The s4 mini, as opposed to the s4, does not seem to support usb OTG. OTG isn,t just a software feature, it needs the connector to be physically connected to something that controls the connected usb device in hardware AND a kernel that supports USB OTG:
http://en.wikipedia.org/wiki/USB_On-The-GoMany cheaper phones don't have these physical connections built into their hardware (for cost saving reasons I guess). Some phones and tablets have the hardware, but no OTG support built into their kernel. In this case you might be able to find a custom kernel that does, but I'm afraid that the s4 mini doesn't have the hardware built into it. It is kind of hard to tell though, because there is no comprehensive list for all android devices and that some devices have the hardware but no kernel support further complicates the issue.
I found some pretty useful reference though, sandisk tells us which devices are compatible to their usb flash readers and therefore must have usb OTG:
http://kb.sandisk.com/app/detect/