math biostatistics boot camp 1

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Mathematical Biostatistics Boot Camp 1 Homework Week 3 1. (1 point) A web site (www.medicine.ox.ac.uk/bandolier/band64/b64-7.html) for home pregnancy tests cites the following: “When the subjects using the test were women who collected and tested their own samples, the overall sensitivity was 75%. Specificity was also low, in the range 52% to 75%.” Suppose a subject has a negative test. Assume the lower bound for the specificity. What number is closest to the multiplier of the pre-test odds of pregnancy to obtain the post-test odds of pregnancy given a negative test result? a. 1.5 b. 0.5 c. 2 d. 1 1. b. Solution: DLR - = 1 - sensitivity specificity = 1 - 0.75 0.52 = 0.25 0.52 =0.4808 0.5 2. (1 point) A web site (www.medicine.ox.ac.uk/bandolier/band64/b64-7.html) for home pregnancy tests cites the following: “When the subjects using the test were women who collected and tested their own samples, the overall sensitivity was 75%. Specificity was also low, in the range 52% to 75%.” Assume the lower value for specificity. Suppose a subject has a negative test and that 30% of women taking pregnancy tests are actually pregnant. What number is closest to the probability of pregnancy given a negative test? a. 30% b. 60% c. 90% d. 20% e. 80% f. 50% g. 40% h. 70% i. 10% 2. d. Solution: P (preg|-)= P (-|preg)P (preg) P (-|preg)P (preg)+ P (-|preg c )P (preg c ) = (1 - P (+|preg))P (preg) (1 - P (+|preg))P (preg)+ P (-|preg c )(1 - P (preg)) = (1 - 0.75)(0.30) (1 - 0.75)(0.30) + (0.52)(0.7) = 0.075 0.439 =0.1708 20% 3. (1 point) Suppose that hospital infection counts are models as Poisson with mean μ. Recall the Poisson mass function with mean μ is μ x e -μ x! for x =0, 1,... Three independent hospitals are observed for one year and their infection counts were 5, 4, and 6, respectively. What is the ML estimate for μ? a. 0 b. 1 c. 4 d. 4.5 e. 5 f. 5.5 g. 6 3. e. Solution: 3 Y i=1 μ x k e -μ x k ! = μ x1 e -μ μ x2 e -μ μ x3 e -μ x 1 !x 2 !x 3 ! = μ x1+x2+x3 e -3μ x 1 !x 2 !x 3 ! = μ 5+4+6 e -3μ x 1 !x 2 !x 3 ! = μ 15 e -3μ x 1 !x 2 !x 3 ! We will take the derivative of this function and set it to zero. It will be easier first to take logarithm of the function and then find the maximum: ˚ f = log f = 15 log(μ) - 3μ - log(x 1 !x 2 !x 3 !) Now, we find the maximum: d dμ (15 log(μ) - 3μ - log(x 1 !x 2 !x 3 !)) = 15 μ - 3=0 and solve for μ to get μ =5. Page 1 of 3

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Homework with answers (and solutions)Week 3

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Page 1: Math Biostatistics Boot Camp 1

Mathematical Biostatistics Boot Camp 1Homework

Week 3

1. (1 point) A web site (www.medicine.ox.ac.uk/bandolier/band64/b64-7.html) for home pregnancy tests citesthe following: “When the subjects using the test were women who collected and tested their own samples,the overall sensitivity was 75%. Specificity was also low, in the range 52% to 75%.” Suppose a subject hasa negative test. Assume the lower bound for the specificity. What number is closest to the multiplier of thepre-test odds of pregnancy to obtain the post-test odds of pregnancy given a negative test result?

a. 1.5 b. 0.5 c. 2 d. 1

1. b.

Solution:

DLR− =1− sensitivityspecificity

=1− 0.75

0.52=

0.25

0.52= 0.4808 ≈ 0.5

2. (1 point) A web site (www.medicine.ox.ac.uk/bandolier/band64/b64-7.html) for home pregnancy tests citesthe following: “When the subjects using the test were women who collected and tested their own samples, theoverall sensitivity was 75%. Specificity was also low, in the range 52% to 75%.” Assume the lower value forspecificity. Suppose a subject has a negative test and that 30% of women taking pregnancy tests are actuallypregnant. What number is closest to the probability of pregnancy given a negative test?

a. 30% b. 60% c. 90% d. 20% e. 80% f. 50% g. 40% h. 70% i. 10%

2. d.

Solution:

P (preg|−) =P (−|preg)P (preg)

P (−|preg)P (preg) + P (−|pregc)P (pregc)

=(1− P (+|preg))P (preg)

(1− P (+|preg))P (preg) + P (−|pregc)(1− P (preg))

=(1− 0.75)(0.30)

(1− 0.75)(0.30) + (0.52)(0.7)=

0.075

0.439= 0.1708 ≈ 20%

3. (1 point) Suppose that hospital infection counts are models as Poisson with mean µ. Recall the Poisson mass

function with mean µ isµxe−µ

x!for x = 0, 1, . . . Three independent hospitals are observed for one year and their

infection counts were 5, 4, and 6, respectively. What is the ML estimate for µ?

a. 0 b. 1 c. 4 d. 4.5 e. 5 f. 5.5 g. 6

3. e.

Solution:3∏i=1

µxke−µ

xk!=µx1e−µµx2e−µµx3e−µ

x1!x2!x3!=µx1+x2+x3e−3µ

x1!x2!x3!=µ5+4+6e−3µ

x1!x2!x3!=

µ15e−3µ

x1!x2!x3!

We will take the derivative of this function and set it to zero. It will be easier first to take logarithm of thefunction and then find the maximum: f̊ = log f = 15 log(µ)− 3µ− log(x1!x2!x3!)

Now, we find the maximum:d

dµ(15 log(µ)− 3µ− log(x1!x2!x3!)) =

15

µ−3 = 0 and solve for µ to get µ = 5.

Page 1 of 3

Page 2: Math Biostatistics Boot Camp 1

4. (1 point) Let X1, . . . , Xn be iid exponential(β). That is, having density1

βe−x/β for x > 0. What is the ML

estimate for β?

a.n∑i=1

xin

b.n∑i=1

xi c.

(n∑i=1

log(xi)

n

)−1

d.

(n∑i=1

xi

)−1

e.n∑i=1

log(xi)

nf.

(n∑i=1

xin

)−1

4. a.

Solution:n∏i=1

1

βe−xi/β =

1

βne−(x1+···+xn)/β =

1

βne−

∑ni=1 xi

We will take the derivative of this function and set it to zero. It will be easier first to take logarithm of the

function and then find the maximum: f̊ = log f = − 1

β

n∑i=1

xi − n log β

Now, we find the maximum:d

(− 1

β

n∑i=1

xi − n log β

)=

1

β2

n∑i=1

xi −n

β= 0 and solve for β to get

β =

n∑i=1

xin

5. (1 point) Let X be a geometric random variable. That is X counts the number of coin flips until one obtainsthe first head. The mass function is P (X = x) = p(1− p)x−1 for x = 1, 2, . . . What is the maximum likelihoodestimate for p if one observes a geometric random variable?

a. 1/(x− 1) b. 1/x c. 1/2 d. 1/(x+ 1)

5. b.

Solution:

We want to find the derivative of this function and set it to zero. It will be easier to take the logarithm ofthe function first:

log f = log(p) + (x− 1) log(1− p)and now we take the derivative:

d

dp(log(p) + (x− 1) log(1− p)) =

1

p− x− 1

1− p= 0

and solve for p to get p =1

x.

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Page 3: Math Biostatistics Boot Camp 1

6. (1 point) Let X be a Poisson count with mean µ. Recall the Poisson mass function with mean µ isµxe−µ

x!for

x = 0, 1, . . . What is the maximum likelihood estimate for µ?a. µ b. 1/x c. 1/2 d. x2 e. x

6. e.

Solution:

First, we take the logarithm of the function to get:

log f = x log(µ)− µ− log(x!)

now we take the derivative and set it to zero to get maximum:d

dµ(x log(µ)− µ− log(x!)) =

x

µ− 1 = 0

and solve for µ to get µ = x.

Question: 1 2 3 4 5 6 Total

Points: 1 1 1 1 1 1 6

Score:

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