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I am trying to understand the statistical calculations described in a letter to a journal. It describes the annual mortality rate of patients currently being seen by a health clinic.  To calculate the mortality rate two denominators are used: The "total number of individual patients" and "the total number of patient-years: the sum of the number of years spent by each individual as a patient".

From 2010 to 2020, the four deaths equate to 0.03% of the 15,032 patients. Taking the denominator as 30,080 patient-years, the annual mortality rate is calculated as 13 per 100,000 (95% confidence interval: 4 to 34 per 100,000).

Patient Mortality Rate: 100,000 / 30,080 x 4 = 13.3

This mortality rate is then compared against the general population. The ONS has mortality rates for 10 to 14-year-olds and 15 to 19-year-olds. This unfortunately doesn't reflect the patient population which includes 3 to 17-year-olds where the median age is 14.

A better comparison is therefore the overall mortality rate for adolescents aged from 14 to 17 (available only for the entire United Kingdom for 2015–2017), which was 2.7 per 100,000.

Age # Male Mortalities 2015/17 # Female Mortalities 2015/17 Total Mortality UK Pop. 2015 UK Pop. 2016 UK Pop. 2017
14 11 7 18 722,259 707,888 693,304
15 19 21 40 745,697 727,076 713,265
16 48 27 75 755,727 750,567 732,627
17 77 28 105 778,221 762,441 757,787
Total 155 83 238 3,001,904 2,947,972 2,896,983

Population Mortality Rate: (100,000 / (3,001,904 + 2,947,972 + 2,896,983) ) x 238 = 2.7

The final part is where I am lost at as I don't know how the final "5.5 times" figure is calculated

Comparison should also account for the difference between the sexes, because males are more likely to die than females. Of the patients, 69% were female. Adjusting for sex, the patients were 5.5 times more likely to die than the overall population of adolescents aged 14 to 17.

Can anyone help show how the final figure is calculated?

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  • $\begingroup$ Yes that's that source of the question that I tried to simplify, the population data is from the ONS "Estimates of the population for the UK, England, Wales, Scotland and Northern Ireland" in the "Mid-2012 to mid-2016 edition of this dataset" and the deaths are read from figure 1 from another journal article doi.org/10.1192/bjo.2020.33 $\endgroup$ Commented Feb 17 at 14:06
  • $\begingroup$ The 3 million figure is the population of the14 to 17 year olds in the UK which is then used to create a death rate for that age group and is then compared with the patient death rate $\endgroup$ Commented Feb 17 at 14:12
  • $\begingroup$ By simplifying the question to the core concept of the problem I hoped it would prevent it from negatively impacting people who may be vulnerable samaritans.org/about-samaritans/media-guidelines/… $\endgroup$ Commented Feb 17 at 14:24
  • $\begingroup$ Good point, I have updated my question to use the term "mortality". $\endgroup$ Commented Feb 17 at 14:34
  • $\begingroup$ Let us continue this discussion in chat. $\endgroup$ Commented Feb 17 at 14:34

1 Answer 1

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To adjust for sex in the vulnerable population, I make two assumptions:

  • The proportion of females in the general population is 50%.
  • The relative risk for males compared to females is the same in the general and vulnerable populations.

Under these assumptions, the risk to male patients is 155 / 83 times higher than the risk to female patients. We also know that the patients' mortality rate per 100,000 is 13 and that 69% of the patients are female. We put this information together in the following equation:

# x is the rate for females, (155/83) * x is the rate for males
# 0.31 * (155 / 83) * x + 0.69 * x = 13

We solve for x, the mortality rate for females in the vulnerable population:

x <- 13 / (0.31 * (155 / 83) + 0.69)

Next we adjust for the sex ratio so that we can compare the mortality rate of the vulnerable population to that of the general population of adolescents:

0.5 * (155 / 83) * x + 0.5 * x
#> [1] 14.68857

How does the sex-adjusted mortality rate in the vulnerable population compare to the mortality rate of 2.7 in the general population?

14.68857 / 2.7
#> [1] 5.44

It's not quite 5.5 times higher but the input numbers have been rounded along the way, so that can explain the difference.

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    $\begingroup$ Note: The Biggs paper states in the abstract: "Compared to the United Kingdom population of similar age and sexual composition, the (...) rate for patients (...) was 5.5 times higher." The rate of 13 is for similar age, so then it remains to normalize for sex composition. $\endgroup$
    – dipetkov
    Commented Feb 17 at 16:18
  • $\begingroup$ I also make an assumption that the relative risk for males & females is the same in both groups. (The RR is 155 / 83 in the general population but we don't know it's exactly this in the sub-population.) I'll update my answer. $\endgroup$
    – dipetkov
    Commented Feb 17 at 23:33

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