Two-Sample T-Test vs. Paired T-Test: Analyzing Differences in Means with Dependent Data - , Study notes of Data Analysis & Statistical Methods

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Two-Sample Inference for Location Parameters II

FCSM Chapter 6 Dependent samples: 6.4Sample size: 6.

Two Sample Methods

BTRY 6010 & ILRST 6100

Example:

Growth Hormones

Example:

Growth Hormones

The paper “Growth Hormone Increase DuringSleep After Daytime Exercise” (1974,

J. of

Endocrinology, 473-

) reports results of an

experiment involving 6 healthy male subjectsexperiment involving 6 healthy male subjects.Blood samples were drawn from each participantd^ i^

l^

t^ diff

t^

i ht^

i

during sleep on two different nights, using avenous catheter.Night #1: no strenuous exercise the day before;Night #2: strenuous exercise the day before.

BTRY 6010 & ILRST 6100

Two Sample Methods

Two^ sample

t^ test shows no Two-sample

t^ test shows no difference, whether or notwe assume equal variances:( p^ = 0 2183

p^ = 0 2156) ( p^ 0.2183,

p^ 0.2156). Right?? What other assumptions

BTRY 6010 & ILRST 6100

Two Sample Methods

p have we made???

W^ !!

C^ i

i^

d h^

b^

th

^ Wrong!!

Care is required here because the

observations within each subject are certainlydependent – the observations from the exercisedependent

the observations from the exercise

“groups” (night #1 vs night #2) being compared arefrom the same subjects, thus far from independent.  In fact: notice that there is an increase in hormonelevel from night #1 to night

within^

each subject.

^ Relationship between observations within a subjectmust be accounted for in the analysis. ^ Appropriate analysis in this situation:a^ paired t-test

or^ matched pairs t-test

BTRY 6010 & ILRST 6100

Two Sample Methods

Basic idea:Basic

idea:

^ Suppose we have

2n^ observations that can be

sensibly paired with each other (e.g., pre- vspost-tests, measurements on both eyes,measurements on twins etc

measurements on twins, etc…)  Let^ y^ represent the ij^

th^ j observation for the

th^ i pair,

h^ i^

d^ j 1 2

where^

i=1,…,n

and^ j

^ Consider the difference in sample means:

1 2

1

2 1

1 1

n^

n i^

i i^

i

y^ y^

y^

y n^

n ^

^ ^

^ ^

^

^ ^

^

^ ^

^

BTRY 6010 & ILRST 6100

Two Sample Methods

1

1 i^

i n^

n ^

 ^

^ ^

Using a little bit of algebra: Using^ a little bit of algebra:

^

 1

n^

n

y^ y^

d y^ y^

d

^

^

 2 1 2

1

1 (^1) i i^ i

i i^ d

i

y^ y^

n^

d n y^ y^

d ^



^

^

^ ^

^ With paired data: a difference between sampleaverages can be

naturally

expressed as the

average of the differences between theaverage of the differences between theobservations on the same pair.A^

th t di ti

t^ i

i d

d^

t Th

^ Assume that distinct

pairs^ are independent. Then

a paired t-test amounts to using a standard one-sample t-test on the differences

dd^1

d

sample t test on the differences

d, d^ , …, d^12

. n^

BTRY 6010 & ILRST 6100

Two Sample Methods

^ Compare

Difference

(^

) vs.^ Mean

(^ ) on

y^ y ^

d

^ Compare

Difference

(^

) vs.^ Mean

(^ ) on

Slides 4 & 9 – these are equal (slide 8).  But:^ Std Err Dif

and^ Std Err Mean

on Slides 4 &

y^ y^^1

d

^ But:^

Std^ Err Dif

and^ Std

Err Mean

on^ Slides 4 &

9 are actually quite different: 3.303 vs. 1.346Note: one-sample t-test is performed using

d^ d^

d^ ; so

Note: one-sample t-test is performed using

d, d, …, d^1

;^ so n (^) 2 1

=^ , whe

1 r^

( 1 e^

n ) d

d^

i i

Std Err Me

s^

s^

d^ dn

a^ n^ n

^

 

^ Accounting for pairing here leads to a 2.5-foldreduction in variance (compared to usual two

(^1) n^1 i n^



reduction in variance (compared to usual two-sample t-test). The resulting test statistics are very^ different: 1.32 vs. 3.24 very^ different: 1.32 vs. 3.

BTRY 6010 & ILRST 6100

Two Sample Methods

Aside:

Var( X

-^ Y )^

for dependent

X & Y

Aside:

Var(^ X

Y )^

for^ dependent

X^ & Y

^ Let^ X

&^ Y be two random variables.

In general: the

variance of

X^ Y^

or^ Var(X

Y)^ is given by

variance of

X^ – Y , or

Var(X^

- Y),^ is

given by

Var( X – Y ) = Var(X) + Var(Y) – 2 Cov( X,Y ) where^ Cov( X ,Y )

denotes covariance of

X^ &^ Y.

^ We say

X^ &^ Y^

are uncorrelated if

Cov( X ,Y )

y^

(^ )

correlated if

Cov( X ,Y )

^0

positively correlated if

Cov( X ,Y )

negatively correlated if

Cov( X ,Y )

^ Independence of

X^ &^ Y^

implies

Cov( X ,Y )

p^

p^

(^ ,^ )

(but not

vice versa

) BTRY 6010 & ILRST 6100

Two Sample Methods

Result: Exact Sampling Distribution of

p^ g

Normally Distributed DifferencesL t^ b^

th^

l^

diff^

f t

Let^ be the sample mean difference of two D observations taken on a SRS of

n^ independent

pairs. Assume the differences follow a

N (^ ,^  dd

D^ n p

(^ ,^ ) dd^

distribution. Then: n^ d ~ D^  t

If^30

Dn^ d n^

 N

^

. (^1) n t sd n

If 

30,^

d n^

N

 s n The result above forms the basis for CIs andhypothesis tests in a paired data setting. Two Sample Methods

BTRY 6010 & ILRST 6100 hypothesis tests in a paired data setting.

Testing:

paired data,

^ unknown

Testing:

paired

data,

^ unknown *^

0

Test Statistic:

d^  dt  sd n As usual, we have three possible sets of hypotheses:

n

(i)^ H :^ ^0

≤^  vs Hd d^

:^  >^  a d^ d

^ RR is

*^ t > t^ n-1,^ ^

& p = P (^

t^ > t*^ ) n-^

(ii) H :^ ^0

≥^  vs Hd d^

:^  <^  a d^ d

^ RR is

*^ t < - t^ n-1,

& p = P  (^ t^ < t*^ ) n-^

0 d^ d

a^ d^ d^

n 1,^ ^

n 1

(iii) H^ :^ ^0

=^  vs Hd d^

:^ ^   add^

^ RR is^

*  t | > t^ n-1,^ 

& p = 2P (^ t^ > |t*|n-^

)

Usual comments apply: If

n^ ≥^ 30, we

can use normal critical points in

place of^ t.

If the sample size is very small and normality of differences is suspect, use the t-test with skepticism or use nonparametricTwo Sample Methods

14

BTRY 6010 & ILRST 6100 p^ ,^

p^

p

methods (Wilcoxon signed-rank).

Example:

Growth Hormones

Example:

Growth Hormones

Obligatory check: no “obvious” deviations from normality

BTRY 6010 & ILRST 6100 Obligatory check: noTwo Sample Methods

obvious

deviations from normality.

Example:

Fertilized Tomatoes

Example:

Fertilized Tomatoes

Does a new fertilizer improve tomato yield?Concerns about impact of differences in soil type,light, moisture, … led to following experiment: ^ 30 plots of tomato plants, 10 plants per plot. ^ New & existing fertilizer applied within each plot,

g^

pp^

p^ ,

5 plants each, order randomized. Average yieldfor each fertilizer is computed within each plot.  Should we use a paired or unpaired t-test toevaluate difference in yields (in pounds)?

BTRY 6010 & ILRST 6100

Two Sample Methods

Comments:Comments: ^ Pairing

( matching

) can be useful in observational studies as a way to control for

confounding

: the impact of

as a way to control for

confounding

: the impact of

measured (and unmeasured) variables that may beassociated with both “response” and “group” variable. Itt

d^

t^ ll d^

f^ i bilit

serves to reduce uncontrolled sources of variability.  Pairing^

is an example of

blocking

, a term originating from

experimental design

Blocking serves to reduce impact experimental design. Blocking serves to reduce impactof uncontrolled sources of variability on a comparison oftreatments (e.g., new vs. existing fertilizer; post- vs. pre-i^

) b^ fi^ t

ti^

bl^ k^ (i

f^ i^ il

exercise) by first creating

blocks^ (i.e., groups of similar,

or relatively homogeneous, units, such as tomato plants,plots, or subjects) and then assessing treatment effects

j^ )^

g

by using “within-block” differences.

BTRY 6010 & ILRST 6100

Two Sample Methods

^ Settings involving pairs of measurements represent a^ Settings

involving pairs of measurements represent a special case of the more general problem of

repeated

measurements

(two or more measurements per it/bl^ k)

S^ h d t

i^ i^

lti l

unit/block). Such data can arise in multiple ways, e.g.,multiple treatments per block, longitudinal data on eachsubject, and so on.  As in the paired setting, one expects the measurementson one unit/block to be more correlated with each otherth^ ith

t^ diff

t^

it /bl^ k

than with measurements on different units/blocks.  Methods of analysis must deal with the various levels ofcorrelation that may exist; otherwise one can easilycorrelation that may exist; otherwise, one can easilyobtain incorrect assessments of sampling variability,leading to impaired statements of statistical significanced/^

fid^

l^ l^ (^

th^

bl^ )

and/or confidence levels (among other problems).

BTRY 6010 & ILRST 6100

Two Sample Methods