Do we have to use a covariance (at quantitative scale) that is a normal distribution in linear regression when running a GWAS

Question

[wenqianglongli]

Hi all,

Hope this finds you well.

I am adding a covariance which is a quantitative variable (~600 samples) in my linear regression when I run a GWAS. The data turn out not normally distributed and I have tried many different ways to normalize it and outliers have been removed as well. Although the histograms and QQ plots of a few that were treated with cube root, Tukey ladder, box cox transformation looked fine. The normality tests still suggested that there are not normally distributed.

So my questions, is it ok to use one of them (after the normalizing) as a cavariance even it is not normally distributed (but close to)?

Thank you for your time!

Aydem

[Kevin Blighe]

Did you try natural log and squaring? For example:

log(cov)
cov^2

Can you plot the histogram of the covariate?

Another option is to categorise it.

[wenqianglongli]

Hi Kevin,

Many thanks for your reply. Yes, I did try them. However, Iike I said, even some of the histograms look fine, the normality tests still suggest that there are not normally distributed.

Please find the histograms of the data following the link below.

https://www.dropbox.com/s/usz23ftwkkwbeay/my_plots_remove_outliers2.pdf?dl=0

[Kevin Blighe]

Which normality tests are you using? What are the results? Only synthetic data will follow a perfect normal distribution. Looking at the histograms, they all visually look fine... like I said, no distribution of real data will follow a perfect normal.

[wenqianglongli]

Hi Kevin,

Thanks for your reply! Yes, I do get your points that wont be a perfect normal distribution in real data especially in some cases of diseases. Not only the histograms look fine to me, but also the qq plots look good as well.

I tried shapiro test first, and the results are: 1) data: d$Question (without data transformation) W = 0.92559, p-value < 2.2e-16 2) data: d$T_sqrt W = 0.98555, p-value = 1.551e-05 3) data: d$T_cub W = 0.99179, p-value = 0.002566 4) data: d$T_log W = 0.97304, p-value = 6.868e-09 5)data: d$T_tuk W = 0.99192, p-value = 0.002894 6) data: d$T_box W = 0.99192, p-value = 0.002894

As you can see, none of them are normally distributed.

I thought that this test might only apply to small sample size, so I tried One-sample Kolmogorov-Smirnov test, and the results are: 1) data: d$Question (without data transformation) D = 0.079553, p-value = 0.001232 2) data: d$T_sqrt D = 0.088076, p-value = 0.0002323 3) data: d$T_cub D = 0.075268, p-value = 0.002675 4) data: d$T_log D = 0.064056, p-value = 0.01658 5) data: d$T_tuk D = 0.073257, p-value = 0.003791 6) data: d$T_box D = 0.074277, p-value = 0.00318

However, they are still not normally distributed.

So back to my origin question, even the normality tests do not suggest they are normally distributed. Some of the histograms and qq plots do look fine. Is it ok to use one of the data sets?

[Kevin Blighe]

You may want to read the first answer in this very interesting thread: Is normality testing 'essentially useless'?

In light of everything, and based on the fact that the qq plots look fine, I'd say that you can go ahead.

[wenqianglongli]

Hi Kevin,

That answer is very useful!! Thanks for sharing and for your help.