1

Testing Statistical Hypothesisfor Dependent Samples

2

Testing Hypotheses about Two Dependent Means

Dependent Groups t-test Paired Samples t-test Correlated Groups t-test

3

Steps in Test of Hypothesis

1. Determine the appropriate test 2. Establish the level of significance:α3. Determine whether to use a one tail or

two tail test4. Calculate the test statistic5. Determine the degree of freedom6. Compare computed test statistic against

a tabled/critical value

Same as Before

4

1. Determine the appropriate test

1. When means are computed for the same group of people at two different points in time (e.g., before and after intervention)

2. When subjects in one group are paired to subjects in the second group on the basis of some attribute. Examples:

Husbands versus wives First-born children versus younger siblings AIDS patients versus their primary caretakers

5

1. Determine the appropriate test

3. Researchers sometimes deliberately pair-match subjects in one group with unrelated subjects in another group to enhance the comparability of the two groups.

For example, people with lung cancer might be pair-matched to people without lung cancer on the basis of age, education, and gender, and then the smoking behavior of the two groups might be compared.

6

Example: Two Interventions in Same Patients

Suppose that we wanted to compare direct and indirect methods of blood pressure measurement in a sample of trauma patients. Blood pressure values (mm Hg) are obtained from 10 patients via both methods: X1 = Direct method: radial arterial catheter X2 = Indirect method: the bell component of

the stethoscope

7

2. Establish Level of Significance

α is a predetermined value The convention

α = .05 α = .01 α = .001

8

3. Determine Whether to Use a One or Two Tailed Test

H0 : µD = 0

Ha : µD 0

Meanof

differencesacross patients

Two Tailed Test if no direction is specified

9

3. Determine Whether to Use a One or Two Tailed Test

H0 : µD = 0

Ha : µD 0

One Tailed Test if direction is specified

10

4. Calculating Test StatisticsAverage of

differences

Stan

dard

Er

ror o

f d

iffer

ence

s

Standa

rd

Deviat

ion of

differ

ence

s

Sample sizeHow to calculate

standard deviationof differences

11

Defining Formula Calculating Formula

4. Calculating Test Statistics

12

4. Calculating Test Statistics

Obs

erva

tion

s1

and

2 on

sam

e pa

tien

t

Sq

uar

edd

iffe

ren

ces

Dif

fere

nce

of

obse

rvat

ions

13

4. Calculating Test Statistics

Calculate

totals

14

4. Calculating Test Statistics

15

Calculate t-statistic from average of differences and standard error of differences

90.1

68.0

3.1 t68.0

n

SD S

S tc cD

D

D

4. Calculating Test Statistics

16

5. Determine Degrees of Freedom

Degrees of freedom, df, is value indicating the number of independent pieces of information a sample can provide for purposes of statistical inference.

Df = Sample size – Number of parameters estimated

Df for paired t-test is n minus 1

17

6. Compare the Computed Test Statistic Against a Tabled Value

α = .05 Df = n-1 = 9

tα(df = 9) = 2.26 Two tailed

tα(df = 9) = 1.83 One tailed

Reject H0 if tc is greater than tα

18

Alternative Approach

Estimating Standard deviation of differences from sample

standard deviations

19

Variance / Covariance matrix

S21 S12

S12 S22

X1 X2

X1

X2

Variance ofthe first measure

Variance ofthe second measure

Co-varian

ce of

Measures

of 1 an

d 2

))((

))((

211212

21

1212

SSrS

SS

Sr

Correlat

ion of

measu

res 1 an

d 2

20

Variance / Covariance matrix

X1 X2

S21

S22

X1

X2 ))(( 2112 SSr

))(( 2112 SSr

Standard error of differencecan be calculated from above table

21

Alternative Approach for Calculating standard ErrorStandard error of

Differences

Var

ianc

e of

s

econ

d m

easu

re

Sta

ndar

d de

viat

ion

of s

econ

d m

easu

re

Var

ianc

e of

Fir

st m

easu

re

Sta

ndar

d de

viat

ion

of F

irst

mea

sure

Cor

rela

tion

betw

een

two

mea

sure

s

22

Correlation Matrix

Direct IndirectDirect Pearson Correlation 1 .996(**) Sum of Squares and

Cross-products 4496.100 4611.00 Covariance 499.567 512.333 N 10 10

Indirect Pearson Correlation .996(**) 1 Sum of Squares and

Cross-products 4611.00 4768.00 Covariance 512.333 529.778 N 10 10

23

Alternative Approach for Calculating standard Error

0.68

10

554.4

10

)017.23)(351.22)(996.0(2778.529567.499

n

))()((2

21

21

21

21

x- x

x- x

x- x

21122

22

1x- x

S

S

S

SSrSSS

Same value as before

24

Paired Samples Statistics

Direct Method129.30 10 22.351 7.068

Mean NStd.

DeviationStd. Error

Mean

Pair 1

Indirect Method

128.00 10 23.017 7.279 Paired Samples Correlations

N Correlation Sig.

Pair 1

Direct Method & Indirect Method

10 .996 .000

 Paired Samples Test

Paired Differences tdf

Sig. Level(p-value)

MeanStd.

DeviationStd. Error

Mean95% Confidence Interval

of the Difference

Lower Upper

Pair 1

Direct Method - Indirect Method 1.300 2.163 .684 -.247 2.847 1.901 9 .090

SPSS output for Paired Sample t-test 

25

Take Home Lesson

How to compare means of paired dependent samples