Statistics
Psych 231: Research
Methods in Psychology
XA
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XB
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
The generic test statistic - is a ratio of sources of
variability
Computed
Observed difference
TR + ID + ER
=
=
test statistic
Difference from chance
ID + ER
“Generic” statistical test
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1 factor with two groups
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T-tests
• Between groups: 2-independent samples
• Within groups: Repeated measures samples (matched, related)
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1 factor with more than two groups
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Analysis of Variance (ANOVA) (either between groups or
repeated measures)
Multi-factorial
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Factorial ANOVA
Some inferential statistical tests
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Design
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2 separate experimental conditions
Degrees of freedom
• Based on the size of the sample and the kind of t-test
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Formula:
Observed difference
T=
X1 - X2
Diff by chance
Computation differs for
between and within t-tests
T-test
Based on sample error
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Reporting your results
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The observed difference between conditions
Kind of t-test
Computed T-statistic
Degrees of freedom for the test
The “p-value” of the test
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“The mean of the treatment group was 12 points higher than the
control group. An independent samples t-test yielded a significant
difference, t(24) = 5.67, p < 0.05.”
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“The mean score of the post-test was 12 points higher than the
pre-test. A repeated measures t-test demonstrated that this
difference was significant significant, t(12) = 5.67, p < 0.05.”
T-test
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Designs
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XA
XB
XC
More than two groups
• 1 Factor ANOVA, Factorial ANOVA
• Both Within and Between Groups Factors
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Test statistic is an F-ratio
Degrees of freedom
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Several to keep track of
The number of them depends on the design
Analysis of Variance
XA
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XB
XC
More than two groups
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Now we can’t just compute a simple difference score since
there are more than one difference
So we use variance instead of simply the difference
• Variance is essentially an average difference
Observed variance
F-ratio =
Variance from chance
Analysis of Variance
XA
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XB
XC
1 Factor, with more than two levels
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Now we can’t just compute a simple difference score since
there are more than one difference
• A - B, B - C, & A - C
1 factor ANOVA
Null hypothesis:
XA
XB
XC
H0: all the groups are equal
XA = XB = XC
Alternative hypotheses
HA: not all the groups are equal
XA ≠ XB ≠ XC
XA = XB ≠ XC
1 factor ANOVA
The ANOVA
tests this one!!
Do further tests to
pick between these
XA ≠ XB = XC
XA = XC ≠ XB
Planned contrasts and post-hoc tests:
- Further tests used to rule out the different
Alternative hypotheses
XA ≠ XB ≠ XC
Test 1: A ≠ B
Test 2: A ≠ C
Test 3: B = C
XA = XB ≠ XC
XA ≠ XB = XC
XA = XC ≠ XB
1 factor ANOVA
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Reporting your results
The observed differences
 Kind of test
 Computed F-ratio
 Degrees of freedom for the test
 The “p-value” of the test
 Any post-hoc or planned comparison results
“The mean score of Group A was 12, Group B was 25, and
Group C was 27. A 1-way ANOVA was conducted and the
results yielded a significant difference, F(2,25) = 5.67, p < 0.05.
Post hoc tests revealed that the differences between groups A
and B and A and C were statistically reliable (respectively t(1) =
5.67, p < 0.05 & t(1) = 6.02, p <0.05). Groups B and C did not
differ significantly from one another”
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1 factor ANOVA
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We covered much of this in our experimental design lecture
More than one factor
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Factors may be within or between
Overall design may be entirely within, entirely between, or mixed
Many F-ratios may be computed
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An F-ratio is computed to test the main effect of each factor
An F-ratio is computed to test each of the potential interactions
between the factors
Factorial ANOVAs
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Reporting your results
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The observed differences
• Because there may be a lot of these, may present them in a table
instead of directly in the text
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Kind of design
• e.g. “2 x 2 completely between factorial design”
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Computed F-ratios
• May see separate paragraphs for each factor, and for interactions
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Degrees of freedom for the test
• Each F-ratio will have its own set of df’s
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The “p-value” of the test
• May want to just say “all tests were tested with an alpha level of
0.05”
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Any post-hoc or planned comparison results
• Typically only the theoretically interesting comparisons are
presented
Factorial ANOVAs