# ガシアンラインプロファイルの統計的に有意なレベルを計算する方法

It is difficult to say, there isn't a description in the paper as to how they arrive at those numbers. Given that the equivalent widths of the lines have signal to noise levels that mean they are only greater than zero at significance levels of 2-3, then it isn't clear these are meaningful numbers at all.

However, here is how it might have been done.

You fit a continuum model to the spectrum, get a minimum chi-squared $$\chi^2_1$$ (or maximum [log] likelihood value); you then fit a continuum plus absorption line model and get a new, lower value, of chi-squared $$\chi^2_2$$ (or higher value of max log likelihood)

You then get the difference in chi-squared $$\Delta \chi^2 = \chi^2_1 - \chi^2_2$$ (or form the likelihood ratio) of the two models, which should be distributed as chi-squared with the number of degrees of freedom being equal to the number of new parameters in the more advanced model (i.e. 3 for a line centre, width and depth). You then check a chi-squared table for the corresponding p-value and significance.

In other words, if $$\Delta \chi^2 = 6.25$$, that corresponds to p=0.1, or if $$\Delta \chi^2 = 11.35$$ for p=0.01 etc.

To convert to a number of Gaussian sigma, it is the stuff of standard statistics that $$n\sigma = \sqrt{2} {\rm erf}^{-1} (1 - p)$$ which in the case above suggests a significance of $$4.26\sigma$$.