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A simultaneous-equation regression model of language complexity trade-offs.

1. Introduction

A language complexity trade-off is a situation in which a higher level of complexity for a certain language component appears in correspondence to a lower level of complexity for another component. The literature about this topic can be divided between papers that show that languages usually exhibit complexity trade-offs, and papers that show that such trade-offs do not exist. Among the first group of papers we can cite contributions such as Nettle (1995) and Fenk-Oczlon and Fenk (2008), while in the second group there are articles like Shosted (2006) and Nichols (2009).

In general, the way in which the different authors assess the possible existence and significance of complexity trade-offs is some version of correlation analysis. Under that approach, two different measures of complexity (e.g., phonological and morphological complexity) are supposed to display a trade-off if they are negatively correlated between themselves, and the way to find that correlation (or its absence) is to calculate a coefficient based on the values of the different complexity measures in a sample of languages. (1)

If we look at the main methodological difference between the literature that finds statistically significant complexity trade-offs and the literature that does not find them, we see that one important point is the type of measures that they use. While in the first group authors generally rely on "empirical measures" (i.e., measures of complexity calculated using data from actual words or texts written in different languages), in the second group they generally use theoretical or "typological" measures (i.e., measures obtained from the grammars of the sample languages).

Another feature that we have found in previous work (Coloma 2014, 2016) is that language complexity trade-offs seem to be more important and statistically more significant if we measure them using partial correlation coefficients instead of standard correlation coefficients. This is related to the fact that, when we use a partial correlation coefficient, we are also including information from factors besides the two correlating variables. It is also linked to the idea that complexity variables can be determined by a system in which there are interactions among them, so each partial measure of complexity can be correlated to several other measures at the same time.

This reference to a system of relationships between the different complexity measures can be related to a branch of the theoretical literature that sees language as a self-organizing and self-regulating system whose properties come from the interaction of several constitutive, forming and control requirements. That literature is known as "synergetic linguistics", and its origins can be traced back to Kohler (1986, 1987). It is also related to another branch of the linguistic literature that sees language as a complex adaptive system (e.g., Beckner et al. 2009).

The aim of this paper is to look for the existence of trade-offs in a context where language complexity is measured using theoretical variables (which is the one in which they have been harder to find), through a synergetic approach in which different factors interact. To do that we use a statistical methodology based on simultaneous-equation regressions, whose results allow to calculate different types of partial correlation coefficients between complexity measures. The analysis will be performed using the so-called "100-language sample" from the World Atlas of Language Structures (WALS), and complexity will be measured using binary variables that represent different concepts of phonological, morphological, syntactic and lexical complexity.

2. Description of the data

The WALS is a large database that compiles information about structural features from the grammars of the world's languages. In its current online version (Dryer and Haspelmath 2013), it contains data from 2679 languages and dialects, corresponding to 192 features that belong to different components of language structure.

The editors of the WALS have selected a sample of 100 languages for which they ask the authors of the different chapters of the atlas to include in their reports "if at all possible", and those languages are supposed to form a relatively balanced sample of genealogical and areal diversity. (2) Making use of the fact that we have more information about the languages that belong to this sample than the one available for the remaining languages, in this paper we use the 100-language WALS sample for a series of statistical analyses aimed at the detection of possible complexity trade-offs. To do that, we define four binary variables whose values can alternatively be "simple" or "complex", and those variables are built using information from certain features. (3)

The definitions of the abovementioned complexity variables are the following:

a) Phonology: A language is considered to be complex if it has more than 25 consonant phonemes, more than 6 vowel qualities, or uses tone as a distinctive phonological feature. This generates a division in which 60 languages are complex, and the remaining 40 languages are simple.

b) Morphology: A language is considered to be complex if it is polysynthetic, and simple if it is not. This implies that 32 languages in the sample are complex, and the remaining 68 ones are simple.

c) Syntax: A language is considered to be complex if it has no dominant word order for subject, object and verb, or if it uses relative pronouns to build relative clauses. Under this definition, 22 languages are complex and the remaining 78 ones are simple.

d) Lexicon: A language is considered to be complex if it has definite articles and uses different verbs for nominal and locational predication. This implies that 33 languages are complex, and the remaining 67 languages are simple. (4)

The easiest way to detect possible trade-offs between these binary complexity variables is to calculate standard (product-moment) correlation coefficients, like the ones that appear on table 1. In that table there are five negative correlation coefficients and one positive correlation coefficient, but none of them is statistically significant at a 5% probability level. (5)

3. Simultaneous equation regressions

The standard correlation coefficients reported on table 1 are in all cases calculated using information that covers two variables for each coefficient. In this case, however, it is possible to consider that our measures of phonological, morphological, syntactic and lexical complexity are somehow interrelated, in the sense that the relationship between any pair of those measures can be influenced by the other complexity variables.

One way to model a situation like the one described in the previous paragraph is to build a system of simultaneous equations like the following:

Phonology = c(1) + c(2) * Morphology + c(3) * Syntax + c(4) * Lexicon (1);

Morphology = c(5) + c(6) * Phonology + c(7) * Syntax + c(8) * Lexicon (2);

Syntax = c(9) + c(10) * Phonology + c(11) * Morphology + c(12) * Lexicon (3);

Lexicon = c(13) + c(14) * Phonology + c(15) * Morphology + c(16) * Syntax (4);

where Phonology, Morphology, Syntax and Lexicon are the complexity variables defined for the 100-language WALS sample, whose values can either be equal to 0 (if the language is simple in the corresponding domain) or equal to 1 (if the language is complex in that domain). Additionally, coefficients c(1) to c(16) are the values of the parameters that relate each complexity measure with the other measures.

One easy way to estimate coefficients c(1) to c(16) is to run a set of four separate ordinary least-square (OLS) regressions. If we do that, we get the following results:

Phonology = 0.7286 - 0. 1572 * Morphology - 0. 0749 * Syntax - 0.1872 * Lexicon (5);

Morphology = 0.4300 - 0.1462 * Phonology + 0.0389 * Syntax - 0.0935 * Lexicon (6);

Syntax = 0.2649 - 0.0560 * Phonology + 0.0313 * Morphology - 0.0644 * Lexicon (7);

Lexicon = 0.4826 - 0.1749 * Phonology - 0.0939 * Morphology - 0.0804 * Syntax (8).

With these results, it is possible to calculate new (partial) correlation coefficients, defined as the square roots of the products of the corresponding pairwise regression coefficients. (6) For example, for the relationship between phonological and morphological complexity, this is equal to the square root of "-0.1572" (which is the regression coefficient of Morphology as a determinant of Phonology) times "-0.1462" (which is the regression coefficient of Phonology as a determinant of Morphology). As both regression coefficients are negative, we must assign a negative sign to the corresponding correlation coefficient (i.e., to the corresponding square root), whose value is "r = -0.1516". If we make similar calculations for all the possible pairwise relationships that appear in our system, we will have a set of numbers like the ones reported on table 2.

The procedure used to calculate the regression coefficients that appear in equations 5 to 8 (which is the basis for the calculation of the partial correlation coefficients reported on table 2) regresses each equation independently. However, if we use a truly simultaneous procedure in which the four equations are regressed at the same time, we can also use the correlation coefficients between the residuals of the different equations, and derive a new set of regression coefficients like the following:

Phonology = 0.8580 - 0.3149 * Morphology - 0.1532 * Syntax - 0.3 743 * Lexicon (9);

Morphology = 0.5499 - 0.2928 * Phonology + 0.0586 * Syntax - 0.2032 * Lexicon (10);

Syntax = 0.3174 - 0.1146 * Phonology + 0.0472 * Morphology - 0.1324 * Lexicon (11);

Lexicon = 0.6414 - 0.3496 * Phonology - 0.2040 * Morphology - 0.1652 * Syntax (12).

This new set of regression coefficients comes from a statistical method known as "seemingly unrelated regression" technique (SUR), originally proposed by Zellner (1962). This method is not very common in linguistics, but it is relatively widespread in other social sciences such as economics (where it is standard for applications like demand estimation). In this case, however, its use generates an important increase in the magnitude of the estimated negative partial correlation coefficients, which can now be approximated by the numbers reported on table 3.

An additional variation that can be introduced is the elimination of the only positive correlation coefficient that we have obtained (which relates morphological and syntactic complexity), provided that its sign is counterintuitive and its absolute value (r = 0.0526) is small and statistically insignificant. If we do that, we can regress a new restricted system of equations, whose results (using SUR) are the following:

Phonology = 0.8591 - 0.3161 * Morphology - 0.1563 * Syntax - 0.3745 * Lexicon (13);

Morphology = 0.5686 - 0.2989 * Phonology - 0.2099 * Lexicon (14);

Syntax = 0.3421 - 0.1263 * Phonology - 0.1404 * Lexicon (15);

Lexicon = 0.6424 - 0.3498 * Phonology - 0.2053 * Morphology - 0.1671 * Syntax (16).

The new partial correlation coefficients implied by this system of regression equations appear on table 4, in which three out of the five estimated coefficients (phonology vs. morphology, phonology vs. lexicon, and morphology vs. lexicon) are now statistically significant at a 5% probability level.

4. Instrumental variables

The logic behind the equations used to estimate the partial correlation coefficients between phonological, morphological, syntactic and lexical complexity has to do with the idea that those complexity levels come from a system that generates them as the outcome of some unified procedure. That procedure may consist of the interaction between several constraints such as the ones proposed by the synergetic linguistics' literature (e.g., Kohler 2005), or some kind of iterative learning mechanism like the one proposed by Smith, Kirby and Brighton (2003).

Those theoretical approaches share the common assumption that languages emerge in environments that can be influenced by a series of non-linguistic factors. Among those factors, the ones that are easier to analyze in empirical work are the geographic, phylogenetic and population characteristics of the different languages. For example, as any language originated in a certain point in space, it can be classified as belonging to a certain region or area (e.g., one of the six large macro-areas that the WALS defines).

The other major classification used by the linguistic literature is the phylogenetic one, which groups languages into families that share a common ancestor (e.g., Indo-European, Afro-Asiatic, Niger-Congo, etc.). A third element that we can use to classify languages is their relative size in terms of population, which is related to the geographic expansion that each language has had in history, and its alternative use as a first or second language by different people. Those characteristics have been used to analyze the relationship between language complexity and population, in papers such as Dahl (2011) or Bentz et al. (2015).

One general way to think about the relationship between linguistic and nonlinguistic variables is to assume that the latter are part of the environment in which the former arise. This implies that linguistic variables may be influenced by non-linguistic factors, buy not the other way round. If we use this type of reasoning, we may represent our relationships by a graph like the one that appears in figure 1. In it we see that each linguistic variable (phonological, morphological, syntactic and lexical complexity) is the outcome of a language system which has in turn been influenced by phylogenetic, geographic and population factors.

The application of this view to the statistical explanation of the levels of language complexity implies the possibility of running a system of equations where those levels of complexity are the dependent variables, and the non-linguistic factors are the independent variables (on which the language complexity levels depend). In order to do that, we first need to encode the phylogenetic, geographic and population factors into numerical values, and the simplest way to do it is to create binary variables that take a value equal to one when a language belongs to a certain (geographic, phylogenetic or population) group, and zero otherwise.

Using the six WALS macro-areas and three additional divisions for those areas, we have created nine binary geographic variables that correspond to Eurasia, South East Asia, Africa, Papunesia, Australia, North America, Mesoamerica, the Amazon basin, and (the rest of) South America. Due to the fact that in the 100-language WALS sample there are relatively many observations that belong to three particular families (Austronesian, Indo-European, and Niger-Congo), we have also created three variables related to those phylogenetic factors. Finally, we have classified languages according to their relative size, considering the ones with more than 5 million native speakers as "major", and the other ones as "minor". (7)

The next step in the estimation of the effect of non-linguistic factors on language variables was to run a system of OLS equations in which each of the four complexity measures used in the previous sections was regressed against the thirteen non-linguistic binary variables. As the sum of the geographic variables completely covers the whole sample of languages, we have used one region (Eurasia) as the default one (constant), and included the remaining geographic variables as explanatory variables. The results for each of the four regressions appear on table 5.

The outcome of this regression analysis shows results that are in line with received linguistic knowledge. We can see, for example, that Australian and Austronesian languages tend to have simpler phonologies, that North American languages tend to have more complex morphologies, that Indo-European languages tend to have more complex levels of syntax, and that major languages tend to have a simpler lexicon (but a more complex phonology). These results may also be combined with the analysis that we performed in section 3, since the newly obtained regression coefficients can be the basis to build variables that are "instrumental" for that analysis.

The way to build these instrumental variables is to recover the predictions of the different regressions for each of the dependent variables of those regressions. With that we obtain four new variables (Phonology, Morphology, Syntax and Lexicon), which are linear combinations of the values of our thirteen binary non-linguistic variables (multiplied by their respective regression coefficients). These instrumental variables have the property that they can replace the original variables of the regression systems run in section 3, and are at the same time completely exogenous to those systems.

Instrumental variables are a useful device to solve a statistical problem known as the "endogeneity problem". This arises when we run a regression in which we know that both the dependent variable and (at least one of) the independent variables are somehow determined by the same mechanism. When this is the case, the obtained regression coefficients can be biased or inconsistent. If, however, we replace the endogenous independent variables by other variables that serve as exogenous instruments to approximate the value of those variables, then the estimation may turn less precise but more consistent and unbiased. (8)

In the system of equations introduced in section 3, all variables seem to be endogenous in the sense described in the previous paragraph. This is because they are at the same time dependent variables in one equation and independent variables in other equations, and all the relationships are supposed to be generated by the same mechanism. If we add the idea that such mechanism is somehow influenced by non-linguistic factors like the ones represented by the set of phylogenetic, geographic and population variables included in the regressions performed in this section, we can think of those variables as good candidates to act as exogenous instruments to replace the original (endogenous) linguistic variables.

The statistical method that uses a set of instruments to estimate instrumental variables, and then uses those instrumental variables to replace the original endogenous variables in a context of a simultaneous-equation regression estimation is known as "three-stage least squares" (3SLS). It was originally proposed by Zellner and Theil (1962), and is widely used in other social sciences such as economics (where it is standard for problems such as supply and demand estimation). If we use this method to run the system formed by equations 13 to 16, what we find is the following:

Phonology = 1.1276 - 0.5125 * Morphology - 0.6073 * Syntax - 0.6969 * Lexicon (17);

Morphology = 0.7413 - 0.5560 * Phonology - 0.2657 * Lexicon (18);

Syntax = 0.5538 - 0.4416 * Phonology - 0.2088 * Lexicon (19);

Lexicon = 0.7729 - 0.5569 * Phonology - 0.1947 * Morphology - 0.2112 * Syntax (20).

With these regression coefficients, we can now derive new partial correlation coefficients, which are the ones that appear on table 6. There we can see that the five estimated coefficients are now negative and statistically significant at a 5% probability level, since all of them are higher than 0.2 in absolute value.

5. Concluding remarks

The analysis performed in this paper about possible complexity trade-offs in the 100-language WALS sample can be seen as a particular statistical exercise whose outcome is likely to change if we use other language samples or other definitions for the different types of language complexity. The message that we get from that analysis, however, is probably of a more general nature, since it seems to reconcile some contradictory results from previous literature.

In the very beginning, our analysis generates the standard result that, if we measure trade-offs using product-moment correlation coefficients between typological complexity measures, what we get is a set of statistically insignificant values which imply that language complexity trade-offs are either inexistent or unimportant (and this is equivalent to the conclusions of papers such as Shosted 2006). When we use non-linguistic variables related to phylogenetic, geographic and population factors, conversely, we obtain results that indicate that some complexity variables may indeed be influenced by those factors, and this seems to be in line with some contributions from sociolinguistic typology (e.g., Trudgill 2009).

What we do not get, if we restrict ourselves to standard correlation and regression techniques, is anything related to the logic behind the idea of language as a complex adaptive system, since that idea implies that language should evolve to be at the same time "compressed" (i.e., relatively simple and easy to learn) and "expressive" (i.e., relatively complex and capable to convey meanings for multiple concepts). (9) To reconcile these two requirements, we need to find some kind of trade-off between different levels of language complexity, such as the ones that typically appear in the literature that uses empirical measures of complexity (e.g., Fenk-Oczlon and Fenk 2008).

In one contribution that belongs to that literature (Coloma 2016) we got a result that shows that language complexity trade-offs seem to be more significant if we measure them using partial correlation coefficients instead of standard correlation coefficients, and they get even more significant if we use simultaneous-equation regression methods such as SUR. We therefore decided to apply the same logic to study the possible trade-offs between typological complexity measures, since simultaneous-equation regression methods have been designed to deal with statistical problems in which the different equations that we want to regress are generated by the same mechanism. And this is precisely the case here, where we are supposing that language complexity variables come from some kind of unified generating process.

But, as we also have variables related to non-linguistic factors that may influence the language system from outside, we can use those variables to solve a statistical problem that simultaneous-equation models usually have, which is the endogeneity problem. To solve this we use non-linguistic factors to create instrumental variables, and then we use those instrumental variables as part of a 3SLS procedure. In this case, this can be seen as the statistical representation of a model in which non-linguistic factors are able to influence the system in which language is produced, and such system is in turn the one that generates the (interrelated) levels of complexity that correspond to its different sub-systems (i.e., phonology, morphology, syntax and lexicon).

When we did this, our results changed dramatically. Except for the coefficient that relates morphology and syntax, which is always insignificant, all the other correlation coefficients are negative and statistically significant when we estimate them using 3SLS. Moreover, their statistical significance increases when we move from standard to partial (OLS) coefficients, and the same occurs when we move from OLS to SUR, and from SUR to 3SLS coefficients.

This behavior may obey to different causes, but one plausible one is the idea that the statistical sophistications included in our calculations are related to an increasing consideration of the interactions between language complexity variables. When we only use standard correlation coefficients, those interactions are computed pairwise, while the calculation of partial correlation coefficients through an OLS procedure implies considering multiple interactions as well. Using the SUR method is in turn equivalent to introduce relationships between the errors that arise when we estimate the different complexity equations, whereas 3SLS implies considering the effect of non-linguistic factors (and their influence on the system that is producing the different levels of language complexity).

As a final conclusion, therefore, we can say that language complexity trade-offs may be more pervasive than what it seems when we measure them using simple statistical tools such as standard correlation coefficients or univariate regression equations. This is because there may be some interferences from other factors, whose effects have to be taken into account using more sophisticated statistical procedures. But that is indeed the message implied by the theoretical approach that sees language as a complex adaptive system, and the use of simultaneous-equation regression models can be a way to interpret the available data which is compatible with that approach.

References

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Appendix 1: List of languages in the WALS sample

Code   Language                Region          Family

1      Abkhaz                  Eurasia         Northwest Caucasian
2      Acoma                   North America   Keresan
3      Alamblak                Papunesia       Sepik
4      Amele                   Papunesia       Trans-New Guinea
5      Apurina                 Amazonia        Arawakan
6      Arabic (Egyptian)       Eurasia         Afro-Asiatic
7      Arapesh (Mountain)      Papunesia       Kombio
8      Asmat                   Papunesia       Trans-New Guinea
9      Bagirmi                 Africa          Nilo-Saharan
10     Barasano                Amazonia        Tucanoan
11     Basque                  Eurasia         Vasconic
12     Berber (Middle Atlas)   Africa          Afro-Asiatic
13     Burmese                 South East      Sino-Tibetan
14     Burushaski              Eurasia         Burushaskian
15     Canela-Kraho            Amazonia        Macro-Ge
16     Chamorro                Papunesia       Austronesian
17     Chukchi                 Eurasia         Chukotkan
18     Cree (Plains)           North America   Algic
19     Daga                    Papunesia       Dagan
20     Dani (Lower Valley)     Papunesia       Trans-New Guinea
21     English                 Eurasia         Indo-European
22     Fijian                  Papunesia       Austronesian
23     Finnish                 Eurasia         Uralic
24     French                  Eurasia         Indo-European
25     Georgian                Eurasia         Kartvelian
26     German                  Eurasia         Indo-European
27     Gooniyandi              Australia       Bunuban
28     Grebo                   Africa          Niger-Congo
29     Greek (Modern)          Eurasia         Indo-European
30     Greenlandic (West)      Eurasia         Eskimo-Aleut
31     Guarani                 South America   Tupian
32     Hausa                   Africa          Afro-Asiatic
33     Hebrew (Modern)         Eurasia         Afro-Asiatic
34     Hindi                   Eurasia         Indo-European
35     Hixkaryana              Amazonia        Cariban
36     Hmong Njua              South East      Hmong-Mien
37     Imonda                  Papunesia       Border
38     Indonesian              Papunesia       Austronesian
39     Jakaltek                Mesoamerica     Mayan
40     Japanese                Eurasia         Japonic
41     Kannada                 Eurasia         Dravidian
42     Karok                   North America   Karokian
43     Kayardild               Australia       Tangkic
44     Kewa                    Papunesia       Trans-New Guinea
45     Khalkha                 Eurasia         Altaic
46     Khoekhoe                Africa          Khoisan
47     Kiowa                   North America   Tanoan
48     Koasati                 North America   Muskogean
49     Korean                  Eurasia         Koreanic
50     Koyraboro Senni         Africa          Nilo-Saharan
51     Krongo                  Africa          Kaduglian
52     Kutenai                 North America   Salish
53     Lakhota                 North America   Siouan
54     Lango                   Africa          Nilo-Saharan
55     Lavukaleve              Papunesia       East Papuan
56     Lezgian                 Eurasia         Nakh-Daghestanian
57     Luvale                  Africa          Niger-Congo
58     Makah                   North America   Wakashan
59     Malagasy                Africa          Austronesian
60     Mandarin                South East      Sino-Tibetan
61     Mangarrayi              Australia       Mangarrayian
62     Mapudungun              South America   Araucanian
63     Maricopa                North America   Hokan
64     Martuthunira            Australia       Pama-Nyungan
65     Maung                   Australia       Iwaidjan
66     Maybrat                 Papunesia       West Papuan
67     Meithei                 South East      Sino-Tibetan
68     Mixtec (Chalcatongo)    Mesoamerica     Oto-Manguean
69     Ngiyambaa               Australia       Pama-Nyungan
70     Oneida                  North America   Iroquoian
71     Oromo (Harar)           Africa          Afro-Asiatic
72     Otomi (Mezquital)       Mesoamerica     Oto-Manguean
73     Paiwan                  Papunesia       Austronesian
74     Persian                 Eurasia         Indo-European
75     Piraha                  Amazonia        Mura
76     Quechua (Imbabura)      South America   Quechuan
77     Rama                    Mesoamerica     Chibchan
78     Rapanui                 Papunesia       Austronesian
79     Russian                 Eurasia         Indo-European
80     Sango                   Africa          Niger-Congo
81     Sanuma                  Amazonia        Yanomam
82     Slave                   North America   Na-Dene
83     Spanish                 Eurasia         Indo-European
84     Supyire                 Africa          Niger-Congo
85     Swahili                 Africa          Niger-Congo
86     Tagalog                 Papunesia       Austronesian
87     Thai                    South East      Tai-Kadai
88     Tiwi                    Australia       Tiwian
89     Tukang Besi             Papunesia       Austronesian
90     Turkish                 Eurasia         Altaic
91     Vietnamese              South East      Austro-Asiatic
92     Warao                   South America   Waraoan
93     Wari                    Amazonia        Chapacuran
94     Wichita                 North America   Caddoan
95     Wichi                   South America   Matacoan
96     Yagua                   Amazonia        Peba-Yaguan
97     Yaqui                   Mesoamerica     Uto-Aztecan
98     Yoruba                  Africa          Niger-Congo
99     Zoque (Copainala)       Mesoamerica     Mixe-Zoque
100    Zulu                    Africa          Niger-Congo

Appendix 2: Complexity variables

Code   Language        Phonology   Morphology   Syntax   Lexicon

1      Abkhaz                 1            1        0         1
2      Acoma                  1            1        1         0
3      Alamblak               1            0        0         1
4      Amele                  0            1        0         0
5      Apurina                0            1        0         0
6      Arabic                 1            0        0         0
         (Egyptian)
7      Arapesh                1            0        0         1
         (Mountain)
8      Asmat                  0            1        0         0
9      Bagirmi                1            0        0         1
10     Barasano               1            0        1         0
11     Basque                 0            0        0         1
12     Berber                 1            0        0         0
13     Burmese                1            0        0         0
14     Burushaski             1            0        0         0
15     Canela-Kraho           1            1        0         0
16     Chamorro               0            0        0         1
17     Chukchi                0            1        1         0
18     Cree (Plains)          0            0        1         1
19     Daga                   0            1        0         0
20     Dani                   1            1        0         0
         (Lower
         Valley)
21     English                1            0        1         0
22     Fijian                 0            0        1         0
23     Finnish                1            0        1         0
24     French                 1            0        1         0
25     Georgian               1            0        1         0
26     German                 1            0        1         0
27     Gooniyandi             0            0        1         0
28     Grebo                  1            0        0         0
29     Greek                  0            0        1         0
         (Modern)
30     Greenlandic            0            1        0         0
         (West)
31     Guarani                0            1        0         1
32     Hausa                  1            0        0         1
33     Hebrew                 0            0        0         0
         (Modern)
34     Hindi                  1            0        0         0
35     Hixkaryana             0            1        0         0
36     Hmong Njua             1            0        0         0
37     Imonda                 1            0        0         0
38     Indonesian             0            0        0         1
39     Jakaltek               1            1        0         0
40     Japanese               1            0        0         0
41     Kannada                1            0        0         0
42     Karok                  1            1        1         1
43     Kayardild              0            1        1         0
44     Kewa                   1            0        0         0
45     Khalkha                1            0        0         0
46     Khoekhoe               1            0        0         1
47     Kiowa                  1            1        0         0
48     Koasati                1            1        0         1
49     Korean                 1            0        0         0
50     Koyraboro              0            0        0         1
         Senni
51     Krongo                 1            0        0         0
52     Kutenai                1            1        1         1
53     Lakhota                0            0        0         1
54     Lango                  1            0        0         0
55     Lavukaleve             0            0        0         1
56     Lezgian                1            0        0         0
57     Luvale                 1            0        0         0
58     Makah                  1            1        0         1
59     Malagasy               1            0        0         1
60     Mandarin               1            0        0         0
61     Mangarrayi             0            1        0         0
62     Mapudungun             0            1        0         1
63     Maricopa               0            1        0         0
64     Martuthunira           0            0        0         1
65     Maung                  0            1        0         0
66     Maybrat                0            0        0         1
67     Meithei                1            0        0         1
68     Mixtec                 1            0        0         0
69     Ngiyambaa              0            0        0         0
70     Oneida                 1            1        1         1
71     Oromo                  1            0        0         0
         (Harar)
72     Otomi                  1            0        0         0
         (Mezquital)
73     Paiwan                 0            0        1         1
74     Persian                0            0        0         0
75     Piraha                 1            1        0         0
76     Quechua                1            0        0         0
         (Imbabura)
77     Rama                   0            0        0         0
78     Rapanui                0            0        0         1
79     Russian                1            0        1         0
80     Sango                  1            0        0         1
81     Sanuma                 1            1        0         0
82     Slave                  1            1        0         0
83     Spanish                0            0        1         1
84     Supyire                1            0        0         1
85     Swahili                1            0        0         0
86     Tagalog                0            0        0         0
87     Thai                   1            0        0         0
88     Tiwi                   0            1        0         0
89     Tukang Besi            0            0        0         1
90     Turkish                1            0        0         0
91     Vietnamese             1            0        0         1
92     Warao                  0            0        1         0
93     Wari                   0            0        0         0
94     Wichita                0            1        1         0
95     Wichi                  0            0        0         1
96     Yagua                  1            1        1         0
97     Yaqui                  1            1        0         0
98     Yoruba                 1            0        0         0
99     Zoque                  0            1        0         1
         (Copainala)
100    Zulu                   1            0        0         0
Tot    Total                 60           32       22        33


German Coloma, CEMA University; Av. Cordoba 374, Buenos Aires, C1054AAP, Argentina. Telephone: 54-11-63143000. E-mail: gcoloma@cema.edu.ar. I thank Gabriel Altmann, Damian Blasi, Guiomar Ciapuscio, Veronica Nercesian and Frans Plank for their comments to a previous version of this paper. The opinions expressed in this publication are my own, and are not necessarily the ones of CEMA University.

(1) An alternative way to do that is to run a regression between the two variables under analysis. In that case, the relevant coefficient is the slope of the regression line obtained, which should also be negative if we are looking for evidence of a complexity trade-off.

(2) This sample was used by us in previous work (Coloma, 2015). The complete list of languages is reproduced in appendix 1.

(3) The WALS features used are: 1A (Consonant inventories), 2A (Vowel quality inventories), 13A (Tone), 20A (Fusion of selected inflectional formatives), 26A (Prefixing vs. suffixing in inflectional morphology), 37A (Definite articles), 81A (Order of subject, object and verb), 119A (Nominal and locational predication) and 122A (Relativization on subjects).

(4) The value of each complexity variable for each language is reported in appendix 2, where "simple" is denoted as "0" and "complex" is denoted as "1".

(5) For any two variables whose correlation is calculated using 100 observations, correlation coefficients are statistically significant at a 5% probability level (i.e., the probability that the true correlation is zero is less than 5%) if they are greater than 0.2 in absolute value.

(6) For a more thorough explanation of the concept of partial correlation, and the available alternatives for its calculation, see Prokhorov (2002).

(7) The 33 major languages in the 100-language WALS sample are the following: Mandarin, English, Spanish, Hindi, Arabic, Russian, Japanese, German, French, Indonesian, Korean, Turkish, Vietnamese, Persian, Kannada, Hausa, Burmese, Tagalog, Yoruba, Swahili, Oromo, Thai, Malagasy, Greek, Zulu, Quechua, Berber, Hebrew, Khalkha, Finnish, Guarani, Georgian and Hmong Njua. The remaining 67 ones are considered to be "minor languages". To see which languages belong to the different geographic and phylogenetic groups, see appendix 1.

(8) For a more complete explanation of the endogeneity problem, see Kennedy (2008), chapter 9.

(9) For an interesting analysis of this dichotomy, see Kirby et al. (2015).

Table 1: Standard correlation coefficients between complexity
variables

Variables    Phonology   Morphology   Syntax    Lexicon

Phonology      1.0000
Morphology    -0.1400       1.0000
Syntax        -0.0591       0.0497     1.0000
Lexicon       -0.1650      -0.0711    -0.0647    1.0000

Table 2: Partial correlation coefficients between complexity
variables

Variables    Phonology   Morphology   Syntax    Lexicon

Phonology      1.0000
Morphology    -0.1516       1.0000
Syntax        -0.0648       0.0349     1.0000
Lexicon       -0.1809      -0.0937    -0.0720    1.0000

Table 3: Partial correlation coefficients using SUR

Variables    Phonology   Morphology   Syntax    Lexicon

Phonology      1.0000
Morphology    -0.3036       1.0000
Syntax        -0.1325       0.0526     1.0000
Lexicon       -0.3618      -0.2036    -0.1479    1.0000

Table 4: Partial correlation coefficients using a restricted
version of SUR

Variables    Phonology   Morphology   Syntax    Lexicon

Phonology      1.0000
Morphology    -0.3074       1.0000
Syntax        -0.1405       0.0000     1.0000
Lexicon       -0.3620      -0.2076    -0.1532    1.0000

Table 5: Regression coefficients for the language complexity
variables

Explanatory       Phonology   Morphology   Syntax    Lexicon
variables

Constant            0.5919       0.2839     0.2471    0.3461
Africa              0.2445      -0.1906    -0.2413    0.3149
South East Asia     0.2117      -0.1674    -0.1817    0.2827
Papunesia          -0.1318       0.0834    -0.2281    0.0316
Australia          -0.5919       0.2876     0.0386   -0.2032
North America       0.0747       0.5495     0.2529    0.2373
Mesoamerica         0.0747       0.2161    -0.2471   -0.1794
South America      -0.4862       0.1720    -0.0157    0.3958
Amazon              0.0331       0.4661     0.0029   -0.3461
Austronesian       -0.4706      -0.2806     0.2621    0.4699
Indo-European      -0.2026      -0.1441     0.5814    0.1335
Niger-Congo         0.0626      -0.0334     0.0279   -0.2233
Major Language      0.2357      -0.1398    -0.0785   -0.3546

Table 6: Partial correlation coefficients using 3SLS

Variables    Phonology   Morphology   Syntax    Lexicon

Phonology      1.0000
Morphology    -0.5338       1.0000
Syntax        -0.5179       0.0000     1.0000
Lexicon       -0.6230      -0.2275    -0.2100    1.0000
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Date:Oct 1, 2016
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