How to calculate Hm-Index?

Question

Michael Schreiber proposed the Hm-Index. It is related to the h-index but it adjusts for the number of authors on a paper. So, for instance, if you had two people with a h-index of 10, but one had all sole-author papers and the other had multi-author papers, the one with the sole-author papers would have a higher hm-index.

However, I can't find a concise statement of the formula on the internet. While a close reading of the paper provides the answer, I think there should be a concise statement of the method quickly available on the interent.

Thus, how do you calculate the hm-index?

Schreiber, M. (2008). A modification of the h-index: The hm-index accounts for multi-authored manuscripts. Journal of Informetrics, 2(3), 211-216.

Answer 1

This appears to be the algorithm:

1. Order publications by decreasing number of citations
2. Assign a weight to each publication as the inverse of the number of authors (e.g., 1 author paper = 1; 2 author paper = 0.5; 3 author paper = 0.33, etc.)
3. Take the cumulative sum of weights while this cumulative sum remains less than or equal to the number of citations for a given paper. This is the hm-index.

So, in R code, it might look like this:

hmindex <- function(citations, authors) {
    dat <- data.frame(citations = citations, authors = authors)
    dat <- dat[order(dat$citations, decreasing = TRUE), ]
    dat$weights <- 1/dat$authors
    dat$cweights <- cumsum(dat$weights)
    list(
        citetab = dat,
        hmindex =  dat[sum(dat$cweights <= dat$citations), "cweights"])
}

And here's an example using the data from the paper of an academic with 8 papers each with a certain number of citations and a certain number of authors.

citetab <- data.frame(
    citations = c(16,15,14,12, 10, 3,3,2),
    authors = c(2,2,3,3,2,2,3,1))

hmindex(citetab$citations, citetab$authors)

It produces the following output:

> hmindex(citetab$citations, citetab$authors)
$citetab
  citations authors   weights cweights
1        16       2 0.5000000 0.500000
2        15       2 0.5000000 1.000000
3        14       3 0.3333333 1.333333
4        12       3 0.3333333 1.666667
5        10       2 0.5000000 2.166667
6         3       2 0.5000000 2.666667
7         3       3 0.3333333 3.000000
8         2       1 1.0000000 4.000000

$hmindex
[1] 3

I.e., the researcher's h-index is 5 (because they have 5 papers with at least 5 citations. But their hm-index is 3 (i.e., largest cumulative weight that is greater than or equal to paper citation count).

Some comments:

• So it does not distinguish between different author positions (i.e., 1st or last author are weighted more in some metrics).
• If a researcher only publishes sole author papers, h-index equals hm-index. In all other cases hm-index will be less than h-index.