{"id":639,"date":"2018-06-11T08:30:00","date_gmt":"2018-06-11T12:30:00","guid":{"rendered":"http:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/?p=639"},"modified":"2018-06-10T11:50:16","modified_gmt":"2018-06-10T15:50:16","slug":"causal-inference-in-ecology-the-rubin-causal-model-in-ecology","status":"publish","type":"post","link":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/blog\/2018\/06\/11\/causal-inference-in-ecology-the-rubin-causal-model-in-ecology\/","title":{"rendered":"Causal inference in ecology – The Rubin causal model in ecology"},"content":{"rendered":"

Causal inference in ecology – links to the series<\/a><\/p>\n

Evaluating the claim that viewing of the X Files<\/em> caused women to have more positive beliefs about science illustrated how the Rubin causal model can be used to make causal influences from observational data. The basic idea is that you make the observational sample similar to a randomized experiment by using statistical adjustments to make the \u201ctreatment\u201d and \u201ccontrol\u201d conditions as similar as possible – except for the \u201ctreatment\u201d difference.1<\/a><\/sup> Several weeks ago<\/a>, I promised to describe how we might use the Rubin causal model in ecology, drawing on data from a paper in PLoS One<\/em> that I\u2019m reasonably happy with. After playing with that data a bit, I changed gears. I\u2019m going to use data from a more recent paper (Carlson et al., Annals of Botany<\/em> 117:195-207; 2016 (doi: https:\/\/dx.doi.org\/10.1093\/aob\/mcv146<\/a>).<\/p>\n

I\u2019ll focus on a subset of the data that explores the relationship between stomatal density of Protea repens<\/em> seedlings grown in an experimental garden at Kirstenbosch National Botanical Garden<\/a> and three principal components associated with the environment in the populations from which seed was collected. You\u2019ll find the details of the analysis, an <tt>R<\/tt> notebook, and the data in Github<\/a>. The HTML produced by the R notebook showing the results is at http:\/\/darwin.eeb.uconn.edu\/pages\/Protea-causal-analysis.nb.html<\/a>. To run the analyses from the code you can download there, you\u2019ll need to retrieve the CSV from Github: https:\/\/github.com\/kholsinger\/Protea-causal-analysis\/blob\/master\/traits-environment-pca.csv<\/a>.<\/p>\n

Here\u2019s the bottom line. If we run a simple regression (treating year of observation as a random effect), we get the following results for the regression coefficients:<\/p>\n\n\n\n\n\n\n
<\/th>\nMean<\/th>\n2.5%tile<\/th>\n97.5%tile<\/th>\n<\/tr>\n
PCA 1 (annual temperature)<\/td>\n2.422<\/td>\n1.597<\/td>\n3.216<\/td>\n<\/tr>\n
PCA 2 (summer rainfall)<\/td>\n-2.125<\/td>\n-2.980<\/td>\n-1.277<\/td>\n<\/tr>\n
PCA 3 (annual rainfall)<\/td>\n1.317<\/td>\n0.538<\/td>\n2.099<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

All three principal components are strongly associated with stomatal density. We\u2019ve all been told repeatedly that \u201ccorrelation does not equal causation,\u201d but it\u2019s still very tempting to conclude that warmer climates favor higher stomatal densities (PCA 1), more summer rainfall favors lower stomatal densities (PCA 2), and more annual rainfall favors higher stomatal densities (PCA 3). Given what I wrote last week<\/a> about the Rubin causal model, we might even feel justified in reaching this conclusion, since we\u2019ve statistically controlled for relevant differences among populations (other than those that we measured). But go back and read that post again, and pay particular attention to this sentence:<\/p>\n

The degree to which you can be confident in your causal inference depends (a) on how well you\u2019ve done at identifying and measuring plausible causal factors and (b) how closely your two groups are matched on those other causal factors.<\/p><\/blockquote>\n

Notice (a) in particular. We have good evidence for the associations noted above,2<\/a><\/sup> but the principal components we identified were based on only 7 environmental descriptors, six from the South African Atlas of Agrohydrology and Climatology<\/a> and elevation (from a NASA digital elevation model<\/a>). There could easily be other environmental factors correlated with one (or all) of the principal components we identified that drive the association we observe. Now if similar associations had been observed in worldwide datasets involving many different groups of plants, it might not unreasonable to conclude that there is a causal relationship between the principal components we analyzed and stomatal density, but<\/strong><\/em> that conclusion wouldn\u2019t be based solely on the data and analysis here. It would depend on seeing the same pattern repeatedly in different contexts, which gives us something analogous to haphazard (not random) assignment to experimental conditions.<\/p>\n

There is, however, a further caveat.<\/p>\n

In Carlson et al., we obtained the following results for the mean and 95% credible interval on the association between stomatal density and each of the three principal component axes:<\/p>\n\n\n\n\n\n\n
<\/th>\nMean<\/th>\n2.5%tile<\/th>\n97.5%tile<\/th>\n<\/tr>\n
PCA 1 (annual temperature)<\/td>\n0.258<\/td>\n0.077<\/td>\n0.441<\/td>\n<\/tr>\n
PCA 2 (summer rainfall)<\/td>\n-0.216<\/td>\n-0.394<\/td>\n-0.040<\/td>\n<\/tr>\n
PCA 3 (annual rainfall)<\/td>\n0.155<\/td>\n-0.043<\/td>\n0.349<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Don\u2019t worry about the difference in magnitude of the coefficients. In Carlson et al. we transformed the response variables to a mean of 0 and a standard deviation of 1 before the analysis. Focus on the credible intervals. Here the credible interval for PCA 3 overlaps zero. In a conventional interpretation, we\u2019d say that we don\u2019t have evidence for a relationship between annual rainfall and stomatal density. 3<\/a><\/sup>I\u2019d prefer to say that the relationship with annual rainfall appears to be positive, but the evidence is weaker than for the relationships with annual temperature or summer rainfall. However you say it though, there seems to be a difference in the results. Why would that be?<\/p>\n

Because in Carlson et al. we analyzed stomatal density as one of a suite of leaf traits (length-width ratio, stomatal density, stomatal pore index, specific leaf area, and leaf area) that are correlated with one another. In particular, leaf area and stomatal density are associated with one another, perhaps because of the way that leaves develop. Leaf area is associated with annual rainfall. Thus, the association between leaf area and stomatal density intensifies the observed relationship between annual rainfall and stomatal density.<\/p>\n

In short, we should modify that sentence from last week to add a condition (c):<\/p>\n

The degree to which you can be confident in your causal inference depends (a) on how well you\u2019ve done at identifying and measuring plausible causal factors, (b) how closely your two groups are matched on those other causal factors, and (c) whether or not your response variable is associated with something else (measured or not) that is influenced by the causal factors you\u2019re studying.<\/p><\/blockquote>\n

Bottom line: For the types of observations I make4<\/a><\/sup> the Rubin causal model doesn\u2019t seem likely to help me make causal inferences. It does, however, illuminate the ways in which additional data from different systems could be combined (informally) with the data I collect5<\/a><\/sup> to make plausible causal inferences. At least they should be plausible enough to motivate careful experimental or observational tests of those inferences (if the causal processes are interesting enough to warrant those tests).<\/p>\n

    \n
  1. Implementing this approach in analysis of a real data set can become very complicated. There\u2019s a large literature on the Rubin causal model in social science. I\u2019ve read almost none of it. What I\u2019ve learned about the Rubin causal model comes from reading Gelman and Hill\u2019s regression modeling book and from reading Imbens and Rubin. ↩<\/a><\/li>\n
  2. That\u2019s overstating it a bit. See the discussion that follows this paragraph. ↩<\/a><\/li>\n
  3. There are serious problems with this kind of interpretation. See Andrew Gelman\u2019s post<\/a> explaining why \u201cthe difference between \u2018significant\u2019 and \u2018not significant\u2019 is not itself statistically significant. ↩<\/a><\/li>\n
  4. Remember, when I write \u201cI make\u201d I really mean \u201cmy students, postdocs, and collaborators make.\u201d I just follow along and help with the statistics. ↩<\/a><\/li>\n
  5. Remember what I wrote in that last footnote. ↩<\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Causal inference in ecology – links to the series Evaluating the claim that viewing of the X Files caused women to have more positive beliefs about science illustrated how the… Read more ><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-639","post","type-post","status-publish","format-standard","hentry","category-statistics"],"_links":{"self":[{"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/posts\/639","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/comments?post=639"}],"version-history":[{"count":3,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/posts\/639\/revisions"}],"predecessor-version":[{"id":641,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/posts\/639\/revisions\/641"}],"wp:attachment":[{"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/media?parent=639"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/categories?post=639"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/darwin.eeb.uconn.edu\/uncommon-ground\/wp-json\/wp\/v2\/tags?post=639"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}