# Data detective work: work out the numerator or denominator given a percentage

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Here’s some fun I had today. If you are looking at some published stats and they tell you a percentage but not the numerator & denominator, you can still work them out. That’s to say, you can get your computer to grind through a lot of possible combinations and find which are compatible with the percentage. Usually you have some information about the range in which the numerator or denominator could lie. For example, I was looking at a paper which followed 63 people who had seen a nurse practitioner when they attended hospital, and the paper told me that 18.3% *of those who responded* had sought further healthcare. But not everyone had answered the question; we weren’t told how many but obviously it was less than or equal to 63. It didn’t take long to knock an R function together to find the compatible numerators given a range of possible denominators and the percentage, and later I did the opposite. Here they are:

# deducing numerator from percentage and range of possible denominators whatnum<-function(denoms,target,dp) { nums<-rep(NA,length(denoms)) for (i in 1:(length(denoms))) { d<-denoms[i] lo<-floor(target*d) hi<-ceiling(target*d) if(round(lo/d, digits=dp)==target) { nums[i]<-lo if(round(hi/d, digits=dp)==target) { warning(paste("More than one numerator is compatible with denominator ",d,"; minima are returned",sep="")) } } else if(round(hi/d, digits=dp)==target) nums[i]<-hi } res<-cbind(nums[!is.na(nums)],denoms[!is.na(nums)]) res<-cbind(res,round(res[,1]/res[,2],digits=dp)) colnames(res)<-c("numerator","denominator","proportion") return(res) } # and the opposite whatdenom<-function(nums,target,dp) { denoms<-rep(NA,length(nums)) for (i in 1:(length(nums))) { n<-nums[i] lo<-floor(n/target) hi<-ceiling(n/target) if(round(n/lo, digits=dp)==target) { denoms[i]<-lo if(round(n/hi, digits=dp)==target) { warning(paste("More than one denominator is compatible with numerator ",n,"; minima are returned",sep="")) } } else if(round(n/hi, digits=dp)==target) denoms[i]<-hi } res<-cbind(nums[!is.na(denoms)],denoms[!is.na(denoms)]) res<-cbind(res,round(res[,1]/res[,2],digits=dp)) colnames(res)<-c("numerator","denominator","proportion") return(res) }

By typing

whatnum(denoms=(30:63),target=0.183,dp=3)

I could find straight away that the only possibility was 11/60.

That particular paper also had a typo in table 4 ("995.3%") which meant it could be 99.5% or 99.3% or 95.3%. I could run each of those through and establish that it could only possibly have been 95.3%. Handy for those pesky papers that you want to stick in a meta-analysis but are missing the raw numbers!

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