#### Date of Award

5-1-2023

#### Degree Type

Thesis

#### Degree Name

Honors Thesis

#### Department

Mathematics

#### First Advisor

Dr. Mohammad Patwary

#### Second Advisor

Dr. Chris Wilson

#### Abstract

Missing a single observation, or more, very commonly occurs in observational and designed studies. Estimating a single missing observation and analysis of these types of data is found in literature (Montgomery, 2020). But the estimation and analysis of data becomes more complicated when the study dataset becomes imbalanced due to multiple missing. Though the case of two missing values is the simplest case of multiple missing values, the analytic estimation and analysis will not be as straight forward as in the one missing value case, because two missing can occur in various ways. This thesis will be exploring mainly the idea of multiple missing values in two-way classified data. In Azadeh et al (2008) it is stated that missing values are incredibly common. In order to continue working with the dataset, those missing values must be estimated. There are two different types of missing values: missing at random (MAR) and missing not at random (MNAR) (Efromovich, 2018). This thesis will be focusing specifically on values that are MAR. Values that are MAR are relatively convenient to deal with because each value in the data set has the same probability of being missing. Also, in Gomer and Ke-Hai (2021) it is stated that the cause of missingness being unobserved makes MNAR valuables difficult to deal with, but it is found in literature (Efromovich, 2018). With a MAR in datasets, the missing value can be in any treatment and in any block. With two missing values, each missing value can appear in any treatment and any block which gives three separate cases that will be discussed later. This thesis will also be looking at multiple missing values instead of just one. Estimating for one missing value has been researched extensively, but estimating more than one still has plenty of room for exploration. In Tang \& Ishwaran (2017) and Montgomery (2020), the authors mentioned that their method for estimating missing values is iterative because the estimation of one missing value takes into account the value of the other missing value. This means for the first estimation one of the missing values is given a random number and that is used to compute the second missing value. That second missing value is then plugged in to find the estimation for the first missing value. This process is done over and over until the estimates have stabilized. The method proposed is an analytic method which will provide closed form solution for missing values and will not require the iterative process. Having the analytic solution will allow us to explore the inferential, statistical properties of the estimators.

#### Recommended Citation

Marshall, Aaron Christopher, "Special Cases in Estimating Multiple Missing Values in Linear Models" (2023). *Undergraduate Honors Thesis Collection*. 666.

https://digitalcommons.butler.edu/ugtheses/666