Comparing Two MeansDependent and Independent T-Tests

Class 14

Logic of Inferential Stats

Detective Althype: “Tony 'Trout Eyes' Nullhype was at the murder scene.”

Tony “Trout Eyes” Nullhype: “No fuggin way! I was at duh church rummage sale!”

Dataville Witness Reports

Witness 1: Saw Tony at scene

Witness 2: Saw Tony at scene

Witness 3: Not sure

Dataville Witness Reports

Witness 1: Saw Tony at scene

Witness 2: Saw Tony at scene

Witness 3: Not sure

Witness 4: Not sure

Witness 5: Not sure

Witness 6: Not sure

Witness 7: Not sureError

Logic of Inferential Stats

Degree of CertaintyAll Observations

2 witnesses ID’d Tony = 0.66 confirmation rate 3 witnesses total

2 witness ID’d Tony = 0.29 confirmation rate 7 witnesses total

Generating Anxiety—Photos vs. Reality:

Within Subjects and Between Subjects Designs

Problem Statement: Are people as aroused by photos of threatening things as by the physical presence of threatening things?

Hypothesis: Physical presence will arouse more anxiety than pictures.

Expt’l Hypothesis: Seeing a real tarantula will arouse more anxiety than will spider photos.

Spider Photos

WUNDT!!!!

WITHIN SUBJECTS DESIGN1. All subjects see both spider pictures and real tarantula

2. Counter-balanced the order of presentation. Why?

3. DV: Anxiety after picture and after real tarantula

Data (from spiderRM.sav)

Subject Picture (anx. score) Real T (anx. score)

1 30 40

2 35 35

3 45 50

--- --- ---

12 50 39

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35

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45

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60

Picture Real T

Anx

iety

Results: Anxiety Due to Pictures vs. Real Tarantula

Do the means LOOK different? Are they SIGNIFICANTLY DIFFERENT?

YesNeed t-test

WHY MUST WE LEARN FORMULAS?

Don’t computers make stat formulas unnecessary

1. SPSS conducts most computations, error free

2. In the old days—team of 3-4 work all night

to complete stat that SPSS does in .05 seconds.

Fundamental formulas explain the logic of stats

1. Gives you more conceptual control over your work

2. Gives you more integrity as a researcher

3. Makes you more comfortable in psych forums

)+ ( X (5) X (365 X3y)

=

TODDLER FORMULA

Point: Knowing the formula without understanding concepts leads to impoverished understanding.

Logic of Testing Null Hypothesis

Inferential Stats test the null hypothesis ("null hyp.")

This means that test is designed to CONFIRM that the null hyp is true.

In WITHIN GROUPS t-test (AKA "dependent" t-test) null hyp. is that responses in Cond. A and in Cond. B come from same population of responses. Null hyp.: Cond A and Cond B DON'T differ.

In BETWEEN GROUPS t-test (AKA "independent" t-test) null hyp. is that responses from Group A and from Group B DON’T differ.

If tests do not confirm the null hyp, then must accept ALT. HYPE.

Alt. hyp. within-groups: Cond A differs from Cond BAlt. hyp. between-groups Group A differs from Group B

Null Hyp. and Alt. Hyp in Pictures vs. Reality Study

Within groups design: Cond. A (all subjs. see photos), then Cond. B (all subs. see actual tarantula)

Null hyp? No differences between seeing photos (Cond A) and seeing real T (Cond B)

Anxiety ratings

Alt. hyp? There is a difference between seeing photos (Cond A) and seeing real T (Cond B)

QUIZ 2 POSTPONED TO NOV. 12

MID-TERM GRADE ADJUSTMENT

2 PTS. ADDED TO ALL SCORES

(i.e., 84 originally now = 86)

Key to T-Test is:Central Tendency (i.e. Mean) Relative to

Random Distribution (i.e., SD or SE)

Diffs Btwn Means

Distribution Distribution

T-Test as Measure of Difference Between Two Means

1. Two data samples—do means of each sample differ significantly?

2. Do samples represent same underlying population (null hyp: small diffs) or two distinct populations (alt. hyp: big diffs)?

3. Compare diff. between sample means to diff. we’d expect if null hyp is true

4. Use Standard Error (SE) to gauge variability btwn means. a. If SE small & null hyp. true, sample diffs should be smaller

b. If SE big & null hyp. true, sample diffs. can be larger

5. If sample means differ much more than SE, then either: a. Diff. reflects improbable but true random difference w/n true pop. b. Diff. indicates that samples reflect two distinct true populations.

6. Larger diffs. Between sample means, relative to SE, support alt. hyp.

7. All these points relate to both Dependent and Independent t-tests

Logic of T-Test

observed difference between sample

means

expected difference between population means

(0 if null hyp. is true)t = −

SE of difference between sample means

Note: Logic the same for Dependent and Independent t-tests. However, the specific formulas differ.

If Difference Between Means Relative to SE (overlap) is Small: Null Hyp. Supported

If Difference Between Means Relative to SE (overlap) is Large: Alternative Hyp. Supported

SD: The Standard Error of

Differences Between MeansSampling Distribution: The spread of many sample means around a true mean.

SE: The average amount that sample means vary around the true mean. SE = Std. Deviation of sample means.

Formula for SE: SE = s/√n, when n > 30

If sample N > 30 the sampling distribution should be normal.

Mean of sampling distribution = true mean.

SD = Average amount Var. 1 mean differs from Var. 2 mean in Sample 1, then in Sample 2, then in Sample 3, ---- then in Sample N

Note: SD is differently computed in Between-subs. designs.

SD: The Standard Error of Differences Between Means

TARANTULA PICTURE D MEAN MEAN (T mean – P mean)

Study 1 6 3 3Study 2 5 3 2Study 3 4 2 2Study 4 5 3 2 .

Ave. 2.25

SD: The Standard Error of Differences Between Means

TARANT. PICT. D D - D (D-D)2

Sub. 1 6 3 3 -. 75 .56Sub. 2 5 3 2 .25 .07Sub. 3 4 2 2 .25 .07Sub. 4 5 3 2 .25 .07

X Tarant = 5 X Pic = 2.75 D = 2.25 Σ (D-D)2 = .77

SD2 = Sum (D -D)2 / N - 1; = .77 / 3 = .26

SD = √SD2 = √.26 = .51

SE of D = σD = SD / √N = .51 / √4 = .51 / 2 = .255

t = D / SE of D = 2.25 / .255 = 8.823

Small SD indicates that average difference between pairs of variable means should be large or small, if null hyp true?

Small SD will therefore increase or decrease our chance of confirming experimental prediction, if actual difference is real?

Small

Increase it.

Understanding SD and Experiment Power

Power of Experiment: Ability of expt. to detect actual differences.

Assumptions of Dependent T-Test

1. Samples are normally distributed

2. Data measured at interval level (not ordinal or categorical)

Conceptual Formula for Dependent Samples T-Test

t =D − μD

sD / √N

D = Average difference between mean Var. 1 – mean Var. 2. It represents systematic variation, aka experimental effect.

μD = Expected difference in true population = 0 It represents random variation, aka the the null effect.

sD / √N = Estimated standard error of differences between all potential sample means.

It represents the likely random variation between means.

= Experimental Effect

Random Variation

Dependent (w/n subs) T-Test SPSS Output

t = expt. effect / error

t = X / SE

t = -7 / 2.83 = -2.473

SE = SD / √n2.83 = 9.807 / √12

Note:

Mean = mean diff pic anx - real anx.= 40 - 47 = - 7

Independent (between-subjects) t-test1. Subjects see either spider pictures OR real tarantula

2. Counter-balancing less critical (but still important). Why?

3. DV: Anxiety after picture OR after real tarantula

Data (from spiderBG.sav)

Subject Condition Anxiety

1 1 30

2 2 35

3 1 45

22 2 50

23 1 60

24 2 39

Assumptions of Independent T-Test

DEPENDENT T-TEST

1. Samples are normally distributed

2. Data measured at least at interval level (not ordinal or categorical)

INDEPENDENT T-TESTS ALSO ASSUME

3. Homogeneity of variance

4. Scores are independent (b/c come from diff. people).

Logic of Independent Samples T-Test (Same as Dependent T-Test)

observed difference between sample

means

expected difference between population means

(if null hyp. is true)t = −

SE of difference between sample means

Note: SE of difference of sample means in independent t test differs from SE in dependent samples t-test

Conceptual Formula for Independent Samples T-Test

t =(X1 − X2) − (μ1 − μ2)

Est. of SE

(X1 − X2) = Diffs. btwn. samples

It represents systematic variation, aka experimental effect.

(μ1 − μ2) = Expected difference in true populations = 0 It represents random variation, aka the the null effect.

Estimated standard error of differences between all potential sample means.

It represents the likely random variation between means.

= Experimental Effect

Random Variation

Computational Formulas for Independent Samples T-Tests

t = X1 − X2

2

N1 N2

( ) s1 s2

2

+√

When N1 = N2

t = X1 − X2

sp sp

2

+√2

n1 n2

When N1 ≠ N2

sp2

= (n1 -1)s1 + (n2 -1)s22 2

n1 + n2 − 2

Weighted average of each groups SE=

Independent (between subjects) T-Test SPSS Output

t = expt. effect / error

t = (X1 − X2) / SE

t = -7 / 4.16 = - 1.68

Dependent (within subjects) T-Test SPSS Output

t = expt. effect / error

t = X / SE

t = -7 / 2.83 = -2.473

SE = SD / √n2.83 = 9.807 / √12

Note:

Mean = mean diff pic anx - real anx.= 40 - 47 = - 7

Dependent T-Test is Significant; Independent T-Test Not Significant.

A Tale of Two Variances

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Picture Real T

Anx

iety

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Picture Real T

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Dependent T-Test Independent T -Test

SE = 2.83 SE = 4.16