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dependentt-tests
Factors affecting statistical power in the t-test
• Statistical power• ability to identify a statistically significant
difference when a difference between means actually exists
Decision Table: Correct
Ho TRUE Ho FALSE
Ho TRUE
Ho FALSE POWER
DECISION
REALITY
Truth is everlasting, but our ideas about truth are interchangeable
Factors affecting statistical power in the t-test
level• how much risk are YOU willing to take in
making a Type I error• Frank & Huck (1986, RQES): Why does
everyone use the 0.05 level of significance?
0.01
conservative
0.10
liberal
Power
Factors affecting statistical power in the t-test
level• df (number of subjects)
• affects variability associated with the sample mean & variability within the sample
• limited by time & money• GREATER n = GREATER POWER
(point of diminishing return)
Statistics Humour One day there was a fire in a wastebasket in the
Dean's office and in rushed a physicist, a chemist,
and a statistician. The physicist immediately
starts to work on how much energy would have to be
removed from the fire to stop the combustion. The
chemist works on which reagent would have to be
added to the fire to prevent oxidation. While they are
doing this, the statistician is setting fires to all the other
wastebaskets in the office. "What are you doing?" they
demanded. "Well to solve the problem, obviously you
need a large sample size" the statistician replies.
Factors affecting statistical power in the t-test
level• df (number of subjects)• magnitude of the mean difference
• how different are the treatments imposed• measurement errors• sampling errors• SIZE OF THE TREATMENT EFFECT
Factors affecting statistical power in the t-test
level• df (number of subjects)• magnitude of the mean difference• variability
• how specified is your population• control of extraneous variables
Estimated Standard Error of the Difference between 2 independent means
SESEs mymxdm
22
t-test for independent samples
Smaller is better
stdm
obs
YX
Comparing paired (correlated) measures instead of group (uncorrelated) measures
• Match subjects• what factors (variables) might affect time to
exhaustion on the exercise bike• daily diet? Fitness level? Genetics?• Height? Weight? Age?• Regular training program?
Comparing paired (correlated) measures instead of group (uncorrelated) measures
• Match subjects• Repeated measures
• measure the SAME subject under both protocols
• test & retest• pre & posttest• condition 1 & condition 2
Comparing paired (correlated) measures instead of group (uncorrelated) measures
• Match subjects• Repeated measures
Subject
serves as own
Control
Comparing paired (correlated) measures instead of group (uncorrelated) measures
• Match subjects• Repeated measures
Subject serves as own Control
Intra-subject variability
should be LESS than
Inter-subject variability
Dependent t-test(paired or correlated t-
test)
• Pairs of scores are matched• same subject in 2 conditions or matched
subjects
• Question: Does ankle bracing affect load during landing?• IV: brace condition• DV: Vertical GRF
Steps to dependent t-test
• Set (0.05)• Set sample size
• One randomly selected group• n = 7
• condition 1: Brace• condition 2: No brace
• Set Ho (null hypothesis)
Set statistical hypotheses
• Ho
• Null hypothesis• Any observed
difference between the two conditions will be attributable to random sampling error.
• HA
• Alternative hypothesis
• If Ho is rejected, the difference is not attributable to random sampling error
• perhaps brace???
Steps to independent t-test
• Set (0.05)• Set sample size (n = 7)• Set Ho
• Test each subject in both conditions with a standardized protocol (drop landings)• Note: condition performance order is
randomized across subjects
GRF data
Steps to dependent t-test
• Set (0.05)• Set sample size (n = 7)• Set Ho
• Test each subject in both conditions• Calculate descriptive statistics of each
condition• scattergram• mean, SD, n
No Brace
1614121086420
An
kle
Bra
ce
20
18
16
14
12
10
8
6
Figure 1. Scattergram of vertical GRF during
landing in different brace conditions (N/kg)
Descriptive statistics for atble401.sav data
Group n Mean SD
No brace 7 8.0 4.3
brace 7 10.9 3.5
Steps to dependent t-test
• Set (0.05)• Set sample size (n = 7)• Set Ho
• Test each subject in both conditions• Calculate descriptive statistics of each
condition• compare the condition means
How to compare the condition means
• Even if the two conditions were the same (samples drawn from the same population), would not expect the statistics to be the same
• Need a measure of expected variability against which the mean of the difference between paired scores (Xi - Yi) could be compared
Paired scores, so the data are somewhat correlated
• Calculate the difference between the two conditions for each case (Xi - Yi)
• Calculate the Mean Difference• Use the correlation among the pairs of
scores to reduce the error term (denominator) used to evaluate the difference between the means
t-test for dependent (paired)
samples
t =
Mdiff
SEMD
GRF data
Subject No brace Brace X - Y1 5 8 -32 10 12 -23 11 10 14 6 9 -35 15 18 -36 7 11 -47 2 8 -6
= -20
Mean Diff = -2.9
t-test for dependent (paired)
samples
t =
Mdiff
SEMD
Standard error
of the
Mean difference for Paired Scores
Estimated Standard Error of the Difference between 2 dependent means
SESESESEs mymxdmrmymx 2
22
?
Estimated Standard Error of the Difference between 2 dependent means
SESESESEs mymxdmrmymx 2
22
If r = 0, this term reduces
to the same equation as
for independent groups
t-test for dependent (paired)
samples
t =
Mdiff
SEMD
df = ??
t-test for dependent (paired)
samples
t =
Mdiff
SEMD
df = npairs - 1
Running the dependent t-test with SPSS
• Enter the data as pairs • atble401.sav
Reporting paired t-test outcome
Group n Mean SD
No brace 7 8.0 4.3
brace 7 10.9 3.5
Table 1. Descriptive statistics of vertical ground reaction
force (in N/kg) for the two conditions (n = 7)
Reporting t-test outcome
0
5
10
15
20
No Brace Braced
Brace Condition
Ve
rtic
al G
RF
(N/k
g)
*
Figure 1. Mean vertical GRF in the two conditions
(* p 0.05)
Reporting t-test in textDescriptive statistics of the vertical ground reaction force
(VGRF) data during landing in the two braced conditions
are presented in Table 1 and graphically in Figure 1. A
paired t-test indicated that the mean VGRF of 10.9
(SD = 3.5) N/kg in the braced condition was significantly
higher ( = 0.05) than the mean VGRF of 8.0 (4.3) N/kg
in the unbraced condition (t6 = 3.57, p = 0.012). The
mean difference of 2.9 N/kg represents a 36% higher
VGRF during the landings with a brace compared to
without a brace.
What if you set = 0.01?
Descriptive statistics of the vertical ground reaction force
(VGRF) data during landing in the two braced conditions
are presented in Table 1 and graphically in Figure 1. A
paired t-test indicated that the mean VGRF of 10.9
(SD = 3.5) N/kg in the braced condition was ...
What if you set = 0.01?Descriptive statistics of the vertical ground reaction force
(VGRF) data during landing in the two braced conditions
are presented in Table 1 and graphically in Figure 1. A
paired t-test indicated that the mean VGRF of 10.9
(SD = 3.5) N/kg in the braced condition was significantly
higher ( = 0.01) than the mean VGRF of 8.0 (4.3) N/kg
in the unbraced condition (t6 = 3.57, p = 0.012). The
mean difference of 2.9 N/kg represents a 36% higher
VGRF during the landings with a brace compared to
without a brace.
not
Statistics Humour
A student set forth on a quest
To learn which of the world’s beers was best
But his wallet was dried out
At the first pub he tried out
With two samples he flunked the means test
Gehlbach, SH (2002)
Interpreting the medical literature
Summary: both t-tests are of the form:
t = Standard Error
Mean Difference
To increase statistical power
t = Standard Error
Mean Difference
Maximize
Minimize
Choosing which t-test to use
• Independent• no correlation
between the two groups
• Dependent• two sets of data (pair
of scores) from matched subjects or from the same subject (repeated measures)
• data are correlated
Time for Lunch