals for the geometric means. I would recommend looking at histograms of your log-transformed data to see if this is a reasonable approach.

If it's not reasonable to use the geometric mean, you might just go with the medians or five number summary * for summary statistics . You could add confidence intervals for the medians.

If you are looking to model the data or conduct hypothesis tests --- if the log-normal data aren't appropriate for common OLS ** techniques --- you might look at Gamma regression *** or non parametric approaches.

__________

* https://en.wikipedia.org/wiki/Five-number_summary

** https://en.wikipedia.org/wiki/Ordinary_least_squares

*** https://en.wikipedia.org/wiki/Gamma_distribution

Dear @Salvatore,

Thank you very much for your kind answer. Yes, I am looking for summary statistics and just want to show seasonal and spatial distribution/ occurrence in the study area. The histograms for log normal data do not follow the bell shape for all congeners. I am going to follow your recommendations. I have a question. Is there another way to normalize the data?

Nushrat Chowdhury ,

There some pretty powerful techniques to transform data to a normal distribution. The inverse normal scores transformation (INST) works very well. Varieties go by names like Elfving, Blom, van der Waerden, Tukey, and rankit. And Tukey's ladder of powers transformation applies a transformation similar in approach to, say, a log transformation or a square-root transformation, but will find the best power to use for a power transformation. Data can be back-transformed for a power transformation. But data cannot be back-transformed for an INST because it is a rank-based transformation.

But the big question is, Why do want to transform your data to a normal distribution? What do you hope to accomplish?

Professor @Salvatore S. Mangiafico,

Thank you for your informative answer. A peer of my work has advised to normalize the data. I do have the same questions just like yours. As far as I understand....as my goal is not conducting any statistical analysis e.g., PCA, correspondence analysis etc., this data do not have to be normally distributed or significant just for presenting summary of PCBs (Correct me, if I am wrong). I have asked these questions few of the peers before and did not get any satisfactory answer.

When presenting your data, it's most useful for the reader if the summary statistics are in the original units of the measurements. That is, it easier to interpret a mean of 10 ng/g than to interpret a mean of log(concentration) of 2.3.

Often for environmental contaminants, we use a log transformation, calculate the mean or run statistical analyses, and then back-transform the results to the original units. Since this yields e.g. the geometric mean, and since this is what we use with e.g. E. coli concentrations, this is easy for the reader to understand.

However, taking the mean of a skewed distribution may not represent that data the best way. This is the case whether you are taking the mean of the original data or the mean of the log concentrations.

Especially for skewed or unusual distributions, it's helpful to present as much data as possible for the reader. One idea is to present all the data as individual dots ( https://r-graphics.org/R-Graphics-Cookbook-2e_files/figure-html/FIG-DISTRIBUTION-DOTPLOT-MULTI-1.png ). If this is too much, box plots are useful: ( https://www.mathworks.com/help/examples/stats/win64/CreateBoxPlotsForGroupedDataExample_01.png ).

For skewed distributions, mean and sd might not be the best. Depending on the method used, confidence intervals for the mean may be useful. Otherwise, the median and five number summary may work well.

As to statistical analyses, you do mention in your pre-print that you will be using anova to compare among seasons. If you are interested in a one-way test, you will have many options that may be applicable if the distribution of your data is troublesome. These might include rank-based tests or permutation-based tests, or others.

Sal Mangiafico,

I am going to apply your recommendations for summary statistics. I have conducted One-Way ANOVA test to find the variations among the concentrations of PCBs and seasonal differences. In both case, there are no statistical significant differences between the independent variables. Here I have attached the test result here. Should I conduct more test for significance or just mention the results of ANOVA test in the text?

Interpretation of Shapiro-Wilk test of normality is reverse as compared to other inferential from an ANOVA table. If P-value 0.05 then we know that the data is normally distributed. We can also plot normal Q-Q plot to cross-check. I hope this makes sense. Best wishes.