IRIII.2 Quantitative Methods in the Study of International Relations
Bivariate Relations
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# '.,,/'.,, Lab 3: Bivariate Relations
If it’s not installed, install it.
library(tidyverse)
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#> ✔ ggplot2 3.4.2 ✔ tibble 3.2.1
#> ✔ lubridate 1.9.2 ✔ tidyr 1.3.0
#> ✔ purrr 1.0.1
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#> ✖ dplyr::filter() masks stats::filter()
#> ✖ dplyr::lag() masks stats::lag()
#> ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(stevemisc)
#>
#> Attaching package: 'stevemisc'
#>
#> The following object is masked from 'package:lubridate':
#>
#> dst
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#> The following object is masked from 'package:dplyr':
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#> tbl_df
Reminder: Please have read this:
Load and prepare the data
On Athena, I uploaded the data as an Excel file. You can download that data to your working directory and load into your session as follows. Here, I have to impress upon you that we have only so much time together so I have to assume some preliminary knowledge with file structures.
#Data <- readxl::read_excel("Lipset59.xlsx")
# ALTERNATIVELY, which I'll do here:
Data <- readr::read_csv("http://svmiller.com/extdata/democracy-income-1950.csv")
#> Rows: 48 Columns: 8
#> ── Column specification ────────────────────────────────────────────────────────
#> Delimiter: ","
#> chr (3): country, iso3c, cat
#> dbl (5): wbgdp2011est, wbpopest, v2x_polyarchy, polity2, xm_qudsest
#>
#> ℹ Use `spec()` to retrieve the full column specification for this data.
#> ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.
# Or...
# Data <- stevedata::Lipset59
Let’s see what these data look like.
Data
#> # A tibble: 48 × 8
#> country iso3c cat wbgdp2011est wbpopest v2x_polyarchy polity2 xm_qudsest
#> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 Australia AUS EE: … 1.12e11 8.07e6 0.782 10 1.87
#> 2 Belgium BEL EE: … 7.75e10 8.29e6 0.685 10 1.85
#> 3 Canada CAN EE: … 1.79e11 1.33e7 0.75 10 1.52
#> 4 Denmark DNK EE: … 4.66e10 4.09e6 0.896 10 1.58
#> 5 Ireland IRL EE: … 1.77e10 2.86e6 0.801 8 1.20
#> 6 Luxembourg LUX EE: … 4.16e 9 2.85e5 0.832 10 1.75
#> 7 Netherlan… NLD EE: … 9.28e10 9.64e6 0.791 10 1.71
#> 8 New Zeala… NZL EE: … 2.48e10 1.82e6 0.795 10 1.85
#> 9 Norway NOR EE: … 3.44e10 3.15e6 0.803 10 1.71
#> 10 Sweden SWE EE: … 7.87e10 6.70e6 0.767 10 1.79
#> # ℹ 38 more rows
I want to keep things nice and simple for this exercise and avoid the somewhat convoluted file names (for beginners). Here is where I have to impress upon you to read the above-linked blog post that’s on my website for a description of what the democracy indicator is communicating. One thing I’ll know I’ll have to do is create a variable for GDP per capita. I have data on GDP, and I have data on population size. I just have to divide one over the other.
Ideally, you can tease out what’s happening in this sequence of commands.Any volunteers to elaborate what’s happening here?
Data %>%
mutate(gdppc = wbgdp2011est/wbpopest) %>%
select(country:cat, xm_qudsest, gdppc) %>%
rename(demest = xm_qudsest) -> Data
Data
#> # A tibble: 48 × 5
#> country iso3c cat demest gdppc
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Australia AUS EE: Stable Democracies 1.87 13919.
#> 2 Belgium BEL EE: Stable Democracies 1.85 9358.
#> 3 Canada CAN EE: Stable Democracies 1.52 13413.
#> 4 Denmark DNK EE: Stable Democracies 1.58 11384.
#> 5 Ireland IRL EE: Stable Democracies 1.20 6192.
#> 6 Luxembourg LUX EE: Stable Democracies 1.75 14589.
#> 7 Netherlands NLD EE: Stable Democracies 1.71 9633.
#> 8 New Zealand NZL EE: Stable Democracies 1.85 13602.
#> 9 Norway NOR EE: Stable Democracies 1.71 10916.
#> 10 Sweden SWE EE: Stable Democracies 1.79 11755.
#> # ℹ 38 more rows
Basic descriptive statistics
I refer you again to the blog post, but Lipset (1959) groups these 48 countries into four categories. These are admittedly clumsy, but the four groups are stable democracies in Europe and the English-speaking world (EE), unstable democracies and dictatorships in the same, democracies and unstable dictatorships among Latin American Nations (LAN), and stable dictatorships in the same.
# As verification:
Data %>%
distinct(cat)
#> # A tibble: 4 × 1
#> cat
#> <chr>
#> 1 EE: Stable Democracies
#> 2 EE: Unstable Democracies and Dictatorships
#> 3 LAN: Democracies and Unstable Dictatorships
#> 4 LAN: Stable Dictatorships
# To see who is grouped into what:
Data %>%
group_split(cat)
#> <list_of<
#> tbl_df<
#> country: character
#> iso3c : character
#> cat : character
#> demest : double
#> gdppc : double
#> >
#> >[4]>
#> [[1]]
#> # A tibble: 13 × 5
#> country iso3c cat demest gdppc
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Australia AUS EE: Stable Democracies 1.87 13919.
#> 2 Belgium BEL EE: Stable Democracies 1.85 9358.
#> 3 Canada CAN EE: Stable Democracies 1.52 13413.
#> 4 Denmark DNK EE: Stable Democracies 1.58 11384.
#> 5 Ireland IRL EE: Stable Democracies 1.20 6192.
#> 6 Luxembourg LUX EE: Stable Democracies 1.75 14589.
#> 7 Netherlands NLD EE: Stable Democracies 1.71 9633.
#> 8 New Zealand NZL EE: Stable Democracies 1.85 13602.
#> 9 Norway NOR EE: Stable Democracies 1.71 10916.
#> 10 Sweden SWE EE: Stable Democracies 1.79 11755.
#> 11 Switzerland CHE EE: Stable Democracies 1.01 16188.
#> 12 United Kingdom GBR EE: Stable Democracies 1.90 11696.
#> 13 United States USA EE: Stable Democracies 1.27 17171.
#>
#> [[2]]
#> # A tibble: 15 × 5
#> country iso3c cat demest gdppc
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Austria AUT EE: Unstable Democracies and Dictatorships 1.79 6330.
#> 2 Bulgaria BGR EE: Unstable Democracies and Dictatorships -0.460 3245.
#> 3 Czechoslovakia CSK EE: Unstable Democracies and Dictatorships -0.504 6242.
#> 4 Finland FIN EE: Unstable Democracies and Dictatorships 1.58 7503.
#> 5 France FRA EE: Unstable Democracies and Dictatorships 1.58 8769.
#> 6 West Germany DEU EE: Unstable Democracies and Dictatorships 1.69 6933.
#> 7 Greece GRC EE: Unstable Democracies and Dictatorships 0.606 3432.
#> 8 Hungary HUN EE: Unstable Democracies and Dictatorships -0.485 4670.
#> 9 Iceland ISL EE: Unstable Democracies and Dictatorships 1.74 9339.
#> 10 Italy ITA EE: Unstable Democracies and Dictatorships 1.78 5558.
#> 11 Poland POL EE: Unstable Democracies and Dictatorships -0.411 4713.
#> 12 Portugal PRT EE: Unstable Democracies and Dictatorships -0.634 3519.
#> 13 Romania ROU EE: Unstable Democracies and Dictatorships -0.485 2276.
#> 14 Spain ESP EE: Unstable Democracies and Dictatorships -0.590 4040.
#> 15 Yugoslavia YUG EE: Unstable Democracies and Dictatorships -0.512 2893.
#>
#> [[3]]
#> # A tibble: 7 × 5
#> country iso3c cat demest gdppc
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Argentina ARG LAN: Democracies and Unstable Dictatorships 0.266 5750.
#> 2 Brazil BRA LAN: Democracies and Unstable Dictatorships 0.797 2485.
#> 3 Chile CHL LAN: Democracies and Unstable Dictatorships 0.694 5825.
#> 4 Colombia COL LAN: Democracies and Unstable Dictatorships -0.116 3835.
#> 5 Costa Rica CRI LAN: Democracies and Unstable Dictatorships 1.01 3809.
#> 6 Mexico MEX LAN: Democracies and Unstable Dictatorships -0.155 4827.
#> 7 Uruguay URY LAN: Democracies and Unstable Dictatorships 0.827 7669.
#>
#> [[4]]
#> # A tibble: 13 × 5
#> country iso3c cat demest gdppc
#> <chr> <chr> <chr> <dbl> <dbl>
#> 1 Bolivia BOL LAN: Stable Dictatorships -0.128 3020.
#> 2 Cuba CUB LAN: Stable Dictatorships 0.780 3726.
#> 3 Dominican Republic DOM LAN: Stable Dictatorships -0.719 2173.
#> 4 Ecuador ECU LAN: Stable Dictatorships 0.665 2805.
#> 5 El Salvador SLV LAN: Stable Dictatorships -0.195 1527.
#> 6 Guatemala GTM LAN: Stable Dictatorships 0.605 3156.
#> 7 Haiti HTI LAN: Stable Dictatorships -0.100 1918.
#> 8 Honduras HND LAN: Stable Dictatorships 0.0495 2525.
#> 9 Nicaragua NIC LAN: Stable Dictatorships -0.547 3648.
#> 10 Panama PAN LAN: Stable Dictatorships 0.283 2928.
#> 11 Paraguay PRY LAN: Stable Dictatorships -0.181 2689.
#> 12 Peru PER LAN: Stable Dictatorships 0.0455 3191.
#> 13 Venezuela VEN LAN: Stable Dictatorships -0.119 9750.
Lipset (1959) observes that there is something he thinks is interesting among these countries. The exact values are going to necessarily vary (because we’re using different data than him) but he observes the poorest stable democracy in the EE group is just about as rich as the richest unstable democracy in the same group. He finds the overlap among the Latin American states to be much closer, but that the more democratic states are on the balance richer.
We can use the following code to see what he saw.
Data %>%
summarize(min = min(gdppc),
median = median(gdppc),
mean = mean(gdppc),
max = max(gdppc), .by=cat)
#> # A tibble: 4 × 5
#> cat min median mean max
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 EE: Stable Democracies 6192. 11755. 12294. 17171.
#> 2 EE: Unstable Democracies and Dictatorships 2276. 4713. 5297. 9339.
#> 3 LAN: Democracies and Unstable Dictatorships 2485. 4827. 4886. 7669.
#> 4 LAN: Stable Dictatorships 1527. 2928. 3312. 9750.
# Alternatively, if you were feeling fancy...
group_split(Data, cat) %>%
setNames(unique(Data$cat)) %>%
map(., ~summary(., gdppc))
#> $`EE: Stable Democracies`
#> country iso3c cat demest
#> Length:13 Length:13 Length:13 Min. :1.005
#> Class :character Class :character Class :character 1st Qu.:1.519
#> Mode :character Mode :character Mode :character Median :1.713
#> Mean :1.615
#> 3rd Qu.:1.848
#> Max. :1.899
#> gdppc
#> Min. : 6192
#> 1st Qu.:10916
#> Median :11755
#> Mean :12294
#> 3rd Qu.:13919
#> Max. :17171
#>
#> $`EE: Unstable Democracies and Dictatorships`
#> country iso3c cat demest
#> Length:15 Length:15 Length:15 Min. :-0.6344
#> Class :character Class :character Class :character 1st Qu.:-0.4945
#> Mode :character Mode :character Mode :character Median :-0.4109
#> Mean : 0.4450
#> 3rd Qu.: 1.6327
#> Max. : 1.7880
#> gdppc
#> Min. :2276
#> 1st Qu.:3476
#> Median :4713
#> Mean :5297
#> 3rd Qu.:6631
#> Max. :9339
#>
#> $`LAN: Democracies and Unstable Dictatorships`
#> country iso3c cat demest
#> Length:7 Length:7 Length:7 Min. :-0.15521
#> Class :character Class :character Class :character 1st Qu.: 0.07506
#> Mode :character Mode :character Mode :character Median : 0.69395
#> Mean : 0.47433
#> 3rd Qu.: 0.81168
#> Max. : 1.00812
#> gdppc
#> Min. :2485
#> 1st Qu.:3822
#> Median :4827
#> Mean :4886
#> 3rd Qu.:5788
#> Max. :7669
#>
#> $`LAN: Stable Dictatorships`
#> country iso3c cat demest
#> Length:13 Length:13 Length:13 Min. :-0.71942
#> Class :character Class :character Class :character 1st Qu.:-0.18118
#> Mode :character Mode :character Mode :character Median :-0.10025
#> Mean : 0.03375
#> 3rd Qu.: 0.28265
#> Max. : 0.78015
#> gdppc
#> Min. :1527
#> 1st Qu.:2525
#> Median :2928
#> Mean :3312
#> 3rd Qu.:3191
#> Max. :9750
It would be useful to visualize this to get a sense of the distribution. There are (only slightly) more complicated techniques here that may get more out of the visualization, but this should suffice.
ggplot(Data, aes(cat, gdppc)) +
geom_boxplot(fill="#619cff") +
# put your thing down flip it and reverse it...
coord_flip() +
theme_minimal() +
# Be mindful of the coord_flip(), though...
scale_y_continuous(labels = scales::dollar_format()) +
labs(x = "", y="GDP per Capita Quartiles",
caption = "Data: Lipset (1959) [for categories/countries], Anders et al. [for GDP per capita estimates]")
Box plots are kind of blech, but you’re seeing basically what Lipset (1959) saw. Democracies are almost always richer than non-democracies in Europe. The difference is not as stark in Latin America—indeed, there aren’t a lot of democracies out there at this point in time—but the center of gravity is clearly different. Those two anomalous observations you see—which box plot defaults identify as “extreme” observations—are Ireland and Venezuela. I will defer to you to know why that’s the case.
Correlation (Pearson’s r)
You were introduced to correlation in the last lecture, so let’s talk about what correlation does. Correlation statistics are tools to help you describe how closely two things travel together. There are a few correlation statistics out there, and you might even encounter Kendall’s tau or Spearman’s rho in the wild. However, the most common correlation statistic for students in your position is Pearson’s r (which is used largely for interval-level variables that you see here).
In lecture, you also saw the formula for calculating Pearson’s r. Understand that its formula reveals three things about it.
Symmetry
The first is that it’s symmetrical. The main action that happens in the formula is multiplying (transformed, more on that later) values of x with (transformed) values of y. You may remember from gymnasium that multiplying x and y is equal to multiplying y and x, and that order only matters for division and subtraction.
Let’s see what this looks like when applied. Of note: I don’t think Swedes can easily access a dollar sign operator (I think it’s Alt Gr + 4, though). So, I’m doing it this way.
Data %>% summarize(cor = cor(demest, gdppc, use='complete.obs'))
#> # A tibble: 1 × 1
#> cor
#> <dbl>
#> 1 0.701
# Alternatively:
# cor(Data$demest, Data$gdppc, use='complete.obs')
In the above sequence, I’m starting with the data frame we created and
summarizing it into a single column data frame using, importantly, the
cor() function. In this function, democracy is treating as the x
variable and GDP per capita is treated as the y variable. The use
argument is optional in this context, but you may want to know about it.
It tells the cor() function to omit missing data if it finds it and
just use what’s available. That doesn’t matter in this simple case,
though it’s good to know about it.
We’ll unpack this statistic later, but I just want to emphasize one thing here. In applied statistics, we often use “x” as a shorthand for some type of causal variable and “y” as a shorthand for some kind of response variable that is a function of the causal variable. Alternatively: democracy causes GDP per capita. As I mention in the blog post, this is a huge debate in the field because there is no agreement about what causes what. It’s also amid this debate that we should emphasize that correlation gives you no answer. Observe:
Data %>% summarize(cor = cor(gdppc, demest, use='complete.obs'))
#> # A tibble: 1 × 1
#> cor
#> <dbl>
#> 1 0.701
^ same statistic. Correlation is symmetrical and does not care about the data-generating process that leads to x and y. y could be a function of x, as an objective fact, or be artifacts of some other different processes. Pearson’s r won’t care about the order.
Bounds
The other feature of Pearson’s r that is worth belaboring is that it is hard-bound to be between -1 and 1. If you calculated a Pearson’s r of 1.04, you screwed up. The computer won’t make that mistake for you, though.
One somewhat dissatisfying feature about the construction of Pearson’s r is that there are very few objective rules for summarizing the result of this statistic. What follows will omit a discussion of the correlation test of significance (i.e. could you rule out 0) and focus on the following objective rules for the summary of Pearson’s r.
1) Direction: Pearson’s r could be either positive or negative. Positive means that we expect y to increase (decrease) as x increases (decreases). Negative means we expect y to increase (decrease) as x decreases (increases). 2) Perfection: Pearson’s r statistics of -1 or 1 communicate perfect relationships. If Pearson’s r is 1, for example, an increase of 1 in x will result in some constant increase in y 100% of the time. Here, the data would look like some slope-intercept equation of y = mx + b, which you might remember from gymnasium. It’s an effective impossibility that you will find that in the wild, though. Noise is all around you. 3) Zero: Pearson’s r statistics of 0 indicate no change whatsoever in y for any change in x. These are also rare in as much that we are talking about 0 with precision and ignoring a conversation about a test of significance against 0.
So where does that leave us with the Pearson’s r we got? Karina offered you a guide for summarizing Pearson’s r statistics you see. I will only caution this is a useful framework for students and beginners, but it is not a rule. What you see is a Pearson’s r of .701 or so. That’s clearly positive, but it’s not perfect and it’s clearly not 0. I would call it a strong positive relationship, but here you have to use you words to describe anything that’s not perfect and not 0. Use your head, but use your words.
Standardization
Lecture gave you a preview of this, but one of the niftier things about the Pearson’s r formula is that the numerator side of the equation is creating z-scores, or standardizing both the x variable and y variable to have a mean of 0 and a standard deviation of 1. Did you create z-scores in gymnasium? I remember doing it in my AP Stats class in the United States. Ever seen the standard normal distribution with the bounds communicating 68-90-95-99-percent coverage? Also standardization.
To drive the point home, let’s create a standardized version of the democracy and GDP per capita variable.
Data %>%
r1sd_at(c("demest", "gdppc")) -> Data
Data
#> # A tibble: 48 × 7
#> country iso3c cat demest gdppc s_demest s_gdppc
#> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 Australia AUS EE: Stable Democracies 1.87 13919. 1.32 1.73
#> 2 Belgium BEL EE: Stable Democracies 1.85 9358. 1.30 0.651
#> 3 Canada CAN EE: Stable Democracies 1.52 13413. 0.944 1.61
#> 4 Denmark DNK EE: Stable Democracies 1.58 11384. 1.01 1.13
#> 5 Ireland IRL EE: Stable Democracies 1.20 6192. 0.600 -0.0948
#> 6 Luxembourg LUX EE: Stable Democracies 1.75 14589. 1.20 1.88
#> 7 Netherlands NLD EE: Stable Democracies 1.71 9633. 1.16 0.716
#> 8 New Zealand NZL EE: Stable Democracies 1.85 13602. 1.30 1.65
#> 9 Norway NOR EE: Stable Democracies 1.71 10916. 1.15 1.02
#> 10 Sweden SWE EE: Stable Democracies 1.79 11755. 1.24 1.22
#> # ℹ 38 more rows
In the above console output, the s_ prefix precedes the name of the variable that was standardized into a new column. We know our correlation coefficient is .701, suggesting a pretty strong, positive relationship. We are expecting to see that most of the standardized variables share the same sign. If it’s above the mean in democracy, we expect it to be above the mean in per capita income. If it’s below the mean in democracy, we expect it to be below the mean in per capita income. Looking at just the first 10 observations, incidentally all in Europe, we see exactly that. Nine of these 10 states are consistent with this positive correlation, with just the one obvious exception of Ireland. We kind of expected to observe that, and we did.
It’s helpful to unpack what this mean visually by way of our scatterplot. We know that correlation creates z-scores of x and y underneath the hood, so let’s draw a vertical line at the mean of democracy and a horizontal line at the mean of per capita income to gather more information about our data.
mean_gdppc <- mean(Data$gdppc)
mean_demest <- mean(Data$demest)
ggplot(Data, aes(demest, gdppc)) +
theme_minimal() +
geom_point() +
geom_smooth(method ='lm') +
geom_vline(xintercept = mean_demest, linetype = 'dashed') +
geom_hline(yintercept = mean_gdppc, linetype = 'dashed') +
scale_y_continuous(labels = scales::dollar_format())
#> `geom_smooth()` using formula = 'y ~ x'
This is effectively breaking our bivariate data into quadrants. The bottom-left and top-right quadrants are so-called positive correlation quadrants. They are above (below) the mean in x and above (below) the mean in y and their placement in this quadrant is consistent with a positive correlation. The top-left and bottom-right quadrants are so-called negative correlation quadrants. They are above (below) the mean in x and below (above) the mean y, so observations here are inconsistent with a positive correlation and consistent with a negative correlation. The correlation coefficient we get implies that we should expect to see the bulk of observations in the top-right and bottom-left quadrants, which we incidentally do.
One thing this implies is that if we kept just those observations in the top-left and bottom-right quadrants, the ensuing Pearson’s r would be misleadingly (sic?) negative. Observe:
Data %>%
mutate(quadrant = case_when(
s_demest > 0 & s_gdppc > 0 ~ "Positive",
s_demest < 0 & s_gdppc < 0 ~ "Positive",
TRUE ~ "Negative"
)) -> Data
Data %>%
filter(quadrant == "Negative") %>%
summarize(cor = cor(demest, gdppc))
#> # A tibble: 1 × 1
#> cor
#> <dbl>
#> 1 -0.211
It’s helpful here to identify the observations that are inconsistent with the positive correlation we observe. There are any number of ways of doing this. The tibble gives us a preview, and helps us find Ireland, though let’s find the whole gang of off-quadrant observations.
Data %>%
filter(quadrant == "Negative")
#> # A tibble: 9 × 8
#> country iso3c cat demest gdppc s_demest s_gdppc quadrant
#> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> <chr>
#> 1 Ireland IRL EE: Stable Democracies 1.20 6192. 0.600 -0.0948 Negative
#> 2 Austria AUT EE: Unstable Democrac… 1.79 6330. 1.24 -0.0624 Negative
#> 3 Italy ITA EE: Unstable Democrac… 1.78 5558. 1.23 -0.244 Negative
#> 4 Brazil BRA LAN: Democracies and … 0.797 2485. 0.155 -0.968 Negative
#> 5 Chile CHL LAN: Democracies and … 0.694 5825. 0.0428 -0.181 Negative
#> 6 Costa Rica CRI LAN: Democracies and … 1.01 3809. 0.386 -0.656 Negative
#> 7 Cuba CUB LAN: Stable Dictators… 0.780 3726. 0.137 -0.676 Negative
#> 8 Ecuador ECU LAN: Stable Dictators… 0.665 2805. 0.0116 -0.893 Negative
#> 9 Venezuela VEN LAN: Stable Dictators… -0.119 9750. -0.845 0.743 Negative
What you make of the observations requires some subject domain expertise. You gotta know Ireland’s back story to understand why it’s at this point in the 1950s. Go read about what Venezuela was doing at this point in time (i.e. it was the Saudi Arabia of its day). Just because those two observations are what they are doesn’t invalidate the overall relationship we observe. Democracy and wealth are relatively robust and stable equilibrium. However, we don’t know what causes what.
Bonus: calculate your own Pearson’s r. It’s not hard!
sum(Data$s_demest*Data$s_gdppc)/(nrow(Data) - 1)
#> [1] 0.7008677
cor(Data$demest, Data$gdppc, use='complete.obs')
#> [1] 0.7008677
# ^ insert soyjaks pointing meme here.