Bausch & Lomb has a big problem. It has suspended sales of its ReNu with MoistureLoc contact lens solution after reports that some customers acquired a fungal infection.

The data: There have been 109 reported cases of the infection. The CDC has investigated 30 cases, and 21 of those involved people using ReNu, while another five were using ReNu plus another manufacturer’s lens solution.

So, here is the question: What is the statistical likelihood that this happened by chance alone? What, in short, is the likelihood that in looking at the 30 samples out of 109 taken from a larger population you will see 21 people using ReNu with MoistureLoc if the true proportion of ReNu with MoistureLoc in fungal infection cases is actually in-line with the company’s overall market share?

To answer that question you need to know, among other things, the yearly incidence (and variance) of fungal keratitis among soft contact lens wearers, and the market share (and population numbers) of ReNu with MoistureLoc.

I don’t have all the answers, but until recently this fungal keratitis was actually rare. While bacterial infections among contact lens wearers is nothing new, fungal infections like this one are more dire (sometimes leading to vision loss and a corneal transplant) and much more unusual. With respect to market share numbers, I can only find non-MoistureLoc ReNu stats: It had something like 21% of the market in late 2004, making it the market leader.

It will be interesting to see if it turns out Bausch & Lomb is actually at fault here, or if it’s just large numbers conspiring against the company and its shareholders. My quick and dirty look at the above numbers would suggest that it’s likely that the issue *is* tied to B&L’s product.

**[Update]** Roger Bohn demonstrates B&L’s problem analytically here.

The incidence of fungal whatsit in the overall population is not needed. All we need is (21/30) versus 21% market share. This ignores the 5 cases that were mixed, which is probably optimistic, but never mind.

Let’s suppose their market share has risen to 25%. So with p = .25, n = 30, what is chance of drawing 21 out of 30? Standard deviation is (n*p*(1-p))^.5 = 2.3.

The expected number is .25*30 = 7.5

So 21 actual cases is (14/2.3) = 6 standard deviations away from the mean.

We can DECISIVELY reject the hypothesis that B&L’s solution is causing the same disease incidence as the average product on the market.

Conclusion: They have a problem.

Roger — I bow in your statistical direction. And it’s even less likely, considering that ReNu with MoistureLoc is actually much lower marketshare than 25%.