comparing two means dependent and independent t-tests class 14
TRANSCRIPT
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
40
45
50
55
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