update Moderation example to best practice and v4 (#297)

* create truncated regression example

* delete truncated regression example from main branch

* create truncated regression example

* delete truncated regression example from main branch

* create truncated regression example

* delete truncated regression example from main branch

* start the update of moderation analysis notebook

* bring back target_accept to resolve divergences

* suggested updates

* add explanation to category

Co-authored-by: Oriol Abril-Pla <[email protected]>

* fix reference: article -> book

* fix a date to match up with references

* attempt to add cross references between moderation and mediation notebooks

* remove pm. tags on mediation notebook while I'm here

* fix notebook cross references

Co-authored-by: Benjamin T. Vincent <[email protected]>
Co-authored-by: Oriol Abril-Pla <[email protected]>

Author: 3 people

examples/case_studies/mediation_analysis.ipynb

@@ -9,14 +9,16 @@
     "# Bayesian mediation analysis\n",
     "\n",
     ":::{post} February, 2022\n",
-    ":tags: mediation, path analysis, pymc3.ConstantData, pymc3.Deterministic, pymc3.HalfCauchy, pymc3.Model, pymc3.Normal, regression\n",
+    ":tags: mediation, path analysis, regression\n",
     ":category: beginner\n",
     ":author: Benjamin T. Vincent\n",
     ":::\n",
     "\n",
     "This notebook covers Bayesian [mediation analysis](https://en.wikipedia.org/wiki/Mediation_(statistics) ). This is useful when we want to explore possible mediating pathways between a predictor and an outcome variable.\n",
     "\n",
-    "It is important to note that the approach to mediation analysis has evolved over time. This notebook was heavily influenced by the approach of {cite:t}`hayes2017introduction` as set out in his textbook \"Introduction to Mediation, Moderation and Conditional Process Analysis.\""
+    "It is important to note that the approach to mediation analysis has evolved over time. This notebook was heavily influenced by the approach of {cite:t}`hayes2017introduction` as set out in his textbook \"Introduction to Mediation, Moderation and Conditional Process Analysis.\"\n",
+    "\n",
+    "Readers should be aware that mediation analysis is commonly confused with moderation analysis for which we have a separate example ({ref}`moderation_analysis`)."
    ]
   },
   {

examples/case_studies/moderation_analysis.ipynb

@@ -9,6 +9,16 @@ @article{ando2007bayesian
   publisher     = {Oxford University Press},
   doi           = {10.1093/biomet/asm017}
 }
+@article{bauer2005probing,
+  title         = {Probing interactions in fixed and multilevel regression: Inferential and graphical techniques},
+  author        = {Bauer, Daniel J and Curran, Patrick J},
+  journal       = {Multivariate behavioral research},
+  volume        = {40},
+  number        = {3},
+  pages         = {373--400},
+  year          = {2005},
+  publisher     = {Taylor \& Francis}
+}
 @book{breen1996regression,
   title         = {Regression models: Censored, sample selected, or truncated data},
   author        = {Breen, Richard and others},
@@ -117,6 +127,26 @@ @misc{hogg2010data
   archiveprefix = {arXiv},
   primaryclass  = {astro-ph.IM}
 }
+@article{iacobucci2016mean,
+  title         = {Mean centering helps alleviate ``micro'' but not ``macro'' multicollinearity},
+  author        = {Iacobucci, Dawn and Schneider, Matthew J and Popovich, Deidre L and Bakamitsos, Georgios A},
+  journal       = {Behavior research methods},
+  volume        = {48},
+  number        = {4},
+  pages         = {1308--1317},
+  year          = {2016},
+  publisher     = {Springer}
+}
+@article{iacobucci2017mean,
+  title         = {Mean centering, multicollinearity, and moderators in multiple regression: The reconciliation redux},
+  author        = {Iacobucci, Dawn and Schneider, Matthew J and Popovich, Deidre L and Bakamitsos, Georgios A},
+  journal       = {Behavior research methods},
+  volume        = {49},
+  number        = {1},
+  pages         = {403--404},
+  year          = {2017},
+  publisher     = {Springer}
+}
 @book{ivezić2014astroMLtext,
   author        = {\v{Z}eljko Ivezi\'{c} and Andrew J. Connolly and Jacob T. VanderPlas and Alexander Gray},
   doi           = {10.1515/9781400848911},
@@ -187,6 +217,12 @@ @article{kruschke2013
   title         = {Bayesian estimation supersedes the t test.},
   journal       = {Journal of Experimental Psychology: General}
 }
+@book{kruschke2014doing,
+  title         = {Doing Bayesian data analysis: A tutorial with R, JAGS, and Stan},
+  author        = {Kruschke, John},
+  year          = {2014},
+  publisher     = {Academic Press}
+}
 @misc{kucukelbir2015automatic,
   title         = {Automatic Variational Inference in Stan},
   author        = {Alp Kucukelbir and Rajesh Ranganath and Andrew Gelman and David M. Blei},
@@ -218,6 +254,16 @@ @book{martin2021bayesian
   publisher     = {Chapman and Hall/CRC},
   doi           = {10.1201/9781003019169}
 }
+@article{mcclelland2017multicollinearity,
+  title         = {Multicollinearity is a red herring in the search for moderator variables: A guide to interpreting moderated multiple regression models and a critique of Iacobucci, Schneider, Popovich, and Bakamitsos (2016)},
+  author        = {McClelland, Gary H and Irwin, Julie R and Disatnik, David and Sivan, Liron},
+  journal       = {Behavior research methods},
+  volume        = {49},
+  number        = {1},
+  pages         = {394--402},
+  year          = {2017},
+  publisher     = {Springer}
+}
 @book{mcelreath2018statistical,
   title         = {Statistical rethinking: A Bayesian course with examples in R and Stan},
   author        = {McElreath, Richard},
@@ -294,6 +340,16 @@ @article{silver2016masteringgo
   year          = {2016},
   url           = {https://doi.org/10.1038/nature16961}
 }
+@article{spiller2013spotlights,
+  title         = {Spotlights, floodlights, and the magic number zero: Simple effects tests in moderated regression},
+  author        = {Spiller, Stephen A and Fitzsimons, Gavan J and Lynch Jr, John G and McClelland, Gary H},
+  journal       = {Journal of marketing research},
+  volume        = {50},
+  number        = {2},
+  pages         = {277--288},
+  year          = {2013},
+  publisher     = {SAGE Publications Sage CA: Los Angeles, CA}
+}
 @misc{szegedy2014going,
   title         = {Going Deeper with Convolutions},
   author        = {Christian Szegedy and Wei Liu and Yangqing Jia and Pierre Sermanet and Scott Reed and Dragomir Anguelov and Dumitru Erhan and Vincent Vanhoucke and Andrew Rabinovich},
@@ -302,6 +358,12 @@ @misc{szegedy2014going
   archiveprefix = {arXiv},
   primaryclass  = {cs.CV}
 }
+@online{vandenbergSPSS,
+  author        = {van den Berg, R. G},
+  title         = {SPSS Moderation Regression Tutorial},
+  url           = {https://www.spss-tutorials.com/spss-regression-with-moderation-interaction-effect/},
+  urldate       = {2022-03-20}
+}
 @inproceedings{vanderplas2012astroML,
   author        = {{Vanderplas}, J.T. and {Connolly}, A.J. and {Ivezi{\'c}}, {\v Z}. and {Gray}, A.},
   booktitle     = {Conference on Intelligent Data Understanding (CIDU)},
@@ -330,3 +392,13 @@ @article{yuan2009bayesian
   year          = {2009},
   publisher     = {American Psychological Association}
 }
+@article{zhang2017moderation,
+  title         = {Moderation analysis with missing data in the predictors.},
+  author        = {Zhang, Qian and Wang, Lijuan},
+  journal       = {Psychological methods},
+  volume        = {22},
+  number        = {4},
+  pages         = {649},
+  year          = {2017},
+  publisher     = {American Psychological Association}
+}

examples/references.bib

@@ -15,7 +15,7 @@ kernelspec:
 # Bayesian mediation analysis
 
 :::{post} February, 2022
-:tags: mediation, path analysis, pymc3.ConstantData, pymc3.Deterministic, pymc3.HalfCauchy, pymc3.Model, pymc3.Normal, regression
+:tags: mediation, path analysis, regression
 :category: beginner
 :author: Benjamin T. Vincent
 :::
@@ -24,6 +24,8 @@ This notebook covers Bayesian [mediation analysis](https://en.wikipedia.org/wiki
 
 It is important to note that the approach to mediation analysis has evolved over time. This notebook was heavily influenced by the approach of {cite:t}`hayes2017introduction` as set out in his textbook "Introduction to Mediation, Moderation and Conditional Process Analysis."
 
+Readers should be aware that mediation analysis is commonly confused with moderation analysis for which we have a separate example ({ref}`moderation_analysis`).
+
 ```{code-cell} ipython3
 import arviz as az
 import matplotlib.pyplot as plt