Comparison of 2 or more means

Acknowledgement: Thanks to Professor Pagano (Harvard School of Public Health) for lecture material

e.g. n=16, df=15, 0.05 points are ±2.13

Pr 2.131 2.131 0.95S/ nX mì ü-ï ïï ï- £ £ =í ýï ïï ïî þ

Is = 0 ? -- considerx

ts/ n

m0-= versus

ors s

C.I. : x 2.131 , x 2.13116 16

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One sample t test

Population (Normal)

T-treatment C-control

T C

T C

Sample

nT nC

sT sC

H0 : T = C

Two sample t

1.Samples dependent (Paired)

2.Samples independent

Two situations:

Example – Paired study

• Each subject is tested under 2 conditions: – Time to angina when exposed to plain

air

– Time to angina when exposed to air + CO

Perc

ent d

ecre

ase

.5 1 2 2.5

-30

-20

-10

0

10

20

30

40

Plain air Carbon monoxide

Example – Paired study

Plain air

Carbon monoxide

Example – Paired study

Perc

ent d

ecre

ase

.5 1 2 2.5

-30

-20

-10

0

10

20

30

40

Plain air Carbonmonoxide

Example – Paired study

Example – Paired study

• Response Error Model for a Subject:

sk s skY y E

At time 1

1 1sk s skY y E

At time 2

2 2sk s skY y E

Measured under same conditions!

Example – Paired study

• Response Error Model for a Subject:

sk s skY y E

At time 1

1 1sk s skY y E

At time 2

2 2sk s s skY y E

Measured under different conditions!

s = The condition effect

Example – Paired study

• Take Difference

-At time 1 1 1sk s skY y E

At time 2 2 2sk s s skY y E

Take Sample of Subjects, Test whether

0d

2 1 2 1

2 1

*

sk sk s s sk s sk

s s sk sk

s s s

Y Y y E y E

D E E

D E

So

Look at the differences:

So treat the d’s as the data and perform a one-sample t-test:

T-test

Average change in time to angina = -6.63SD of change in time to angina = 20.29 Calculate p value for H0: μ=0

n= 63

Example - Paired study (Hypothesis test)

Example - Paired Study (One sided CI)

Obtain an UPPER BOUND for mean difference

In order to decide the s12 and S1

2 and the degrees of freedom we need to know whether , or not, T = C

For two independent samples:

If T C (recommended)

and degrees of freedom, :

Heteroscedastic

If T = C (which can be tested)we can use a common value:

Homoscedastic

Two samples (groups):

Treatment Control 4 7 6 6 2 9 5 10

Example

Stata Output paired t

. ttest var1 = var2

Paired t test

------------------------------------------------------------------------------

Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]

---------+--------------------------------------------------------------------

var1 | 4 4.25 .8539126 1.707825 1.532469 6.967531

var2 | 4 8 .9128709 1.825742 5.094837 10.90516

---------+--------------------------------------------------------------------

diff | 4 -3.75 1.493039 2.986079 -8.501518 1.001518

------------------------------------------------------------------------------

Ho: mean(var1 - var2) = mean(diff) = 0

Ha: mean(diff) < 0 Ha: mean(diff) ~= 0 Ha: mean(diff) > 0

t = -2.5117 t = -2.5117 t = -2.5117

P < t = 0.0434 P > |t| = 0.0868 P > t = 0.9566

Stata Output unpaired t

. ttest var1 = var2, unpaired

Two-sample t test with equal variances

-----------------------------------------------------------------------------

Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]

---------+--------------------------------------------------------------------

var1 | 4 4.25 .8539126 1.707825 1.532469 6.967531

var2 | 4 8 .9128709 1.825742 5.094837 10.90516

---------+--------------------------------------------------------------------

combined | 8 6.125 .9149063 2.587746 3.96159 8.28841

---------+--------------------------------------------------------------------

diff | -3.75 1.25 -6.80864 -.6913601

------------------------------------------------------------------------------

Degrees of freedom: 6 Ho: mean(var1) - mean(var2) = diff = 0

Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0

t = -3.0000 t = -3.0000 t = -3.0000

P < t = 0.0120 P > |t| = 0.0240 P > t = 0.9880

Stata Output unpaired unequal

. ttest var1 = var2, unpaired unequal

Two-sample t test with unequal variances

------------------------------------------------------------------------------Variable | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval]---------+-------------------------------------------------------------------- var1 | 4 4.25 .8539126 1.707825 1.532469 6.967531 var2 | 4 8 .9128709 1.825742 5.094837 10.90516---------+--------------------------------------------------------------------combined | 8 6.125 .9149063 2.587746 3.96159 8.28841---------+-------------------------------------------------------------------- diff | -3.75 1.25 -6.811938 -.6880619------------------------------------------------------------------------------Satterthwaite's degrees of freedom: 5.97345

Ho: mean(var1) - mean(var2) = diff = 0

Ha: diff < 0 Ha: diff ~= 0 Ha: diff > 0 t = -3.0000 t = -3.0000 t = -3.0000 P < t = 0.0121 P > |t| = 0.0241 P > t = 0.9879

Summary

• Paired test• Hypothesis test• CI

• 2 independent samples:– Hypothesis test for equal/unequal

variance– CI under equal/unequal variance