the independent samples t-test
DESCRIPTION
A Classic!. The Independent Samples t-Test. PG-17. Feared by Graduate Students Everywhere!. Independent Samples. Random Selection : Everyone from the Specified Population has an Equal Probability Of being Selected for the study (Yeah Right!) Random Assignment : - PowerPoint PPT PresentationTRANSCRIPT
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Independent Samples
1. Random Selection:Everyone from the Specified Population has an Equal ProbabilityOf being Selected for the study (Yeah Right!)
2. Random Assignment:Every participant has an Equal Probability of being in the TreatmentOr Control Groups
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The Null Hypothesis
•Both groups from Same PopulationNo Treatment Effect
•Both Sample Means estimate Same Population MeanDifference in Sample Means reflect Errors of Estimation of Mu
X-Bar1 + e1 = Mu (Mu – X-Bar1 = e1)X-Bar2 + e2 = Mu (Mu – X-Bar2 = e2)
Errors are Random and hence Unrelated
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Expectation
If Both Samples were selected from the Same Population:
How much should the Sample Means Disagree about Mu?X-Bar1 – X-Bar2
•Errors of Estimation decrease with N•Errors of Estimation increase with Population Heterogeneity
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The Expected Disagreement
The Standard Error of a Difference:SEX-Bar1-X-Bar2
The Average Difference between two Sample MeansThe Expected Difference between two Sample Means
•When they are Estimating the Same Mu•68% chance of this much Or Less•95% chance of (this much x 2) Or Less
Actually this much x 1.96, if you know sigmaRounded up to 2
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Expectation: The Standard Error of the Difference
The Expected Disagreement between two Sample Means (if H0 true)
T for Treatment GroupC for Control Group
SEM for Treatment Group
SEM for Control Group
Add the Errors and take the Square Root
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Evaluation
Compare the Difference you Got to the Difference you would ExpectIf H0 true
What you Got
What you Expect
?
df = n1 + n2 - 2
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Evaluation
Compare the Difference you Got to the Difference you would ExpectIf H0 true
What you Got
What you Expect
?a) If they agree: Keep H0
b) If they disagree: Reject H0
Is TOO DAMN BIG!
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Burn This!
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Power
The ability to find a relationship when it exists
•Errors of Estimation and Standard Errors of the Difference decrease with N
Use the Largest sample sizes possible
•Errors of Estimation increase with Population Heterogeneity
Run all your subjects under Identical Conditions (Experimental Control)
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Power
Case Number
10987654321
Val
ue
40
30
20
10
0
Pre-Test
Post-Test
What if your data look like this?Everybody increased their score (X-bar1 – X-Bar2),but heterogeneity among subjects (SEM1 & SEM2) is large
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Power
Correlated Samples Designs:
•Natural Pairs: E.G.: Father vs. SonMeasuring liberal attitudes
•Matched Pairs: Matching pairs of students on I.Q.One of each pair gets treatment (e.g., teaching with technology
•Repeated Measures:Measure Same Subject Twice (e.g., Pre-, Post-therapy)
Look at differences between Pairs of Data Points, ignoring BetweenSubject differences
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Correlated Samples
Same as usual
Minus strength of Correlation
Smaller denominatorMakes t bigger, henceMore Power
If r=0, denominator is the same, but df is smaller
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Effect Size
•What are the Two Ts of Research?•What is better than computing Effect Size?
A weighted average ofTwo estimates of Sigma
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Confidence Interval
Use 2-tailed t-value at95% confidence levelWith N1 + N2 –2 df
N-1 df
Does the Interval cross Zero?
Best Estimate
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1020N =
SEX
mf
Me
an
+-
2 S
E H
EIG
HT
76
74
72
70
68
66
64
62
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Group Statistics
20 64.9500 2.45967 .55000
10 72.3000 1.82878 .57831
SEXf
m
HEIGHTN Mean Std. Deviation
Std. ErrorMean
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Independent Samples Test
1.352 .255 -8.338 28 .000 -7.3500 .88151 -9.15568 -5.54432
-9.210 23.527 .000 -7.3500 .79809 -8.99893 -5.70107
Equal variancesassumed
Equal variancesnot assumed
HEIGHTF Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
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18111N =
HAIR
nb
Me
an
+-
2 S
E H
EIG
HT
72
70
68
66
64
62
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Independent Samples Test
.748 .395 -1.527 27 .139 -2.4242 1.58807 -5.68268 .83420
-1.573 23.314 .129 -2.4242 1.54102 -5.60972 .76123
Equal variancesassumed
Equal variancesnot assumed
HEIGHTF Sig.
Levene's Test forEquality of Variances
t df Sig. (2-tailed)Mean
DifferenceStd. ErrorDifference Lower Upper
95% ConfidenceInterval of the
Difference
t-test for Equality of Means
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Assumptions of the t-Test
Both (if more than one) population(s):1. Normally distributed2. Equal variance
Violations of Assumptions:Robust unless gross
Transform scores (e.g. take log of each score)
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Power
Power = 1 – BetaTheoretical (Beta usually unknown)
Reject H0:Decision is clear, you have a relationship
Fail to reject H0:Decision is unclear, you may have failed to find a Relationshipdue to lack of Power
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Power
1. Increases with Effect Size (Mu1 – Mu2)
2. Increases with Sample SizeIf close to p<0.05 add N
3. Decreases with Standard Error of the Difference (denominator)Minimize by
• Recording data correctly• Use consistent criteria• Maintain consistent experimental conditions (control)• (Increasing N)