Data collection.

An important aspect of meta-analyses has to do with the selection of the studies to be included in the analysis. The selection criteria should be broad enough to allow a large number of studies to be considered, yet systematic enough to be reproducible.

In a previous meta-analysis of Zipf’s estimates, V. Nitsch [8] used the search words “rank AND size AND cities” in the EconLit database. He then filtered out the non-empirical studies and added some references which were absent from the EconLit results but present in the bibliographies of many studies found that way. He ended up with a list of 29 references. Although the first selection criterion is systematic, the largest flaw with this approach is that it introduces a very large bias towards studies published in economics journals, ignoring the contributions from social sciences which are not included in the EconLit database. Furthermore, the second round of selection appears non systematic. Finally, the resulting database used for the meta-analysis is not available to the reader.

In this study, I take advantage of new technologies of interactive online tools and crowdsourcing to establish a pool of studies as exhaustive as possible in relation to Zipf’s law. All studies are included in the meta-analysis as long as:

1. They contain at least one estimate of the rank-size exponent based on population
2. The regression is made on empirical urban data
3. The regression model is bivariate (i.e. relating populations and ranks or ranks—1/2, but not to any other instrumental variable).

At the time of writing, I had collected 86 such studies, and for each of them, recorded the series of specifications which characterises the estimation of Zipf’s law and could influence its value. The analysis is reproducible with this selection, although the database will continue to grow.

As shown in Fig 1, the data collection involves reading through the paper and recording the relevant pieces of information into a consistent and machine-readable format. Put together, the 86 studies give 1962 estimates of the rank-size coefficient, which correspond to 1962 different regressions and their specifications.The meta-elements associated with each value of estimate are the following:

• REFERENCE: the reference ID, which links an estimate to the study where it was published. Each study contains from 1 to 255 empirical estimates.
• ALPHA: the value of the estimate. Some were produced using the Lotka form (Eq 1) and other using the Pareto form (Eq 5). In any case, the value for the other form can be easily computed as α′ = 1/α. In the present paper, I choose to express all results using the Lotka form, thus interpreting the coefficient α as an index of unevenness of size in a given urban system.
• R²: the r-squared value of the regression. The information was available only half of the time.
• ESTIMATION: the method used to fit the regression model. OLS refers to the Ordinary Least Squares method, GI to Gabaix and Ibragimov’s trick of using rank—1/2, MC stands for Markov Chains and ML for Maximum Likelihood estimations.
• DATE: the year to which the population of cities refer (which is different from the publication year). The distribution of these dates goes from 1600 to 2014.
• TERRITORY: the administrative boundaries within which cities are drawn.
• URBANDEF: the way cities are defined in the study, using the original and heterogenous phrasing of the authors.
• N: the number of cities used in the regression. One third of that information is not reported.
• TRUNCATION: the minimum population of cities used in the regression. A quarter of that information is not reported.

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Fig 1. Metadata as data.

The information relating to the estimation of the slope of the rank-size curve in the study is recorded and formatted in a consistent way to serve as data in the meta-analysis.

https://doi.org/10.1371/journal.pone.0183919.g001

In addition to the specifications of the estimations, which can vary within a single study, I recorded some particularities of the study itself, in the same way as in [8]. I used the year of publication, the number of estimates published alongside the one under enquiry, as well as the period covered and the number of territories investigated.

Additional data.

Because of the heterogeneity of the data and the way it is reported, an additional variable was constructed. URBANSCALE is the simplification of the URBANDEF categories. It has only four values which correspond to a conceptual way of defining cities. It takes the value “LocalUnits” when the URBANDEF corresponds to municipalities, communes or places, i.e. administrative way of delineating cities. It takes the value “MorphoCities” for definitions using the continuity of the built environment as a criterion to define cities (urban areas or urban agglomerations for example). It takes the value “MetroArea” when a functional criterion is used to define cities (MSA in the Unites States, FUA in Europe for example). It takes the value “VariaMixed” in all other cases.

Finally, and this is how this new meta-analysis innovates, four topical variables were added to the technical variables classically used. For estimates obtained within national boundaries, I matched the meta-database with historical datasets from the United Nations and the World Bank to add the total population [19], GDP per capita [20] and urbanisation level (% of the population which is urban) [21] at the date corresponding to the data from the study. I also added a binary variable describing the age of the urbanisation. It is set as “old” in areas that were urbanised more than a millennium ago: in Europe, Asia and the Middle East. It is set as “recent” for the American, Oceanian and African continents. The dichotomy is pretty crude but has the merit of enabling to test the hypothesis according to which ‘old urban systems’ are more even in terms of city sizes.

Overview of a century of publication.

The resulting database is composed of 1962 estimates covering all the continents and spanning over 400 years. They come from 86 studies published in a variety of journals (the highest provider being Urban Studies with 8 studies and the Journal of Regional Science with 6). There seems to have been a fashion in publishing empirical estimates over the last two decades, with 54 of the 86 studies included published since 2000, although I detect a peak in publication about empirical estimations of Zipf’s law in the 1980s too (Fig 2A). Consequently, estimations relating to the year 2000 or 2001 are dominant in the set (Fig 2C), and in the majority of cases, estimates refer to urban data collected after the 1950s. The distribution of estimates by studies is itself skewed and if I estimate the relation between its rank and its size with the Lotka form (Fig 2B), I find a coefficient of 1.32, i.e. showing more unevenness than in city systems! Indeed, a handful of authors have published a large mass of estimates whereas most of them only published a handful of estimates in each study.

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Fig 2. Temporal overview of the metadata.

A: Distribution of studies publishing Zipf’s estimates over time. B: rank size distribution of the number of estimates per study. C: Distribution of estimates of Zipf by year of the data used in the estimation.

https://doi.org/10.1371/journal.pone.0183919.g002

The studies with the most estimates published are comparative studies with measures for many countries in the world (255 estimates in [13], 99 in [22]; 61 in [23], 56 in [16]). Another type of studies with a large number of estimates corresponds to sensitivity analyses on a single territory, with variations of dates and cutoff values (110 estimates in the United States in [24], 45 estimates in China in [25]). Those studies are over-represented in the meta-analysis when each estimate counts as one observation. However, they have the advantage of providing comparable cases to better isolate the effect of unique factors (minimum population, year, etc.), everything else being equal. I therefore weight estimates equally, irrespective of the study in which they were published. However, as in Nitsch’s study, I control for study effects with fixed-effects and panel models.

In terms of geographical coverage, the picture is one of large urban unevenness in the world. Some visual trends arise from Fig 3: American and Australasian cities seem more unevenly distributed in each countries, whereas Asian cities seem more evenly distributed in terms of size. The trend in European and African countries is much less clear-cut.

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Fig 3. Geographical overview of the metadata.

Mean value of alpha reported by country in the literature reviewed.

https://doi.org/10.1371/journal.pone.0183919.g003

Ordinary linear models.

For the static analysis, I use ordinary bivariate then multivariate linear models (Eq 5). (5) with: X_i,j,t a vector of variables describing the estimate α in the observation i in study j at time t, and ϵ a randomly distributed error.

In the bivariate models, the correlation between a single variable and the value of α is estimated. In multivariate models, I draw variables from three sets representing different regimes of explanation (Table 1):

1. Estimation variables refer to the way the estimation i is performed.
2. Study variables refer to the attributes of the publication j where i was reported.
3. Territory variables refer to the attributes of the urban system and its territorial envelope at the time t of its observation.

The first two sets correspond to the technical drivers of variation whereas variables of the territorial set are used to test theoretical hypotheses of change in the urban hierarchy as measured by Zipf’s law.

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Table 1. Three sets of static variables.

https://doi.org/10.1371/journal.pone.0183919.t001

In order to control for biases such as correlated errors within the same publication, I also present results estimated with a fixed-study effects model where I simply add an error term u_j to Eq 5.

Dynamic meta-analysis.

The models for a static meta-analysis can include the date of observation, but they are not really suited for looking at dynamic and longitudinal evolutions. For that purpose, I use two strategies. Firstly, I regress the static value of α with a random effect panel model (Eq 6). This method helps controlling for correlated errors within units and over time. (6) with: U_i,j,t the between-entity error and ϵ_i,j,t the within-entity error.

Secondly, I then compute the average annual growth rate of α values estimated under the same specifications with three sets of variables (Table 2):

1. Territory variables refer to the attributes of the urban system and its territorial envelope at the time t₁ of initial observation.
2. Dynamic variables refer to the growth of territorial variables between t₁ and t₂.
3. Event variables refer the presence or absence of territorial events between t₁ and t₂.

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Table 2. Three sets of dynamic variables.

https://doi.org/10.1371/journal.pone.0183919.t002

All these models were estimated using R and all their results are reproducible on the MetaZipf application, which is free and open: https://clementinegeo.shinyapps.io/MetaZipf/.

Estimation variables.

Although city definitions have various frequencies (1209 out of 1962 values of α were estimated using local units as “cities”), there is evidence that built-up areas (or MorphoCities) are more uneven on average than local units. This pattern has been observed empirically when comparing the two definitions within the same territory [1, 16, 22]. The reason for this is that morphological agglomerations are aggregates of local units and than on average, large built-up areas concentrate more units than small areas where the two definitions match. While comparing ‘natural’ cities defined as agglomeration of street intersections and urban areas in the United States, [10] found that more than a simple difference in values of α, the choice of city definition also determines if the model is robust at all or not.

The population cutoff used to select cities is an important factor affecting the value of Zipf’s estimate. Unfortunately, in 25% of all cases, the value of this cutoff is not reported in the publication. Lower cutoffs tend to be associated with larger inequality of city sizes, and larger cutoffs with more evenness. This is linked to the fact that rank-size curves are not always perfectly straight (i.e. not truly scale-free [10]) but show some convexity [12, 13, 16, 22]. Observing a small set of large cities thus leads to underestimate artificially the index of urban hierarchy.

The number of cities used in the regression is somehow dependent on the cutoff, but also on the size of the country. It is also massively under-reported (728 missing values). It shows a suspiciously high average α for data sets of less than 30 observations, which can be regarded as too small and inconsistent to be interpreted [16, 26]. The extreme values I recorded all correspond to these very small sets. However, sets of 30 to 300 cities show a low degree of inequality, which matches the previous observation: the top of the urban distribution is more even than the whole set. This has been interpreted as a lack of integration in large systems [17, 18].

A factor that has not been tested systematically so far is the impact of the estimation method on the value of the estimate, although the vast majority of published results were obtained using OLS. I find a significantly lower average value of α when authors have used the method of Gabaix and Ibragimov [9]. Additionally, I find a significant and worrying effect of the regression form used, with Pareto estimations giving more uneven results on average.

Study variables.

Using variables describing the study where estimates were published, I find that:

• Recent studies tend to publish, on average, higher values of α (1.04), whereas studies published before 2000 have average estimates of 0.98 (unlike [8]).
• Studies reporting more than 10 estimates tend also to report, on average, higher values of α (1.03), whereas studies with less than 10 estimates have an average value of 0.96 (like [8]).
• Studies reporting estimates for more than one date tend also to report, on average, higher values of α (1.03), whereas studies for one year only have an average value of 0.93 (like [8]).
• Studies reporting estimates for 2 to 4 countries tend also to report, on average, lower values of α (0.97), whereas studies for one country only (1.06) or more than five of them (1.01) report higher unevenness (like [8]).

Territory variables.

Finally, my interest lies in the last set of variables describing urban systems and their territorial envelope.

In terms of evolution, I find that estimates for city systems of 1940 and before are reported with significantly lower values on average compared to estimates for cities in the 1940s to 1990s. This result could be interpreted in two ways, which this simple model cannot disentangle. Firstly, this could confirm theories about the tendency of city systems to grow more uneven with time [8, 14]. Second, it could correspond to the different mix of countries studied over time, as developed countries of the western world (and their colonial empire) form the vast majority of available data for periods prior to 1940. Therefore, the higher average unevenness might not be a consequence their own evolution but could result from the addition to the literature of case studies from the developing world.

Although some scholars have argued that cities of smaller countries should be more unevenly distributed because powers are relatively more concentrated in the primate city [15, 16], I find no evidence of such relationship: estimates from large countries (≥ 100 million inhabitants) are on average very similar to the estimates in small countries (≤ 10 million), although a third of data are missing.

Richer countries tend to be associated with more unevenness in city sizes, although the information is missing in 875 cases, and the R² is low. Nonetheless, the relationship is ordered systematically, with countries with less than 1,000 current US dollars per capita having an average value of α equal to 0.976, whereas the average is 1.049 for intermediate countries and 1.073 for countries with more than 10,000 dollars per capita.

Cities in territories urbanised early on appear more evenly distributed in terms of size (α = 0.993), compared to newly urbanised systems (α = 1.094). One explanation from Evolutionary Urban Geography [13, 14] is that because the transport networks available at the time were slower than today, a larger amount of small cities were necessary. In areas urbanised with railways and highways, these small cities tend to have been short-circuited, leading to more uneven distribution of sizes.

Finally, city systems are more uneven when urbanisation is still low (≤ 20% of the national population lives in a UN defined city) compared to intermediate levels of urban population (from 20 to 60%). Although not significantly, this value tends to increase again above 60% suggesting a phenomenon opposite to Kuznets’ curve of income inequality. According to Kuznets [27], countries start their economic development with a low level of income inequality. The rising income due to development is at first unevenly distributed among citizens, but this inequality then tends to lessen as progress increases everyone’s situation. In the case of urban inequality, I find that countries start the urban transition from a situation where there is high unevenness between city sizes (usually one large city and a few emerging towns). Rural migrations and urban growth then pushed second-tier cities to even the distribution of city size, whereas saturated territories encounter a new rise of Zipf’s coefficient. Metropolises keep growing in interaction with other world cities whereas small cities stagnate or shrink [28].