Independent samples t-test · 2012-03-27 · Bigger difference – larger t-test Size of sample...
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Two-Group Designs Independent samples t-test & paired samples t-test Chapter 10
Transcript of Independent samples t-test · 2012-03-27 · Bigger difference – larger t-test Size of sample...
Two-Group Designs Independent samples t-test
& paired samples t-test
Chapter 10
Previous tests (Ch 7 and 8)
Z-test t-test
(one-sample)
N
Mz
N
s
Mt
M = standard error of the mean
p. 198-9
sM = estimated standard error
p. 211-2
Remember: s2 = variance n
ssM
2
n
ssM
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:
Total score ≠ 0
t-test example
Lateralization study (N = 173); 2-tailed hypothesis
H0: totscore = 0; H1: totscore ≠ 0
Totscore M = -11.7, SD = 16.6, S2 = 276
t-test: -11.7 – 0 = -11.7 = - 9.28
sqrt(276/173) 1.26
-9.28?? Look up tcv or t* in table…
df = 172, but use df = 120:
tcv = 1.98 @ p = .05; tcv = 2.617 @ p = .01; tcv = 3.373 @ p = .001
Reject the null hypothesis
t(172) = -9.28, p < .001
95% CI = M +/- t*(sM) = -11.7 +/- 1.98 (1.26) = -9.20 to -14.19
N
s
Mt
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
expected by chance (M = 0).
New parametric tests (Ch 10)
Independent-samples t-test
Paired-samples t-test
)(
21
21
)(
mms
MMt
DM
DD
s
Mt
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
Independent t-test
Independent-measures or between-subject design
Null hypothesis
H0: µ1 - µ2 = 0 (or µ1= µ2)
Alternative hypothesis
H1: µ1 - µ2 ≠ 0 (or µ1 ≠ µ2 or µ1 < µ2 or µ1 > µ2)
t-test: double elements of single t-test formula
Compare mean difference (top) with difference
expected by chance (bottom)
ms
Mt
)(
2121
21
)()(
mms
MMt
= 0
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 n1 ≠ n2, need to calculate pooled variance
Let SPSS do it!
n
ssM
2
2
2
2
1
2
1)( 21 n
s
n
ss MM
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 variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
DRWEEK
NDRWEEK
PDRWEEK
NPDRWEEK
F Sig.
Lev ene's Test for
Equality of Variances
t df Sig. (2-tailed)
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
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
# d
rin
ks
Calculations
Calculate top:
Calculate bottom:
Calculate t:
Calculate df = (n1 – 1)+(n2-1)
Look up critical t*
Does t value exceed critical value?
2
2
2
1
2
1)( 21 n
s
n
ss MM
)(
21
21
)(
mms
MMt
)( 21 MM
)(
21
21
)(
mms
MMt
t table
For independent
t-test
df = (n1 – 1)+(n2-1)
Does the
calculated t at or
above the t*?
Given df and α
level selected
If yes, then
“significant”
difference
between means
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
Step-by-Step Calculation
1. Information given for problem:
Mm = 14.9, s2m = 124.7, nm = 89
Mf = 9.6, s2f = 124.7, nf = 94
2. Calculate s(m1-m2) SE of M for the difference
124.7 124.7
89 94SE 1.65
3. Calculate the t-statistics
14.9 9.6 5.3
1.65 1.65t
= 3.2
Find t* or tcv
Go to the t table!
)(
2121
21
)()(
mms
MMt
4. Make a decision
t table
df = df1+df2
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
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)
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 (r2) 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 = #.##.”
22
2
2
2
1
21
ss
MMd
dft
tr
2
22
Effect sizes: Alcohol data
t(181) = 3.23, p < .001, d = 0.47
While statistic is significant, effect size is
small.
22
2
2
2
1
21
ss
MMd
475.0
2
7.124
2
7.124
6.99.14
d
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 = M1 – M2 +/- t* (sm1-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.
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”
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 variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
Equal variances
assumed
Equal variances
not assumed
DRWEEK
NDRWEEK
PDRWEEK
NPDRWEEK
F Sig.
Lev ene's Test for
Equality of Variances
t df Sig. (2-tailed)
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
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
2
N
MXs
2
2
2
1
2
1)( 21 n
s
n
ss MM
)(
21
21
)(
mms
MMt
means
variance
se of difference
mean
diff
t-test
x - M x - M x-M2 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)
L M N O P Q
