Because I have a very imbalanced dataset (9% positive outcomes), I decided a precision-recall curve was more appropriate than an ROC curve. I obtained the analogous summary measure of area under the P-R curve (.49, if you're interested) but am unsure of how to interpret it. I've heard that .8 or above is what a good AUC for ROC is, but would the general cutoffs be the same for the AUC for a precision-recall curve?
There is no magic cut-off for either AUC-ROC or AUC-PR. Obviously, higher is better, and the closer you are to 1.0, the closer you are to solving the problem.
However, the meaning of "close" is entirely application dependent. For example, if you could reliably identify profitable investments with an AUC of 0.7 or, for that matter anything distinguishable from chance, I would be very impressed and you would be very rich. On the other hand, classifying handwritten digits with an AUC of 0.95 is still substantially below the current state of the art.
Furthermore, while the best possible AUC-ROC is guaranteed to be in [0,1], this is not true for precision-recall curves because there can be "unreachable" areas of P-R space, depending on how skewed the class distributions are. This may render a "large" AUC-PR value less impressive than it might otherwise seem. See this paper by Boyd et al (2012) for details.
A random estimator would have a PR-AUC of 0.09 in your case (9% positive outcomes), so your 0.49 is definitely a substantial increase.
If this is a good result could only be assessed in compariso to other algorithms, but you didn't give detail on the method/data you used.
Additionally, you might want to assess the shape of your PR-curve. An ideal PR-curve goes from the topleft corner horizontically to the topright corner and straight down to the bottomright corner, resulting in a PR-AUC of 1. In some applications, the PR-curve shows instead a strong spike at the beginning to quickly drop again close to the "random estimator line" (the horizontal line at 0.09 precision in your case). This would indicate a good detection of "strong" positive outcomes, but poor performance on the less clear candidates.
If you want to find a good threshold for your algorithm's cutoff parameter, you might consider the point on the PR-curve that's closest to the topright corner. Or even better, consider cross validation if possible. You might achieve precision and recall values for a specific cutoff parameter that are more interesting for your application than the value of the PR-AUC. The AUCs are most interesting when comparing different algorithms.
.49 is not great, but its interpretation is different than the ROC AUC. For ROC AUC, if you obtained a .49 using a logistic regression model, I would say you are doing no better than random. For .49 PR AUC, however it might not be that bad. I would consider looking at individual precision and recall, perhaps one or the other is what is driving down your PR AUC. Recall will tell you how much of that 9% positive class you are actually guessing correct. Precision will tell you how many you guessed positive that were not. (False Positives). 50% recall would be bad meaning you're not guessing many of your imbalanced class, but perhaps 50% precision wouldn't bad. Depends on your situation.