I'm not sure what "lifetime manic features at age 22 – 23 years" means. Lifetime, or between ages 22 and 23?

But the numbers:

There was a positive association between IQ at age 8 years and lifetime manic features at age 22–23 years (Pearson’s correlation coefficient 0.159 (95% CI 0.120–0.198), P>0.001).

I shall be generous and take the upper end of their range for the correlation, and round it up to c = 0.2.

The shared variance is c^2 = 0.04. That is childhood IQ "explains" (in the technical sense of that word) 4% of the variance of "lifetime manic features at age 22 – 23 years".

For the following calculations I assume, for no reason other than mathematical simplicity, that we are dealing with a bivariate normal distribution. However, I doubt the overall message would be very different for whatever the real distribution is.

The mutual information between the variables, is log2( 1/sqrt(1-c^2) ) = 0.0294 bits.

What can you do with 30 millibits? You might try to use IQ at age 8 to predict "lifetime manic features at age 22 – 23 years". How much will knowing the former narrow your estimate of the latter? The ratio (standard deviation conditional on that information)/(unconditional standard deviation) is sqrt(1-c^2) = 0.980. That is, the spread is 2% smaller.

Suppose you try to predict from IQ at age 8, whether their "manic features" will be above or below the average? By random guessing you will be right 50% of the time. By using that information, you will be right (1/π)acos(−c) of the time = 56%.

Perhaps, if the IQ is really high, the "manic features" will be more significantly above the average? In principle, yes, but in practice, not enough to matter. The probability that an individual has an IQ high enough to be 95% sure that they will be above average for "manic features" is 7.5 x 10^-14. Of course, the bivariate normal approximation cannot be observably accurate so far out, but I think it gives an indication of the scale of the matter.

The mathematics underlying the calculations can be found here. The figures at the end include a scatterplot of what c=0.2 looks like. That was the lowest correlation for which I thought it worth while to include in the tabulations.