Advanced Methods in Biostatistics I

• Course title: Advanced Biostatistics Methods I
• Instructor: Rafael Irizarry. Phone 410-614-5157, email: rafa at jhu dot edu.
• TA: Jeff Goldsmith. Phone 410-502-3365, email: jgoldsmi at jhsph dot edu.
• Class: Tuesday & Thursday 10:30-11:50am at Biostat library (W3513).
• Lab: Time and location TBD.
• TA office hour: Drop by, make appointment, or email question
• Grading: 4 Quizzes, 1 exam, homeworks and class presentations.

Data-sets:

9/04	Medical Expenditures	csv
9/04	US Heights Subsample	csv file

1. Textbook: Stat Labs (ebook). The link has been updated; you should have free access to the text.
2. Biostat 651-652 (Methods in Biostatistics I-II) lecture notes can be found at here.
3. Ingo's notes (on introduction to mathematical statistics) can be found at here.
4. Lecture stream (files will be updated regularly): pdf file, R script.
5. Here is the R file from the second lab.

Homework

• Helicopter Challenge:
The challenge will be held in class on 10/19 (Tuesday). For the challenge, bring your team's "optimal" helicopter to class. Reports are due by 5:00 pm on 10/20 (Wednesday).

For this challenge, form teams of two. Use this guide and this picture to cut out your helicopters (I drew this in Powerpoint then printed to pdf. For some reason some lines didn't extend to the boundry, but it should be good enough for cutting.); note that each sheet will make two helicopters. You are not to modify the "body" in anyway, only adjust "L" and "W". Also, make sure you put the paperclip at the base of your helicopter. A mixture of research on flight dynamics as well as empirical observation must be summarized into a report, answering the questions below. Although the work is done in pairs, each student must write and submit his/her own report. You are allowed NO MORE than 20 helicopters, and no more than 200 flights TOTAL. For example, you could make 20 helicopters and fly each of them 10 times, recording their L and W and flight time. Flight times should be recorded to the nearest hundredth of a second. Flights are supposed to be drops from a height of 9 feet measured from the ground to the bottom of the copter when being held suspended (preflight). It also may behoove you to annotate your numeric records, i.e., in case a copter hits a wall or gets blown out of the sky by the Apaches or Blackwater.

Good luck.

1. Write a report on how you decided on W, L. Hint: consider measurement error, use more than one analysis step to find the max, use Taylor's approximation to approximate the shape of the time of flight curve.
2. Draw your estimate of the time of flight surface. But remember your goal is to find the maximum not the entire surface.
• Due 10/12 (Tuesday) before 5:00PM:
1. Discuss cross validation. How could this technique be used to assess the ability of a logistic regression model to predict success/failure?
2. Read Chapter 2 of McCullagh and Nelder: Generalized Linear Models.
3. Assume a logistic regression model for binary outcomes Y and binary regressors X₁ with linear predictor β₀ + β₁X₁. What is the interpretation of β₀? of β₁?
4. Propose a logistic regression model for the Kyphosis data. Fit the model and interpret the parameters.
• Due 10/5 (Tuesday) before 5:00PM:
1. Given a sample (x₁, ..., x_n) from a Normal distribution with unknown mean μ and known variance σ², what is the maximum likelihood estimate of μ?
2. Learn about multivariate linear regression.
   • Study the linear algebra notation used in linear regression. A useful text is Seber and Lee, Chapter 3.
   • For the model we defined in class, find the Least Square Estimates (LSE) of β in matrix notation. Show that the LSE is unbiased.
   • Let ε^hat = Y-Xβ. Find E(ε^hat), Var(ε^hat), and show that E{(ε^hat)^T(ε^hat)} = (n-p) σ².
   • Download the data on babies' health. Fit a linear model of birthweight on an intercept, mother's age, and mother's height using the R command lm(). Verify the coefficients using the LSE of β in matrix notation.
3. The Surgeon General says that "Smoking by pregnant women may result in fetal injury, premature birth, and low birthweight." Use the data on babies' health to prepare a short (less than 2 pages) report supporting or opposing the opinion that smoking by pregnant women can cause low birth weights. Notes and hints:
   • You can stick with using the R command lm() for the models you construct here.
   • Write a short introduction discussing the biology and examples of previous studies.
   • Maternal smoking status might be confounded with other variables that might cause low birthweight. The data include various covariates. Assess the relation of these variabes with birth weight and smoking status.
   • If you find smoking has an effect on birthweights, is it the same for different values of the other covariates? e.g. is the effect the same for tall women and short women?
   • Estimate the effect of smoking on birth weight after correcting for any other variable that also appears to cause low birth weight.
   • Which of the covariates has the biggest effect on birth weight?
• Due 9/23 (Thursday) before 5:00PM:
1. DNA Patterns: A biologist hands you a dataset containing locations of palindromes in the CMV DNA. She wants to know if there are clusters of palindromes. Study the data and prepare a short report (2-3 pages of text, doublespaced) for the biologist. Are there any clusters? What location would you recommend she studies first? Is the finding worth pursuing? Hints:
• Learn a bit about Human Cytomegalovirus (CMV) and Palindromes in virus DNA - google will help.
• Teach yourself about the Poisson process: What is the distribution of the number of events in an interval of size L? What is the distribution of the locations of events? What is the distribution of the times between occurrances of a Poisson process?
• Make figures to prove your points.
• Updated Super-hint: check out chapter four of the Stat Labs eBook.
• Due 9/16 (Thursday) before 5:00PM:
1. From the last homework: Compare 95% confidence intervals for the mean constructed using the CLT with 95% confidence intervals constructed using the bootstrap. Keep in mind sample size, the distribution of the observed data, and other factors in your discussion. You may use the medical expenditures data set or simulations to justify your response.
2. What is the difference between "standard deviation" and "standard error"? If you wanted to shrink the SE by half (and reduce the width of your confidence interval by half), what could you do?
3. Find the expected value of sum(xi-xbar)^2. The identity E(x^2)=Var(x) + E(x)^2 will likely be useful at least once. Based on the result, explain why "dividing by 1/(n − 1) is a bit better" (page 29 of the class notes).
   Note: the property you're showing is that the sample variance is "unbiased".
4. What is the Poisson distribution?
5. Show that if X is binomial(N,p), N-> Inf, p->0 and Np -> lambda then the distribution of X converges to Poisson.
   Hint 1: You may use the fact that lim_n (1- lamba/n)^n -> exp(-lambda).
   Hint 2: You may also refer to Professor Rohde's notes on this subject; however, please do not look at Rohde's notes until after trying this problem on your own for 30 min.
• Due 9/9 (Thursday) before 5:00PM:
1. What is a Bernoulli trial?
2. Show that if Sn is the sum of n independent Bernoulli trials with probabiliy of success 0.5 then 2 √n (Sn/n-0.5) tends to N(0,1) (the standard normal). Hint: Either use the Central Limit Theorem, or check out Stirling's approximation if you're feeling technical.
3. Conduct a simulation to examine the performance of the Normal approximation in the previous question.
4. In a survey of the biostatistics department, 48 of 63 respondents could not recognise Tattoo from Fantasy Island. Construct a 95% confidence interval for the proportion of people in the biostat department who cannot recognise Tattoo.
5. Suppose some unknown fraction p of biostat students cannot identify Don Johnson from Miami Vice. What value of p gives the largest population variance?
6. Suppose X follows a standard normal distribution and Y follows a normal distribution with mean m and standard deviation s. What is the slope and intercept line going through the qqplot of Y versus X?
7. Download the Medical Expenditures dataset. Are medical expenditures Normally distributed? Justify your answer with descriptive statistics and plots. If not, how would you describe their distribution?
8. Deferred to next homework. Compare 95% confidence intervals for the mean constructed using the CLT with 95% confidence intervals constructed using the bootstrap. Keep in mind sample size, the distribution of the observed data, and other factors in your discussion. You may use the medical expenditures data set or simulations to justify your response.
• Due 8/31 (Tuesday)

Read Chapter 1 of Stat Labs and do all the exercises. Chapter 1: pdf file