Final Project STAT 3303: Bayesian Analysis and Statistical Decision Making Due: Sunday, April 28, 2024 (submit on Carmen by midnight) INSTRUCTIONS This is an exam and should be treated as such.DO NOT discuss any aspect of this exam with anyone, including MSLC tutors, other than your instructor.This includes, but is not limited to, not discussing report structure, coding problems, instructor expectations, how long the exam took you, whether you finished it, etc. You are responsible for ensuring that other students do not have access to your exam. You may search online for general coding questions (i.e., “how do I pass through my seed to rjags?”) through websites such as StackExchange. Use of AI for coding or to assist with the write-up ishighly inadvisable(see below). Any violation of these instructions constitutes academic misconduct and will be reported to the university’s Committee on Academic Misconduct. PROBLEM STATEMENT For sexually reproducing species, the equilibrium ratio of male and female individuals of reproductive age is a 50/50 ratio. Fisher explained why this would be true – if the ratio deviated from 50/50, it would be reproductively advantageous for a child to be born of the less frequent gender and the ratio would then be restored. At birth, among the general population, this ratio is not met – more boys than girls are born (the proportion of female births is 0.485). Fisher explained this fact is due to higher mortality rates (due to disease and violence) among boys prior to reaching puberty, and so a disproportionate number of boys “need” to be born to maintain the equilibrium. The sex ratio can deviate from the overall average in subpopulations. We saw one example in class, where placenta previa is associated with a male birth. We will study another possible example in this research project. An urban legend exists among U.S. Navy submariners that they disproportionately father girls. 1 The belief is common enough that in 2003 researchers at Bathesda Naval Hospital decided to formally examine whether it was true and attempt to determine a cause. After obtaining national security and research ethics approvals, they sent surveys to all submariners stationed at six submarine bases, asking if they had a child born in the last 12 months, the sex of the child, and a number of work-related demographic questions. They received responses from 1000 sailors. A number of causes have been proposed for why submariners may have more girls (if such an effect exists). The environment is highly stressful, with an erratic and limited sleep schedule. The atmosphere is enclosed and differs from the normal Earth atmosphere – typical oxygen levels kept are 3-4 percentage points lower than normal, CO 2 levels can be high and require chemical treatment to prevent acute harm to sailors, and all 1 At the time of the research this project is based on, submariners were all male. While that has changed, the number of babies born to female submariners is too small for any useful analysis. sorts of potentially harmful things like hydraulic oils and refrigerants can leak into the air and remain in the closed environment. The effects of radiation, either from the nuclear reactor or from the nuclear warheads carried on board some submarines, may also be a cause. To test these hypotheses, the researchers obtained data on the following variables: seabinary indicator where 1 indicates the sailor is currently assigned to a submarine. Sailors will typically alternate between 2-3 years working directly onboard a submarine and 1-2 years in administrative duties on shore. BNbinary indicator where 1 indicates the sailor is assigned to a ballistic missile submarine (if sea=1) or his last command was a ballistic missile submarine (if sea=0). U.S. Navy submarines can be classified into two types: those that carry nuclear-armed ballistic missiles and those that don’t. engineeringbinary indicator where 1 indicates the sailor works directly with the nuclear reactor of a sub- marine (again, if sea=0 this refers to his last assignment) weapsbinary indicator where 1 indicates the sailor works directly on nuclear weapons (again, if sea=0 this refers to his last assignment) time inservicea quantitative variable that indicates time in years since the sailor began submarine service girlbinary indicator where 1 indicates the sailor’s child is a girl ASSIGNMENT: Write a formal research report as if you were one of the researchers at Bethesda reporting your findings to your supervisor. Your report should begin with a general discussion of the research question, a description of your data (don’t forget your exploratory data analysis) including limitations and potential sources of bias. Only then should you move on to describing your model and reaching some conclusions. Be sure to define all variables and interpret model parameters to your audience and state whether you have evidence of an effect and whether or any particular cause can be supported by the data. You will be graded on the level of professionalism in the report as well as the statistics, so obvious use of AI or any “hallucinations” will be severelypenalized. Specific modeling guidelines: 1. You should propose a Bayesian regression model for the probability that a sailor with a certain set of covariates has a girl. For your priors, you should use the guidance from the lecture notes on Bayesian regression – you do not want to use a weakly uninformative prior as it is highly unlikely that there will be very large effect sizes. 2. Specify your model in detail, including conditional independence and prior assumptions (providing just your code is NOT sufficient). 3. Provide details on model fitting (what were your starting values, how many iterations did your algo- rithm run, how did you diagnose convergence of the model fitting algorithm). 2 4. Provide interpretations of the results of your statistical analysis in the context of the problem with particular focus on what parameters have a high posterior probability of differing from zero. 5. You should include at least one posterior predictive estimate for the probability of a girl for a given set of covariates. FORMATTING GUIDELINES: • Your report should be typed. You may use R Markdown if you wish, but DO NOT include any code in the main body of your report. • Again, carefully proofread and spell check your report. Write in complete sentences and in para- graphs, not bulleted lists. • Define all mathematical notation in the text of the report. • Make sure all figures/tables are straightforward to understand, have captions, and are referenced in the text. • Include commented code in an appendix. • You may assume that the reader is familiar with Bayesian statistics, but not that they are familiar with the content of STAT 3303. For example, do not refer to specific examples that have been discussed in lecture or homework. Your report should beno longer than six pagesdouble-spaced, including figures and tables. (Text, figures, and tables that are after six pages may not be considered by the instructor.) Your appendix with code does not count toward the six page limit. Submit your final exam report as asingle PDF fileon Carmen before the deadline. 3