1. Inferential Statistics
Psych 231: Research Methods in Psychology

2. Testing Hypotheses Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data from your sample(s)
Step 4: Compute your test statistics
Descriptive statistics (means, standard deviations, etc.)
Inferential statistics (t-tests, ANOVAs, etc.)
Step 5: Make a decision about your null hypothesis
Reject H0 “statistically significant differences”
Fail to reject H0 “not statistically significant differences”
Testing Hypotheses

3. “Generic” statistical test
The generic test statistic distribution
To reject the H0, you want a computed test statistics that is large
reflecting a large Treatment Effect (TR)
What’s large enough? The alpha level gives us the decision criterion
TR + ID + ER
ID + ER
Distribution of the test statistic
Test statistic
Distribution of sample means
α-level determines where these boundaries go
“Generic” statistical test

4. Some inferential statistical tests
1 factor with two groups
T-tests
Between groups: 2-independent samples
Within groups: Repeated measures samples (matched, related)
1 factor with more than two groups
Analysis of Variance (ANOVA) (either between groups or repeated measures)
Multi-factorial
Factorial ANOVA
Some inferential statistical tests

5. T-test Design Formula: Observed difference X1 - X2 T =
2 separate experimental conditions
Degrees of freedom
Based on the size of the sample and the kind of t-test
Formula:
Observed difference
T =
X X2
Diff by chance
Based on sample error
Computation differs for
between and within t-tests
T-test

6. T-test Reporting your results
The observed difference between conditions
Kind of t-test
Computed T-statistic
Degrees of freedom for the test
The “p-value” of the test
“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.”
“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

7. Analysis of Variance XB XA XC Designs Test statistic is an F-ratio
More than two groups
1 Factor ANOVA, Factorial ANOVA
Both Within and Between Groups Factors
Test statistic is an F-ratio
Degrees of freedom
Several to keep track of
The number of them depends on the design
Analysis of Variance

8. Analysis of Variance More than two groups F-ratio = XB XA XC
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
Variance from chance
F-ratio =
Analysis of Variance

9. 1 factor ANOVA 1 Factor, with more than two levels XB XA XC
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

10. 1 factor ANOVA The ANOVA tests this one!! XA = XB = XC XA ≠ XB ≠ XC
Null hypothesis:
H0: all the groups are equal
The ANOVA tests this one!!
XA = XB = XC
Do further tests to pick between these
Alternative hypotheses
HA: not all the groups are equal
XA ≠ XB ≠ XC
XA ≠ XB = XC
XA = XB ≠ XC
XA = XC ≠ XB
1 factor ANOVA

11. 1 factor ANOVA Planned contrasts and post-hoc tests:
- Further tests used to rule out the different Alternative hypotheses
XA ≠ XB ≠ XC
Test 1: A ≠ B
XA = XB ≠ XC
Test 2: A ≠ C
XA ≠ XB = XC
Test 3: B = C
XA = XC ≠ XB
1 factor ANOVA

12. 1 factor ANOVA 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 < 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”
1 factor ANOVA

13. We covered much of this in our experimental design lecture
More than one factor
Factors may be within or between
Overall design may be entirely within, entirely between, or mixed
Many F-ratios may be computed
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

14. Factorial ANOVAs Reporting your results The observed differences
Because there may be a lot of these, may present them in a table instead of directly in the text
Kind of design
e.g. “2 x 2 completely between factorial design”
Computed F-ratios
May see separate paragraphs for each factor, and for interactions
Degrees of freedom for the test
Each F-ratio will have its own set of df’s
The “p-value” of the test
May want to just say “all tests were tested with an alpha level of 0.05”
Any post-hoc or planned comparison results
Typically only the theoretically interesting comparisons are presented
Factorial ANOVAs