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Question edited with the correct code - apologies

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

set.seed(123)
N=1000
R1=vector('double',N)
R2=vector('double',N)
x=rnorm(100)
for (k in 1:N){
  y1=x+rnorm(100)
  y2=x+rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

    Pearson's product-moment correlation

data:  R1 and R2
t = -7.2284, df = 998, p-value = 9.704e-13
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2811533 -0.1633130
sample estimates:
       cor 
-0.2230479 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

\begin{align} & x_i \sim \operatorname N(0,1),\quad i=1,\ldots,100 \quad (\sim\text{indicates i.i.d.}) \\[6pt] & y_{1,i} = x_i + u_i, \quad u_i\sim\operatorname N(0,1) \\[6pt] & y_{2,i} = x_i + v_i, \quad v_i\sim\operatorname N(0,1) \end{align}

We can arrange these random variables as vectors:

\begin{align} & \mathbf x = [x_i]_{i=1,\ldots,100} \\[6pt] & \mathbf y_1 = [y_{1,i}]_{i=1,\ldots,100} \\[6pt] & \mathbf y_2 = [y_{2,i}]_{i=1,\ldots,100} \end{align}

And we now create the concatenated vectors as follows:

\begin{align} & \mathbf X = [\mathbf x, \mathbf x] \\[6pt] & \mathbf Y = [\mathbf y_1, \mathbf y_2] \end{align}

It turns out that the Pearson's correlation between y₁ and y₂ is very close to Var(X)/Var(Y). However, the two are negatively correlated when we repeat the process of realising y_1,i, and y_i while maintaining x_i constant across realisations.

Would you know why?

Question edited with the correct code - apologies

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

set.seed(123)
N=1000
R1=vector('double',N)
R2=vector('double',N)
x=rnorm(100)
for (k in 1:N){
  y1=x+rnorm(100)
  y2=x+rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

    Pearson's product-moment correlation

data:  R1 and R2
t = -7.2284, df = 998, p-value = 9.704e-13
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2811533 -0.1633130
sample estimates:
       cor 
-0.2230479 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

\begin{align} & x_i \sim \operatorname N(0,1),\quad i=1,\ldots,100 \quad (\sim\text{indicates i.i.d.}) \\[6pt] & y_{1,i} = x_i + u_i, \quad u_i\sim\operatorname N(0,1) \\[6pt] & y_{2,i} = x_i + v_i, \quad v_i\sim\operatorname N(0,1) \end{align}

We can arrange these random variables as vectors:

\begin{align} & \mathbf x = [x_i]_{i=1,\ldots,100} \\[6pt] & \mathbf y_1 = [y_{1,i}]_{i=1,\ldots,100} \\[6pt] & \mathbf y_2 = [y_{2,i}]_{i=1,\ldots,100} \end{align}

And we now create the concatenated vectors as follows:

\begin{align} & \mathbf X = [\mathbf x, \mathbf x] \\[6pt] & \mathbf Y = [\mathbf y_1, \mathbf y_2] \end{align}

It turns out that the Pearson's correlation between $\mathbf y_1$ and $\mathbf y_2$ is very close to $\operatorname{Var}(X)/\operatorname{Var}(Y).$ However, the two are negatively correlated when we repeat the process of realising $y_{1,i}$ and $y_{2,i}$ while maintaining $x_i$ constant across realisations.

Would you know why?

Question edited with the correct code - apologies

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

set.seed(123)
N=1000
R1=vector('double',N)
R2=vector('double',N)
x=rnorm(100)
for (k in 1:N){
  y1=x+rnorm(100)
  y2=x+rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

    Pearson's product-moment correlation

data:  R1 and R2
t = -7.2284, df = 998, p-value = 9.704e-13
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2811533 -0.1633130
sample estimates:
       cor 
-0.2230479 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

x_i ~ N(0,1), i=1,...,100 (~ indicates i.i.d.)

y_1,i = x_i+u_i, u_i ~ N(0,1)

y_2,i = x_i+v_i, v_i ~ N(0,1)

We can arrange these random variables as vectors:

x=[x_i]_i=1,...,100

y₁=[y_1,i]_i=1,...,100

y₂=[y_2,i]_i=1,...,100

And we now create the concatenated vectors as follows:

X=[x,x]

Y=[y₁,y₂]

It turns out that the Pearson's correlation between y₁ and y₂ is very close to Var(X)/Var(Y). However, the two are negatively correlated when we repeat the process of realising y_1,i, and y_i while maintaining x_i constant across realisations.

Would you know why?

Question edited with the correct code - apologies

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

set.seed(123)
N=1000
R1=vector('double',N)
R2=vector('double',N)
x=rnorm(100)
for (k in 1:N){
  y1=x+rnorm(100)
  y2=x+rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

    Pearson's product-moment correlation

data:  R1 and R2
t = -7.2284, df = 998, p-value = 9.704e-13
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2811533 -0.1633130
sample estimates:
       cor 
-0.2230479 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

\begin{align} & x_i \sim \operatorname N(0,1),\quad i=1,\ldots,100 \quad (\sim\text{indicates i.i.d.}) \\[6pt] & y_{1,i} = x_i + u_i, \quad u_i\sim\operatorname N(0,1) \\[6pt] & y_{2,i} = x_i + v_i, \quad v_i\sim\operatorname N(0,1) \end{align}

We can arrange these random variables as vectors:

\begin{align} & \mathbf x = [x_i]_{i=1,\ldots,100} \\[6pt] & \mathbf y_1 = [y_{1,i}]_{i=1,\ldots,100} \\[6pt] & \mathbf y_2 = [y_{2,i}]_{i=1,\ldots,100} \end{align}

And we now create the concatenated vectors as follows:

\begin{align} & \mathbf X = [\mathbf x, \mathbf x] \\[6pt] & \mathbf Y = [\mathbf y_1, \mathbf y_2] \end{align}

It turns out that the Pearson's correlation between y₁ and y₂ is very close to Var(X)/Var(Y). However, the two are negatively correlated when we repeat the process of realising y_1,i, and y_i while maintaining x_i constant across realisations.

Would you know why?

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

x=rnorm(100)
for (k in 1:1000){
  y1=x+0.5*rnorm(100)
  y2=x+0.5*rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

    Pearson's product-moment correlation

data:  R1 and R2
t = -4.8844, df = 998, p-value = 1.208e-06
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.21277453 -0.09167219
sample estimates:
       cor 
-0.1527969 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

x_i ~ N(0,1), i=1,...,100 (~ indicates i.i.d.)

y_1,i = x_i+u_i, u_i ~ N(0,1)

y_2,i = x_i+v_i, v_i ~ N(0,1)

We can arrange these random variables as vectors:

x=[x_i]_i=1,...,100

y₁=[y_1,i]_i=1,...,100

y₂=[y_2,i]_i=1,...,100

And we now create the concatenated vectors as follows:

X=[x,x]

Y=[y₁,y₂]

It turns out that the Pearson's correlation between y₁ and y₂ is very close to Var(X)/Var(Y). However, the two are negatively correlated when we repeat the process of realising y_1,i, and y_i while maintaining x_i constant across realisations.

Would you know why?

Question edited with the correct code - apologies

I have observed a somewhat puzzling negative correlation.

In code (R) (mathematical formulation below)

set.seed(123)
N=1000
R1=vector('double',N)
R2=vector('double',N)
x=rnorm(100)
for (k in 1:N){
  y1=x+rnorm(100)
  y2=x+rnorm(100)
  X=c(x,x)
  Y=c(y1,y2)
  R1[k]=cor(y1,y2)
  R2[k]=var(X)/var(Y)  
}
cor.test(R1,R2)

 
    Pearson's product-moment correlation

data:  R1 and R2
t = -7.2284, df = 998, p-value = 9.704e-13
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.2811533 -0.1633130
sample estimates:
       cor 
-0.2230479 

It turns out that R1 and R2 are very close to each other but negatively correlated. This is only the case when x is kept out of the loop (i.e. fix across experiments).

Would you know why?

Mathematical formulation

Let

x_i ~ N(0,1), i=1,...,100 (~ indicates i.i.d.)

y_1,i = x_i+u_i, u_i ~ N(0,1)

y_2,i = x_i+v_i, v_i ~ N(0,1)

We can arrange these random variables as vectors:

x=[x_i]_i=1,...,100

y₁=[y_1,i]_i=1,...,100

y₂=[y_2,i]_i=1,...,100

And we now create the concatenated vectors as follows:

X=[x,x]

Y=[y₁,y₂]

It turns out that the Pearson's correlation between y₁ and y₂ is very close to Var(X)/Var(Y). However, the two are negatively correlated when we repeat the process of realising y_1,i, and y_i while maintaining x_i constant across realisations.

Would you know why?