A study collected data comparing treatments for kidney stones. Two of the treatments studied were open surgery and percutaneous nephrolithotomy. Treatment was deemed "successful" if, after three months, the kidney stones were either eliminated or less than $2 \mathrm{~mm}$. The latter treatment (nephrolithotomy) is cheaper and less invasive, but is it as successful? Results are shown in Figure 2.97, first overall and then broken down by stone size. (If the answers are not obvious visually, in each case you can calculate the proportion of successes using the numbers shown on the graph. )

(a) When all stone sizes are considered, which treatment is more successful?

(b) When only small kidney stones are considered, which treatment is more successful?

(c) When only large kidney stones are considered, which treatment is more successful?

All Stones

Small Stones

Large Stones

Figure 2.97 Success rates, first for all stones, and then broken down by stone size

(d) Which stone size results in higher success rates, regardless of treatment type?

(e) Which treatment is more commonly used for small stones?

(f) Which treatment is more commonly used for large stones?

(g) This is an example of Simpson's Paradox. Use your answers to parts (d) to (f) to explain how one treatment can be better for both small AND large stones, yet the other treatment appears to be better overall.

(h) Do you think this was a randomized experiment, with treatment randomly assigned? Why or why not?

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350 300 All Stones 250 Small Stones 250 Large Stones Success Failure 200 200 250 273 200 289 150 234 150 T T 192 150 100 100 100 50 50 81 50 T 55 50 71 77 61 36 25 0 0 0 Open Percutaneous Surgery Nephrolithotomy Open Percutaneous Surgery Nephrolithotomy Open Percutaneous Surgery Nephrolithotomy