STAT 408/608 Guided Exercise 1

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1 STAT 408/608 Guided Exercise Be sure to: Please submit your answers in a Word file to Sakai at the same place you downloaded the file Remember you can paste any Excel or JMP output into a Word File (use Paste Special for best results). We prefer you submit only one file for the assignment. It is ok to do problems by hand. However, you will need to scan or take a picture of your work. Pictures can also be pasted into the assignment. Put your name and the Assignment # on the file name that you submit: e.g. Ilvento Guided.doc Answer as completely as you can and show your work. We do not grade Guided Assignments. However, we do check that you completed the work. The answers to each Guided Exercise are posted on Sakai.. I love data controversies! Body Mass Index is a widely-used measure of health and obesity. It is easy to calculate and nonobtrusive to patients. It was developed over 60 years for a different purpose, but it is heavily used now to measure obesity. Read this article on Body Mass Index (BMI) and then answer some questions at the end. Here is the link: Why BMI is inaccurate and misleading Written by Christian Nordqvist Published: Sunday 25 August 203 BMI (body mass index), which is based on the height and weight of a person, is an inaccurate measure of body fat content and does not take into account muscle mass, bone density, overall body composition, and racial and sex differences, say researchers from the Perelman School of Medicine, University of Pennsylvania. Every few months the same comment is made by experts "BMI is flawed". The news hits the headlines, everybody agrees, and then all goes quiet for a while. You are of normal weight if your BMI is between 8.5 and 25, overweight if it is between 25 and 30. Anybody with a BMI of 30 or more is obese. Mitchell Lazar, MD, PhD, Professor of Medicine and Genetics and Director of the Institute of Diabetes, Obesity, and Metabolism, and Rexford Ahima, MD, PhD, Professor of Medicine and Director of the Obesity Unit in the Institute for Diabetes, Obesity and Metabolism, discuss the challenges health professionals face when studying the mortality risks and health of obese people in the journal Science. We all know that obesity increases the risk of developing heart diseases, type 2 diabetes, cancer, sleep apnea and other diseases and conditions. However, according to recent studies, obesity may also protect against death from all causes, as well as death due to stroke, heart failure and diabetes. In the Science article - "The Health Risk of Obesity - Better Metrics Imperative" - the authors refer to the pros-and-cons of obesity as the "obesity-mortality paradox". A paradox that generates a great deal of controversy. Dr. Ahima said: "There is an urgent need for accurate, practical and affordable tools to measure fat and skeletal muscle, and biomarkers that can better predict the risks of diseases and mortality. Advances to improve the measurement of obesity and related factors will help determine the optimal weight for an individual, taking into account factors such as age, sex, genetics, fitness, preexisting diseases, as well novel blood markers and metabolic parameters altered by obesity." People with a BMI of 30 or more, i.e. obese individuals, have a significantly higher risk of eventually becoming diabetic, developing cancer, cardiovascular diseases, osteoarthritis, and liver and gallbladder diseases. Being obese heightens the risk of premature death. However, a number of studies have demonstrated that some obese individuals have lower cardiovascular risk and an improved metabolic profile, while a subset of "normal-bmi"

2 people are metabolically unhealthy and have increased mortality risk. A team of researchers at the University of Virginia, Charlottesville, found better post-surgical short-term survival rates among obese people than patients of normal weight3. Patients with a BMI of 23. or less were more than twice as likely to die within 30 days of surgery than those with a BMI of 35.3 or more. Drs. Lazar and Ahima point out that the true impact of obesity may not be fully understood, because population studies focus on the link between BMI, health and mortality risks, without taking into account how unintentional/intentional weight loss/gain may affect these outcomes. Dr. Lazar noted "Future research should be focused more on molecular pathways, especially how metabolic factors altered by obesity change the development of diabetes, heart diseases, cancer and other ailments, and influence the health status and mortality." BMI exaggerates thinness in short people and fatness in tall people Nick Trefethen, Professor of Numerical Analysis at Oxford University's Mathematical Institute, in a letter to The Economist explained that BMI leads to confusion and misinformation. BMI = weight in kilograms divided by height in meters squared. Professor Trefethen believes that the BMI height 2 /weight term divides the weight by too much in short people and too little in tall individuals. This results in tall people believing they are fatter than they really are, and short people thinking they are thinner. BMI was devised in the 830s by Lambert Adolphe Jacques Quetelet ( ), a Belgian mathematician, sociologist, statistician and astronomer. Trefethen explained that during Quetelet's time there were no calculators, computers or electronic devices - which is probably why he opted for a super-simple system. Trefethen wonders why institutions today on both sides of the Atlantic continue using the same flawed-bmi formula. "Perhaps nobody wants to rock the boat", Trefethen 2 added. Trefethen believes a better calculation than the present weight/height 2 for BMI would be weight/height 2.5. "Certainly if you plot typical weights of people against their heights, the result comes out closer to height 2.5 than height 2." Waist size linked to diabetes risk, regardless of BMI Researchers from the Medical Research Council (MRC) Epidemiology Unit, UK, reported in PLoS Medicine that waist circumference is strongly and independently associated with type two diabetes risk5, even after accounting for BMI. Study leader, Dr Claudia Langenberg and team suggested that waist circumference should be measured more widely for estimating type 2 diabetes risk. They pointed out that a non-obese, overweight male with a waist circumference of at least 40.2 inches (02cm) has the same or higher risk of type 2 diabetes as an obese male. The same applies for females with a waist of 34.6 inches (88cm) or more. A study published by the RAND Corporation showed that waist size explained the higher type 2 diabetes rate in the USA than UK, not BMI6. Co-author, James P. Smith said "Americans carry more fat around their middle sections than the English, and that was the single factor that explained most of the higher rate of diabetes seen in the United States, especially among American women. Waist size is the missing new risk factor we should be studying." Waist-to-height ratio better than BMI Dr Margaret Ashwell, an independent consultant and former science director of the British Nutrition Foundation, explained at the 9th Congress on Obesity in Lyon, France, May 202, that waist-toheight ratio is a superior predictor than BMI7 of type 2 diabetes and cardiovascular diseases. Dr. Ashwell said "Keeping your waist circumference to less than half your height can help increase life expectancy for every person in the world." Thus a 6ft-tall man should have a waist circumference of 36 inches or less, while a 5ft 4in woman's waist should not exceed 32 inches. The waist-to-height ratio should be considered as a screening tool, Ashwell added. Ashwell explained that BMI does not take into account the distribution of fat around the body. Abdominal fat affects organs

3 like the kidney, liver and heart more severely than fat around the bottom or hips. Waist circumference gives an indication of abdominal fat levels. Dr. Ashwell and colleagues believe that the thought "keep your waist circumference to less half your height" is an easier one to hold on to that BMI. An example of the biggest flaw in using BMI Body Mass Index' biggest flaw is that it does not take into account the person's body fat versus muscle (lean tissue) content. Muscle weighs more than fat (it is denser, a cubic inch of muscle weighs more than a cubic inch of fat). Therefore, BMI will inevitably class muscly, athletic people as fatter than they really are. A 6ft-tall Olympic 00 meter sprinter weighing 90kg (200lbs) may have the same BMI (26) as a couch potato of the same height and weight. A BMI calculation would class both of them as overweight. That calculation is probably right for the sedentary couch potato, but not for the athlete. The athlete's waist circumference, at 34ins, is well within "healthy weight" - if his height is 72 inches, his waist is less than half his height. However, the sedentary person's waist of 40 inches is more than half his height. Here is some additional information The formula for BMI is: Metric Formula: weight (kg)/[height (m)] 2 Example: Weight = 68 kg, Height = 65 cm (.65 m) Calculation: 68 (.65) 2 = Pounds/inches Formula: weight (lbs)/[height (in.)] 2 * 703 Example: Weight = 50 lbs, Height = 5 5 (65") Calculation: [50 (65) 2 ] x 703 = The National Institutes of Health uses the following BMI Categories: Underweight = <8.5 Normal weight = Overweight = Obesity = BMI of 30 or greater So, what do you think? Is the BMI a useful indicator of how overweight people are? Is the notion of being overweight too highly politicized? Should we be about labeling who is or is not overweight or obese? There are no right or wrong answers here, just your opinions! 3

4 2. Each year the Academy of the Screen Actors Guild gives an award for the best actor and actress in a motion picture. We have recorded the name and age of each since 996. The data for males and females is given below (the sample size, n =20). The sum of their age as well as the sum of age squared are also given. YEAR ACTOR MALE AGE ACTRESS FEMALE AGE 996 Geoffrey Rush 45 Frances McDormand Jack Nicholson 60 Helen Hunt Roberto Benigni 46 Gwyneth Paltrow Kevin Spacey 40 Hilary Swank Russell Crowe 36 Julia Roberts Denzel Washington 47 Halle Berry Adrien Brody 29 Nicole Kidman Sean Penn 43 Charlize Theron Jamie Foxx 37 Hilary Swank Philip Seymour Hoffman 38 Reese Witherspoon Forest Whitiker 45 Helen Mirren Daniel Day-Lewis 50 Marion Cotillard Sean Penn 48 Kate Winslet Jeff Bridges 60 Sandra Bullock Colin Firth 50 Natalie Portman Jean Dujardin 39 Meryl Streep Daniel Day-Lewis 55 Jennifer Lawrence Matthew McConaughey 44 Cate Blanchett Eddie Redmayne 32 Julianne Moore Leonardo DiCaprio 4 Brie Larson Casey Affleck 4 Emma Stone 28 Sum X 926 Sum X 750 Sum X-squared 42,46 Sum X-squared 29,382 Here is the Stem and Leaf plot for each group to compare the distributions. Stem and Leaf Plot of Actors Winning Academy Award Since 996 Males Females Stem Leaf Stem Leaf

5 represents represents 60 a. Calculate the measures of central tendency and variability for each group. Males Females Mean 926/2 = /2 = 35.7 Median The th observation in ordered data = 44 The th observation in ordered data = 33. Mode Not a unique mode Not a unique mode Range = = 40 Variance [42,46 (926) 2 /2]/(2-) [42,46 40,832.9]/ /20 = [29,382 (750) 2 /2]/(2-) [29,382 26, ]/ /20 = Standard Deviation SQRT(65.69) = 8.0 SQRT(29.84) =.39 Coefficient of Variation CV = 8.0/44.09 *00 = 8.38% CV =.55/36.0 *00 = 3.90% b. Briefly compare the two distributions with an emphasis on the measures of Central Tendency and Variability. For males, the distribution is symmetric and centered around the mean of There are no obvious outliers. The median is very close to the mean at The values vary from 29 to 60 for a range of 3 years. The standard deviation is 8.0 years, which is relatively small compared with the mean (CV = 8.38%). For females, the mean is lower at 35.7, which is higher that the median of 33. Two large outliers influence the distribution for females at 6 and 62, which pulled the mean up. Otherwise the spread for females is centered in the mid 20s to mid 30s. The range is larger for females compared with that for males (62-22 = 40), as is the standard deviation (.39 for females). The higher standard deviation is also a reflection of the outliers. The CV for females is much higher than that of males at 3.90%. c. For both men and women there are a few outliers. For men there are two individuals with a value of 60. For women there is one winner aged 6 and another aged 62. Calculate z-scores for these values and interpret their meaning. Z m = ( )/8.0 =.99 Z f = ( )/.39 = 2.3 Z f2 = (6-35.7)/.39 =

6 Z-score represents the distance between Xi and the mean X-bar, expressed in standard deviation. For Z m =.99, it means that there is distance of.99 standard deviation between 60 and the mean d. Suppose we wanted to remove the two female outliers from the data. Calculate the new mean, median, variance, standard deviation, and CV for women winners for the remaining 9 winners. Hint: subtract the values from the old sum and divide by 9. Did the outliers influence the mean and median age much? What about the variance and standard deviation? ( ) = /9 = The mean for females decreased from 35.7 to by removing the two outliers. This is a 7.6% decrease. Both the variance and the standard deviation declined dramatically to and Answer the following questions about variability of data sets: a. How would you describe the variance and standard deviation in words, rather than a formula? Think of what you are calculating and how it might be useful in describing a variable. The Variance is the average Squared deviation around the center (in this case the center is the mean). The standard deviation is the average deviation around the center (in this case the center is the mean). b. What is the primary advantage of using the inter-quartile range compared with the range when describing the variability of a variable? The range only uses two values - the maximum and the minimum - to calculate the range. It can be very sensitive to outliers. The inter-quartile range shows the range of the middle 50% of the values. c. Can the standard deviation ever be larger than the variance? Explain. In most cases the standard deviation is less than the variance since it is a square root of the variance. However, in the special case where the variance is between 0 and, the standard deviation will be more than the variance. For example, if S 2 =.5, then s =.7 d. Can the variance ever be negative? Why or why not? Since the variance is based on a squared measure, no, it cannot be negative e. Show the formula for the Coefficient of Variation and explain what it is and how it can be useful in comparing the variability of different variables. The ratio of the standard deviation to the absolute value of the mean, usually multiplied by 00. It expresses the standard deviation in relation to the mean. It makes it easier to compare the spread of different 5. Below is the data for infant mortality for 34 OECD countries. The Organization for Economic Co-operation and Development (OECD) is an international economic organization of 34 countries, founded in 96 to stimulate economic progress and world trade. It is a forum of countries describing themselves as committed to democracy and the market economy, providing a platform to compare policy experiences, seeking answers to common problems, identify good practices and coordinate domestic and international policies of its members. OECD s web site provided some data on infant mortality for 34 countries. Infant mortality (the rate of death of children under year of age per,000 live births) is a measure of development. The Histogram and the Stem and Leaf Plot for this data is given below. Use the stem and leaf values for some calculations, such as the min and max. For other calculations, the Sum of (x) is 28.2 and the Sum of (x 2 ) is The two outliers are Turkey (0.2) and Mexico (3.0). 6

7 Stem and Leaf Stem Leaf Count represents.3 a. Calculate the: Mean = 3.77 (28.20/34 = 3.77) Median = 3.2 = (3.+3.3)/2 Mode = undefined Maximum = 3 Minimum =.3 Range =.7 Variance = ( /34)/33=5.49 Standard Deviation = 2.34 Coefficient of Variation = 2.34/3.77=0.62 b. What is the position of the median value for this data? Q2 c. Does the mode make sense as a measure of Central Tendency for this data? Yes. It is pulled by the outliers. d. Calculate a z-score for an infant mortality rate of rate of 3. (3-3.77)/2.34=3.94 7