Linear Correlation Demo

Consider the data in the 3 waves:

data1data2data3
95.478.4
8.785.848.8
9.025.799
8.295.338.3
8.375.498.4
9.065.188.3
8.695.678.7
8.535.538.5
8.785.758.8
9.15.889.1
8.615.958.6
9.266.249.2

We can calculate the (linear) correlation coefficients between the waves and test various hypotheses in one operation. To run the test for waves data1 and data2 execute the command:

StatsLinearCorrelationTest/T=1/Q data1,data2

The test results are displayed in the Linear Correlation Test table:

n12
r0.422481
sr0.28662
rc10.497265
rc20.575983
t_Value1.47401
tc11.81246
tc22.22814
F2.46309
Fc12.97824
Fc23.71679
Power10.38787
Power20.268562

There are 12 data points in each wave. The correlation coefficient r=0.422481 with a standard error sr. For the hypothesis H0: the correlation coefficient is zero, the t_Value is below the critical value so H0 can't be rejected. On the other hand, if we test data2 and data3 we get different results. To run the test execute the command:

StatsLinearCorrelationTest/T=1/Q data2,data3

n12
r0.876562
sr0.152197
rc10.497265
rc20.575983
t_Value5.7594
tc11.81246
tc22.22814
F15.2025
Fc12.97824
Fc23.71679
Power10.992761
Power20.982695

In this case the correlation coefficient r=0.876562 and both t_Value and the F exceed their respective critical values and H0 must be rejected.

The operation can also test the specific hypothesis ρ=ρ0 agains the alternative ρ!=ρ0. Using the same data as in the last example execute the command:

StatsLinearCorrelationTest/T=1/Q/RHO=0.5 data2,data3

n12
r0.876562
sr0.152197
rc10.497265
rc20.575983
t_Value5.7594
tc11.81246
tc22.22814
F15.2025
Fc12.97824
Fc23.71679
FisherZ1.36073
zeta0.549306
sigmaZ0.333333
Zstatistic2.43427
Zc11.64485
Zc21.95996
Power10.787354
Power20.679027

Since the Zstatistic=2.43427 exceeds the two-tailed critical value Zc2=1.95996, H0: ρ=0.5 must be rejected. However, if we test ρ=0.75, we find:

StatsLinearCorrelationTest/T=1/Q/RHO=0.75 data2,data3

n12
r0.876562
sr0.152197
rc10.497265
rc20.575983
t_Value5.7594
tc11.81246
tc22.22814
F15.2025
Fc12.97824
Fc23.71679
FisherZ1.36073
zeta0.972955
sigmaZ0.333333
Zstatistic1.16332
Zc11.64485
Zc21.95996
Power10.317867
Power20.210131

In this case Zstatistic=1.16332<Zc2=1.95996 and so H0 can't be rejected.