Statistics 431: Paired Samples and Two-Sample Inference, Study notes of Statistics

Download Statistics 431: Paired Samples and Two-Sample Inference and more Study notes Statistics in PDF only on Docsity!

Statistics 431:

Statistical Inference

Lecture 8: More on two-sample inference

Paired samples: basics

• (^) Our two samples X 1 ,... , Xm and Y 1 ,... , Yn up to now have been unpaired :

there’s no correspondence between observations in the first sample and observations in the second. (The number of Xi ’s is not even the same as the number of Yi ’s, in general.)

• (^) In some situations where m = n, there is a natural matching of each Xi with

one Yi , to form the n pairs (Xi , Yi ).

• (^) Examples:
  • n patients in a clinical trial are given a sedative. The drowsiness of each patient i is measured one hour (Xi ) and two hours (Yi ) after treatment.
  • n houses are selected at random in Philadelphia. At each house i, the radon level is measured in the basement (Xi ) and on the highest floor (Yi ).
  • For each of n sets of identical twins, we measure the body mass index of the older (Xi ) and younger (Yi ) twin.
• (^) Instead of comparing the sampled Xi ’s to the sampled Yi ’s, as for unpaired

data, we should focus on comparing the pairs (Xi , Yi ) to each other.

Why look at pairs?

• (^) Include pairing

Time by Patient

Drowsiness

+1h +2h

89

1011

12 l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

+1h +2h

l l

P

l l

P

l l

P

l l

P

l l

P l l

P l l

P l l

P

l l

P

89

1011

12 l l

P

89

1011

12 l l

P19 (^) l P20l l l

P

l l

P

l l

P l l

P

l l

P

l l

P l l

P

l l

P l l

P

l l

P l l

P

l l

P

l l

P

l l

P

l l

P

89

1011

12 l l

P

89

1011

12

l l

P

l l

P

l l

P

l l

P l l

P

l l

P

l l

P

l l

P

l l

P

Difference between means: paired test

• (^) The data: (X 1 , Y 1 ),... , (Xn, Yn) ∼ p(x, y), IID. We are not assuming the

Xi ’s are independent of the Yi ’s as we did with unpaired data.

• (^) E Xi = μ 1 , EYi = μ 2.
• (^) Define the within-pair differences D 1 = X 1 − Y 1 ,... , Dn = Xn − Yn.
• (^) Let μD = E Di (we know μD = μ 1 − μ 2 ), and let σ (^) D^2 = Var(Di ).
• (^) Under H 0 : μD = 10 (the same null as with unpaired data), and substituting

in the sample variance, we get

T =

D¯ − 10

SD/

n

≈ N ( 0 , 1 )

for large samples. For small samples, T ∼ tn− 1 , once we add the assumption that each Di is normal.

• (^) As before: HA : μD 6 = 10 ⇒ reject when |T | > cα

HA : μD > 1 0 ⇒ reject when T > cα HA : μD < 1 0 ⇒ reject when T < cα.

• (^) As before, set cα using N ( 0 , 1 ) in large samples and tn− 1 in small samples.

Collect paired or unpaired data?

• (^) We can be more precise about the benefits of paired data.
• (^) Just like Var( X¯) = σ (^) X^2 /m and Var( Y¯ ) = σ (^) Y^2 /n, we know that Var( D¯) = σ (^) D^2 /n.
• (^) But σ (^) D^2 = Var(Di ) = Var(Xi − Yi ) = σ (^) X^2 + σ (^) Y^2 − 2 · Cov(Xi , Yi ).
• (^) So: when Xi and Yi are positively correlated, Var( D¯) is smaller than when

they are uncorrelated.

• (^) Very often, the within-pair correlation is positive, rather than negative. In a

paired analysis, the positive correlation is reflected in S^2 D, which usually turns out smaller than S^2 X + S Y^2.

• (^) However: with n Xi ’s and n Yi ’s, we have 2 n unpaired obsvns, but only n pairs. So the reference t distrn in a small sample will have 2 n − 1 obsvns for unpaired data, which is better than n − 1 for paired data.
• (^) Despite this, for moderate n, the (usually) smaller SE of D¯ leads to

increased testing power via a larger T statistic, and narrower CIs.

• (^) If Xi and Yi are negatively correlated in a paired dataset, you still need to

do a paired analysis, but you would have been better off without pairing!

Inference about two population proportions

• (^) Recall the one-sample setup: p is the proportion of “successes” in a

population, i.e. the fraction of the population possessing some characteristic.

• (^) Now we have two populations: the success fraction in population “A” is p 1 ,

and in “B” it is p 2.

• (^) We draw m times independently from population “A”, and call X the number

of successes in the sample. Similarly, we make n independent draws from population “B”, and call Y the number of successes.

• (^) Then X ∼ Bin(m, p 1 ) and Y ∼ Bin(n, p 2 ). We assume the two samples are

drawn independently, so X is independent of Y.

• (^) Example: in 1954, a large-scale randomized controlled experiment was

conducted to study Salk’s polio vaccine. Randomizing a child to the control (placebo) group is like drawing from a population having probability p 1 of “success” (contracting polio). Randomizing a child to the treatment (vaccination) group is like drawing from a population with probability p 2 of contracting polio.

• (^) In 1954, people were very interested in p 1 − p 2.

• (^) If p 1 = p 2 = p, then X ∼ Bin(m, p), and, independently, Y ∼ Bin(n, p). But

then X + Y ∼ Bin(m + n, p).

• (^) So a natural estimator of p under H 0 : p 1 = p 2 = p is just

p ˆ = (X + Y )/(m + n).

• (^) This results in the large-sample test statistic

Z =

pˆ 1 − ˆp 2 √ pˆ( 1 − ˆp)( 1 /m + 1 /n)

and the procedures HA : p 1 − p 2 6 = 0 ⇒ reject when |Z | > zα/ 2 HA : p 1 − p 2 > 0 ⇒ reject when Z > zα HA : p 1 − p 2 < 0 ⇒ reject when Z < −zα.

Two population proportions: large-sample CIs

• (^) With both m and n large, and substituting sample proportions in the

denominator,

Z =

pˆ 1 − ˆp 2 − ( p 1 − p 2 ) √ pˆ 1 ( 1 − ˆp 1 )/m + ˆp 2 ( 1 − ˆp 2 )/n

≈ N ( 0 , 1 ).

• (^) Try this at home: write down the 100 ( 1 − α)% pivoting confidence statement

for p 1 − p 2 , using Z. Rearrange it to obtain the confidence interval

p ˆ 1 − ˆp 2 ± zα/ 2

pˆ 1 ( 1 − ˆp 1 ) m

pˆ 2 ( 1 − ˆp 2 ) n

• (^) A suggested correction when m and n are not huge: replace pˆ 1 = X/m with

p ˜ 1 = (X + 1 )/(m + 2 ) and pˆ 2 with the analogous p˜ 2. This can improve the quality of the normal approximation, hence the correctness of the CI.

• (^) When m or n is quite small? A different approach is needed, which we won’t cover. (Take more statistics courses.)