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Expand Up @@ -3,16 +3,18 @@ The ecologocal fallacy: Thorndike’s (1939) example

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*](https://www.jstor.org/stable/2087176?origin=crossref&seq=1#page_scan_tab_contents)
William S. Robinson’s ([1950](#ref-robinsonEcologicalCorrelations1950))
paper, [*Ecological correlations and the behavior of
individuals*](https://www.jstor.org/stable/2087176?origin=crossref&seq=1#page_scan_tab_contents),
and the name itself first appeared in sociologist Hanan C. Selvin’s
(1958) paper, [*Durkheim’s suicide and problems of empirical
([1958](#ref-selvinDurkheimSuicideProblems1958)) paper, [*Durkheim’s
suicide and problems of empirical
research*](https://s3.amazonaws.com/academia.edu.documents/33024288/Durkheim-suicide_empirical-research-problems.pdf?response-content-disposition=inline%3B%20filename%3DDurkheims_Suicide_and_Problems_of_Empiri.pdf&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAIWOWYYGZ2Y53UL3A%2F20191014%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20191014T151247Z&X-Amz-Expires=3600&X-Amz-SignedHeaders=host&X-Amz-Signature=4e9eeb342fa0332cd0f7ed00a4769661cb7c6921e07dd19a763595f037d35dae).
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
back at least as far as E. L. Thorndike’s
([1939](#ref-thorndikeFallacyImputingCorrelations1939)) paper, [*On the
fallacy of imputing the correlations found for groups to the individuals
or smaller groups composing
them*](https://www.jstor.org/stable/1416673?seq=1#page_scan_tab_contents).
The purpose of this project is to walk out Thorndike’s original
examples.
Expand All @@ -34,7 +36,10 @@ 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
[tribble](https://tibble.tidyverse.org/reference/tribble.html)s.
[tribble](https://tibble.tidyverse.org/reference/tribble.html)s (see
Müller & Wickham, [2020](#ref-R-tibble); also Wickham,
[2019](#ref-R-tidyverse); Wickham et al.,
[2019](#ref-wickhamWelcomeTidyverse2019)).

``` r
library(tidyverse)
Expand Down Expand Up @@ -514,7 +519,7 @@ t13 %>%
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 11),
show.legend = F) +
scale_fill_viridis_c(option = "C") +
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") +
Expand Down Expand Up @@ -563,7 +568,7 @@ t13 %>%
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 20),
show.legend = F) +
scale_fill_viridis_c(option = "C") +
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") +
Expand All @@ -573,7 +578,7 @@ t13 %>%
![](README_files/figure-markdown_github/unnamed-chunk-5-1.png)

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
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.

Expand Down Expand Up @@ -604,7 +609,7 @@ t13 %>%
geom_tile(aes(fill = n)) +
geom_text(aes(label = n, color = n < 2),
show.legend = F) +
scale_fill_viridis_c(option = "C") +
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") +
Expand Down Expand Up @@ -643,7 +648,10 @@ right where we started.
That is, the results from group-level data will not necessarily
correspond to the results of subgroup- or individual-level data. Proceed
with caution.
with caution. To learn more about this phenomena, see the
([2012](#ref-hamakerWhyResearchersShould2012)) chapter by Hamaker or the
paper by Kievit et al.
([2013](#ref-kievitSimpsonParadoxPsychological2013)).

Session info
------------
Expand All @@ -652,9 +660,9 @@ Session info
sessionInfo()
```

## R version 3.6.0 (2019-04-26)
## R version 3.6.3 (2020-02-29)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.6
## Running under: macOS Catalina 10.15.3
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
Expand All @@ -667,25 +675,61 @@ sessionInfo()
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] forcats_0.4.0 stringr_1.4.0 dplyr_0.8.3 purrr_0.3.2
## [5] readr_1.3.1 tidyr_1.0.0 tibble_2.1.3 ggplot2_3.2.1
## [9] tidyverse_1.2.1
## [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_0.2.5 xfun_0.10 haven_2.1.0
## [4] lattice_0.20-38 colorspace_1.4-1 vctrs_0.2.0
## [7] generics_0.0.2 htmltools_0.4.0 viridisLite_0.3.0
## [10] yaml_2.2.0 utf8_1.1.4 rlang_0.4.0
## [13] pillar_1.4.2 glue_1.3.1 withr_2.1.2
## [16] modelr_0.1.4 readxl_1.3.1 lifecycle_0.1.0
## [19] munsell_0.5.0 gtable_0.3.0 cellranger_1.1.0
## [22] rvest_0.3.4 evaluate_0.14 labeling_0.3
## [25] knitr_1.23 fansi_0.4.0 broom_0.5.2
## [28] Rcpp_1.0.2 scales_1.0.0 backports_1.1.5
## [31] jsonlite_1.6 hms_0.4.2 digest_0.6.21
## [34] stringi_1.4.3 grid_3.6.0 cli_1.1.0
## [37] tools_3.6.0 magrittr_1.5 lazyeval_0.2.2
## [40] crayon_1.3.4 pkgconfig_2.0.3 zeallot_0.1.0
## [43] xml2_1.2.0 lubridate_1.7.4 assertthat_0.2.1
## [46] rmarkdown_1.13 httr_1.4.0 rstudioapi_0.10
## [49] R6_2.4.0 nlme_3.1-139 compiler_3.6.0
## [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

References
----------

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>

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