Introduction

During a recent panel of scientific journal editors, all participating editors proudly announced that their journals maintained high quality standards.Footnote 1 They all claimed that they only accepted the best articles for publication, as was “proven” by their high rejection rates that varied from as much as 70 to 85 %.Footnote 2 Nevertheless, later on in the discussion, one of the editors complained that there were too many journals in their field and that almost all articles, including the ones of low quality, got published. Thus, the question arose which of the two statements is true and how the number of journals might influence the judgement.

A base model with two generalizations

To analyse this question, we start with a base model of the cumulative acceptance or cumulative rejection of a single scientific manuscript, which is as simple as possible. Hence, in first instance, we do away with most complications that arise in reality, and introduce more realistic assumptions later on.

Assume a single journal, with a single editor, who has no memory, and a single manuscript that is accepted with a probability a, and rejected with a probability r, with no third option: thus a + r = 1. Assume further that this article, when rejected, is resubmitted endlessly, unless it is accepted. This last assumption obviously cannot be true, but reality comes close.Footnote 3 With these assumptions, the probability of acceptance in the second round (i.e., after the first rejection) will be r a. In the third round it will be r 2 a. In the fourth round it will be r 3 a, and so on. The question then is whether the sum of this series converges to 1 or not, that is, whether the article ultimately will be published or not. The answer is that it will, as the ultimate acceptance rate equals

$$ u = a + r\,a + r^{2} a + r^{3} a + \cdots = \left( {1 + r + r^{2} + r^{3} + \cdots } \right)a = \left( {1{-}r} \right)^{ - 1} a = a^{ - 1} a = 1, $$
(1)

because r < 1.Footnote 4 Note that the ultimate acceptance of this article does not depend on the rejection rate r, as long as that rejection rate does not increase with the number of resubmissions. This raises the side question, whether there is a type of increase in r that would prevent the ultimate acceptance of this article.

The only thing that does depend on r is the speed of convergence, i.e., after how many rounds the cumulative acceptance rate will reach which values. Table 1 gives the answer for a set of ordinary rejection rates of, respectively, 90, 80, 70, 60 and 50 %.Footnote 5 With a rejection rate of only 50 % almost all articles, in fact 97 %, will be published in the 5th round, that is, after four resubmissions. But even with a rejection rate as high as 90 %, we still see that 41 % of all articles is published after 4 resubmissions.

Table 1 Cumulative acceptance of articles (in %) with the acceptance rates in the first row
Full size table

Note that our wording has changed from a single article to a population of identical articles facing identical rejection rates in each (re)submission round. Obviously, this minor generalisation does not change (1), nor the conclusion. It does, however, raise the question whether our conclusion would change if we make our model much more realistic by allowing a multitude of journals.

Therefore, next, assume, without loss of generality, that we have J journals, instead of only one, and assume that each journal has its own rejection rate r j , rank-ordered in a column vector r or a diagonal matrix \( {\hat{\mathbf{r}}} \) with r on its main diagonal. Again assume that there is no third option. Then the sum of each journal´s acceptance rate, \( a_{j} \in {\mathbf{a}} \), and rejection rate, \( r_{j} \in {\mathbf{r}} \), still equals one, \( 1 \in {\mathbf{i}} \), that is, a + r = i. Since, in this second model, there are more journals, there is no more need of the unrealistic assumption of a single editor without memory.

Instead, we assume that a manuscript that is rejected by journal i will be resubmitted to a different journal j with a transfer probability \( p_{ij} \in {\mathbf{P}} \). The resubmission to a different journal, of course, implies that p ii  = 0. We, however, maintain the assumption that each rejected article is resubmitted, i.e., that authors do not withdraw their articles from the publication process. This implies that the matrix with the transfer probabilities has row sums that are equal to one, that is, \( {\mathbf{P}}\,{\mathbf{i}} = {\mathbf{i}} \).

With these new, more realistic and more general assumptions, Eq. (1) changes into a matrix equation that defines the value of the ultimate acceptance rate of a manuscript that was originally submitted to journal i, as follows:

$$ u_{i} \in {\mathbf{u}} = \left[ {{\mathbf{I}} + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right) + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{2} + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{3} + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{4} + \cdots } \right]\;{\mathbf{a}} = \left( {{\mathbf{I}} - {\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{ - 1} {\mathbf{a}} $$
(2)

where \( {\mathbf{I}} = {\hat{\mathbf{i}}} \), i.e., the unity matrix with ones on its main diagonal, and ()−1 = the inverse of the matrix (), for which holds that ()−1 () = I.Footnote 6

The proof that all u i in (2) are equal to one, as was the case with the single u in (1), only requires the assumption that rejected articles are not withdrawn from the publication process, i.e., that Pi = i, along with a + r = i:Footnote 7

$$ {\mathbf{u}} = \left[ {{\mathbf{I}} + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right) + \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{2} + \cdots } \right]\;({\mathbf{i}} - {\mathbf{r}}) = {\mathbf{i}} + {\mathbf{r}} + {\hat{\mathbf{r}}}\,{\mathbf{Pr}} + \cdots - {\mathbf{r}} - {\hat{\mathbf{r}}}\,{\mathbf{Pr}} - \left( {{\hat{\mathbf{r}}}\,{\mathbf{P}}} \right)^{2} {\mathbf{r}} - \cdots = {\mathbf{i}} $$
(3)

Thus, also in this more general and more realistic case, all articles will still be published in the end, irrespective of the various rejection rates of the journals that they pass through. This is a rather strong statement.

The realism of this statement, of course, partly depends on the earlier side question whether there exist round-by-round increases in the rejection rates r that prevent the convergence of (2). The obvious cause for a higher rejection rate of the next journal in the series is that the content of the manuscript may become outdated, increasingly, either theoretically, methodologically, or empirically. The most important cause for a lower rejection rate, on the other hand, is that wise authors will use the referee reports of each earlier journal to improve its quality before each next submission.

Besides, in part to counteract the possible increase in the likelihood of a next rejection, authors usually react by choosing a new journal with a lower rejection rate, reflecting a lower status in the quality hierarchy of journals of which most authors are usually well aware. This implies that, besides r, also P will change with each resubmission round. Assume that authors strictly follow this strategy, then P becomes a triangular matrix, as we have rank-ordered r from journals with high rejections rates to journals with low rejection rates. With this assumption a more general version of (2) with varying r and P can still be solved relatively easily, as one journal drops out after each resubmission round. With these new assumptions, our third model for the ultimate acceptance rates becomes

$$ {\mathbf{u}} = {\mathbf{a}}_{1} + {\hat{\mathbf{r}}}_{1} {\mathbf{P}}_{1} \,{\mathbf{a}}_{2} + {\hat{\mathbf{r}}}_{1} {\mathbf{P}}_{1} \,{\hat{\mathbf{r}}}_{2} {\mathbf{P}}_{2} \,{\mathbf{a}}_{3} + {\hat{\mathbf{r}}}_{1} {\mathbf{P}}_{1} \,{\hat{\mathbf{r}}}_{2} {\mathbf{P}}_{2} \,{\hat{\mathbf{r}}}_{3} {\mathbf{P}}_{3} \,a_{4} + \cdots + {\hat{\mathbf{r}}}_{1} {\mathbf{P}}_{1} \,{\hat{\mathbf{r}}}_{2} {\mathbf{P}}_{2} \cdots {\hat{\mathbf{r}}}_{J - 1} {\mathbf{P}}_{J - 1} \,{\mathbf{a}}_{J} , $$
(4)

where the subscript indicates the number of the (re)submission round, and where the number of journals J determines the maximum number of (re)submission rounds.

The question now becomes whether u reaches unity before or after the maximum number of (re)submission rounds J is reached. The answer can no longer be given analytically, but needs to be based on empirical values of \( {\hat{\mathbf{r}}}_{j} \) and P j .Footnote 8 Nevertheless, it is clear that scientific areas with smaller numbers of journals will have a higher probability of having an ultimate acceptance rate smaller than 1, than areas with many competing journals.

Conclusion and evaluation

In summary, we developed three, increasingly realistic theoretical models that all show that the dissenting editor of the panel, mentioned in the introduction, is right on both counts. High rejection rates may well go together with the ultimate acceptance of at least the majority of the initially submitted articles, while a large number of journals increases the probability of ultimate acceptance.

Adding an empirical foundation to our third and last theoretical model requires information on the rejection rates of journals and the resubmission behaviour of authors. The average rejection rates by journal are readily available, but how rejection rates change with resubmissions is difficult to establish, as editors mostly have no information about how many other journals, if any, rejected the article earlier, i.e., before it reaches their own desk. Further, information about the resubmission behaviour of authors is practically absent. A survey done via the editors of scientific journals probably leads to strategic and thus false answers.Footnote 9 Interviewing authors directly, without the help of editors or publishers, may provide a way out.

Anyhow, a conclusive empirical proof of the validity of the above three models, will need time to develop, as the data needed for such a proof are not readily available. Still, Kochar (1986) found two studies that report ultimate acceptance rates of 85 % for manuscripts that were rejected by The New England Journal of Medicine and the Journal of Clinical Investigation, which have high own rejection rates of, respectively, 85 and 70 %.Footnote 10

Our conclusion that most manuscripts will ultimately be accepted, therefore, not only has the theoretical backing given in this short article, but also has some initial empirical backing. This conclusion not necessarily, but most likely implies that too many manuscripts do get published in the end. The adjective “too” is justified because our conclusion implies that manuscripts of low quality also get published, unless one believes that the repeated refereeing process increases the quality of each manuscript sufficiently to warrant its ultimate publication.