BIOS6040 Mathematics for Applied Statistics Assignment 1

This assignment will be marked out of 20 and is worth 20% of the total marks for BIOS6040. It covers material in Modules 1 and 2.

Attempt all questions. Show necessary working and justify your answers with appropriate reasoning. It is recommended that you typeset your answers using Microsoft Word equation editor or LATEX or similar software. Submit your answers as a .doc .docx or .pdf file through Blackboard.

Due date: Sunday 19 March 2017, 11:59pm (your time zone, wherever you live)

Question 1 [4 marks]

a) Alice is 10 years old and weighs 25kg and Bob is 20 years old and weighs 75kg. If these two data are plotted on a graph of ‘weight vs. years’, what is the slope of the line that passes through both points? [1 mark]

b) If the line in part (a) perfectly predicts weight at a given age, what would Eve (an 18 year old) weigh? [1 mark]

c) Eve actually weighs 60kg. Explain why there is no line that perfectly predicts these three people’s weights from their ages. [1 mark]

d) To what domain should we restrict the 𝑥-variable age so that the equation produces a nonnegative answer for the 𝑦-variable weight? [1 mark]

Question 2 [6 marks]

The ASSIGN score is an equation developed by the University of Dundee that estimates a person’s 10 year risk of cardiovascular disease and is used to identify those high at risk. A score above 20 is considered high risk. The risk factors are:

cigs – number of cigarettes smoked per day

diabetes – variable equals 1 if diagnosed with diabetes, 0 otherwise sysbp – systolic blood pressure mg/Hg tchol – total cholesterol mmol/L

hdl – HDL cholesterol mmol/L

For females the calculation is:

𝑥 = -5.2 + 0.02𝑐𝑖𝑔𝑠 + 0.82𝑑𝑖𝑎𝑏𝑒𝑡𝑒𝑠 + 0.01𝑠𝑦𝑠𝑏𝑝 + 0.22𝑡𝑐h𝑜𝑙 - 0.54h𝑑𝑙 + 0.06𝑎𝑔𝑒

𝑦 = 100(1 - 0.83𝑒𝑥)

For males the calculation is:

𝑥 = -5.2 + 0.03𝑐𝑖𝑔𝑠 + 0.98𝑑𝑖𝑎𝑏𝑒𝑡𝑒𝑠 + 0.01𝑠𝑦𝑠𝑏𝑝 + 0.13𝑡𝑐h𝑜𝑙 - 0.56h𝑑𝑙 + 0.07𝑎𝑔𝑒

𝑦 = 100(1 - 0.93𝑒𝑥)

a) Freya is a 50 year old healthy female non-smoker with diabetes. Her measurements are: systolic BP 120mg/Hg, total cholesterol 5, and HDL cholesterol 1.5. Calculate her ASSIGN score by substituting her values into the appropriate equations above. Is Freya a high risk patient? [2 marks]

b) Micah is a 58 year old male smoker (25 cigarettes per day) with no diabetes diagnosis but he has high cholesterol and high blood pressure. His measurements are systolic BP 150 mm/Hg, total cholesterol 6.3, and HDL 0.5. Micah’s doctor tells him that his ASSIGN score puts him at higher risk and advises him that quitting smoking alone will drastically reduce his risk (even if his other measurements don’t change). If Micah quits smoking, by how much does he reduce his ASSIGN score? [2 marks]

c) Let 𝑓(𝑥) = 𝑒𝑥 and 𝑔(𝑥) = 100(1 - 0.83𝑥). What is the composition 𝑔(𝑓(𝑥))? [1 mark]

d) Suppose that biological constraints ensure that the largest value 𝑥 could take is around 4. Calculate 𝑔(𝑓(4)). What do you think the largest possible ASSIGN score is if it can be expressed as a function like 𝑔(𝑓(𝑥))? [1 mark]

Question 3 [4 marks]

𝑥 𝑒𝑥

The logit function 𝑓(𝑥) = ln (1 -𝑥) and the logistic function 𝑔(𝑥) = 𝑒 𝑥+1 are important in logistic regression. The domain of 𝑓 is restricted to 0 𝑥 1. Show that 𝑓 and 𝑔 are inverse functions.

Question 4 [2 marks]

Suppose 𝛽0 and 𝛽1 are constants and we have a function defined by: odds(𝑥) = 𝑒𝛽0+𝛽1𝑥 for 𝑥 ? (-8, 8)

What is the effect of a 1 unit change in the variable 𝑥? In other words, find the ratio of the odds: odds(𝑥 + 1)/odds(𝑥)

Question 5 [4 marks]

A researcher is collecting measurement data from males and females over time. A line of best fit is drawn separately for the male and female data and the equations of the lines are:

𝐹𝑒𝑚𝑎𝑙𝑒: 𝑦 = 𝛽0 + 𝛽1 + (𝛽2 + 𝛽3)𝑎𝑔𝑒 𝑀𝑎𝑙𝑒: 𝑦 = 𝛽0 + 𝛽2𝑎𝑔𝑒

where 𝛽0 and 𝛽1 are assumed to be positive values. The slope of the line can be interpreted as the rate of change in the measured variable over time as the subjects age. Consider the two graphs below.

Graph A Graph B

Which one graph (A or B) best illustrates the following possible situations? Give a brief reason.

a) 𝛽2 0 [1 mark]

b) 𝛽3 = 0 [1 mark]

c) Females improved over time at the same rate as males [1 mark]

d) Over time, the rate of change in measurements for females was greater than for males [1 mark]

This assignment will be marked out of 20 and is worth 20% of the total marks for BIOS6040. It covers material in Modules 1 and 2.

Attempt all questions. Show necessary working and justify your answers with appropriate reasoning. It is recommended that you typeset your answers using Microsoft Word equation editor or LATEX or similar software. Submit your answers as a .doc .docx or .pdf file through Blackboard.

Due date: Sunday 19 March 2017, 11:59pm (your time zone, wherever you live)

Question 1 [4 marks]

a) Alice is 10 years old and weighs 25kg and Bob is 20 years old and weighs 75kg. If these two data are plotted on a graph of ‘weight vs. years’, what is the slope of the line that passes through both points? [1 mark]

b) If the line in part (a) perfectly predicts weight at a given age, what would Eve (an 18 year old) weigh? [1 mark]

c) Eve actually weighs 60kg. Explain why there is no line that perfectly predicts these three people’s weights from their ages. [1 mark]

d) To what domain should we restrict the 𝑥-variable age so that the equation produces a nonnegative answer for the 𝑦-variable weight? [1 mark]

Question 2 [6 marks]

The ASSIGN score is an equation developed by the University of Dundee that estimates a person’s 10 year risk of cardiovascular disease and is used to identify those high at risk. A score above 20 is considered high risk. The risk factors are:

cigs – number of cigarettes smoked per day

diabetes – variable equals 1 if diagnosed with diabetes, 0 otherwise sysbp – systolic blood pressure mg/Hg tchol – total cholesterol mmol/L

hdl – HDL cholesterol mmol/L

For females the calculation is:

𝑥 = -5.2 + 0.02𝑐𝑖𝑔𝑠 + 0.82𝑑𝑖𝑎𝑏𝑒𝑡𝑒𝑠 + 0.01𝑠𝑦𝑠𝑏𝑝 + 0.22𝑡𝑐h𝑜𝑙 - 0.54h𝑑𝑙 + 0.06𝑎𝑔𝑒

𝑦 = 100(1 - 0.83𝑒𝑥)

For males the calculation is:

𝑥 = -5.2 + 0.03𝑐𝑖𝑔𝑠 + 0.98𝑑𝑖𝑎𝑏𝑒𝑡𝑒𝑠 + 0.01𝑠𝑦𝑠𝑏𝑝 + 0.13𝑡𝑐h𝑜𝑙 - 0.56h𝑑𝑙 + 0.07𝑎𝑔𝑒

𝑦 = 100(1 - 0.93𝑒𝑥)

a) Freya is a 50 year old healthy female non-smoker with diabetes. Her measurements are: systolic BP 120mg/Hg, total cholesterol 5, and HDL cholesterol 1.5. Calculate her ASSIGN score by substituting her values into the appropriate equations above. Is Freya a high risk patient? [2 marks]

b) Micah is a 58 year old male smoker (25 cigarettes per day) with no diabetes diagnosis but he has high cholesterol and high blood pressure. His measurements are systolic BP 150 mm/Hg, total cholesterol 6.3, and HDL 0.5. Micah’s doctor tells him that his ASSIGN score puts him at higher risk and advises him that quitting smoking alone will drastically reduce his risk (even if his other measurements don’t change). If Micah quits smoking, by how much does he reduce his ASSIGN score? [2 marks]

c) Let 𝑓(𝑥) = 𝑒𝑥 and 𝑔(𝑥) = 100(1 - 0.83𝑥). What is the composition 𝑔(𝑓(𝑥))? [1 mark]

d) Suppose that biological constraints ensure that the largest value 𝑥 could take is around 4. Calculate 𝑔(𝑓(4)). What do you think the largest possible ASSIGN score is if it can be expressed as a function like 𝑔(𝑓(𝑥))? [1 mark]

Question 3 [4 marks]

𝑥 𝑒𝑥

The logit function 𝑓(𝑥) = ln (1 -𝑥) and the logistic function 𝑔(𝑥) = 𝑒 𝑥+1 are important in logistic regression. The domain of 𝑓 is restricted to 0 𝑥 1. Show that 𝑓 and 𝑔 are inverse functions.

Question 4 [2 marks]

Suppose 𝛽0 and 𝛽1 are constants and we have a function defined by: odds(𝑥) = 𝑒𝛽0+𝛽1𝑥 for 𝑥 ? (-8, 8)

What is the effect of a 1 unit change in the variable 𝑥? In other words, find the ratio of the odds: odds(𝑥 + 1)/odds(𝑥)

Question 5 [4 marks]

A researcher is collecting measurement data from males and females over time. A line of best fit is drawn separately for the male and female data and the equations of the lines are:

𝐹𝑒𝑚𝑎𝑙𝑒: 𝑦 = 𝛽0 + 𝛽1 + (𝛽2 + 𝛽3)𝑎𝑔𝑒 𝑀𝑎𝑙𝑒: 𝑦 = 𝛽0 + 𝛽2𝑎𝑔𝑒

where 𝛽0 and 𝛽1 are assumed to be positive values. The slope of the line can be interpreted as the rate of change in the measured variable over time as the subjects age. Consider the two graphs below.

Graph A Graph B

Which one graph (A or B) best illustrates the following possible situations? Give a brief reason.

a) 𝛽2 0 [1 mark]

b) 𝛽3 = 0 [1 mark]

c) Females improved over time at the same rate as males [1 mark]

d) Over time, the rate of change in measurements for females was greater than for males [1 mark]

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