1. Psych 231: Research Methods in Psychology
Statistics (cont.)
Psych 231: Research Methods in Psychology

2. Announcements Quiz 10 is due on Friday at midnight
Class experiment final drafts due in labs this week
Announcements

3. Statistics 2 General kinds of Statistics Descriptive statistics
Used to describe, simplify, & organize data sets
Describing distributions of scores
Inferential statistics
Used to test claims about the population, based on data gathered from samples
Takes sampling error into account. Are the results above and beyond what you’d expect by random chance?
Population
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
Inferential statistics used to generalize back
Statistics

4. Inferential Statistics
Two approaches
Hypothesis Testing
“There is a statistically significant difference between the two groups”
Confidence Intervals
“The mean difference between the two groups is between 4% ± 2%”
Population
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
Inferential statistics used to generalize back
Inferential Statistics

5. Using Confidence intervals
CI: μ = (X) ± (tcrit) (diff by chance)
What DOES “confident” mean?
“90% confidence” means that 90% of the interval estimates of this sample size will include the actual population mean
9 out of 10 intervals contain μ
Actual population mean μ
Using Confidence intervals

6. Using Confidence intervals
CI: μ = (X) ± (tcrit) (diff by chance)
Distribution of the test statistic
The upper and lower 25%
Confidence interval uses the tcrit values that identify the top and bottom tails
2.5%
2.5%
A 95% CI is like using a “two-tailed” t-test with with α = 0.05
95% of the sample means
Using Confidence intervals

7. Using Confidence intervals
CI: μ = (X) ± (tcrit) (diff by chance)
Note: How you compute your standard error will depend on your design
Using Confidence intervals

8. Error bars Two types typically Standard Error (SE)
diff by chance
Confidence Intervals (CI)
A range of plausible estimates of the population mean
CI: μ = (X) ± (tcrit) (diff by chance)
Note: Make sure that you label your graphs, let the reader know what your error bars are
Error bars

9. 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

10. T-test Design Formulae: 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
Formulae:
Observed difference
T =
X X2
Diff by chance
Based on sampling error
Computation differs for
between and within t-tests
CI: μ=(X1-X2)±(tcrit)(Diff by chance)
T-test

11. 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, 95% CI [7.62, 16.38]”
“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) = 7.50, p < 0.05, 95% CI [8.51, 15.49].”
Dep Var
Error bars are 95% CIs
T-test

12. Analysis of Variance (ANOVA)XB XA XC
More than two groups
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 (ANOVA)

13. Analysis of Variance (ANOVA)XB XA XC
Designs
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 (ANOVA)

14. 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

15. 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

16. 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

17. 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

18. 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

19. Factorial design example
Consider the results of our class experiment
Main effect of cell phone ✓
Main effect of site type ✓
An Interaction between cell phone and site type ✓
-0.78
0.04
Factorial design example
Resource: Dr. Kahn's reporting stats page

20. 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