Another post, another probability distribution. This time it is Weibull‘s turn.
Everything breaks down over time. Suppose that over the lifespan of those things, the failure rate is proportional to the time it’s been around raised to some power k. If that power is one, then over any given time-span X% of the population will kick the bucket. This is similar to the Poisson distribution, except here we’re assuming there’s a finite population that isn’t being replaced. The number of failures we observe goes down, because there are a dwindling number of things left to fail.
k could also be greater or less than one, however. In the former case, that means the failure rate starts off around zero and increases as time goes on. The typical result is a period of tranquility, then an increasing number of observed failures, then after a peak they tail off until everything’s broken.
This distribution is used quite a bit in cancer research, where it’s been found to be an excellent model of tumor behavior. In the cell phone paper, k = 3 and k = 6 were good matches for the data.
The paper doesn’t say a lot about rat mortality, oddly. At the end of the Discussion section, however, the authors slip in this critical paragraph:
The survival of the control group of male rats in the current study (28%) was relatively low compared to other recent NTP studies in Hsd:Sprague Dawley® SD® (Harlan) rats (average 47%, range 24-72%). 
Set aside the implication that rats can be trademarked, and remember that 72% of the control group was dead by the time of the “terminal sacrifice” at week 105. If the rate of death due to malignant gliomas follows a Weibull distribution, with k between 3 and 6, then some simple math shows that we’d expect half of the cancers to appear sometime after week 88 or week 95. Roughly, of course, as the Weibull distribution points out that mortality can play a big role in the observed cancer rate. If a number of rats died off early, that could make a big difference on the observed number of cancers. The authors even admit this, in that same paragraph.
If malignant gliomas or schwannomas are late-developing tumors, the absence of these lesions in control males in the current study could conceivably be related to the shorter longevity of control rats in this study.
On top of that, cancer isn’t a binary; sometimes, it can be tough to tell the difference between a benign and malignant tumour, yet that boundary is critical to studies like this. The paper mentions it took two rounds of expert opinion to settle on a diagnostic criteria, which raised the eyebrows of more than one reviewer. As one of them put it, in point-form:
- Difficulty in achieving diagnostic consensus in lesions classifications of rare, unusual, and incompletely understood lesion association
- Document appears to indicate that the second Pathology Working Group (PWG) empaneled to review and obtain lesion classification consensus, following the inability of the initial PWG to do so, may have reviewed different lesions sets
- No record of clinical disease manifestations due to lesions involving heart and brain [note lesions in heart and brain are mutually exclusive; affected rats have either one or the other and do not appear to have the involvement of both organs together (appendix E)]
- Lesions, including malignancies, do not appear to materially shorten lifespan, except for a subgroup of rats (less than 1/3 of affected rats) with malignant Schwannomas in heart
- Lack of shortened lifespan as a consequence of malignancy for the majority of affected rats contrasts with shortened lifespan of male control rats for which there is absence of attributable cause of death. The survival of the control group of male rats in the current study (28%) was relatively low compared to other recent NTP studies (avg 47%, range 24 to 72%). 
Malignant tumours take time to develop, and in the early stages may look benign. Benign tumors are surprisingly common, which means they also vary widely. Take breast cancer in human beings; it’s estimated that 20%-50% of detected breast tumors are “overdiagnosed,” or benign but detected by early screening and treated as possibly malignant. This is why the recent change in screening guidelines called for fewer tests in low-to-medium risk patients.
The flip side of this is “underdiagnosis,” where early screening misses tumours that would have become malignant later. Every rat in this study had a biopsy done when they died, but if an unusual number of controls died early their biopsies would be more likely to miss a developing tumour. Conversely, the rats who survived the entire study were given plenty of time to let their symptoms develop, making them easier to spot and catalog.
Appendix E lists the time on study for each animal with a malignant glioma or heart schwannoma. Most of the gliomas were observed in animals that died late in the study, or at the terminal sacrifice.
This is entirely compatible with tumours caused by radio exposure, but it’s also entirely compatible with a set of healthy rats who naturally developed tumours late in life. The real question is, which of those two hypothesis is more compatible with the observed data? You can’t answer that by nitpicking study design, except in cases of gross incompetence; instead, you’ve got to roll up your sleeves and do a statistical analysis.
 Portier, Christopher J., John C. Hedges, and David G. Hoel. “Age-Specific Models of Mortality and Tumor Onset for Historical Control Animals in the National Toxicology Program’s Carcinogenicity Experiments.” Cancer Research 46, no. 9 (1986): 4372–4378.
 Moon, Hojin, Hongshik Ahn, Ralph L. Kodell, and J. Jack Lee. “Estimation of K for the Poly-K Test with Application to Animal Carcinogenicity Studies.” Statistics in Medicine 22, no. 16 (2003): 2619–2636.
 Wyde, Michael, Mark Cesta, Chad Blystone, Susan Elmore, Paul Foster, Michelle Hooth, Grace Kissling, et al. “Report of Partial Findings from the National Toxicology Program Carcinogenesis Studies of Cell Phone Radiofrequency Radiation in Hsd: Sprague Dawley® SD Rats (Whole Body Exposure),” May 26, 2016. http://biorxiv.org/lookup/doi/10.1101/055699.
 Oeffinger, Kevin C., et al. “Breast cancer screening for women at average risk: 2015 guideline update from the American Cancer Society.” JAMA 314.15 (2015): 1599-1614.