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

2. Quiz 9 is due on Friday at midnight
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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. 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
Step 5: Make a decision about your null hypothesis
“Reject H0”
“Fail to reject H0”
Testing Hypotheses

6. Type I error: concluding that there is an effect (a difference between groups) when there really isn’t.
Sometimes called “significance level”
We try to minimize this (keep it low)
Pick a low level of alpha
Psychology: 0.05 and 0.01 most common
For Step 5, we compare a “p-value” of our test to the alpha level to decide whether to “reject” or “fail to reject” to H0
Type II error: concluding that there isn’t an effect, when there really is.
Related to the Statistical Power of a test
How likely are you able to detect a difference if it is there
Error types
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error

7. 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.
“Statistically significant differences”
Essentially this means that the observed difference is above what you’d expect by chance (standard error)
Testing Hypotheses

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

9. 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
Two possibilities in the “real world”
H0 is true (no treatment effect)
H0 is false (is a treatment effect)
Two populations XA XB XB XA
76%
80%
76%
80%
People who get the treatment change,
they form a new population
(the “treatment population)
Step 4: “Generic” statistical test

10. Step 4: “Generic” statistical testXB XA
ER: Random sampling error
ID: Individual differences (if between subjects factor)
TR: The effect of a treatment
Why might the samples be different?
(What is the source of the variability between groups)?
Step 4: “Generic” statistical test

11. 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

12. Step 4: “Generic” statistical test
The generic test statistic distribution
Observed difference
TR + ID + ER
ID + ER
Computed
test statistic = =
Difference from chance
Distribution of sample means
Step 4: “Generic” statistical test

13. Step 4: “Generic” statistical test
The generic test statistic distribution
TR + ID + ER
ID + ER
Distribution of the test statistic
Test statistic
Transformation could be a z-score, t-test, F-ratio (ANOVA)
Distribution of sample means
Step 4: “Generic” statistical test

14. 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
TR + ID + ER
ID + ER
Distribution of the test statistic
Test statistic
Distribution of sample means
α-level determines where these boundaries go
Step 4: “Generic” statistical test

15. 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
Obs test stat in here Reject H0
2.5%
2.5%
“two-tailed” with α = 0.05
Observed test statistic in here
Fail to Reject H0
Step 4: “Generic” statistical test

16. 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
Obs test stat in here 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
Observed test statistic in here
Fail to Reject H0
Step 4: “Generic” statistical test

17. 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
TR + ID + ER
ID + ER
TR + ID + ER
ID + ER XB XA XB XA
Step 4: “Generic” statistical test

18. 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 XB XA XB XA
TR + ID + ER
ID + ER
Step 4: “Generic” statistical test

19. 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)
Error bars

20. 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

21. 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

22. 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].”
T-test

23. 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)

24. 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)

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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