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.

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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−

1β

∑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

dβ

(− 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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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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