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

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

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

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

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

Evaluation

Compare the Difference you Got to the Difference you would ExpectIf H0 true

What you Got

What you Expect

?

df = n1 + n2 - 2

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!

Burn This!

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)

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

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

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

Effect Size

•What are the Two Ts of Research?•What is better than computing Effect Size?

A weighted average ofTwo estimates of Sigma

Confidence Interval

Use 2-tailed t-value at95% confidence levelWith N1 + N2 –2 df

N-1 df

Does the Interval cross Zero?

Best Estimate

1020N =

SEX

mf

Me

an

+-

2 S

E H

EIG

HT

76

74

72

70

68

66

64

62

Group Statistics

20 64.9500 2.45967 .55000

10 72.3000 1.82878 .57831

SEXf

m

HEIGHTN Mean Std. Deviation

Std. ErrorMean

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

18111N =

HAIR

nb

Me

an

+-

2 S

E H

EIG

HT

72

70

68

66

64

62

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

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)

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

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)