Is this true that in some analyses, especially Microbial diversity, linear correlations are weak, but still biologically significant by weak I mean 0.27 or -0.14, 0.07,-0.08? I dont agree that this true
if a correlation is weak but significant would it mean it is correlated having above values.
Statistical significance of a correlation does not mean that the correlation is relivant or important. Statistical significance basically implies that the value of the correlation is probably a true relationship. And even correlations of .27 or -.14 can be significant, but the amount of the relationship between the measurements isn't very strong.
Jim provided an excellent response. This is just a little added advice: make sure you are checking your data/model for non-linearity, outliers, unaccounted variables of time, temperature or mixtures.
Open this link from University of Florida Health and scroll down to the plot of Fuel Used vs. Speed, it is a picture of a strong relationship, but a small correlation negative linear correlation.
Of course a classic example can be found in the Anscombe data. Run the script below. Y1 is what is expected for a 0.66 RSquare ( approx 0.816 linear correlation). Y2 and Y3 have a much stronger relationships: Y3 is weakened by an outlier and Y2 has a quadratic relationship with X (wrong model). Then look at Y4, the data results look more like an outlier, the result is inconclusive. Diagnostic plots and residual anayses are recommended procedures for and "modeling" of relationships.
dt = open("$sample_data/Anscombe.jmp"); dt << run script("The Quartet"); qwin = Window("The Quartet"); xx= (qwin<<child) << Xpath("//OutlineBox[@helpKey='Bivariate Report']") ; yy = xx << get scriptable object; yy << Show Points;
I encourage you to think of correlation as a descriptive statistic...not an inferential statistic. Once you start talking about 'significance' you are in inferential statistics land...not descriptive statistics land.