The ecological fallacy occurs when results from group-level analyses are wrongly extended to individuals. The fallacy is often attributed to William S. Robinson’s (1950) paper, Ecological correlations and the behavior of individuals, and the name itself first appeared in sociologist Hanan C. Selvin’s (1958) paper, Durkheim’s suicide and problems of empirical research. However, my fellow psychologists might be happy to learn the idea goes back at least as far as E. L. Thorndike’s (1939) paper, On the fallacy of imputing the correlations found for groups to the individuals or smaller groups composing them. The purpose of this project is to walk out Thorndike’s original examples.
Here’s how Thorndike began his paper:
If the correlation between two traits, A and B (say, poverty and delinquency), in n groups (say, the residents of w districts) has a certain value, K, the correlation between A and B in the individuals or the families composing the groups need not be K and will not be, save in very special circumstances. (p. 122)
Thorndike referred to an example of this mistake from the literature of his time, which we won’t get into, here. He then worked through the fallacy with 12 simulated data sets. We’ll cover those in detail. In each of his synthetic data sets, “A is supposed to be intelligence quotient and B is supposed to be the fraction of a room or number of rooms per person” (p. 124). That is, he was covering the correlation between IQ and the crowdedness of one’s living conditions.
Here we’ll copy down the data into a series of tribbles (see Müller & Wickham, 2020; also Wickham, 2019; Wickham et al., 2019).
library(tidyverse)
# table I
t1 <-
tribble(
~a, ~b, ~n,
-1, -3, 1,
0, -3, 1,
1, -3, 1,
-3, -2, 1,
-2, -2, 1,
-1, -2, 4,
0, -2, 6,
1, -2, 4,
2, -2, 1,
3, -2, 1,
-4, -1, 1,
-3, -1, 2,
-2, -1, 4,
-1, -1, 7,
0, -1, 15,
1, -1, 7,
2, -1, 4,
3, -1, 2,
4, -1, 1,
-4, 0, 2,
-3, 0, 2,
-2, 0, 6,
-1, 0, 8,
0, 0, 20,
1, 0, 8,
2, 0, 6,
3, 0, 2,
4, 0, 2,
-4, 1, 1,
-3, 1, 2,
-2, 1, 4,
-1, 1, 7,
0, 1, 15,
1, 1, 7,
2, 1, 4,
3, 1, 2,
4, 1, 1,
-3, 2, 1,
-2, 2, 1,
-1, 2, 4,
0, 2, 6,
1, 2, 4,
2, 2, 1,
3, 2, 1,
-1, 3, 1,
0, 3, 1,
1, 3, 1
)
# table II
t2 <-
tribble(
~a, ~b, ~n,
-2, -3, 1,
-1, -3, 1,
0, -3, 1,
1, -3, 1,
-3, -2, 1,
-2, -2, 2,
-1, -2, 4,
0, -2, 4,
1, -2, 2,
2, -2, 1,
-3, -1, 1,
-2, -1, 3,
-1, -1, 6,
0, -1, 6,
1, -1, 3,
2, -1, 1,
-3, 0, 1,
-2, 0, 2,
-1, 0, 4,
0, 0, 4,
1, 0, 2,
2, 0, 1,
-2, 1, 1,
-1, 1, 1,
0, 1, 1,
1, 1, 1,
)
# table III
t3 <-
tribble(
~a, ~b, ~n,
-1, -2, 1,
0, -2, 1,
1, -2, 1,
2, -2, 1,
-3, -1, 1,
-2, -1, 1,
-1, -1, 2,
0, -1, 5,
1, -1, 5,
2, -1, 2,
3, -1, 1,
-3, 0, 1,
-2, 0, 3,
-1, 0, 5,
0, 0, 10,
1, 0, 10,
2, 0, 5,
3, 0, 3,
4, 0, 2,
-3, 1, 1,
-2, 1, 3,
-1, 1, 5,
0, 1, 10,
1, 1, 10,
2, 1, 5,
3, 1, 3,
4, 1, 2,
-3, 2, 1,
-2, 2, 1,
-1, 2, 2,
0, 2, 5,
1, 2, 5,
2, 2, 2,
3, 2, 1,
-1, 3, 1,
0, 3, 1,
1, 3, 1,
2, 3, 1
)
# table IV
t4 <-
tribble(
~a, ~b, ~n,
-3, -3, 1,
-2, -3, 1,
-1, -3, 1,
0, -3, 1,
-4, -2, 1,
-3, -2, 2,
-2, -2, 4,
-1, -2, 4,
0, -2, 2,
1, -2, 1,
-4, -1, 1,
-3, -1, 3,
-2, -1, 6,
-1, -1, 6,
0, -1, 3,
1, -1, 1,
-4, 0, 1,
-3, 0, 2,
-2, 0, 4,
-1, 0, 4,
0, 0, 2,
1, 0, 1,
-3, 1, 1,
-2, 1, 1,
-1, 1, 1,
0, 1, 1,
)
# table V
t5 <-
tribble(
~a, ~b, ~n,
-3, -4, 1,
-2, -4, 1,
-1, -4, 1,
-4, -3, 1,
-3, -3, 2,
-2, -3, 4,
-1, -3, 2,
0, -3, 1,
-4, -2, 1,
-3, -2, 3,
-2, -2, 6,
-1, -2, 3,
0, -2, 1,
-4, -1, 1,
-3, -1, 2,
-2, -1, 4,
-1, -1, 2,
0, -1, 1,
-3, 0, 1,
-2, 0, 1,
-1, 0, 1
)
# table VI
t6 <-
tribble(
~a, ~b, ~n,
-2, -4, 1,
-1, -4, 1,
0, -4, 1,
-3, -3, 1,
-2, -3, 2,
-1, -3, 4,
0, -3, 2,
1, -3, 1,
-3, -2, 1,
-2, -2, 3,
-1, -2, 6,
0, -2, 3,
1, -2, 1,
-3, -1, 1,
-2, -1, 2,
-1, -1, 4,
0, -1, 2,
1, -1, 1,
-2, 0, 1,
-1, 0, 1,
0, 0, 1
)
# table VII
t7 <-
tribble(
~a, ~b, ~n,
-2, -3, 1,
-1, -3, 1,
0, -3, 1,
-3, -2, 1,
-2, -2, 2,
-1, -2, 4,
0, -2, 2,
1, -2, 1,
-3, -1, 1,
-2, -1, 3,
-1, -1, 6,
0, -1, 3,
1, -1, 1,
-3, 0, 1,
-2, 0, 2,
-1, 0, 4,
0, 0, 2,
1, 0, 1,
-2, 1, 1,
-1, 1, 1,
0, 1, 1
)
# table VIII
t8 <-
tribble(
~a, ~b, ~n,
-1, -4, 1,
0, -4, 1,
1, -4, 1,
-2, -3, 1,
-1, -3, 2,
0, -3, 4,
1, -3, 2,
2, -3, 1,
-2, -2, 1,
-1, -2, 3,
0, -2, 6,
1, -2, 3,
2, -2, 1,
-2, -1, 1,
-1, -1, 2,
0, -1, 4,
1, -1, 2,
2, -1, 1,
-1, 0, 1,
0, 0, 1,
1, 0, 1
)
# table IX
t9 <-
tribble(
~a, ~b, ~n,
-1, -2, 1,
0, -2, 1,
1, -2, 1,
-2, -1, 1,
-1, -1, 2,
0, -1, 4,
1, -1, 2,
2, -1, 1,
-2, 0, 1,
-1, 0, 3,
0, 0, 6,
1, 0, 3,
2, 0, 1,
-2, 1, 1,
-1, 1, 2,
0, 1, 4,
1, 1, 2,
2, 1, 1,
-1, 2, 1,
0, 2, 1,
1, 2, 1
)
# table X
t10 <-
tribble(
~a, ~b, ~n,
0, -1, 1,
1, -1, 1,
2, -1, 1,
-1, 0, 1,
0, 0, 2,
1, 0, 4,
2, 0, 2,
3, 0, 1,
-1, 1, 1,
0, 1, 3,
1, 1, 6,
2, 1, 3,
3, 1, 1,
-1, 2, 1,
0, 2, 2,
1, 2, 4,
2, 2, 2,
3, 2, 1,
0, 3, 1,
1, 3, 1,
2, 3, 1
)
# table XI
t11 <-
tribble(
~a, ~b, ~n,
1, 0, 1,
2, 0, 1,
3, 0, 1,
0, 1, 1,
1, 1, 2,
2, 1, 4,
3, 1, 2,
4, 1, 1,
0, 2, 1,
1, 2, 3,
2, 2, 6,
3, 2, 3,
4, 2, 1,
0, 3, 1,
1, 3, 2,
2, 3, 4,
3, 3, 2,
4, 3, 1,
1, 4, 1,
2, 4, 1,
3, 4, 1
)
# table XII
t12 <-
tribble(
~a, ~b, ~n,
2, 1, 1,
3, 1, 1,
4, 1, 1,
1, 2, 1,
2, 2, 2,
3, 2, 4,
4, 2, 2,
5, 2, 1,
1, 3, 1,
2, 3, 3,
3, 3, 6,
4, 3, 3,
5, 3, 1,
1, 4, 1,
2, 4, 2,
3, 4, 4,
4, 4, 2,
5, 4, 1,
2, 5, 1,
3, 5, 1,
4, 5, 1
) We’ll combine them, here.
t13 <-
bind_rows(
t1 %>% mutate(t = 1),
t2 %>% mutate(t = 2),
t3 %>% mutate(t = 3),
t4 %>% mutate(t = 4),
t5 %>% mutate(t = 5),
t6 %>% mutate(t = 6),
t7 %>% mutate(t = 7),
t8 %>% mutate(t = 8),
t9 %>% mutate(t = 9),
t10 %>% mutate(t = 10),
t11 %>% mutate(t = 11),
t12 %>% mutate(t = 12)
) In Thorndike’s example, the data from each of his 12 tables corresponded to “scores for sample persons in each of twelve districts into which a city is divided” (p. 124). For simplicity, he gave the values in a standardized (i.e., z-score) metric. Here’s a plot of all those data, disaggregated by district. It corresponds directly to the way Thorndike presented the tables in his article.
t13 %>%
mutate(table = t) %>%
ggplot(aes(x = a, y = b)) +
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 11),
show.legend = F) +
scale_fill_viridis_c(expression(italic(n)), option = "C", limits = c(0, NA)) +
scale_color_manual(values = c("black", "white")) +
scale_x_continuous(breaks = -4:4, position = "top") +
scale_y_continuous(breaks = -4:4, trans = "reverse") +
theme(panel.grid = element_blank()) +
facet_wrap(~table, ncol = 2, labeller = label_both)
Thorndike asserted that “within each of the districts the correlation between A and B is zero” (p. 124). Let’s check.
t13 %>%
uncount(n) %>%
group_by(t) %>%
summarise(r = cor(a, b))## # A tibble: 12 x 2
## t r
## <dbl> <dbl>
## 1 1 0
## 2 2 0
## 3 3 0
## 4 4 0
## 5 5 0
## 6 6 0
## 7 7 0
## 8 8 0
## 9 9 0
## 10 10 0
## 11 11 0
## 12 12 0
Yep. Those correlations are zero for each. Here’s what the data look like if you naïvely combine the numbers from all districts.
t13 %>%
uncount(n) %>%
group_by(a, b) %>%
count() %>%
ggplot(aes(x = a, y = b)) +
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 20),
show.legend = F) +
scale_fill_viridis_c(expression(italic(n)), option = "C", limits = c(0, NA)) +
scale_color_manual(values = c("black", "white")) +
scale_x_continuous(breaks = -4:5, position = "top") +
scale_y_continuous(breaks = -4:5, trans = "reverse") +
theme(panel.grid = element_blank())
This corresponds to Thorndike’s Table XIII. If you look closely at his original, you’ll see he miscounted a few cells. No worry–the overall findings are still sound. Here’s the bivariate correlation for the naïvely combined data.
t13 %>%
uncount(n) %>%
summarise(r = cor(a, b))## # A tibble: 1 x 1
## r
## <dbl>
## 1 0.454
But what if we look at aggregated data? To do so, we’ll take the mean of
a and b from each of the districts. Here’s what that plot might look
like.
t13 %>%
uncount(n) %>%
group_by(t) %>%
summarise(a = mean(a),
b = mean(b)) %>%
count(a, b) %>%
ggplot(aes(x = a, y = b)) +
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 2),
show.legend = F) +
scale_fill_viridis_c(expression(italic(n)), option = "C", breaks = 1:2) +
scale_color_manual(values = c("black", "white")) +
scale_x_continuous(breaks = -4:5, position = "top") +
scale_y_continuous(breaks = -4:5, trans = "reverse") +
theme(panel.grid = element_blank())
That plot corresponds directly to Thorndike’s Table XIV. Here’s the correlation.
t13 %>%
uncount(n) %>%
group_by(t) %>%
summarise(mu_a = mean(a),
mu_b = mean(b)) %>%
summarise(r = cor(mu_a, mu_b))## # A tibble: 1 x 1
## r
## <dbl>
## 1 0.914
To recap, within each district the correlation is exactly 0. When we naïvely combine all the data, the correlation is about .45. When we aggregate the data, the correlation goes to .91. We might as well finish right where we started.
If the correlation between two traits, A and B (say, poverty and delinquency), in n groups (say, the residents of w districts) has a certain value, K, the correlation between A and B in the individuals or the families composing the groups need not be K and will not be, save in very special circumstances. (p. 122)
That is, the results from group-level data will not necessarily correspond to the results of subgroup- or individual-level data. Proceed with caution. To learn more about this phenomena, see the (2012) chapter by Hamaker or the paper by Kievit et al. (2013).
sessionInfo()## R version 3.6.3 (2020-02-29)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS Catalina 10.15.3
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] forcats_0.5.0 stringr_1.4.0 dplyr_1.0.1 purrr_0.3.4
## [5] readr_1.3.1 tidyr_1.1.1 tibble_3.0.3 ggplot2_3.3.2
## [9] tidyverse_1.3.0
##
## loaded via a namespace (and not attached):
## [1] tidyselect_1.1.0 xfun_0.13 haven_2.2.0 lattice_0.20-38
## [5] colorspace_1.4-1 vctrs_0.3.4 generics_0.0.2 htmltools_0.5.0
## [9] viridisLite_0.3.0 yaml_2.2.1 utf8_1.1.4 rlang_0.4.7
## [13] pillar_1.4.6 glue_1.4.2 withr_2.2.0 DBI_1.1.0
## [17] dbplyr_1.4.2 modelr_0.1.6 readxl_1.3.1 lifecycle_0.2.0
## [21] munsell_0.5.0 gtable_0.3.0 cellranger_1.1.0 rvest_0.3.5
## [25] evaluate_0.14 labeling_0.3 knitr_1.28 fansi_0.4.1
## [29] broom_0.5.5 Rcpp_1.0.5 scales_1.1.1 backports_1.1.9
## [33] jsonlite_1.7.0 farver_2.0.3 fs_1.4.1 hms_0.5.3
## [37] digest_0.6.25 stringi_1.4.6 grid_3.6.3 cli_2.0.2
## [41] tools_3.6.3 magrittr_1.5 crayon_1.3.4 pkgconfig_2.0.3
## [45] ellipsis_0.3.1 xml2_1.3.1 reprex_0.3.0 lubridate_1.7.8
## [49] assertthat_0.2.1 rmarkdown_2.1 httr_1.4.1 rstudioapi_0.11
## [53] R6_2.4.1 nlme_3.1-144 compiler_3.6.3
Hamaker, E. L. (2012). Why researchers should think "within-person": A paradigmatic rationale. In Handbook of research methods for studying daily life (pp. 43–61). The Guilford Press. https://www.guilford.com/books/Handbook-of-Research-Methods-for-Studying-Daily-Life/Mehl-Conner/9781462513055
Kievit, R., Frankenhuis, W. E., Waldorp, L., & Borsboom, D. (2013). Simpson’s paradox in psychological science: A practical guide. Frontiers in Psychology, 4. https://doi.org/10.3389/fpsyg.2013.00513
Müller, K., & Wickham, H. (2020). tibble: Simple data frames. https://CRAN.R-project.org/package=tibble
Robinson, W. S. (1950). Ecological correlations and the behavior of individuals. American Sociological Review, 15(3). https://doi.org/10.2307/2087176
Selvin, H. C. (1958). Durkheim’s suicide and problems of empirical research. American Journal of Sociology, 63(6), 607–619. https://doi.org/10.1086/222356
Thorndike, E. L. (1939). On the fallacy of imputing the correlations found for groups to the individuals or smaller groups composing them. The American Journal of Psychology, 52(1), 122–124. https://doi.org/10.2307/1416673
Wickham, H. (2019). tidyverse: Easily install and load the ’tidyverse’. https://CRAN.R-project.org/package=tidyverse
Wickham, H., Averick, M., Bryan, J., Chang, W., McGowan, L. D., François, R., Grolemund, G., Hayes, A., Henry, L., Hester, J., Kuhn, M., Pedersen, T. L., Miller, E., Bache, S. M., Müller, K., Ooms, J., Robinson, D., Seidel, D. P., Spinu, V., … Yutani, H. (2019). Welcome to the tidyverse. Journal of Open Source Software, 4(43), 1686. https://doi.org/10.21105/joss.01686
React
Typescript
Django
Laravel
Facebook
Microsoft
Google
Alibaba
D3
Tencent