## Abstract

One of the main aims of phylogenetics is to reconstruct the “Tree of Life.” In this respect, different methods and criteria are used to analyze DNA sequences of different species and to compare them in order to derive the evolutionary relationships of these species. Maximum parsimony is one such criterion for tree reconstruction, and it is the one which we will use in this paper. However, it is well known that tree reconstruction methods can lead to wrong relationship estimates. One typical problem of maximum parsimony is long branch attraction, which can lead to statistical inconsistency. In this work, we will consider a blockwise approach to alignment analysis, namely the so-called *k*-tuple analyses. For four taxa, it has already been shown that *k*-tuple-based analyses are statistically inconsistent if and only if the standard character-based (site-based) analyses are statistically inconsistent. So, in the four-taxon case, going from individual sites to *k*-tuples does not lead to any improvement. However, real biological analyses often consider more than only four taxa. Therefore, we analyze the case of five taxa for 2- and 3-tuple-site data and consider alphabets with two and four elements. We show that the equivalence of single-site data and *k*-tuple-site data then no longer holds. Even so, we can show that maximum parsimony is statistically inconsistent for *k*-tuple-site data and five taxa.

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## Acknowledgements

The first and second authors thank the University of Greifswald for the Bogislaw studentship and the Landesgraduiertenförderung studentship, respectively, under which this work was conducted. Moreover, we wish to thank two anonymous reviewers for very helpful suggestions on an earlier version of this manuscript.

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## Appendix

### Appendix

All calculations in this manuscript were carried out with Mathematica (Wolfram Research 2017). By way of example, we will demonstrate the respective calculations for 2-tuple-site data and two character states (corresponding to the results presented in Sect. 2.1). To begin with, we implemented both the well-known Fitch algorithm (Fitch 1971) for the calculation of the parsimony score of a character or tuple, as well as the well-known Felsenstein algorithm (Felsenstein 1981) to compute the probabilities of characters and tuples on a given phylogenetic tree. Note that we assumed tree \((T_1,\theta _{T_1})\) (cf. Fig. 5) to be the generating tree on which all characters evolved according to the i.i.d. \(N_2\)-model. Based on these two algorithms, we first calculated the expected parsimony score for 2-tuple-site data and two character states according to Formula (2) for all trees \(T' \in {\mathcal {T}}\), where \({\mathcal {T}}\) is the set of all phylogenetic *X*-trees on five leaves. We summarized the results in a vector \(\mathtt {eps2Tuples}\) containing the expected parsimony score for each tree as entries. These entries were sorted according to Table 1, i.e., the first entry of \(\mathtt {eps2Tuples}\) contained the expected parsimony score of tree \(T_1\) and so on. Recall that in our case the expected parsimony scores depend on two parameters, *p* and *q* (representing the edge lengths of the generating tree), where we have \(0 \le p,q \le \frac{1}{2}\) (as we are considering two character states). To show that MP is statistically inconsistent on 2-tuple-site data, we had to find values for *p* and *q* such that the expected parsimony score of \(T_1\) (i.e., the first entry of the vector \(\mathtt {eps2Tuples}\)) was not the minimum of all values in \(\mathtt {eps2Tuples}\). Thus, we had to find values of *p* and *q* fulfilling the following constraints:

To find an explicit example for such values of *p* and *q* (as for example used in the proof of Theorem 2), we used the predefined Mathematica function \(\mathtt {FindInstance[expr,vars]}\), which (if they exist) finds values for the variables \(\mathtt {vars}\) where the expression \(\mathtt {expr}\) is true. In our example, the expressions are the three Inequalities (5), (6) and (7), and the variables are *p* and *q*. So we used this function in the following way:

The results are explicit values for *p* and *q* such that MP is statistically inconsistent (in our example, i.e., for \(k=2\) and \(r=2\), this yielded the values \(p=\frac{91}{256} \approx 0.35547\) and \(q=0.1\) as already shown in the proof of Theorem 2).

However, we not only wanted to find one explicit example of *p* and *q*, but the set of all values for *p* and *q* such that MP is statistically inconsistent on 2-tuple-site data. To plot all such combinations of *p* and *q*, we used the Mathematica function \(\mathtt {RegionPlot[pred,\{x, x_{min}, x_{max}\},\{y,y_{min}, y_ {max}\}]}\) which shows the region where the predicate \(\mathtt {pred}\) is true. In our example, the predicate was Inequality (5) and the parameters \(\mathtt {x}\) and \(\mathtt {y}\) were our parameters *p* and *q* with \(p_{min}=q_{min}=0\) and \(p_{max}=q_{max}=\frac{1}{2}\) as in Inequalities (6) and (7). Thus, we used this function as follows:

The results are shown in Fig. 8. Note that in this figure we can see that the areas where MP is statistically inconsistent or consistent on 2-tuple-site data are separated by a curve. With the function \(\mathtt {Reduce}\) and the same input as we used for the function \(\mathtt {FindInstance}\), we obtained the set of all values which fulfill Inequalities (5), (6) and (7). The result of this function is a very complicated term, which is why we skip the technical details here. Basically, the problem is that the corresponding curve is not as smooth as it appears at first glance in Fig. 8. This is due to the fact that inconsistency is not everywhere caused by the same tree. For instance, when \(p=\frac{91}{256} \approx 0.35547\) and \(q=0.1\), tree \(T_3\) has a lower expected parsimony score than \(T_1\), but when \(p= \frac{8187}{16384} = 0.49959 \approx \) and \(q=\frac{1967}{4096} \approx 0.48022\), this is not the case. Instead, here \(T_5\) has a lower parsimony score.

To summarize, by implementing algorithms for the calculation of parsimony scores and probabilities of characters and tuples on a phylogenetic tree, as well as by using the three predefined Mathematica functions \(\mathtt {FindInstance}\), \(\mathtt {RegionPlot}\) and \(\mathtt {Reduce}\), we computed all our results for 2-tuple-site data and two character states. Analogously, all other results presented in this manuscript were obtained.

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Galla, M., Wicke, K. & Fischer, M. Statistical Inconsistency of Maximum Parsimony for *k*-Tuple-Site Data.
*Bull Math Biol* **81, **1173–1200 (2019). https://doi.org/10.1007/s11538-018-00552-2

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### Keywords

- Maximum parsimony
- Statistical inconsistency
- Codons
- Long branch attraction
- Felsenstein zone