Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$. Moreover, like $r$ (see above), $R$ is $n$ biased, and if we adjust $R$ for degrees of freedom using adjusted $R^2$, negative $R^2$ values become even more frequent.
Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$.
Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$. Moreover, like $r$ (see above), $R$ is $n$ biased, and if we adjust $R$ for degrees of freedom using adjusted $R^2$, negative $R^2$ values become even more frequent.
Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$.
Finally, asking what a 'good' $R^2$ is, is also a bit ambiguous. Unlike $r^2$, $R^2$ can (surprise, shock and awe) actually take on negative values. So, although $R^2$ is more general than $r^2$, it also has problems that never occur with $r^2$.
Perhaps the best way to communicate how variable the answer is, is to back calculate what the critical $r$ and $r^2$ values are for a $p<0.05$ significance.
First to calculate the t-value from an r-value let us use
$t=\frac{r}{\sqrt{(1-r^2)/(n-2)}}$, where $n\geq 6$
Then $r=\frac{t}{n-2+t^2}$, where $n\geq 6$ and using the t-significance tables
the critical two-tailed values of $r$ for significance are:
n r r^2 6 0.9496 0.9018 7 0.8541 0.7296 8 0.7827 0.6125 9 0.7267 0.5281 10 0.6812 0.4640 11 0.6434 0.4140 12 0.6113 0.3737 13 0.5836 0.3405 14 0.5594 0.3129 15 0.5377 0.2891 16 0.5187 0.2690 17 0.5013 0.2513 18 0.4857 0.2359 19 0.4715 0.2223 20 0.4584 0.2101 21 0.4463 0.1992 22 0.4352 0.1894 23 0.4249 0.1806 24 0.4152 0.1724 25 0.4063 0.1650 26 0.3978 0.1582 27 0.3899 0.1520 28 0.3824 0.1462 29 0.3753 0.1408 30 0.3685 0.1358 40 0.3167 0.1003 50 0.2821 0.0796 60 0.2568 0.0659 70 0.2371 0.0562 80 0.2215 0.0491 90 0.2086 0.0435 100 0.1977 0.0391 Note that the explained fraction ($r^2$) need for a significant $r$-value varies from 90% for $n=6$ to 3.9% for $n=100$. Nor does it stop there, the higher the value of $n$, the less explained fraction is needed for significance.
Perhaps the best way to communicate how variable the answer is, is to back calculate what the critical $r$ and $r^2$ values are for a $p<0.05$ significance.
First to calculate the t-value from an r-value let us use
$t=\frac{r}{\sqrt{(1-r^2)/(n-2)}}$, where $n\geq 6$
Then $r=\frac{t}{n-2+t^2}$, where $n\geq 6$ and using the t-significance tables
the critical two-tailed values of $r$ for significance are:
n r r^2 6 0.9496 0.9018 7 0.8541 0.7296 8 0.7827 0.6125 9 0.7267 0.5281 10 0.6812 0.4640 11 0.6434 0.4140 12 0.6113 0.3737 13 0.5836 0.3405 14 0.5594 0.3129 15 0.5377 0.2891 16 0.5187 0.2690 17 0.5013 0.2513 18 0.4857 0.2359 19 0.4715 0.2223 20 0.4584 0.2101 21 0.4463 0.1992 22 0.4352 0.1894 23 0.4249 0.1806 24 0.4152 0.1724 25 0.4063 0.1650 26 0.3978 0.1582 27 0.3899 0.1520 28 0.3824 0.1462 29 0.3753 0.1408 30 0.3685 0.1358 40 0.3167 0.1003 50 0.2821 0.0796 60 0.2568 0.0659 70 0.2371 0.0562 80 0.2215 0.0491 90 0.2086 0.0435 100 0.1977 0.0391 Note that the explained fraction ($r^2$) need for a significant $r$-value varies from 90% for $n=6$ to 3.9% for $n=100$. Nor does it stop there, the higher the value of $n$, the less explained fraction is needed for significance.
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