The shorter your anchor rope, the more the force pulls straight down, as the "downward force" is coming directly at the angle of the anchor rope.
The faster the current, the more force that is being exerted down the anchor rope angle.
The more weight in the back of the boat, the less force that is required to pull the stern under.
Can't really math it out, per se, as there are a lot of variables within those three main ones...but the moral is to not overweight your boat, don't anchor in heavy water, and more anchor rope will lead to less downward force on your boat.
The relationship between load and angle/direction of force applied fits under the umbrella of vector force analysis, something that comes into play in the climbing/rescue world all of the time. Mathing it out is definitely possible, but to do that you would need to collect a lot of super boring information such as forward speed of the boat, current speed, weight of the boat and occupants, coefficient of friction between anchor and substrate, angle that water impacts the stern, on and on. The common sense method is a lot easier and more practical.
As has been stated, a steeper angle is achieved with less scope, or anchor rope in service. The steeper the angle, the more the force is applied in a downward direction. The amount of force is a significant factor, as is the direction in which it is being applied. Take the same load or force factor and apply it at two drastically different vector angles, and you will get very different force values in the critical area, in this case the stern of the boat. The same amount of force exists in each equation, but it is dissipated and shared between the components very differently based on the vector angle.
So, as Todd said, consider your force factors or force potential. Weight of the boat, speed of the current, weight of the anchor and manner in which is dropped, etc. The perfect storm is a heavy load in the boat, poor job of weighting and balancing (favoring the rear), heavy flow, and a heavy anchor that is dropped quickly to the bottom and hangs up immediately. If all of these things happened to combine in one event, well, we have all seen how that can go.
So, how do you mitigate? One, don’t overload the boat. Two pay attention to weight and balance issues. Be very cautious about dropping the rock in heavy flows where there is the potential for the anchor to stick before an appropriate amount of anchor line is out. I anchor in heavy water all of the time and have never had an issue, due in large part to paying attention to what has been described above. Additionally, rather than drop the rock immediately to the bottom at full speed, I will often grab a few hard oar strokes to slow my forward speed, slowly lower the anchor to the bottom, and then pay out line slowly until an effective scope/angle has been achieved, sometimes dragging the anchor on the bottom gradually and/or gradually transferring load to the anchor while paying out line should the rock stick immediately.