Two-Group Designs Independent samples t-test & paired samples t-test. Chapter 10

(1)

Two-Group Designs

Independent samples t-test

& paired samples t-test

(2)

(3)

Previous tests (Ch 7 and 8)

Z-test

t-test

(one-sample)

N

M

z









N

s

M

t







M

=

standard error of the mean

p. 198-9

s

M

= estimated standard error

p. 211-2

Remember: s

2

= variance

n

s

M

2



n

s

M



(4)

Lateralization effect: Perception of Emotion

Two chimeric faces – which

one is happier (or younger

example)?

Emotion is processed in the right

hemisphere

Dependent variable:

Total score: -36 to +36

where 0 = no lateralization

What are the hypotheses?

Null hypothesis:

Total score = 0

Alternative hypothesis:

(5)

t-test example

Lateralization study (N = 173); 2-tailed hypothesis

H

₀

: totscore = 0; H

1

: totscore ≠ 0

Totscore

M

= -11.7,

SD

= 16.6,

S

2

= 276

t-test: -11.7 – 0

=

-11.7

= - 9.28

sqrt(276/173)

1.26

-9.28?? Look up

t

cv

or t* in table…

df = 172, but use df = 120:

t

cv

= 1.98 @

p

= .05;

t

cv

= 2.617 @

p

= .01;

t

cv

= 3.373 @

p

= .001

Reject the null hypothesis

t

(172) = -9.28,

p

< .001

95% CI = M +/- t*(s

M

) = -11.7 +/- 1.98 (1.26) = -9.20 to -14.19

N

s

M

t







(6)

Lateralization results

Participants demonstrated a statistically

significant lateralization effect,

t

(172) = -9.28,

p

< .05. Emotion was more influenced by the

right hemisphere (

M

= -11.7,

CI

(.95) = -9.20

to -14.19), as opposed to what would be

(7)

New parametric tests (Ch 10)

Independent-samples t-test

Paired-samples t-test

)

(

2

1

2

1

)

(

m

s

M

t





D

M

D

s

M

t







(8)

Do women and men differ in their

self-reported drinking?

SU male students drink…

14.9

13.8

7.9

9.6

SU female students drink…

Are these means different by an amount that is statistically

significant?

X

s

n

89

94

M

(9)

(10)

Independent t-test

Independent-measures or between-subject design

Null hypothesis

H

₀

: µ

1

- µ

2

= 0 (or µ

1

= µ

2

)

Alternative hypothesis

H

1

: µ

1

- µ

2

≠ 0

(or µ

1

≠ µ

2

or µ

1

< µ

2

or µ

1

> µ

2

)

t-test: double elements of single t-test formula

Compare mean difference (top) with difference

expected by chance (bottom)

m

s

M

t







)

(

2

1

2

1

2 1

)

(

)

(

m

s

M

t







= 0

(11)

Standard error of the difference

between the means

2-sample standard error

Each sample has own population mean and error

When = n’s, add standard errors of the means

together

Equation assumes equal sample size

When n

1

≠ n

2

, need to calculate

pooled variance

Let SPSS do it!

n

s

M

2



2

1

2

1

)

(

₁ ₂

n

s

n

s

M



M





(12)

Alcohol Data

Group Statistics 89 14. 9888 13. 79105 1. 46185 94 9. 6489 7. 95377 .82037 89 23. 3933 12. 45229 1. 31994 94 18. 1383 8. 01558 .82674 89 24. 3034 21. 16193 2. 24316 94 15. 5426 13. 32849 1. 37473 89 37. 7978 24. 49916 2. 59691 93 26. 8495 11. 40266 1. 18240 GENDER 1. 00 2. 00 1. 00 2. 00 1. 00 2. 00 1. 00 2. 00 DRWEEK NDRWEEK PDRWEEK NPDRWEEK

N Mean Std. Dev iation

Std. Error Mean

Independent Samples Test

24. 775 .000 3. 230 181 .001 5. 3398 1. 65340 2. 07741 8. 60224 3. 185 139.101 .002 5. 3398 1. 67631 2. 02549 8. 65416 19. 764 .000 3. 413 181 .001 5. 2550 1. 53986 2. 21657 8. 29335 3. 374 148.905 .001 5. 2550 1. 55748 2. 17734 8. 33258 14. 367 .000 3. 370 181 .001 8. 7608 2. 59986 3. 63088 13. 89075 3. 330 146.908 .001 8. 7608 2. 63090 3. 56151 13. 96012 21. 542 .000 3. 892 180 .000 10. 9483 2. 81309 5. 39741 16. 49917 3. 837 123.204 .000 10. 9483 2. 85342 5. 30022 16. 59636 Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed DRWEEK NDRWEEK PDRWEEK NPDRWEEK F Sig.

Lev ene's Test f or Equality of Varianc es

t df Sig. (2-t ailed)

Mean Dif f erence

Std. Error

Dif f erence Lower Upper 95% Conf idence

Interv al of the Dif f erence t-test f or Equality of Means

(13)

Alcohol write-up

Men were found to drink significantly more (

M

=

14.99,

SD

= 13.8) compared to women (

M

= 9.65,

SD

= 7.9),

t

(181) = 3.23,

p

< .001, two-tailed.

0

5

10

15

20

men

women

# drin

ks

(14)

Calculations

Calculate top:

Calculate bottom:

Calculate t:

Calculate df = (n

1

– 1)+(n

2

-1)

Look up critical t*

Does t value exceed critical value?

2

1

2

1

)

(

1

2

n

s

n

s

M



M





)

(

2

1

2

1

)

(

m

s

M

t





)

(

M

1



M

2

)

(

2

1

2

1

)

(

m

s

M

t





(15)

t table

For independent

t-test

df = (n

1

– 1)+(n

2

-1)

Does the

calculated t at or

above the t*?

Given df and α

level selected

If yes, then

“significant”

difference

between means

(16)

Do women and men differ in their

self-reported drinking?

SU male students drink…

14.9

13.8

7.9

9.6

SU female students drink…

Are these means different by an amount that is statistically

significant?

X

s

n

89

94

(17)

Step-by-Step Calculation

1. Information given for problem:

M

m

= 14.9, s

2

m

= 124.7, n

m

= 89

M

f

= 9.6, s

2

f

= 124.7, n

f

= 94

2. Calculate s

(m1-m2)

SE of M for the difference

124.7

89

94

SE







1.65

3. Calculate the t-statistics

14.9 9.6

5.3

1.65

t







= 3.2

Find t* or t

cv

Go to the t table!

)

(

2

1

2

1

2 1

)

(

)

(

m

s

M

t







4. Make a decision

(18)

t table

df = df

1

+df

2

df = 88+93=181

look up df = 120

t* = 1.98 (for .05)

t* = 2.617 (for .001)

t = 3.2

Decision?

Significant!

t

(191) = 3.2,

p

< .001

(19)

Influencing factors

How do we increase chance for significant

result when there really is an effect (POWER!)

Difference between means

Bigger difference – larger t-test

Size of sample variance

Larger variance – smaller t-test

Sample sizes

Larger sample – higher probability of sig t-test (little

influence on effect size)

(20)

Effect size:

Proportion of variance in DV

accounted for by manipulation of IV

Cohen’s d

Small >.2

Medium >.5

Large > .8

Variance accounted for (r

2

)

Small >.01

Medium >.09

Large > .25

Reporting the statistics

“Group X did more (

M

= #,

SD

= #) than group Y (

M

=

#,

SD

= #).”

“The difference was (or was not) found to be

significant,

t

(df) = #.#,

p

= 0.##,

d

= #.##.”

2

1

2

1

s

M

d







df

t

r





2

(21)

Effect sizes: Alcohol data

t

(181) = 3.23,

p

< .001,

d

= 0.47

While statistic is significant, effect size is

small.

2

1

2

1

s

M

d







₀

.

475

2

7

.

124

2

7

.

124

6

.

9

.

14









d

(22)

Confidence Intervals

95% confident that the interval contains the difference

between the mean scores

If interval does NOT contain 0 then suggests likely a true

difference between groups

Independent groups:

CI

.95

= M

1

– M

2

+/- t* (s

m1-m2

)

Where t* is the critical value given df

(at α=.05, 2-tailed)

Alcohol example:

CI = (14.9 – 9.6) +/- 1.98 (1.65) =

5.3 +/- 3.267 = 2.03 to 8.57

Conclusion: 95% confident that the difference between men and

women in the number of drinks consumed per week from 2.03

to 8.57.

(23)

Assumptions of independent t-test

Data are on interval-ratio scale

Observations must be independent

Population distributions must be normal

Two populations must have equal variances

Average variances only if estimating same

population variance

Called “homogeneity of variance”

Important when sample sizes are different

How do you know with samples?

Difference between sample variances

SPSS: “Equal variances (not) assumed”

(24)

Alcohol Data

Group Statistics 89 14. 9888 13. 79105 1. 46185 94 9. 6489 7. 95377 .82037 89 23. 3933 12. 45229 1. 31994 94 18. 1383 8. 01558 .82674 89 24. 3034 21. 16193 2. 24316 94 15. 5426 13. 32849 1. 37473 89 37. 7978 24. 49916 2. 59691 93 26. 8495 11. 40266 1. 18240 GENDER 1. 00 2. 00 1. 00 2. 00 1. 00 2. 00 1. 00 2. 00 DRWEEK NDRWEEK PDRWEEK NPDRWEEK

N Mean Std. Dev iation

Std. Error Mean

Independent Samples Test

24. 775 .000 3. 230 181 .001 5. 3398 1. 65340 2. 07741 8. 60224 3. 185 139.101 .002 5. 3398 1. 67631 2. 02549 8. 65416 19. 764 .000 3. 413 181 .001 5. 2550 1. 53986 2. 21657 8. 29335 3. 374 148.905 .001 5. 2550 1. 55748 2. 17734 8. 33258 14. 367 .000 3. 370 181 .001 8. 7608 2. 59986 3. 63088 13. 89075 3. 330 146.908 .001 8. 7608 2. 63090 3. 56151 13. 96012 21. 542 .000 3. 892 180 .000 10. 9483 2. 81309 5. 39741 16. 49917 3. 837 123.204 .000 10. 9483 2. 85342 5. 30022 16. 59636 Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed Equal v ariances ass umed Equal v ariances not as sum ed DRWEEK NDRWEEK PDRWEEK NPDRWEEK F Sig.

Lev ene's Test f or Equality of Varianc es

t df Sig. (2-t ailed)

Mean Dif f erence

Std. Error

Dif f erence Lower Upper 95% Conf idence

Interv al of the Dif f erence t-test f or Equality of Means

(25)

Chapter 9: Example of

independent t-test

Data in Table 10.1 p.251

Use Excel to calculate t-test

Data

Excel Formulas

t-test formulas

spaced

massed

x1 - M x2 - M x1-M2 x2-M2

23

15

1

-1.9

1

3.61

18

20

-4

3.1

16

9.61

25

21

3

4.1

9 16.81

22

15

0

-1.9

0

3.61

20

14

-2

-2.9

4

8.41

24

16

2

-0.9

4

0.81

21

18

-1

1.1

1

1.21

24

19

2

2.1

4

4.41

21

14

-1

-2.9

1

8.41

22

17

0

0.1

0

0.01

22

16.9

40 56.90

4.44

6.32

5.1

0.44

0.63

1.08

4.9

1.04

1

)

(

2







N

M

X

s

2

1

2

1

)

(

₁ ₂

n

s

n

s

M



M





)

(

2

1

2 1

)

(

m

s

M

t





means

variance

se of difference

mean

diff

t-tes

t

x - M

x-M2

23

15

=L25-22 =M25-16.9 =N25*N25

=O25*O25

18

20

=L26-22 =M26-16.9 =N26*N26

=O26*O26

25

21

=L27-22 =M27-16.9 =N27*N27

=O27*O27

22

15

=L28-22 =M28-16.9 =N28*N28

=O28*O28

20

14

=L29-22 =M29-16.9 =N29*N29

=O29*O29

24

16

=L30-22 =M30-16.9 =N30*N30

=O30*O30

21

18

=L31-22 =M31-16.9 =N31*N31

=O31*O31

24

19

=L32-22 =M32-16.9 =N32*N32

=O32*O32

21

14

=L33-22 =M33-16.9 =N33*N33

=O33*O33

22

17

=L34-22 =M34-16.9 =N34*N34

=O34*O34

=AVERAGE(L25:L34) =AVERAGE(M25:M34)

=SUM(P25:P34)

=SUM(Q25:Q34)

=P36/9

=Q36/9

=L36-M36

=P37/10

=Q37/10

=SUM(P38+Q38)

=L38/P40

=SQRT(P39)