How to calcualte pearson correlation

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Analysis of correlation and significance of parameters

Correlation

The study of the significance of the impact of input parameters on output parameters should begin with the analysis of the correlation of individual parameters. Three basic dependencies can be checked:

  • monotonic linear
  • monotonic non-linear
  • square

Pearson's correlation coefficient (monotonic linear relationship)

The most basic measure determining whether there is a linear correlation between parametersxi i yiis the Pearson correlation coefficient:

rp=i=1n(xix¯)(yiy¯)i=1n(xix¯)2i=1n(yiy¯)2

wherex¯andy¯mean the mean values ​​of the relevant parameters.

This formula can be simplified to

rp=cov(x,y)var(x)var(y)

wherex=[x1,x2,...],y=[y1,y2,...]

Spearman's correlation coefficient (monotonic non-linear relationship)

Spearman's rank correlation coefficient is more universal because it allows to determine the strength of monotonic correlation, which may be non-linear and is expressed by the relation:

rs=i=1n(RiR¯)(SiS¯)i=1n(RiR¯)2i=1n(SiS¯)2

whereRiis the rank of the observationxi, Si is the rank of the observationyiandR¯ i S¯are the mean values ​​of the respective ranksRi andSi.

Interpretation of the correlation coefficient value

Correlation type:

  • rs> 0 positive correlation – when the value of X increases, so does Y
  • rs= 0 no correlation – when X increases, Y sometimes increases and sometimes decreases
  • rs< 0 negative correlation – when X increases, Y decreases

Correlation strength:

  • |rs|<0.2– no linear relationship
  • 0.2|rs|<0.4- weak dependence
  • 0.4|rs|<0.7– moderate dependency
  • 0.7|rs|<0.9- quite a strong relationship
  • |rs|0.9- very strong dependence

Quadratic correlation coefficient

The quadratic correlation coefficient is determined on the basis of regression analysis.

Error sum of squaresSSEis designated as

SSE=i=1n(yiy^i)2

After performing the approximation with a polynomial of the second degree (i.e. determining the coefficientsa2,a1,a0) y^i is determined by substitutionxito the formula of the approximating function

y^i=a2xi2+a1xi+a0

total sum of squaresSST to

SST=i=1n(yiy¯)2

The correlation coefficient is determined from the relationship

rq=1SSESST

Statistical testing of the significance of the correlation coefficient

To determine whether the determined correlation coefficient is statistically significant, it is necessary to make a null hypothesis

H0:δ=0

meaning that there is no correlation between the parameters. The alternative hypothesis has the form

H1:δ0

It is assumed that the statistic takes the Student's t-distribution o k=n2degrees of freedom and hence, for example, for the Pearson correlation coefficient, the value of the statistics is

t=rpn21rp2

The value of the test statistic cannot be determined whenrp=1 therp=1or whenn<3.

In other cases, the value determined on its basisp (read from the Student's t-distribution) is compared with the assumed significance levelα

  • ifpαwe reject itH0accepting H1
  • ifp>αthere is no reason to reject itH0

Typically, a significance level is selectedα=0.05, agreeing that in 5% of situations we will reject the null hypothesis when it is true.

The same is done for the other correlation coefficients insteadrpsubstitutingrstherq.

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