dependent t-tests

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dependentt-tests
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Factors affecting statistical power in the t-test

• Statistical power
• ability to identify a statistically significant
  difference when a difference between means
  actually exists

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Decision Table Correct
DECISION
R E A L I T Y
Truth is everlasting, but our ideas about truth
are interchangeable
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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?

Power
0.01 conservative
0.10 liberal
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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)

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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.
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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

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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

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Estimated Standard Error of the Difference
between 2 independent means
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t-test for independent samples
Smaller is better
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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?

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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

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Comparing paired (correlated) measures instead of
group (uncorrelated) measures

• Match subjects
• Repeated measures

Subject serves as own Control
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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
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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

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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)

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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???

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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

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GRF data
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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

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Figure 1. Scattergram of vertical GRF during
landing in different brace conditions (N/kg)
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Descriptive statistics for atble401.sav data
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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

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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

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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

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t-test for dependent (paired) samples
Mdiff
t
SEMD
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GRF data
? -20 Mean Diff -2.9
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t-test for dependent (paired) samples
Mdiff
t
SEMD
Standard error of the Mean difference for Paired
Scores
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Estimated Standard Error of the Difference
between 2 dependent means
?
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Estimated Standard Error of the Difference
between 2 dependent means
If r 0, this term reduces to the same equation
as for independent groups
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t-test for dependent (paired) samples
Mdiff
t
SEMD
df ??
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t-test for dependent (paired) samples
Mdiff
t
SEMD
df npairs - 1
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Running the dependent t-test with SPSS

• Enter the data as pairs
• atble401.sav

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Reporting paired t-test outcome
Table 1. Descriptive statistics of vertical
ground reaction force (in N/kg) for the two
conditions (n 7)
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Reporting t-test outcome

Figure 1. Mean vertical GRF in the two
conditions ( p ? 0.05)
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Reporting t-test in text
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 (? 0.05) 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.
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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 ...
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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
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Statistics Humour
A student set forth on a quest To learn which of
the worlds 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
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Summary both t-tests are of the form
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To increase statistical power
Maximize
Minimize
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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

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Time for Lunch