Group challenge 1: Example

Spatial Statistics - BIOSTAT 696/896

Michele Peruzzi

University of Michigan

Introduction

  • We consider a scientific problem in the area of whateverology
  • Data on this topic are available at website https://whateverology.idk
  • Specifically we target the study of ABC, using data https://whateverology.idk/ABC.csv
  • The dataset is a point-referenced dataset of 5000 observations

Data overview

  • The dataset is of dimension 5000, 5; in addition to latitude and longitude, it includes variables x, y, and z
  • A histogram of all variables they are all roughly centered at zero

Data overview

  • Variable Y is missing at 20.3 percent of total locations
  • Variables X and Y are linearly negatively correlated
  • Variable Z seems unrelated to the other two

Data type and locations

  • Data locations cover the spatial domain with uniformity
  • We consider the variables X and Y over the domain D

More visualization

  • We visualize interesting features of the data!

[Here]

Maps of the data

Explaining Y via X

  • By inspecting the maps of X and Y we see that there may be a negative association between them
  • This makes sense because X explains Y in a certain way according to scientific knowledge
  • However, we note that Y is missing in some large areas where X is available
  • Further investigation to figure out whether missingness may impact results
  • Value of Z may relate to whether Y is missing. We explain this with…

Brainstorming models for this dataset

  • We think X \to Y and Z \to Y
  • We consider X and Z as covariates
  • The measurements of Y are reasonably noisy
  • Y has spatial variability that may not be fully explained by X and Z
  • Therefore, the DAG could look like \theta \to Y, \tau \to Y where the former is a vector of unknown parameters describing spatial dependence, \tau is an unknown scalar which refers to how precise our measurements of Y are in this dataset

Preliminary analysis

  • We consider a linear regression of X and Z on Y
  • Coefficients on X and Z are -0.943137 and -0.0409616, respectively
  • P-value for coefficient on X is very small, and the one on Z is statistically significantly different from zero at 5% confidence
  • However, we are unsatisfied with this analysis because the residuals show significant spatial variability
  • Because X and Z alone do not satisfactorily explain all variability in Y, we need to conduct more analyses

Concluding words

  • We have identified a potential route for modeling this dataset by explaining variability of Y via X and Z
  • Preliminary analyses seem to confirm this relationship
  • Simple linear regression is unable to capture all spatial variability
  • We target the development of a more complex model incorporating spatial variability
  • The fact that spatial variability remains important makes sense: Y can be explained by variable W, which is not in this dataset
  • The unobserved variable W has an intuitive spatial pattern that may resemble the spatial pattern of regression residuals
  • Our proposed intuitive model ignores the fact that missingness of Y may be explained by Z but we believe our assumption is reasonable because…

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