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

2. Change in Due date(s) for final draft
Change in Due date(s) for final draft. Instead of Wednesday in class, will turn in during labs that week (so Nov. 29 & 30).
Have a great Fall break.
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
Inferential statistics used to generalize back
Sample
Statistics

4. 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”
Make this decision by comparing your test’s “p-value” against the alpha level that you picked in Step 2.
Testing Hypotheses

5. Step 4: “Generic” statistical test
Tests the question: Are there differences between groups due to a treatment?
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
This is the one that test examines
H0 is true (no treatment effect)
H0 is false (is a treatment effect)
One population
Two populations XA XB
76%
80% XA XB
76%
80%
Step 4: “Generic” statistical test

6. Step 4: “Generic” statistical testXB XA
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
Observed difference
TR + ID + ER
ID + ER
Computed
test statistic = =
Difference from chance
Step 4: “Generic” statistical test

7. Step 4: “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
Real world (‘truth’)
H0 is correct
H0 is wrong
Exp’s concl
Reject H0
Fail to Reject H0
Distribution of the test statistic
Depending on the design: T-scores, F-ratios, r’s
Test statistic
transforms
TR + ID + ER
ID + ER
Distribution of sample means
α-level (from Step 2) determines where these boundaries go
Step 4: “Generic” statistical test

8. Step 4: “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
Distribution of the test statistic
Reject H0
“Two tailed test”: looking for “an effect” – don’t have a theoretical expectation for the direction of the effect
2.5%
2.5%
“two-tailed” with α = 0.05
Fail to reject H0
Step 4: “Generic” statistical test

9. Step 4: “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
Distribution of the test statistic
Reject H0
“One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”)
5.0%
“one-tailed” with α = 0.05
Fail to reject H0
Step 4: “Generic” statistical test

10. Step 4: “Generic” statistical test
Things that affect the computed test statistic
Size of the treatment effect (effect size)
The bigger the effect, the bigger the computed test statistic
Difference expected by chance (standard error)
Variability in the population
Sample size
TR + ID + ER
ID + ER
TR + ID + ER
ID + ER XB XA XB XA
TR + ID + ER
ID + ER
Step 4: “Generic” statistical test

11. Some inferential statistical tests
Statistical tests follow the research design used
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
KahnAcademy: Hypothesis testing and p-values (~11 mins)
Goldstein: Choosing a Statistical Test (~12 mins)

12. 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
StatsLectures: One Sample z-test (~6 mins)
Repeated Measures t Tests Part I Introduction (~14 mins)
Independent vs. Paired samples t-tests (~4 mins)

13. SPSS output for t-tests
Less than α = .05?
Less than α = .05?
Less than α = .05?
SPSS output for t-tests
In SPSS, compare observed p-values with your alpha-level.

14. 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

15. Error bars Two types typically Standard Error (SE)
diff by chance
Confidence Intervals (CI)
A range of plausible estimates of the population mean
Dep Var
Error bars are 95% CIs
CI: μ = (X) ± (tcrit) (diff by chance)
Error bars

16. 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)

17. 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)

18. 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

19. 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

20. 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

21. 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

22. 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

23. Factorial designs Consider the results of our class experiment ✓ ✓ X
Main effect of cell phone ✓
1.19
0.73 X
Main effect of site type
An Interaction between cell phone and site type
Factorial designs
Resource: Dr. Kahn's reporting stats page

24. 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

25. 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

26. 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

27. Using Confidence intervals
CI: μ = (X) ± (tcrit) (diff by chance)
Distribution of the test statistic
The upper and lower 2.5%
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

28. 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

29. 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