Mathematical modeling --- Pearson correlation coefficient hypothesis test conditions

Pearson correlation coefficient hypothesis test conditions

If the data is not normally distributed, the Pearson correlation coefficient cannot be used

How to verify that the data is normally distributed?

Test data is normally distributed

1. Normal distribution JB test (large sample n>30)

• Null hypothesis: Random variables are normally distributed
• Alternative hypothesis: disobedience

Where $pand0.0.\; 5ratiothan$ is: Suppose confidence level$95\%$ , so the significance level is$1-0.95=0.05$

• When $p<Significantlywithsexwaterlevel$ $\iff$ $Pvalue<Significantlysignificantofthe\; waterlevel$ , the null hypothesis is rejected

S = skewness(x)  % 偏度
K = kurtosis(x)  % 峰度

• Where x must be a vector

JB inspection code

Requires sample size greater than 30

[h,p] = jbtext(x,alpha)

• $h=1$ : Reject the null hypothesis;$h=0$ : The null hypothesis cannot be rejected
• x: random variable to be tested — must be a vector
• alpha: represents the significance level
• p: represents the P value in probability theory and mathematical statistics

2. Shapiro-wilk Shapiro-Wilk test (small sample 3<=n<=50)

Operation in SPSS

• Analysis --> Descriptive Statistics --> Exploration --> Graph
  
• Null hypothesis: Random variables are normally distributed
• Alternative hypothesis: disobedience

Where $pand0.0.\; 5ratiothan$ is: Suppose confidence level$95\%$ , so the significance level is$1-0.95=0.05$

• When $p<Significantlywithsexwaterlevel$ $\iff$ $Pvalue<Significantlysignificantofthe\; waterlevel$ , the null hypothesis is rejected
• The significance outlined by the red line represents our $Pvalue$

3. QQ graph to detect normal distribution

qqplot(x)

• x is a vector to check the distribution
  

If the data is not satisfied with the normal distribution

• Pearson correlation coefficient cannot be used
• But you can use Spearman's correlation coefficient

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Reference: Mathematical Modeling Breeze Video