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$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables, when you fit a regression model $\hat y_i = \hat\beta_0 + \hat\beta_1x_i$ with the extremely common methods of least squares.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables, when you fit a regression model $\hat y_i = \hat\beta_0 + \hat\beta_1x_i$ with the extremely common method of least squares.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables, when you fit a regression model $\hat y_i = \hat\beta_0 + \hat\beta_1x_i$ with the extremely common methods of least squares.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$, the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.

$R^2$ has nothing to do with the sign of a correlation. While there are ways of getting $R^2<0$ in a regression model (an indication of a poor fit), the notation comes from the fact that $R^2 = r^2$, where $r$ is the sample correlation between two variables.

However, it looks like you have one variable with the levels in it and one variable with the numerical observations>

$$ X = (1,2,3,4,1,2,3,4,1,2,3,4)\\ Y = (240, 98, 46, 21, 12, 5, 4, 0, 7, 5, 6, 1) $$

Since your levels appear to be ordinal---that is, ordered but with unclear differences between them---Spearman's rank correlation is appropriate here. In R, the line is cor(x, y, method = "spearman").

This gives me a result of about $- 0.5$. However, the plot is not so convincing. When I do a test of the Spearman correlation being nonzero via cor.test(x, y, method = "spearman"), I get a p-value that tends to be considered inconclusive, $p = 0.099$, along with a warning that the exact p-value cannot be computed, due to tied values. I am not sure how serious this is, but, combined with the graph, I am skeptical increasing the level decreases the $Y$ variable.