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Nutrition Project 2 Practice Exam with Answers (49 Solved Questions)

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    project02 March 17, 2019 1 Project 2: Diet and Disease. Due at 11:59pm on Sunday, 3/17. In this project, you will investigate the major causes of death in the world, as well as how one of these causes, heart disease, might be linked to diet! 1.0.1 Logistics Deadline. This project is due at 11:59pm on Sunday, 3/17. It’s much better to be early than late, so start working now. Free Response Questions : The free response questions and plots for the project are optional and ungraded, meaning you do not need to submit a PDF of this notebook to Gradescope. These questions tend to be open-ended. However, these questions will be very good practice for the free-response exam questions, so do give them a good effort. Solutions will be posted after the project late submission deadline. Partners. You may work with one other partner. Your partner must be enrolled in the same lab as you are. Only one of you is required to submit the project. On okpy.org, the person who submits should also designate their partner so that both of you receive credit. Rules. Don’t share your code with anybody but your partner. You are welcome to discuss questions with other students, but don’t share the answers. The experience of solving the prob- lems in this project will prepare you for exams (and life). If someone asks you for the answer, resist! Instead, you can demonstrate how you would solve a similar problem. Support. You are not alone! Come to office hours, post on Piazza, and talk to your classmates. If you want to ask about the details of your solution to a problem, make a private Piazza post and the staff will respond. Tests. Passing the tests for a question does not mean that you answered the question correctly. Tests usually only check that your table has the correct column labels. However, more tests will be applied to verify the correctness of your submission in order to assign your final score, so be careful and check your work! Advice. Develop your answers incrementally. To perform a complicated table manipulation, break it up into steps, perform each step on a different line, give a new name to each result, and check that each intermediate result is what you expect. You can add any additional names or functions you want to the provided cells. All of the concepts necessary for this project are found in the textbook. If you are stuck on a particular problem, reading through the relevant textbook section often will help clarify the concept. To get started, load datascience , numpy , pyplot , and ok . 1

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    In [1]: from datascience import * import numpy as np % matplotlib inline import matplotlib.pyplot as plots plots.style.use( ' fivethirtyeight ' ) from client.api.notebook import Notebook ok = Notebook( ' project02.ok ' ) _ = ok.auth(inline= True ) ===================================================================== Assignment: Project 2: Diet and Disease OK, version v1.13.11 ===================================================================== Successfully logged in as cat028@ucsd.edu 2 Diet and Cardiovascular Disease Death and its many causes are often a disconcerting topic for polite conversation. However, the more we know about it, the better equipped we are to prevent our early demise. As the acclaimed Professor Albus Dumbledore once said, “After all, to the well-organized mind, death is but the next great adventure.” In the following analysis, we will investigate the world’s most dangerous killer: Cardiovascu- lar Disease. Your investigation will take you across decades of medical research, and you’ll look at multiple causes and effects across two different studies. Here is a roadmap for this project: • In Part 1, we’ll investigate the major causes of death in the world during the past century (from 1900 to 2015). • In Part 2, we’ll look at data from the Framingham Heart Study, an observational study into cardiovascular health. • In Part 3, we’ll examine the clinical trials from the Minnesota Coronary Experiment and introduce our second dataset. • In Part 4, we’ll run a hypothesis test on our observed data from the Minnesota Coronary Experiment. • In Part 5, we’ll conclude the experiment and reflect on what we’ve learned about the rela- tionship between diet and cardiovascular disease. 2.1 Part 1: Causes of Death In order to get a better idea of how we can most effectively prevent deaths, we need to first figure out what the major causes of death are. Run the following cell to read in and view the causes_of_death table, which documents the death rate for major causes of deaths over the last century (1900 until 2015). 2

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    In [2]: causes_of_death = Table.read_table( ' causes_of_death.csv ' ) causes_of_death.show(5) <IPython.core.display.HTML object> Each entry in the column Age Adjusted Death Rate is a death rate for a specific Year and Cause of death. The Age Adjusted specification in the death rate column tells us that the values shown are the death rates that would have existed if the population under study in a specific year had the same age distribution as the “standard” population, a baseline. This is so we can compare ages across years without worrying about changes in the demographics of our population. Question 1: What are all the different causes of death in this dataset? Assign an array of all the unique causes of death to all_unique_causes . In [3]: all_unique_causes = causes_of_death.group( ' Cause ' ).column( ' Cause ' ) sorted(all_unique_causes) # This line displays your array in alphabetical order Out[3]: [ ' Accidents ' , ' Cancer ' , ' Heart Disease ' , ' Influenza and Pneumonia ' , ' Stroke ' ] In [4]: _ = ok.grade( ' q1_1 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 2: We would like to plot the death rate for each disease over time. To do so, we must create a table with one column for each cause and one row for each year. Create a table called causes_for_plotting . It should have one column called Year , and then a column with age-adjusted death rates for each of the causes you found in Question 1. There should be as many of these columns in causes_for_plotting as there are causes in Question 1. Hint : Use pivot , and think about how the elem function might be useful in getting the Age Adjusted Death Rate for each cause and year combination. In [5]: def elem(x): return x.item(0) In [6]: causes_for_plotting = causes_of_death.pivot( ' Cause ' , ' Year ' , values= ' Age Adjusted Death causes_for_plotting.plot( ' Year ' ) # Do not change this line 3

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    Let’s examine the graph above. You’ll see that in the 1960s, the death rate due to heart disease steadily declines. Up until then, the effects of smoking, blood pressure, and diet on the cardiovas- cular system were unknown to researchers. Once these factors started to be noticed, doctors were able recommend a lifestyle change for at-risk patients to prevent heart attacks and heart problems. Note, however, that the death rate for heart disease is still higher than the death rates of all other causes. Even though the death rate is starkly decreasing, there’s still a lot we don’t under- stand about the causes (both direct and indirect) of heart disease. 2.2 Part 2: The Framingham Heart Study The Framingham Heart Study is an observational study of cardiovascular health. The initial study followed over 5,000 volunteers for several decades, and followup studies even looked at their descendants. In this section, we’ll investigate some of its key findings about diet, cholesterol, and heart disease. Run the cell below to examine data for almost 4,000 subjects from the first wave of the study, collected in 1956. In [7]: framingham = Table.read_table( ' framingham.csv ' ) framingham Out[7]: AGE | SYSBP | DIABP | TOTCHOL | CURSMOKE | DIABETES | GLUCOSE | DEATH | ANYCHD 39 | 106 | 70 | 195 | 0 | 0 | 77 | 0 | 1 46 | 121 | 81 | 250 | 0 | 0 | 76 | 0 | 0 48 | 127.5 | 80 | 245 | 1 | 0 | 70 | 0 | 0 61 | 150 | 95 | 225 | 1 | 0 | 103 | 1 | 0 46 | 130 | 84 | 285 | 1 | 0 | 85 | 0 | 0 43 | 180 | 110 | 228 | 0 | 0 | 99 | 0 | 1 63 | 138 | 71 | 205 | 0 | 0 | 85 | 0 | 1 45 | 100 | 71 | 313 | 1 | 0 | 78 | 0 | 0 52 | 141.5 | 89 | 260 | 0 | 0 | 79 | 0 | 0 43 | 162 | 107 | 225 | 1 | 0 | 88 | 0 | 0 ... (3832 rows omitted) 4

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    Each row contains data from one subject. The first seven columns describe the subject at the time of their initial medical exam at the start of the study. The last column, ANYCHD , tells us whether the subject developed some form of heart disease at any point after the start of the study. You may have noticed that the table contains fewer rows than subjects in the original study: this is because we are excluding subjects who already had heart disease as well as subjects with missing data. 2.2.1 Section 1: Diabetes and the population Before we begin our investigation into cholesterol, we’ll first look at some limitations of this dataset. In particular, we will investigate ways in which this is or isn’t a representative sample of the population by examining the number of subjects with diabetes. According to the CDC, the prevalence of diagnosed diabetes (i.e., the percentage of the popu- lation who have it) in the U.S. around this time was 0.93%. We are going to conduct a hypothesis test with the following null and alternative hypotheses: Null Hypothesis : The probability that a participant within the Framingham Study has dia- betes is equivalent to the prevalence of diagnosed diabetes within the population. (i.e., any differ- ence is due to chance). Alternative Hypothesis : The probability that a participant within the Framingham Study has diabetes is different than the prevalence of diagnosed diabetes within the population. We are going to use the absolute distance between the observed prevalence and the true pop- ulation prevalence as our test statistic. The column DIABETES in the framingham table contains a 1 for subjects with diabetes and a 0 for those without. Question 1 : What is the observed value of the statistic in the data from the Framingham Study? You should convert prevalences to proportions before calculating the statistic! (re-read the ques- tion if you are not sure what it is asking) In [8]: obs_db = sum(framingham.column( ' DIABETES ' ))/framingham.num_rows observed_diabetes_distance = abs(obs_db - 0.0093) observed_diabetes_distance Out[8]: 0.018029515877147319 In [9]: _ = ok.grade( ' q2_1_1 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 2 : The array diabetes_proportions contains the proportions of the population without and with diabetes. Complete the following code to simulate 5000 values of the statis- tic under the null hypothesis. 5

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    In [10]: diabetes_proportions = make_array(.9907, .0093) diabetes_simulated_stats = make_array() repetitions = 5000 for i in np.arange(repetitions): simulated_stat = sample_proportions(4000, diabetes_proportions).item(1) diabetes_simulated_stats = np.append(diabetes_simulated_stats, simulated_stat) diabetes_simulated_stats Out[10]: array([ 0.00825, 0.0085 , 0.00975, ..., 0.0075 , 0.0075 , 0.00975]) In [11]: _ = ok.grade( ' q2_1_2 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 3 : Run the following cell to generate a histogram of the simulated values of your statistic, along with the observed value. Make sure to run the cell that draws the histogram, since it will be graded. In [12]: Table().with_column( ' Simulated distance to true incidence ' , diabetes_simulated_stats). plots.scatter(observed_diabetes_distance, 0, color= ' red ' , s=30) Out[12]: <matplotlib.collections.PathCollection at 0x7f4541e6eda0> 6

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    Question 4 : Based on the results of the test and the empirical distribution of the test statistic under the null, should you reject the null hypothesis? Yes, the observed value is very different than the simulated values. Question 5 : You know that the study was well-designed to represent the population. Why might there be a difference between the population and the sample? Assign the name framingham_diabetes_explanations to a list of the following explanations that are possible and consistent with the observed data and hypothesis test results. 1. Diabetes was under-diagnosed in the population (i.e., there were a lot of people in the pop- ulation who had diabetes but weren’t diagnosed). 2. Healthy (non-diabetic) people are more likely to volunteer for the study. 3. The relatively wealthy population in Framingham ate a luxurious diet high in sugar (high- sugar diets are a known cause of diabetes). 4. The Framingham Study subjects were older on average than the general population, and therefore more likely to have diabetes. In [13]: framingham_diabetes_possibilities = [1,3,4] framingham_diabetes_possibilities Out[13]: [1, 3, 4] In [14]: _ = ok.grade( ' q2_1_5 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests 7

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    --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed In real-world studies, getting a truly representative random sample of the population is often incredibly difficult. Even just to accurately represent all Americans, a truly random sample would need to examine people across geographical, socioeconomic, community, and class lines (just to name a few). For a study like this, scientists would also need to make sure the medical exams were standardized and consistent across the different people being examined. In other words, there’s a tradeoff between taking a more representative random sample and the cost of collecting all the data from the sample. The Framingham study collected high-quality medical data from its subjects, even if the sub- jects may not be a perfect representation of the population of all Americans. This is a common issue that data scientists face: while the available data aren’t perfect, they’re the best we have. The Framingham study is generally considered the best in its class, so we’ll continue working with it while keeping its limitations in mind. (For more on representation in medical study samples, you can read these recent articles from NPR and Scientific American). 2.2.2 Section 2: Cholesterol and Heart Disease Next, we are going to examine one of the main findings of the Framingham study: an association between serum cholesterol (i.e., how much cholesterol is in someone’s blood) and whether or not that person develops heart disease. We’ll use the following null and alternative hypotheses: Null Hypothesis: In the population, the distribution of cholesterol levels among those who get heart disease is the same as the distribution of cholesterol levels among those who do not. Alternative Hypothesis: The cholesterol levels of people in the population who get heart dis- ease are higher, on average, than the cholesterol level of people who do not. Question 1: From the provided Null and Alternative Hypotheses, what seems more reasonable to use, A/B Testing or the Standard Hypothesis Testing? Assign the variable reasonable_test to one of the following choices. 1. A/B Testing 2. Standard Hypothesis Test In [15]: reasonable_test = 1 #A/B testing: two numerical samples come from the same underlying reasonable_test Out[15]: 1 In [16]: _ = ok.grade( ' q2_2_1 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests 8

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    --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 2: Now that we have a null hypothesis, we need a test statistic. Explain and justify your choice of test statistic in two sentences or less. Hint : Remember that larger values of the test statistic should favor the alternative over the null. Test Statistic: Since ‘ANYCHD’ does not have an equal proportion of 0’s to 1’s, I chose my test statistic to be the absolute difference of the means between the cholesterol levels of people with and without heart disease. Question 3 : Write a function that computes your test statistic. It should take a table with two columns, TOTCHOL and ANYCHD , and compute the test statistic you described above. Hint: think about what does ANYCHD mean in the table In [17]: def compute_framingham_test_statistic(tbl): #f_test_stat1 = np.mean(abs(tbl.where( ' ANYCHD ' , 1).column( ' TOTCHOL ' ) - tbl.where( ' ch_w_hd = tbl.where( ' ANYCHD ' , 1).column( ' TOTCHOL ' ) ch_wo_hd = tbl.where( ' ANYCHD ' , 0).column( ' TOTCHOL ' ) #f_test_stat2 = sum(np.abs(ch_w_hd - ch_wo_hd))/2 #total variation distance #can ' t do f_test_stat 1 or 2 bc ch_w_hd and ch_wo_hd have diff lengths f_test_stat = np.mean(ch_w_hd) - np.mean(ch_wo_hd) #why no abs? return f_test_stat In [18]: _ = ok.grade( ' q2_2_3 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 4 : Use the function you defined above to compute the observed test statistic, and assign it to the name framingham_observed_statistic . In [ ]: framingham_observed_statistic = (compute_framingham_test_statistic(framingham)) framingham_observed_statistic Out[ ]: 16.635919905689406 9

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    In [ ]: _ = ok.grade( ' q2_2_4 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Now that we have defined hypotheses and a test statistic, we are ready to conduct a hypothesis test. We’ll start by defining a function to simulate the test statistic under the null hypothesis, and then use that function 1000 times to understand the distribution under the null hypothesis. Question 5 : Write a function to simulate the test statistic under the null hypothesis. The simulate_framingham_null function should simulate the null hypothesis once (not 1000 times) and return the value of the test statistic for that simulated sample. Hint: * Simulate a new sample (should it be with replacement or not?) * Think about what do you want to change and what do you want to keep the same when you do resampling In [ ]: def simulate_framingham_null(): shuffled_frame = framingham.sample(with_replacement= False ).column( ' TOTCHOL ' ) #check sim_table_frame = framingham.with_column( ' TOTCHOL ' , shuffled_frame) return compute_framingham_test_statistic(sim_table_frame) In [ ]: # Run your function once to make sure that it works. simulate_framingham_null() Out[ ]: -1.3361179346910887 In [ ]: _ = ok.grade( ' q2_2_5 ' ) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Running tests --------------------------------------------------------------------- Test summary Passed: 1 Failed: 0 [ooooooooook] 100.0% passed Question 6 : Fill in the blanks below to complete the simulation for the hypothesis test. Your simulation should compute 1000 values of the test statistic under the null hypothesis and store the result in the array framingham_simulated_stats. Hint : You should use the function you wrote above in Question 3. Note : Warning: running your code might take a few minutes! We encourage you to check your simulate_framingham_null() code to make sure it works correctly before running this cell. 10

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    In [ ]: framingham_simulated_stats = make_array() repetitions = 1000 for i in range(repetitions): sim_stat = simulate_framingham_null() framingham_simulated_stats = np.append(framingham_simulated_stats, sim_stat) The following line will plot the histogram of the simulated test statistics, as well as a point for the observed test statistic. In [ ]: Table().with_column( ' Simulated statistics ' , framingham_simulated_stats).hist() plots.scatter(framingham_observed_statistic, 0, color= ' red ' , s=30); Question 7 : Compute the p-value for this hypothesis test, and assign it to the name framingham_p_value . Hint : One of the key findings of the Framingham study was a strong association between cholesterol levels and heart disease. If your p-value doesn’t match up with this finding, you may want to take another look at your test statistic and/or your simulation. In [ ]: framingham_p_value = np.count_nonzero(framingham_simulated_stats >= framingham_observed framingham_p_value Out[ ]: 0.0 In [ ]: _ = ok.grade( ' q2_2_7 ' ) 11

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