**Instructor: **Eric Slud, Statistics program, Math. Dept.

**Office: ** Mth 2314, x5-5469, email evs@math.umd.edu, **Office Hours:** M11, W1,
or by appointment

**Course Text:** J.-K. Kim and J. Shao, *Statistical Methods for Handling Incomplete Data*, CRC 2013.

**Recommended Texts**:

*Statistical Analysis with Missing Data* (2002),
2nd edition, Wiley. *Handbook of Missing Data Methodology* (2014), Chapman and Hall.

Please fill out the on-line Evaluation form on this
Course and instructor at http://CourseEvalUM.umd.edu.
**Thank you.**

**Overview:** This course covers the statistical analysis of data in which important components
are unobservable or missing. Such data arise frequently in large databases, in sample surveys, and even
in carefully designed experiments. By their nature, such data must be handled through the use of modeling
assumptions, generally of the form that unseen data values or their relationships with observable data
must in some way be similar to corresponding observed data values. So one of the first tasks in studying
the topic of missing data is to understand various statistical models and concepts for mechanisms of
missingness. This is where the well-known terminology of `ignorable' missingness or mechanisms of
`missing at random' come in, but also where modeling concepts of `patterns of missingness' and
`propensities' to be observed are also directly relevant.

**NOTE ON USE OF THEORETICAL MATERIAL. **Both in homeworks and the in-class test, there will
be theoretical material at the level of probability theory needed to apply the law of large numbers and
central limit theorem, along with the `delta method' (Taylor linearization) and other manipulations at
advanced-calculus level.

**Prerequisite: **Stat 420 or Stat 700, plus some computing familiarity.

**Course requirements and Grading:** there will be 5 graded homework sets (one every 2--2.5 weeks)
which together will count 2/3 of the course grade, and a final project or presentation (10-12 page paper)
that will count 1/3 of the grade.

**NOTE ON COMPUTING. **Both in the homework-sets and the course project, you will be required
to do computations on real datasets well beyond the scope of hand calculation or spreadsheet programs.
Any of several statistical-computing platforms can be used to accomplish these: **R**, SAS, Minitab,
Matlab, or SPSS, or others. If you are learning one of these packages for the first time, I recommend
**R** which is free and open-source and is the most flexible and useful for research statisticians.
I will provide links to free online **R** tutorials and will provide examples and scripts and will
offer some **R** help.

**Notes and Guidelines.** Homeworks should be handed in as hard-copy in-class, except for occasional
due-dates on Fridays when you may submit them electronically, via email, in pdf format. Solutions will
usually be posted, and a percentage deduction of the overall HW score will generally be made for late papers.

Assignment 1. (First 2 weeks of course, HW due Mon., Feb. 11). Read about missing-data likelihoods and the definition of Missing at Random and Missing Completely at Random (material in Chapter 2 of Kim & Shao). Then solve and hand in the following problems (counting as 8 problem parts, worth 10 points for each part):

**(1)** Simulate X_{i} ~ *N*(1,1) independent identically distributed
(iid) , i=1,...,200, and Y_{i} = 2 + 0.6 X_{i} + ε_{i}
, where ε_{i} are iid *N*(0,1) and independent of {X_{i}}, and retain as
observations only those data-pairs (X_{i} , Y_{i}) for which X_{i} > 0 .

_{i} ?

_{i}, Y_{i}) satisfy a linear-regression relationship ? (To
justify your answer, find the joint density of the retained observations.)

_{i} in terms of X_{i} has 2 replaced by a and
0.6 by b and you estimate (a, b) by least-squares, then what is the large-sample target
(ie limit) of your estimator b̂ ?

**(2)** Suppose that random variables X_{i} ∈ (0,1] are iid with unknown distribution F, and
that observation X_{i} is observed with probability w(X_{i}) = X_{i}^{2}.

_{i}.

_{i} is discrete with values
{ 1/m, 2/m, ..., m/m }. *Hint: because X _{i} is discrete with m distinct values, you can
view the vector of its probability masses at the first m-1 of them as an unknown finite-dimensional parameter, and
calculate Fisher-information matrices, etc. Note that to prove efficiency, you must either calculate (large-sample
asymptotic) variance of your estimator as being the same as the inverse of Fisher Information, or else show that
your estimator is equivalent to (i.e., differs o(1/sqrt(n)) in probability from) the MLE.*

**(3)** Exercise 7 in Kim and Shao Chapter 2, p.22: Consider a bivariate variable
(Y1,Y2) where (Y1,Y2) takes on possible values (1,1), (1,0), (0,1) and (0,0) with respective
probabilities π_{11}, π_{10}, π_{01} and π_{00},
where π_{11}+π_{10}+π_{01}+π_{00}=1. To answer
the following questions, it may be helpful to define marginal and conditional probabilities
in a 2x2 table by π_{1+}=P(Y1 =1), π_{1|1}=P(Y2 =1|Y1 =1) and
π_{1|0}=P(Y2 =1|Y1 =0). Note that there is a one-to-one correspondence between
the two alternate parameterizations θ_{1} =
(π_{00}, π_{01}, π_{10}) and θ_{2} =
(π_{1+}, π_{1|1}, π_{1|0}). The realized sample observations
are counts n_{ij,H} and n_{i+,K} of configurations (Y1=i,Y2=j) for the
combination of two independent data samples H and K of sizes 300 for H and 100 for K:

_{11,H}=100 , n_{10,H} = 50 , n_{01,H}=75 ,
n_{00,H} = 75 , n_{1+,K} = 40 ,
n_{0+,K}=60

_{2}. _{1}. _{1}.

Assignment 2. (2nd 2 weeks, HW due Fri.,
March 1). Finish Chapter 2, Sec.2.4, and Chapter 3 through Sec.3.5 on EM Algorithm and
Monte Carlo variants in the Kim and Shao book. The problems to solve and hand in are the following
(7 parts total) :

_{1}, λ_{2}) from observed
data (X, R, R T), where X ~ Expon(1),

_{[t≤x]}
exp(-λ_{1} t) + I_{[t>x]} exp(-λ_{1} x - λ_{2}(t-x))
, P(R=1 | T,X) = a I_{[t≤ x]} + b I_{[t>x]}, and λ_{1},
λ_{2} > 0, a, b ∈ (0,1)

*See R code in this handout for coding examples related to the EM algorithms
in the pdf handout.*

Assignment 3. (HW3 due Fri.,
March 15). Finish Chapter 3 (Sec.3.6), plus Ch.4 on Imputation through Sec.4.5 in the Kim and Shao book.
The problems to solve and hand in are the following (8 problem-parts or 80 points total): #4, 5, 7, 12 from
Chapter 3, pp.54-58.*There is a new R script related to
numerical integration that might be used as an alternative to Monte Carlo in Problem #12: although that is not
the way the problem is assigned, you might use the numerical-integration idea given there to check your
work*.

Assignment 4. (HW4 due Fri.,
April 12). Ch.4 on Imputation through Sec.4.5 in the Kim and Shao book.
The problems to solve and hand in are the following (8 problem-parts or 80 points total):
#6, 7, 10 from Chapter 4, pp. 95-97. **See coursemail message for formula hints on
problems 6, 10.**

Assignment 5. (HW5 due Fri.,
May 10 in class). Ch.5 on Propensity Scoring through Sec.5.5 in the Kim and Shao book.
The problems to solve and hand in are the following (8 problem-parts or 80 points total):

*ImputScript.RLog*
(or another similar simulated dataset with different parameters or seed) with monotone missing
pattern.

(a) (15 points) Impute the missing values multiple times using a randomized hot-deck
(within 8 groups defined by cross-classifying observations using the two binary values of X_{1}
and quartiles of X_{2}). Find estimates and (using Rubin's rules) standard errors.

(b)(10 points) Impute the missing values column-wise, multiple times using estimated univariate
(for each Ymat3[,j]) regression models on Xmat, and show that the correlations among complete-data Y
columns are very badly estimated by the multiply imputed data.

(c) (15 points) Show theoretically that the model for Y[,2] on Y[,1] in the data subset consisting
of the first 2 columns **disregarding Xmat** is not MAR. (That is, show that the conditional
distribution of Y[i,2] given Y[i,1] is different for the i's with R_{i} = 1 and those for
R_{i}=0.)

**apipop** and **apiclus2** within R package ** survey**.
Here the

π(X

**Getting Started in R and SAS.** Lots of R introductory materials can be found on my last-year's
STAT 705 website.

Various pieces of information to help you get started in using SAS can be found under an old (F09) course
website Stat430. In particular you can find:

--- an overview of the minimum necessary steps to use SAS from Mathnet.

--- a series of SAS logs with edited outputs for illustrative examples.

FINAL PROJECT ASSIGNMENT, due Friday, May 17, 2019, 5pm. As a final course
project, you are to write a paper including some 5-10 pages of narrative, plus relevant code and graphical or tabular
exhibits, on a statistical journal article related to the course or else a data analysis or case-study based on
a dataset of your choosing. The guideline is that the paper should be 10--12 pages if it is primarily expository
based on an article, but could have somewhat fewer pages of narrative if based on a data-analytic case study.
However, for the latter kind of paper, all numerical outputs should be accompanied by code used to generate them,
plus discussion and interpretation of software outputs and graphical exhibits. For a data-analysis or case study,
the paper should present a coherent and reasoned data analysis with supporting evidence for the model you choose
to fit, the method and approach to handling missing data, and an assessment of the results.

Possible topics for the paper include: *implementation and analysis/interpretation of one or more imputation
methods on a real dataset (e.g., survey public-use data from American Community Survey) using methods and software
discussed in the course; exposition of a journal paper on missing data methods in a subject-matter application,
such as educational statistics; exposition of some other missing-data topic, such as double- or interval-censored
data, from a paper or book-chapter; or some other topic you propose*.

**Good topic choices for the paper include:** (1) Parts of the documentation or related papers on the
**mice** software by van Buuren, as linked under Handouts (5) below; (2) A highly cited paper by Rebecca Andridge,
A Review of Hot Deck Imputation for Survey Nonresponse;
or (3) any subject-matter-related paper on **Propensity Weighting** (e.g. a famous seminal paper by Rosenbaum,
1983 Biometrika, or later papers by various authors) or **Causal Inference** (many possible sources
including a famous 1976 paper by Rubin) or **Inverse Probability Weighting** methodology.

(1) A handout from Stat 705 on ML estimation using the EM (Expectation-Maximization) algorithm along
with another on MCMC (Markov Chain Monte
Carlo) techniques.

(2) ** R scripts related to various topics in the course can be found in the new
web-page directory RScripts.**

(3) A journal paper I wrote related to combining estimators from different samples is related also to the "Generalized Least Squares" method cited in Ch.3 of Kim and Shao in missing-data contexts.

(4) A talk I gave in the UMD Statistics Seminar in March 2019 about Bayesian computing in a
**generalized logistic mixed-model setting** may be of
interest here in the context of MCEM Metropolis-Hastings algorithms.

(5) The journal paper that started the idea of "Chained Imputation". Additional readings on this topic can be found associated with the MICE R-package, and in the "Fully Conditional Specification" Chapter (Ch.13) in the CRC Missing Data Handbook under the authorship of Stef van Buuren. See an online version of it, especially Chapter 4 containing th essence pf the "fully conditional specification idea", expanding on the SRMI idea of Raghunathan et al. You can look at a hands-on introduction to the MICE software in the pdf of a Journal of Statistical Software article of van Buuren at https://www.jstatsoft.org/article/view/v045i03.

(6) A hands-on, purely applied introduction to propensity weighting and matching using a few **R**
packages can be found here.

**Additional Computing Resources. ** There are many
publicly available datasets for practice data-analyses. Many of them are taken from journal articles
and/or textbooks and documented or interpreted. A good place to start is Statlib. Datasets needed in the course
will be either be posted to the course web-page, or indicated by links which will be provided here.

A good set of links to data sources from various organizations including Federal
and international statistical agencies is at Washington
Statistical Society links.

**First Class: Mon., January 28, 2019****Spring Break March 17--24, 2019****Change credit level or drop without W, February 8, 2019****Last schedule-adjustment Date (for Drop/Withdrawal): April 12, 2019****Last day of classes: Mon. May 13, 2019**

**The UMCP Math Department home page.
The University of Maryland home page.
My home page.
© Eric V Slud, May 6, 2019.**