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Research article.
Published in Open Differential Psychology 8th of September, 2014
Submitted 20th of July, 2014
The international general socioeconomic
factor: Factor analyzing international
rankings
Emil O. W. Kirkegaard1
Abstract
Many studies have examined the correlations between national IQs and various country-level indexes
of well-being. The analyses have been unsystematic and not gathered in one single analysis or
dataset. In this paper I gather a large sample of country-level indexes and show that there is a
strong general socioeconomic factor (S factor) which is highly correlated (.86-.87) with national
cognitive ability using either Lynn and Vanhanen’s dataset or Altinok’s. Furthermore, the method of
correlated vectors shows that the correlations between variable loadings on the S factor and cognitive
measurements are .99 in both datasets using both cognitive measurements, indicating that it is the
S factor that drives the relationship with national cognitive measurements, not the remaining variance.
Keywords
: National IQ, social progress index, democracy ranking, intelligence, g-factor, group
differences, general socioeconomic factor, method of correlated vectors, psychoinformatics
1 Introduction
Recently, research has been done on general socioeconomic factors. Gregory Clark argues that there
is a general socioeconomic factor which underlies performance at the individual level.[
1
] Moving one
step up, earlier I found evidence that among 71 Danish immigrant groups ranked on 4 different
measures of socioeconomic variables (crime, use of social benefits, income, education attainment),
there is a large (40% of variance explained) general socioeconomic factor.[
2
] Given that groups can
generally be considered collections of individuals, this leads to the expectation that there may be a
general socioeconomic factor at the country level as well. The general mental ability factor at the
individual level has been termed ”g” (often italicized ”g”[
3
]), while the national-level group equivalent
has been termed ”G” (”big g factor”).[
4
,
5
] Keeping in line with this terminology, one might refer to
the general socioeconomic factor at the individual level as ”s factor” and the group level version ”S
factor” (or ”big s”).
There are by now many different national measures of country well-being. Perhaps the most
common is Human Development Index published by the United Nations, but there are plenty of
others e.g. Social Progress Index, Democracy Ranking, Quality of Life Index, Where-to-be-born
Index, Democracy Index. Previous studies have correlated some of these with national IQs, but
not in a systematic manner (see review in [
6
]). Most of these national indexes have subcomponents
that can be analyzed as well to see whether there is common variance, or whether the presence of
these variables on the index is merely a function of their desirability as judged by the authors. The
index scores are usually not outputs from a factor analysis, but usually some weighted average of the
subcomponents based on how important the authors thought they were.
The goals of this study were thus: 1) gather a large collection of country well-being indexes
1Corresponding author: emil@emilkirkegaard.dk
and their subcomponents, 2) perform factor analyses
2
and other tests to check whether there is an
S factor in the data and how reliably it can be measured, and 3) examine how national cognitive
measurements relate with this.
2 Building the dataset
After I began the task of merging all the datasets into one, the main reason why others had not done
it before became apparent: it is a lot of work to combine datasets manually. There are two reasons.
Firstly, the different datasets do not contain exactly the same countries, so one has to rearrange them
so the two lists match up. Secondly, often the same country is spelled different ways giving problems
with alphabetic sorting of the countries. For instance, South Korea is called ”South Korea”, ”Korea,
Republic of”, ”S. Korea”, ”Korea Rep.”, ”Korea South” and variations in other languages as well.
To avoid this intensive manual labor, I wrote a Python script
3
that can combine two datasets
into one. I then used this script to combine datasets for, among others, GDP per capita (from the
International Monetary Fund, 2014)[
7
], national IQs from Lynn and Vanhanen (2012)[
6
], Human
Development Index, Social Progress Index, Democracy Ranking, Quality-of-Life Index, Where-to-be-
born index, Democracy Index and over 250 other variables concerning countries. The entire dataset
as well as the Python script is available at http://emilkirkegaard.dk/megadataset.
After inspecting the dataset, I decided to limit my analysis to the subcomponents of the Social
Progress Index and Democracy Ranking. This is because both of these are very comprehensive, each
having more than 40 subcomponents that cover a wide spectrum of living. Furthermore, these big
composites usually included many of the more narrow national rankings, so using both would result
in double-sampling some measurements. Furthermore, using two precompiled datasets allows me to
avoid the objection that I put the datasets together in such a way as to bias the results towards the
hypotheses I am testing.
2.1 Social progress Index (SPI) and Democracy Ranking (DR)
The SPI is very comprehensive and is a function of 54 basic subcomponents.[
8
] The structure is
complicated and best shown visually, see Figure 1. A 56 page description of the SPI is found in the
methodological report.[9]
The DR also has a large structure. It has 6 overall dimensions: political system, economy,
environment, gender equality, health and knowledge. These have a total of 42 subcomponents.[10]
The DR sample of subcomponents overlaps heavily with that of SPI. As with the SPI, there is
a 50 page methodological report that explains in further detail how the index works.[
11
] The main
differences between SPI and DR is that DR includes economic variables while the SPI only includes
social and political. The SPI authors explicitly state they want to ”move beyond GDP” (p. 8).
However, the correlation between GDP per capita (IMF, 2013) and SPI is .860. The reader can judge
for themselves how well they succeeded.
2.2 Should one combine the SPI and DR?
One could combine the subcomponents from SPI and DR into one dataset by removing the duplicate
variables. I did not do this for three reasons. Firstly, it is useful to have two separate datasets to
test hypotheses on. If an hypothesis by chance happens to be confirmed in the first dataset, it is
unlikely that it will also be confirmed in the second. Secondly, the indexes are already extremely
comprehensive. Thirdly, it would reduce the sample size and result in a lower case-variable ratio.
The case-variable ratio is already small: 132/54=2.44 in the SPI and 115/42=2.74 in the DR. Small
2
For the purposes of this article, principal components analysis is considered a factor analytic method, even though
some researchers argue that PCA should not be considered as such.
3
Python is a free and versatile yet easy to use general purpose programming language. Learn more at
https://www.python.org/. I used the Anaconda package. https://store.continuum.io/cshop/anaconda/
Figure 1:
The structure of the Social Progress Index. Source: http://socialprogressimperative.org/
case-variable ratios can make factor analytic results unstable. Nathan Zhao has compiled an excellent
overview of recommendations and practices concerning the case-variable ratio.[
12
] He mentions
that in 26% of a sample of studies using principal components analysis (PCA) in PsychINFO, the
case-variable ratio was between 2 and 5 as it is with my two datasets. If the datasets were combined,
the ratio would be much lower.
3 Initial analyses
I used R to do the analyses.4
Since I was primarily interested in the first factor, all analyses extracted only the first factor
unless otherwise noted.
The SPI dataset had missing data for some cases, between 1 and 5 missing values. I decided to
examine the effect of using the reduced sample with complete data vs. imputing the means in the
cases with missing values. The subsample with complete data has N=78, while the full sample has
N=132. In other words, using only the reduced sample implies a sample size decrease of 41%.
I performed PCA on both the reduced and the means-imputed datasets to examine the effect of
the procedure. The Pearson correlation of factor loadings is 0.996, indicating that the procedure does
not alter the structure of the data much. Similarly, the congruence coefficient is 1.0. I performed
KMO tests (a measure of sampling adequacy) on both samples which showed that reducing the
4
R is a free, powerful, and yet easy to use programming language designed for data mining and statistics. See
http://www.r-project.org/
sample reduces the KMO (0.899 to 0.809). In comparison, KMO in the DR dataset is 0.884. All
values are considered ’meritorious’.[13, p. 225] All further analysis uses the means-imputed dataset.
Bartlett’s test (tests whether the data is suitable for factor analysis) are extremely significant in
all three datasets (p<0.00001).
3.1 First factors from different methods
Sometimes factor methods give different results. For this reason, I compared the first factors
extracted using PCA as well as minimum residuals (MinRes/MR), weighted least squares (WLS),
generalized least squares (GLS), principal axis factoring (PAF), minimized chi square (MinChi/MC)
and maximum likelihood estimation (ML). Tables 1, 2 and 3, show factor score intercorrelations,
factor loadings intercorrelation and factor congruence coefficients, respectively.
Method PC1 MR1 WLS1 GLS1 PA1 ML1 MC1
PC1 0.964 0.999 0.999 0.997 0.995 0.964
MR1 0.983 0.968 0.968 0.968 0.978 1
WLS1 0.999 0.985 1 0.998 0.996 0.968
GLS1 0.999 0.985 1 0.998 0.996 0.968
PA1 0.995 0.982 0.997 0.997 0.997 0.968
ML1 0.986 0.995 0.987 0.987 0.986 0.978
MC1 0.983 1 0.985 0.985 0.982 0.995
Table 1: Factor score intercorrelations. SPI below the diagonal, DR above.
Method PC1 MR1 WLS1 GLS1 PA1 ML1 MC1
PC1 0.993 1 1 1 0.999 0.993
MR1 0.997 0.993 0.993 0.994 0.996 1
WLS1 1 0.997 1 1 0.999 0.993
GLS1 1 0.997 1 1 0.999 0.993
PA1 1 0.997 1 1 0.999 0.994
ML1 0.996 1 0.997 0.997 0.997 0.996
MC1 0.997 1 0.997 0.997 0.997 1
Table 2: Factor loadings intercorrelations. SPI below the diagonal, DR above.
Method PC1 MR1 WLS1 GLS1 PA1 ML1 MC1
PC1 0.997 1 1 1 1 0.997
MR1 0.997 0.997 0.997 0.997 0.998 1
WLS1 1 0.997 1 1 1 0.997
GLS1 1 0.997 1 1 1 0.997
PA1 1 0.997 1 1 1 0.997
ML1 0.997 1 0.997 0.997 0.997 0.998
MC1 0.997 1 0.997 0.997 0.997 1
Table 3: Factor congruence coefficients. SPI below the diagonal, DR above.
As can be seen, these are extremely similar across method and dataset.
3.2 Size of the first factor x factor method
Jensen and Weng[
14
] note that PCA tends to overestimate the variance accounted for by the first
factor. Table 4 shows the size of the first factor by factor method. There is not much evidence that
PCA overestimates the size.
Dataset/method PCA MR WLS GLS PA ML MC
SPI 0.42 0.41 0.42 0.42 0.41 0.4 0.41
DR 0.47 0.44 0.47 0.47 0.46 0.46 0.44
Table 4: Variance accounted for by the first factor by factor method.
3.3 Mean loadings x factor method
Major[
15
] report that PCA tends to overestimate factor loadings. Table 5 shows the mean absolute
(because some were negative) loading using each method. There is not much evidence that PCA
overestimates loadings.
Dataset/method PCA MR WLS GLS PA ML MC
SPI 0.603 0.582 0.600 0.600 0.595 0.580 0.582
DR 0.653 0.624 0.650 0.650 0.644 0.641 0.624
Table 5: Mean absolute loading by factor method.
Generally, factor method makes little difference for the first factor in the full analyses.
4 Subset x whole comparisons: How many variables are
necessary for reliable measurement?
Since I find that, regardless of method and dataset used, the first factor is a large factor accounting
for about 40-47% of the variance, it would be interesting to know how many variables one needs to
measure it well. To find out, I sampled subsets of variables at random from the datasets, extracted
the first factor, and then correlated the scores from it with the scores of the first factor using all the
variables. I repeated the sampling 1000 times to reduce sampling error to almost zero. I used the 7
different methods mentioned earlier.
Results are shown in Figures 2 and 3. Each step shows the change in correlation from adding
another random variable. They show that the number of variables necessary to reliably estimate the
first factor from all variables well is small. For PCA, picking one variable at random gives 0.603
which is the same as the average (absolute) loading on the first factor in the full analysis. Using the
first factor of 5 variables yields an average correlation of 0.877.
Since PCA is usually singled out for criticism, I compared PCA with the average of all the other
methods. As one can see, PCA is a bit higher for the small samples. However, even the largest
difference is quite small. The difference in the SPI dataset with 3 variables is 0.042.
Figure 2:
Subset factor correlation with S factor in SPI dataset. The plot shows the average
correlation of the first factor within a sample of X variables with the first factor of all 54 variables.
Figure 3:
Subset factor correlation with S factor in DR dataset. The plot shows the average
correlation of the first factor within a sample of X variables with the first factor of all 54 variables.
5 Subset x subset comparisons
How well does the first factor from a random subset correlate with that of another random subset,
with no overlap in variables? I tested this in both datasets with subset sizes 5 and 10 with PCA.
Results are shown in Table 6.
N subsets / Dataset SPI DR
N=5 0.758 0.804
N=10 0.874 0.902
Table 6: First factors from different and random subsets of the same dataset.
Results are quite high. They show that as long as one picks a reasonable number of them, e.g.
10., it is not so important which subset of national indexes one chooses since they measure to a large
degree a common factor.
6 Factor loadings with different methods x number of vari-
ables
I found before that PCA did not yield higher loadings than other methods. However, that conclusion
was based on analyzing the entire datasets with a large number of variables. There is evidence that
PCA yields higher loadings when the number of variables is small.[
16
,
17
] To see if this true for my
international data, I extracted the mean absolute loading from 1000 analyses of each sample size 2 to
20 with each of 7 different methods. Results are shown in Figures 4 and 5.
Figure 4:
Mean absolute loadings on the first factor with different methods and different sample
sizes of variables. Results for SPI dataset.
Figure 5:
Mean absolute loadings on the first factor with different methods and different sample
sizes of variables. Results for DR dataset.
Results are nearly identical for both datasets. The loadings using PCA are higher than with the
other methods and this is especially the case for smaller samples of variables. Major et al[
16
] seem to
be right. One should not use estimates of variable loadings from PCA when the number of variables
is small.
7 Relationship with national cognitive ability measures
Usually, one finds some interesting country-level variable and then correlates it with national IQ,
perhaps with some controls. Often authors will also argue for a causal connection from national IQ/G
to country-level variables. The typical example of this is wealth (e.g. [
18
,
19
,
20
,
21
,
22
]). Since
it is known that g causes greater wealth at the individual level, and that nations can generally be
considered to be a large group of individuals, it would be very surprising, though not impossible, if
there was no causation at the group level as well.
I don’t want to argue at length for any causal model in this paper, so I merely present the
correlations and leave the interpretation to the reader. I choose to look at two cognitive ability
measures, the total scores from SPI and DR, as well as the S factors (PCA) from each datasets.
For cognitive measures, I use Lynn and Vanhanen’s 2012 national IQ estimates[
6
] and Altinok’s
educational achievement estimates[23]. Results are shown in Table 7.
I expected the correlations to be quite strong. If population differences in G are the main cause of
national differences in many socioeconomic domains, then aggregating measures should increase the
correlation with G, since measurement specificity averages out.
Table 7:
Correlation matrix with Social Progress Index, Democracy Ranking, their S factors, national
IQs and Altinok’s achievements scores.
As expected, the correlations with the aggregated country-level measures have a very strong
correlation with proxies for G. All correlations have P’s
<
0.001 (N’s 100-132). Furthermore, the
correlations between cognitive measures and the S factors from both datasets are stronger than with
the indexes as made by the authors. This indicates either or both of two things: 1) that it is the S
factor that drives the relationship, not the remaining variance, or 2) that the summing procedure
used by the authors obscures the underlying relationship. Global hereditarians[
24
] may interpret
this result as being in line with predictions, while non-hereditarians may interpret it as showing that
national differences in S cause national differences in G, or something else entirely.
I don’t report the results using S extracted from the other methods because they are nearly
identical.
8 The method of correlated vectors and the S factor
In the study of human cognitive abilities, it is known that different tests have different loadings on
the first factor, their g-loading.[
3
] In other words, some tests measure the underlying factor better
than others. Jensen invented the method of correlated vectors (MCV) to test whether it was the g
factor that was related to some other variable, or some other part of the variance. I used the same
method here in a reverse fashion to test whether it is the S factor found in the SPI and DR datasets
that drives the correlation with G proxies or the non-S variance.
I thus calculated the correlation of both G proxies with every variable in both datasets and
correlated these values with the S factor loadings. In both datasets, using both national cognitive
measures, the correlations are 0.99, confirming the indication from before that it is the S factor
that drives the relationships. Nevertheless, it is possible to have a very high correlation and not
have a very linear relationship as demonstrated by Anscombe’s Quartet.[
25
] Inspection of all the
scatter-plots, however, reveal that the relationships really are very linear. Figure 6 shows a plot of
one of these MCV analyses (all the others are similar). Tables 13 and 14 in the Appendix contains
both the factor loadings on the S factor and the correlations of each variable with national IQs (96
variables in total).
Figure 6:
Variable loadings on the S factor (PCA) and their correlation with national IQs in the
SPI dataset.
The very strong correlations resulting from the use of MCV in these analyses give indirect support
for researchers who have been arguing that the MCV results on cognitive data are both spuriously low
and inconsistent due to statistical artifacts. [
26
,
27
,
28
,
29
] see also [
30
]. Probably the two primary
sources of error are: 1) Restriction of range in factor loadings, and 2) sampling error in the loadings
vector due to a small number of variables. Both sources of error seem very small in the present
analysis since I used 54 and 42 variables varying in their loading from near -1 to almost 1. Table 8
shows descriptive statistics of variable loadings in both datasets.
Dataset Min Max Mean SD N
SPI -.90 .92 .10 .64 56
DR -.70 .92 .53 .45 42
Table 8: Descriptive statistics for variable loadings in both datasets using PCA.
There is a question concerning whether it is proper to do the MCV analyses without reversing the
variables that have negative loadings on the S factor first. Using the non-reversed variables means
the variance is higher which increases the correlation. Reversing them would decrease the correlations.
I decided to use the data as they were given by the authors i.e. with no reversal of variables.
If one wanted to reverse them nonetheless, it would not be quite obvious just which ones to
reverse. With cognitive data, there is never any disagreement about which direction of a variable is
associated with better performance. With social, political and economic variables, this is not always
the case. For instance, in the DR dataset, there are two variables (28-29) dealing with expenditure
on health: public and private. The public has a strong positive loading (0.658) and the private has a
small negative loading (-0.210). There doesn’t seem to be much agreement as to whether it is better
for health care to be publicly or privately financed, so which direction should they be in? What about
electric power consumption per capita (17)? It seems best not to get bogged down into discussions of
which of these should be reversed and just use the data as given by authors. Recall that they could
not have biased the data to fit my analyses since they put together the datasets for other purposes.
If, however, one insists on reversing the obvious cases, I have done this for those that I considered
obvious enough. These are variables 1:6,8,13:18,26,28:33,35,42,48,53 from the SPI dataset. Re-doing
the PCA and MCV analysis gives a correlation between S factor loading and national IQ of 0.98.
Very little difference.
9 The S loading and the desirability of the variable: Are
the first factors really general factors?
The mean absolute loading on the S factors is quite high (0.6-0.65). If one inspects the loadings (as
shown in the Appendix, Tables 13 and 14), one may note that what loads positively on the S factor
is generally considered something desirable, and the reverse for negative loadings. For example, the
first variable in the SPI dataset is the percentage of the population that is malnourished. This is
clearly something generally considered undesirable and the loading is strongly negative (-0.71).
There are a few variables, however, where the S-loading and the desirability of the variable are
opposite. In the SPI dataset, obesity rate has a loading of 0.65 on S, but it is generally thought to be
something undesirable, even if it indicates, evolutionarily speaking, that food is plentiful and that
there is an absence of non-intentional starvation. The suicide rate from the same dataset shows the
same pattern (loading 0.19). Death per capita by air pollution has a loading of 0.19. Finally, use of
water resources has a loading of 0.26. There are no variables in the DR dataset like this. All in all,
there are 4 variables out of 96 that have opposite signs on their S loading and desirability. Only in
the case of obesity is the loading on S >0.27.
It is worth noting that group-level correlations need not be the same or even in the same direction
as individual-level correlations. Both suicide and obesity seem to have the opposite directional
relationship at the individual-level.[31, 32]
In cognitive data, if one transforms variables so that a positive value corresponds to better
performance (e.g. by reversing response time variables since longer reaction times indicate worse
performance), then there is a positive manifold: Every variable has a positive correlation with any
other variable. This pattern is seen both in individual-level data with the g factor and in country-level
data with the G factor.
In light of the above, one might wonder whether it is fair to call the first factors ”general factors”
when there are 4 variables that go against the pattern. It comes down to which meaning of ”general”
is used. Dictionary.com gives these two definitions, among others:
1.
of or pertaining to all persons or things belonging to a group or category: a general meeting of
the employees.
2.
of, pertaining to, or true of such persons or things in the main, with possible exceptions; common
to most; prevalent; usual: the general mood of the people.
When I say ”general”, I mean in the second sense, not the first, and in this sense, the S factor
is clearly general since only 4 of 96 variables load in the opposite direction, and only one of them
strongly.
10 Number of factors to extract and their size
All previous analyses used only the first factor. However, one might want to know how many factors
standard methods indicate that one should extract and if doing so changes results. The R package
nFactors includes the nScree() function
5
that calculates the number of factors to extract using 4
different methods: optimal coordinates, acceleration factor (a non-subjective measure based on the
Scree plot), parallel analysis and Kaiser’s Rule (Eigenvalue >1).
I ran the analysis on both datasets. In the SPI dataset, three of the four methods indicate the
number of factors to extract is 9, acceleration factor being the outlier which suggests only to extract
1. The situation is similar in the DR dataset with three methods suggesting 8 factors and acceleration
factor again only suggesting 1.
Extracting the first 9/8 unrotated factors using maximum likelihood estimation, both Pearson
correlations between scores (SPI: 0.98, DR: 0.97), loadings (SPI: 1.00, DR: 1.00) and congruence
5, http://artax.karlin.mff.cuni.cz/r-help/library/nFactors/html/nScree.html
coefficients (SPI: 1.00, DR: 1.00) show that the first factors from the 1-factor analyses are very similar
to or identical with the first factors from the 8-9 factor analyses.
It is also worth mentioning that the percent of the variance accounted for by each factor quickly
decreases from the first factor, and that the first and second factor are not even close in size. Results
are shown in Table 9. The variance accounted for by the first factor is slightly lower than in the 1
factor analysis (0.40 vs 0.39 for SPI, and 0.46 vs 0.43 for DR).
Proportion var. for factor N 1 2 3 4 5 6 7 8 9
SPI 0.39 0.08 0.07 0.04 0.03 0.03 0.02 0.02 0.01
DR 0.43 0.07 0.07 0.04 0.03 0.02 0.02 0.02
Table 9: Proportion of variance accounted for each factor. Both datasets.
The relative size of the first factors compared to the second (SPI: 0.39 to 0.08, ratio = 4.9; DR:
0.43 to 0.07, ratio = 6.1) justifies calling them ”large”. Together with the evidence of their generality
in the previous section and in the following two, this justifies calling them ”large general factors”.
This would not have been the case if they were either not general or that the first two factors were
about equally large.
11 Correlated factors
Another way to test if there is a general factor, is to use repeated factor analysis while allowing for
correlated factors (oblimin rotated). This is sometimes considered the best way to extract g.[14]
The approach taken was this:
1.
Determine number of factors to extract with nScree (choosing the number most criteria agree
on).
2. Extract that number of factors with oblimin rotation using maximum likelihood estimation.
3. Repeat the above steps.
Using this method on cognitive data usually results in a g (general) factor at the second or third
level.[
33
] The same result was found for both datasets with international data. The 3rd order factor
scores correlated 0.97 with the first unrotated factor scores extracted with the same method in both
datasets, indicating robust results. It was not possible to use factor congruence or correlations of
loadings. In the SPI dataset, there were 9 factors at the first level, 3 at the second and 1 at the third.
In the DR dataset, there were 8, 2, and 1, respectively.
11.1 Schmid-Leiman using schmid()
The psych package in R contains the schmid() function
6
which is an implementation of the Schmid-
Leiman transformation[
34
] advocated by e.g. Jensen and Weng. The schmid() function can use three
different extraction methods (MR, ML, PA) and three different rotation methods (oblimin, promax,
simplimax). I ran the schmid() function on both datasets using all possible combinations. Then I
compared the general factor loadings with the loadings from the unrotated first factor extracted using
the same method (for technical reasons it was not possible to use factor congruence). Results are
shown in Tables 10 and 11.
6http://personality-project.org/r/html/schmid.html
Method Oblimin Simplimax Promax
minres 1.00 1.00 1.00
pa .99 .90 .98
ml .98 .85 .97
Table 10:
Comparison table between general factor extracting using schmid() function and the first
unrotated factor using the same extraction method. SPI dataset.
Method Oblimin Simplimax Promax
minres .97 .97 .99
pa .97 1.00 .99
ml .98 .99 .97
Table 11:
Comparison table between general factor extracting using schmid() function and the first
unrotated factor using the same extraction method. DR dataset.
There doesn’t seem to be any overall pattern with regards to factor extraction or rotation method.
Simplimax gave somewhat lower results in the SPI dataset, but the results were high in the DR
dataset. Results were strongly in line with previous results.
12 Other measures of the strength of a general factor
Recent discussion concerning the existence of a general factor in the personality domain has renewed
interest in criteria for considering the strength of a general factor.[
35
,
36
,
37
] Revelle and Wilt
(RW)[
38
] have criticized these studies for using inadequate methods for determining whether a factor
is plausibly interpreted as a strong general factor. Using simulated datasets, RW show that some
of the methods employed by prior studies would indicate a strong general factor even in datasets
where there plainly is none (see their Table 2). For instance, the size of the first factor is not a good
indication of whether there is a strong general factor or not because the datasets with no general
factor can have as large a first factor as those with a strong general factor. They show that a variety
of other methods are able to distinguish between datasets with and without a strong general factor.
One of these methods is the hierarchical omega and its asymptotic value (
ωh
and
ωh∞
). This
method was also recently used to estimate the strength of the g factor in different age groups as a
test of the Dynamic mutualism theory of g.[39]
A second method is to find the eigenvalue of the general factor in the Schmid-Leiman transformation
and divide it by the number of variables (Λ
g
(Λ
/N
)). This is the amount of variance accounted for by
the general factor.
A third method is the explained common variance (ECV).
A fourth method is the squared multiple correlation of regressing the original variables on the
general factor i.e. the amount of factor variance accounted for by the variables (R2).
The omega() function7can be used to calculate these values.
RW used these four methods (and one more which I did not know how to use on my data) on
8 personality datasets and 5 classic cognitive datasets. I have repeated these analyses on my two
datasets. Table 12 shows the average of the personality data, cognitive data and my datasets.
7http://personality-project.org/r/html/omega.html
Method ωhωh∞Λg(Λ/N)ECV R2
Personality data .37 .48 .16 .34 .41
Cognitive data .74 .79 .33 .57 .78
International data SPI .74 .75 .33 (17.7/54) .48 .81
International data DR .77 .78 .37 (15.57/42) .53 .81
Table 12:
Comparison of measures of general factor strength on personality data, cognitive data
and international data. Values for personality and cognitive data from [38].
The strength of the S factor in the international data is quite similar to, perhaps a little stronger
than, the g factor in the classic datasets, while the general factor of personality is clearly weaker.
13 Discussion and conclusion
In the measurement of mental abilities, there has been a long-standing debate on whether g from
one battery of tests was the same g from another battery of tests. The issue now seems to have
been settled by two studies. The authors had access to datasets where the sample took a couple of
different IQ batteries. They then used confirmatory factor analysis to compare latent g factors from
different batteries to each other and found them to be at or very close to unity.[
40
,
41
] The only
exception was the latent g extracted from the Cattell test, which is probably because it only has one
subtest type (non-verbal matrix reasoning) whereas the others had multiple different subtest types.
A second method that has been used is to sample random subsets from the entire battery of tests,
factor analyze the subset with and without a probe test, and then compare the g-loading of this probe
test in different subsets. Studies of this kind found quite stable g-loadings.[42, 43, 44, 17, 15]
The analyses carried out in this paper suggest that the S factor is quite like g in its stability of
measurement. The S factor scores (PCA) extracted from the two international datasets correlated
0.975. Extracting S factors from subsets of variables and comparing them either to each other (subset
x subset) or to the S factor from the complete analysis (subset x whole) suggest that S can be quite
reliably approximated picking a small collection of random variables. It would be interesting to redo
these two analyses on the cognitive data analyzed by Johnson et al.[40, 41]
One difference between cognitive data and international rankings is worth pointing out. The
cognitive test variables are actually themselves some sort of sum of (usually unweighted) the actual
test items (e.g. number of matrices solved). This is not generally the case for the national variables
although some of them, e.g. Freedom House indexes, are some sort of function of subcomponents. It
is not clear in which direction this disanalogy biases results, if any.
Results from both datasets examined in this study were generally extremely similar despite the
fact that the datasets were created by two different sets of researchers with different goals, none of
which was to look for any country-level general socioeconomic factor. This indicates that the results
are robust.
A reviewer pointed to a series of studies concerning a national-level K (of r-K life history theory[
45
])
factor.[
46
,
47
,
48
] They employ some of the variables also found in the SPI and DR datasets and find
a large general factor. From the perspective of this study, it seems that their K factor is likely just
the S factor. Although it is possible that a K factor constitutes a first or second order factor below
the S factor. Further research should investigate this.
The paper is an example of the usefulness of programmatic, as opposed to point-and-click/GUI-
based, statistical tools applied to psychology. This emerging field has been termed ”psychoinforma-
tions” by analogy with ”bioinformatics”.[
49
] The benefits of this approach is that the exact analysis
methods can be inspected instead of only described vaguely by words. They can also quickly be used
on other datasets by other researchers. It is an important step in the push towards open-sourcing
science.[
50
,
51
] The main downside is that the learning curve becomes somewhat steeper but this is a
very small price to pay.
Finally, should we care whether there is an S factor? Yes. The existence of a strong general
socioeconomic factor means that there is a mathematical concept that corresponds to the intuitive
concept of country well-being. This is not a necessary state of affairs. Imagine a two factor solution
where there are a political freedom factor and an economic freedom factor that don’t correlate. That
would be very different from what we see when we look at the real world.
14 Datasets and source code
This paper is open source. Everything necessary to reproduce the PDF and results is available in the
supplementary material at the journal website.
Table 15 in the Appendix contains the S scores for each country with data. They were computed as
the first unrotated factor (PCA). When both datasets covered a country, the average was taken. When
only one dataset had data, that value was used. The data can also be found in the supplementary
material as well as in the Worldwide Megadataset (http://emilkirkegaard.dk/megadataset).
15 Acknowledgments
Thanks to Laird Shaw for help with R and proofreading. Thanks to reviewers of the Open Differential
Psychology journal for numerous suggestions that greatly improved the article.
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Loadings for first factor in Social Progress Index
Subcomponent Loading r x IQ
Malnourishment (% of pop.) -0.71 -0.57
Depth of food deficit (calories/malnourished person) -0.69 -0.55
Maternal mortality rate (deaths/100,000 live births) -0.84 -0.77
Stillbirth rate (deaths/1,000 live births) -0.86 -0.75
Child mortality rate (deaths/1,000 live births) -0.90 -0.80
Deaths from infectious diseases (deaths/100,000) -0.82 -0.82
Access to piped water (% of pop.) 0.88 0.77
Rural vs. urban access to improved water source (absolute difference between % of pop.) -0.70 -0.60
Access to improved sanitation facilities (% of pop.) 0.86 0.74
Availability of affordable housing (% satisfied) 0.25 0.20
Access to electricity (% of pop.) 0.84 0.76
Quality of electricity supply (1=low; 7=high) 0.79 0.74
Indoor air pollution attributable deaths (deaths/100,000) -0.74 -0.57
Homicide rate (1= <2/100,000; 5= >20/100,000 -0.64 -0.71
Level of violent crime (1=low; 5=high) -0.57 -0.57
Perceived criminality (1=low; 5=high) -0.54 -0.46
Political terror (1=low; 5=high) -0.61 -0.48
Traffic deaths (deaths/100,000) -0.66 -0.61
Adult literacy rate (% of pop. aged 15+) 0.82 0.74
Primary school enrollment (% of children) 0.58 0.58
Lower secondary school enrollment (% of children) 0.82 0.67
Upper secondary school enrollment (% of children) 0.88 0.75
Gender parity in secondary enrollment (girls/boys) 0.56 0.40
Mobile telephone subscriptions (subscriptions/100 people) 0.68 0.55
Internet users (% of pop.) 0.92 0.81
Press Freedom Index (0=most free; 100=least free) -0.46 -0.27
Life expectancy (years) 0.89 0.84
Non-communicable disease deaths between the ages of 30 and 70 (probability of dying) -0.72 -0.60
Obesity rate (% of pop.) 0.65 0.44
Outdoor air pollution attributable deaths (deaths/100,000) 0.19 0.32
Suicide rate (deaths/100,000) 0.19 0.29
Greenhouse gas emissions (CO2 equivalents per GDP) -0.36 -0.29
Water withdrawals as a percent of resources 0.26 0.18
Biodiversity and habitat (0=no protection; 100=high protection) 0.25 0.19
Political rights (1=full rights; 7=no rights) -0.63 -0.48
Freedom of speech (0=low; 2=high) 0.33 0.19
Freedom of assembly/association (0=low; 2=high) 0.45 0.27
Freedom of movement (0=low; 4=high) 0.38 0.27
Private property rights (0=none; 100=full) 0.72 0.60
Freedom over life choices (% satisfied) 0.31 0.16
Freedom of religion (1=low; 4=high) 0.14 -0.04
Modern slavery, human trafficking and child marriage (1=low; 100=high) -0.61 -0.50
Satisfied demand for contraception (% of women) 0.79 0.73
Corruption (0=high; 100=low) 0.76 0.62
Women treated with respect (0=low; 100=high) 0.15 0.18
Tolerance for immigrants (0=low; 100=high) 0.15 -0.05
Tolerance for homosexuals (0=low; 100=high) 0.68 0.55
Discrimination and violence against minorities (0=low; 10=high) -0.59 -0.42
Religious tolerance (1=low; 4=high) 0.12 0.00
Community safety net (0=low; 100=high) 0.68 0.51
Years of tertiary schooling 0.65 0.67
Womens’ average years in school 0.87 0.75
Inequality in the attainment of education (0=low; 1=high) -0.80 -0.73
Number of globally ranked universities (0=none; 5= >50) 0.61 0.61
Table 13: Social Progress Index dataset loadings from PCA.
Appendix
Loadings for first factor in Democracy Ranking dataset
Subcomponent Loading r x IQ
Political rights (aggregated scores): Freedom House 0.67 0.43
Civil liberties (aggregated scores): Freedom House 0.74 0.53
Global Gender Gap Report 0.56 0.39
Press Freedom: Freedom House 0.67 0.44
Corruption Perceptions Index (CPI): Transparency International (TI) 0.79 0.64
Change(s) of the head of government (last 13 years, peaceful) 0.23 0.20
Political party change(s) of the head of government (last 13 years, peaceful) 0.39 0.30
GDP per capita, PPP (constant 2005 international $) 0.87 0.72
GDP per capita, PPP (current international $) 0.87 0.72
Central government debt, total (% of GDP) 0.56 0.51
Inflation, consumer prices (annual %) 0.58 0.49
Unemployment, total (% of total labor force) 0.55 0.55
Unemployment, youth total (% of total labor force ages 15-24) 0.59 0.51
CO2 emissions (kg per 2005 PPP $ of GDP) -0.17 -0.26
CO2 emissions (metric tons per capita) -0.59 -0.49
GDP per unit of energy use (constant 2005 PPP $ per kg of oil equivalent) -0.48 -0.50
Electric power consumption (kWh per capita) -0.70 -0.58
Electricity production from hydroelectric sources (% of total) -0.53 -0.54
Labor force, female (% of total labor force) 0.11 0.02
Unemployment, female (% of female labor force) 0.53 0.47
Primary education, pupils (% female) 0.31 0.15
School enrollment, secondary, female (% gross) 0.87 0.76
School enrollment, secondary, female (% net) 0.87 0.73
School enrollment, tertiary, female (% gross) 0.78 0.66
Life expectancy at birth, female (years) 0.86 0.85
Life expectancy at birth, total (years) 0.86 0.84
Health expenditure per capita, PPP (constant 2005 international $) 0.82 0.67
Health expenditure, public (% of GDP) 0.66 0.47
Health expenditure, private (% of GDP) -0.21 -0.28
Hospital beds (per 1,000 people) 0.52 0.58
Physicians (per 1,000 people) 0.68 0.64
Mortality rate, infant (per 1,000 live births) 0.86 0.82
Mortality rate, under-5 (per 1,000 live births) 0.83 0.81
School enrollment, secondary (% gross) 0.89 0.78
School enrollment, secondary (% net) 0.84 0.74
School enrollment, tertiary (% gross) 0.90 0.83
Pupil-teacher ratio, primary 0.59 0.52
Telephone lines (per 100 people) 0.88 0.81
Internet users (per 100 people) 0.92 0.83
Mobile cellular subscriptions (per 100 people) 0.71 0.61
Research and development expenditure (% of GDP) 0.61 0.52
Scientific and technical journal articles (per 1,000 people) 0.78 0.66
Table 14: Democracy Ranking dataset loadings from PCA.
S score ID Names
-1.646905191 AGO Angola
0.007429029 ALB Albania
0.738523723 ARE United Arab Emirates
0.644262065 ARG Argentina
-0.029950119 ARM Armenia
1.558365495 AUS Australia
1.371992315 AUT Austria
0.042892955 AZE Azerbaijan
-1.942758154 BDI Burundi
1.383343184 BEL Belgium
-1.235078464 BEN Benin
-1.602305168 BFA Burkina Faso
-0.873359243 BGD Bangladesh
0.553371357 BGR Bulgaria
-0.116325961 BHR Bahrain
0.066447482 BIH Bosnia and Herzegovina
0.320160291 BLR Belarus
-0.265921951 BOL Bolivia
0.211618534 BRA Brazil
-0.543886157 BTN Bhutan
-0.377329857 BWA Botswana
-2.29869988 CAF Central African Republic
1.439760092 CAN Canada
1.513321679 CHE Switzerland
0.754033701 CHL Chile
-0.248274788 CHN China
-1.350950211 CMR Cameroon
-1.287657542 COG Congo Rep.
0.043145898 COL Colombia
0.625901223 CRI Costa Rica
0.24519397 CUB Cuba
0.627728605 CYP Cyprus
0.955500773 CZE Czech Republic
1.459750885 DEU Germany
-1.134191132 DJI Djibouti
1.534062625 DNK Denmark
-0.114062309 DOM Dominican Republic
-0.071647669 DZA Algeria
0.035434359 ECU Ecuador
-0.332594317 EGY Egypt Arab Rep.
1.210823896 ESP Spain
1.070084848 EST Estonia
1.635287426 FIN Finland
1.276852479 FRA France
1.376690022 GBR United Kingdom
0.040574777 GEO Georgia
-0.888365187 GHA Ghana
-1.865749584 GIN Guinea
0.855402481 GRC Greece
-0.402289169 GTM Guatemala
-0.138027257 GUY Guyana
-0.348985065 HND Honduras
0.62922268 HRV Croatia
-1.271672496 HTI Haiti
0.769771009 HUN Hungary
-0.302354034 IDN Indonesia
-0.744089791 IND India
1.399225147 IRL Ireland
-0.124524871 IRN Iran Islamic Rep.
-0.841668176 IRQ Iraq
1.431614903 ISL Iceland
1.051434987 ISR Israel
0.959591737 ITA Italy
0.199015596 JAM Jamaica
0.279322831 JOR Jordan
1.298028434 JPN Japan
0.2190532 KAZ Kazakhstan
-1.117062455 KEN Kenya
-0.291080922 KGZ Kyrgyz Republic
-0.964596911 KHM Cambodia
1.145562698 KOR Korea Rep.
0.665151534 KWT Kuwait
-0.983513745 LAO Lao PDR
0.070833285 LBN Lebanon
-1.282846945 LBR Liberia
-0.635241651 LBY Libya
-0.185372171 LKA Sri Lanka
-1.066267436 LSO Lesotho
0.777961201 LTU Lithuania
0.593825046 LVA Latvia
-0.368161707 MAR Morocco
0.086260355 MDA Moldova
-1.36754981 MDG Madagascar
0.170060159 MEX Mexico
0.140480755 MKD Macedonia FYR
-1.412397485 MLI Mali
0.406047334 MNE Montenegro
-0.106068729 MNG Mongolia
-1.560046735 MOZ Mozambique
-1.578617978 MRT Mauritania
0.412291654 MUS Mauritius
-1.431907603 MWI Malawi
0.262648808 MYS Malaysia
-0.654597647 NAM Namibia
-1.752581129 NER Niger
-1.51473539 NGA Nigeria
-0.24724988 NIC Nicaragua
1.555502383 NLD Netherlands
1.658886469 NOR Norway
-0.755987594 NPL Nepal
1.429661061 NZL New Zealand
-1.344835442 PAK Pakistan
0.38815429 PAN Panama
0.101986622 PER Peru
-0.123768471 PHL Philippines
-1.040057426 PNG Papua New Guinea
0.837377811 POL Poland
1.049452138 PRT Portugal
-0.291711788 PRY Paraguay
0.37607494 ROU Romania
0.264524615 RUS Russian Federation
-1.147138222 RWA Rwanda
0.351630848 SAU Saudi Arabia
-1.484862655 SDN Sudan
-1.008201142 SEN Senegal
0.92417306 SGP Singapore
-1.736454518 SLE Sierra Leone
-0.052413047 SLV El Salvador
0.368451917 SRB Serbia
0.731005733 SVK Slovak Republic
1.124515502 SVN Slovenia
1.556969725 SWE Sweden
-0.932814898 SWZ Swaziland
-0.826221777 SYR Syrian Arab Republic
-2.346681181 TCD Chad
-1.411308631 TGO Togo
0.094976498 THA Thailand
-0.451511323 TJK Tajikistan
-0.780605433 TLS Timor-Leste
0.513083926 TTO Trinidad and Tobago
0.006151615 TUN Tunisia
0.154421183 TUR Turkey
-1.281883545 TZA Tanzania
-1.37592458 UGA Uganda
0.393375955 UKR Ukraine
0.767864944 URY Uruguay
1.433892248 USA United States
-0.01863086 UZB Uzbekistan
-0.088868721 VEN Venezuela RB
-1.500981407 YEM Yemen Rep.
-0.167586574 ZAF South Africa
-1.199754202 ZMB Zambia
Table 15: S factor scores for all countries found in either SPI, DR or both.
