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

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

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…

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

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

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…