Problems in using Beals' index to detect species trends in incomplete floristic monitoring data (Reply to Bruelheide et al. (2020))

Bruelheide et al. (Diversity and Distributions, 26, 2020, 782) explored repeated habitat mapping data to identify floristic changes over time on the basis of two surveys. Because of the incompleteness of the data, they utilized the Beals' index based on the aggregated data from both surveys as a statistical tool for the analysis. The aim of this note is to illustrate problems of this approach, which in particular is shown to produce a systematic underestimation of species decrease (and—potentially less relevant in practice—increase). A specific set of model cases will be introduced to show the effects of unjustified usage of the Beals' index in this specific form.


| THE R AW DATA AND THE AIM OF B RUELHEIDE E T AL . (2 02 0)
studied repeated habitat mapping data from biotopes in Schleswig-Holstein, Germany (surveys SH 1 and SH4), to identify floristic changes over a period of about 30 years (p. 2).
The authors admit that the statistical examination of this material is a very ambitious task, and they explain in detail the methodological problems. The main obstacles are (a) the incompleteness of the species lists because of shifts in attention between the two mappings, (b) an unknown observer bias (p. 4) and (c) the lacking of an identity function between the polygons of the two data sets (p. 3).
The authors tried to deal with these problems by modifying the raw data and by selecting the intersecting polygons of the first and the second surveys (p. 3).
We do not comment the problems (b) and (c), but focus on (a).

| THE IN COMPLE TENE SS OF DATA AND THE B E AL S' INDE X
The incompleteness of the data was the greatest problem. To identify floristic changes over time, the ratio of relative frequencies of a species i in the second and the first surveys is a suitable quantity to consider. However, a shift in attention between the first and the second inventory of the biotopes generated incompleteness of data, that is gaps in the occurrence matrices and undersized relative frequencies of the affected species.
Thus, Bruelheide et al. searching for a method to fill the gaps in the occurrence matrices used the Beals' index (Beals, 1984), which utilizes species composition of habitats to calculate "occurrence probabilities" (p. 4) for the species, and by this means provides information about the relative frequencies. Not only the presence of species i, but also the co-occurrences with other species determine the Beals' index of species i in polygon p. Even the lack of species i in polygon p can create a Beals' value greater than 0 due to observations in other polygons. Because the Beals' index fills gaps, it seems to be the solution for the problem of incomplete data. However, using this method for the two matrices separately does not solve the problem (see model case 1 and Table 1 "normal Beals").

| THE PROB LEM OF US ING THE INTEG R AL B E AL S' INDE X
In this situation, Bruelheide et al. (2020) proposed the idea of pooling the matrices of both samples as basis for calculating the Beals' index (we will call it "integral Beals' index" in the following). Bruelheide et al. identified the precondition to use integral Beals' index, namely "that the species co-occurrence matrix is static in time" (p. 4), and assumed that this is fulfilled. However, if the co-occurrence matrix is static in time, the relative frequencies of the species must be static in time, too, and therefore there are no trends of species. This creates contradictions to the approach of the authors: • Because Bruelheide et al. found trends, the assumption (or the calculation) must be wrong.
• The assumption does not reflect reality: Transformation in presences and compositions of species in monitored areas with consequences for the co-occurrence matrix is a normal process, at least in a period of decades. Immigration and extinction of species, changes due to environmental conditions (i.e. nitrogen input, climatic changes), competition, succession and diseases, all these have effects on the vegetation and implicate changes of species presences and composition over time. Each species has its own responses to the different environmental factors.
• Bruelheide et al. want to explore trends of species, but they use a procedure which is based on the assumption that there are no trends.
Therefore, there is no mathematical justification of using the Beals' index in this way. All calculations of monitoring data, which are based on the integral Beals' index, are scientifically questionable. The integral Beals' approach gives reason to expect biases in the trends of species.

| WHAT ARE THE CON S EQUEN CE S OF US ING THE INTEG R AL B E AL S' INDE X?
We now describe potential undesirable consequences of an improper application of the integral Beals' index. To this end, we have to explore how the normal vs. the integral Beals' index works. Using real data, these effects can scarcely be detected because they become invisible in the plethora of the many polygons and their various combinations of species. In order to circumvent this problem, we use simple models to understand the driving effects and compare the values of normal and integral Beals' index, if the occurrence matrices have gaps. We will concentrate on these model cases: Note that the Beals' index delivers an estimator for the occurrence probability. The arithmetic mean of normal Beals' values therefore has a certain similarity to the relative frequency of species.
In our model, the arithmetic mean of the normal Beals' values and the relative frequencies of species are equal. Table 1)

| Model case 1: Overlooking a species (see
We first consider two polygons with the same type of habitat including species i in a first survey S1 and the second survey S2, too, but in S1 we assume that species i was overlooked in both polygons. In addition, there are species j1-j29 which occur just in polygon 1 or in polygon 2, resp., or in both polygons, the same in both surveys, all without being overlooked (Table 1). The relative frequencies and normal Beals' values (i.e. the Beals' index separately for S1 and S2) of species i of surveys S1 and S2 are 0% vs. 100%. Because of the assumption that there was no shifting in presences of species i from S1 to S2, these values cannot describe the reality. In contrast, the integral Beals' values are 50% vs. 52.5%, which seems suitable because this indicates that there is nearly no trend, but 50% "occurrence probability" in S2 is not a reasonable measure of a species that appears in all polygons of S2. As expected, using the integral Beals' index lowers the differences due to overlooking of species. Table 1) A different situation is associated with the same figure in Table 1: A new invasive species i appears in the study area, which was not TA B L E 1 Matrix of monitoring the polygons 1 and 2 in surveys S1 and S2 with relative frequencies and normal and integral Beals' values of species i (see text model cases 1 and 2). X: presence S1 S2

| Model case 2: Appearing of a new invasive species (see
Polygon 1  Table 2)

| Model case 3: Extinction of a species (see
To model the situation of the extinction of a species, we reverse Table 1: In S1, species i was present in both polygons, but in the subsequent survey S2, it has died out, and all the other species are assumed to persist. The relative frequency of species i declines from 100% to 0%, so does the normal Beals' index, which seems suitable.
But the integral Beals' index drops from 52.5% to 50% only, with no indication of the extinction of species i in S2.

Model case 3 shows that the integral Beals' index might lead
to a strong underestimation of species decrease. It also shows that even the total loss of a species may lead to a nearly constant integral Beals' index. • If a species shows no trend in reality, but shifts in attention occurred, the integral Beals' values of the two surveys move closer together than the corresponding normal Beals' values. As a consequence, the trends of species flatten (desired), but the estimators of the "occurrence probabilities" are misleading (an undesired feature).
• If a species shows a trend in reality, and shifts in attention occurred, these factors interfere in their impact to integral Beals' index.

| D ISCUSS I ON
The special problem is that the integral Beals' index makes no difference between gaps in occurrence matrices because of overlooking a species, extinction of a species or not yet arriving of a species. Often there are no hints which conditions referring to trend or shift in attention are given. Therefore, the integral Beals' index seems to be unsuitable for most research questions considered in Bruelheide et al. (2020).
Note that the integral Beals' index can deliver biased results even for complete monitoring data sets. Incompleteness of data is a further source of bias.
Thus, the use of the integral Beals' index creates unsecure, often biased and in some cases strange results referring to trends and to the so-called "occurrence probability." If there is no shift in attention, the results will be more suitable, the lesser the data differ from the assumption to be "static in time." The higher the deviation is from the assumption "static in time," the more unsuitable the results will be. The general effect is a flattening of trends, thus an underestimating of trends.
Another point is the "occurrence probability." The reader will expect this to be a theoretical measure which has a connection to concrete presences and their relative frequency. However, the incorrect use of the Beals' index produces "occurrence probabilities" which can distinctly differ from the real relative frequencies of species.

ACK N OWLED G EM ENTS
We give many thanks to Helge Bruelheide for answering our questions and to Michael Breuer (Kiel), Ulf Friedrichsdorf (Eutin), Ulrich Mierwald (Kiel) and Hans-Ulrich Piontkowski (Eckernförde) for reviewing the manuscript and discussions.

PE E R R E V I E W
The peer review history for this article is available at https://publo ns.com/publo n/10.1111/ddi.13276.

DATA AVA I L A B I L I T Y
No data have been used.

B I OS K E TCH E S
Erik Christensen is biologist and mathematician, chairman of the NGO "AG Geobotanik in Schleswig-Holstein & Hamburg," with research interest in descriptive and conceptional models of species-area relationship and models to estimate the real species richness, in recording the flora and its changing, and all aspects of nature conservation. Björn Christensen is professor for statistics and mathematics with a focus on applied quantitative methods. Sören Christensen is professor for probability and statistics mainly interested in stochastic processes with applications, for example in natural resource management.