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- Title
Discussion on "Spatial+: A novel approach to spatial confounding" by Dupont, Wood, and Augustin.
- Authors
Reich, Brian J.; Yang, Shu; Guan, Yawen
- Abstract
Under the PLR assumption that HT <math display="inline" altimg="urn:x-wiley:0006341X:media:biom13651:biom13651-math-0058" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>y</mi><mi>i</mi></msub><mo>=</mo><mi> </mi><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>f</mi><mrow><mo>(</mo><msub><mi mathvariant="bold">t</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>+</mo><msub><mi> </mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">$y {i}=\beta x {i}+f(\mathbf {t} {i})+\epsilon {i}$</annotation></semantics></math> ht , can be interpreted causally. Robinson ([8]) shows that HT <math display="inline" altimg="urn:x-wiley:0006341X:media:biom13651:biom13651-math-0015" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal"> </mi><mo>=</mo><mi>E</mi><mo>[</mo><mrow><mo>{</mo><mi>x</mi><mo>-</mo><mi>E</mi><mrow><mo>(</mo><mi>x</mi><mo>|</mo><mi mathvariant="bold">t</mi><mo>)</mo></mrow><mo>}</mo></mrow><msup><mrow><mo>{</mo><mi>x</mi><mo>-</mo><mi>E</mi><mrow><mo>(</mo><mi>x</mi><mo>|</mo><mi mathvariant="bold">t</mi><mo>)</mo></mrow><mo>}</mo></mrow><mi mathvariant="normal">T</mi></msup><mo>]</mo></mrow><annotation encoding="application/x-tex">$\Phi =E[\lbrace x-E(x\mid \mathbf {t})\rbrace \lbrace x-E(x\mid \mathbf {t})\rbrace ^{\mathrm{T}}]$</annotation></semantics></math> ht being positive definite is a necessary and sufficient condition for to be identified and HT <math display="inline" altimg="urn:x-wiley:0006341X:media:biom13651:biom13651-math-0016" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mover accent="true"><mi> </mi><mo> </mo></mover><annotation encoding="application/x-tex">$\widehat{\beta }$</annotation></semantics></math> ht to be root- I n i consistent.
- Subjects
CONFOUNDING variables; LOGITS; STATISTICS; INSTRUMENTAL variables (Statistics); SMOOTHING (Numerical analysis); STRUCTURAL equation modeling; CAUSAL inference
- Publication
Biometrics, 2022, Vol 78, Issue 4, p1291
- ISSN
0006-341X
- Publication type
Article
- DOI
10.1111/biom.13651