Epidemiology Chapter 2

Question 1

Estimating population proportion, p, by sample proportion, p hat

Answer 1

Number in the sample with the characteristic/ sample size
let X_i=1 if the ith person has the characteristic in question and 0 otherwise 

p_hat = sum from i=1 to n of xi / n

This isa point estimate of p

Question 2

The sampling distribution of p_hat

Answer 2

If each of our samples is large the sampling distribution will be approximately normal with expected value p and variance (p(1-p))/n, where n is the sample size of each sample

P hat is an unbiased estimator of p

Question 3

Standard error of p hat (one sample)

Answer 3

SE(p hat)=sqrt((p(1-p))/n)

Question 4

Inference for a proportion in a one sample case
Confidence interval

Hypothesis

Test statistic

Answer 4

p ̂±z_(α/2) (sqrt(p(1-p)/n))
95% z_α/2 = 1.96
90% z_α/2=1.645
99% z_α/2=2.5758

Null - p=p_0
Alternative
one sides: pp_0
two sides: p≠p_0

Test statistic
z=(p ̂-p_0)/sqrt((p_0 (1-p_0 ))/n)
- compare with the standard normal distribution

Question 5

Inference for a proportion in a two sample case

Notation

Point estimates

Answer 5

Population 1. Population 2
Proportion of successes p_1. p_2
Sample size n_1 n_2
Number of successes m_1. m_2

p ̂_1=m_1/n_1, p ̂_2=m_2/n_2
p ̂_1-p ̂_2 is a point estimate of p_1-p_2

Question 6

Inference for a proportion in a two sample case
Confidence interval

Hypothesis

Test statistic

Answer 6

Confidence interval 
(p ̂_1-p ̂_2)±z_(α/2)SE(p ̂_1-p ̂_2)

SE(p ̂_1-p ̂_2) = sqrt((p ̂_1(1-p ̂_1)/n1) +(p ̂_2(1-p ̂_2)/n2))

Hypothesis 
Null : p1=p2 or p1-p2=0
Alternative
One sides: p1p2 (p1-p2>0)
Two sided: p1≠p2 or p1-p2≠0

Test statistic
Under null, p1=p2=p
z=(p ̂_1-p ̂_2)/sqrt(p bar(1-p ̂)(1/n1 +1/n2))
p ̂=(m1+m2)/(n1+n2)

Question 7

Inference for rates the one sample case
Estimated rate

Standard Error

Confidence interval

Answer 7

Let D be the number of events and Y be the total person time
estimated rate = lama hat = D/Y

standard error (log(lamda hat))=sqrt(1/D)

CI for log rate
log⁡(λ ̂)±z_(α/2) √(1/D)
CI for rate
λ ̂e^(±z_(α/2) √(1/D))

Question 8

Inferences about rate ratio (2 samples)
Point estimates

Confidence interval
Standard Error

Answer 8

Suppose we have two groups to compare.
Let D_0 and D_1 denote the number of events in the unexposed and exposed groups respectively
Let Y_0 and Y_1 denote the person years

Exposed: λ ̂_1=D_1/Y_1
Unexposed:λ ̂_0=D_0/Y_0
Rate Ratio (exposed to unexposed):λ ̂_1/λ ̂_0

Confidence interval 
log rate ratio
log⁡(λ ̂_1/λ ̂_0)±z_(α/2) √(1/D_0+1/D_1)
rate ratio 
(λ ̂_1/λ ̂_0)e^(±z_(α/2) √(1/D_0+1/D_1))

SE(log(λ ̂))=√(1/D_0+1/D_1)

Question 9

Testing for equality of rates ( 2 samples)

Answer 9

Null: λ_1=λ_0 or log(λ_1/λ_0)=0, no difference
Alternative: log(λ_1/λ_0)≠0 or log(λ_1/λ_0)>0 or log(λ_1/λ_0)<0

Test statistic
z=(log⁡(λ ̂_1/λ ̂_0)-0)/SE(log⁡(λ ̂_1/λ ̂_0))
will follow a standard normal distribution under the null hypothesis

Question 10

Inference for odds
Point estimate for odds

SE(log(odds hat))

Confidence intervals

Answer 10

Odds hat = Number in our sample with the disease/ Number in our sample without the disease = m/n-m

SE(log(odds hat)
= sqrt(1/m +1/n-m)

CI
95% confidence interval for the true log odds
log⁡((odds) ̂ )±1.96SE(log⁡((odds) ̂ ))
95% confidence interval for the true odds
e^log⁡((odds) ̂ )±1.96SE(log⁡((odds) ̂ ))

Question 11

Inference for odds ratio

Point estimate

Standard Error

Confidence intervals

Answer 11

Diseased Non-diseased
Exposed. a. b
Unexposed c. d

(OR) ̂=odds of disease for the exposed group/odds of the disease for the unexposed group = (a/b)/(c/d) = ad/bc

SE(log((OR) ̂))=sqrt(1/a+1/b+1/c+1/d)

Confidence interval for log odds ratio
log(OR) ̂±z_(α/2)SE(log(OR) ̂)
for odds ratio
e^log(OR) ̂±z_(α/2)SE(log(OR) ̂)

Question 12

Inferences for Relative Risks
Point estimate

Standard Error

Confidence interval

Answer 12

The ratio of the risk in the exposed group to that in the unexposed group
Diseased Non-diseased
Exposed. a. b
Unexposed c. d

(RR) ̂=risk in exposed group/risk in unexposed group = a(c+d)/c(a+b)

SE(log(RR) ̂)=sqrt(1/a-1/a+b+1/c-1/c+d)

Confidence interval for log relative risk
log(RR) ̂±z_(α/2)SE(log(RR) ̂)
for odds ratio
e^log(RR) ̂±z_(α/2)SE(log(RR) ̂)