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

2. Remember, no formal labs this week
Remember, no formal labs this week. Use the times to work on your group project data analyses.
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. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
What’s the big deal?
Population
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
Example Experiment:
Group A - gets treatment to improve memory
Group B - gets no treatment (control)
After treatment period test both groups for memory
Results:
Group A’s average memory score is 80%
Group B’s is 76%
Is the 4% difference a “real” difference (statistically significant) or is it just sampling error?
Inferential Statistics

5. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
Two approaches based on quantifying sampling error
Hypothesis Testing
“There is a statistically significant difference (4%) 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

6. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
Population
Sample A
Treatment
X = 80%
Sample B
No Treatment
X = 76%
Sampling error is how much a difference you might get between your sample and your population resulting from “chance” (e.g., random sampling)
Factors affecting “chance”
Sample size
Population variability
Inferential Statistics

7. Sampling error: sample size
Population mean
Population
Distribution x
Sampling error
(Pop mean - sample mean)
n = 1
Sampling error: sample size

8. Sampling error: sample size
Population mean
Population
Distribution
Sample mean x x
Sampling error
(Pop mean - sample mean)
n = 2
Sampling error: sample size

9. Sampling error: sample size
Generally, as the sample size increases, the sampling error decreases
Population mean
Population
Distribution
Sample mean x
Sampling error
(Pop mean - sample mean)
Does you sample size matter? Kevin Lyons
Diminishing returns:
n = 10
Amount of reduced error
Sample size increase
1/2
4 times
1/3
9 times
1/4
16 times
1/5
25 times
1/10
100 times
Sampling error: sample size

10. Sampling error: population variability
Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance”
Large population variability
Small population variability
Sampling error: population variability

11. Sampling error “chance”
These two factors combine to impact the distribution of sample means.
The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population
Population
Distribution of sample means XC
Samples of size = n
“Standard
error” (SE) XA XD
Avg. Sampling error XB
Sampling error
“chance”
More info

12. Difference from chance
These two factors combine to impact the distribution of sample means.
Sample s X
Population σ μ
Distribution of sample means
Avg. Sampling error
“chance”
Difference from chance
More info

13. 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

14. 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

15. Testing Hypotheses Step 1: State your hypotheses
This is the hypothesis that you are testing
Null hypothesis (H0)
Alternative hypothesis(ses)
“There are no differences (effects)”
Generally, “not all groups are equal”
You aren’t out to prove the alternative hypothesis (although it feels like this is what you want to do)
If you reject the null hypothesis, then you’re left with support for the alternative(s) (NOT proof!)
Testing Hypotheses

16. Testing Hypotheses Step 1: State your hypotheses
In our memory example experiment
Null H0: mean of Group A = mean of Group B
Alternative HA: mean of Group A ≠ mean of Group B
(Or more precisely: Group A > Group B)
It seems like our theory is that the treatment should improve memory.
That’s the alternative hypothesis. That’s NOT the one the we’ll test with inferential statistics.
Instead, we test the H0
Testing Hypotheses

17. Testing Hypotheses Step 1: State your hypotheses
Step 2: Set your decision criteria
Your alpha level will be your guide for when to:
“reject the null hypothesis”
“fail to reject the null hypothesis”
This could be correct conclusion or the incorrect conclusion
Two different ways to go wrong
Type I error: saying that there is a difference when there really isn’t one (probability of making this error is “alpha level”)
Type II error: saying that there is not a difference when there really is one (probability of making this error is “beta”)
Testing Hypotheses

18. Error types Real world (‘truth’) H0 is correct H0 is wrong
Type I error
Reject H0
Experimenter’s conclusions
Fail to Reject H0
Type II error
Error types

19. Error types: Courtroom analogy
Real world (‘truth’)
Defendant is innocent
Defendant is guilty
Type I error
Find guilty
Jury’s decision
Type II error
Find not guilty
Error types: Courtroom analogy

20. 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

21. 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

22. 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

23. 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

24. 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

25. 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

26. Recall: Sampling error
The distribution of sample means is a distribution of all possible sample means of a particular sample size that can be drawn from the population
Population
Distribution of sample means XC
Samples of size = n XA XD
Avg. Sampling error XB
Difference from chance
Recall: Sampling error

27. 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

28. 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
2.5%
2.5%
“two-tailed” with α = 0.05
Fail to reject H0
Step 4: “Generic” statistical test

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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)

35. 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)

36. 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

37. 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

38. 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

39. 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

40. 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

41. Factorial designs Consider the results of our class experiment X ✓ ✓
Main effect of cell phone X
Main effect of site type ✓
An Interaction between cell phone and site type ✓
-0.50
0.04
Factorial designs

42. 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

43. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
Two approaches based on quantifying sampling error
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

44. 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

45. Distribution of sample means
The following slides are available for a little more concrete review of distribution of sample means discussion.
Distribution of sample means

46. Distribution of sample means
Distribution of sample means is a “theoretical” distribution between the sample and population
Mean of a group of scores
Comparison distribution is distribution of means
Population
Distribution of sample means
Sample
Distribution of sample means

47. Distribution of sample means
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
Assumption: sampling with replacement
Distribution of sample means

48. Distribution of sample means
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
There are 16 of them
mean 2 2 2 6 4 5 6 4 7 8 4 6 8 2 2 4 3 8 4 6 2 4 6 4 8 2 8 2 5 4 2 3 4 4
Distribution of sample means

49. Distribution of sample means2 3 4 5 6 7 8 1
In long run, the random selection of tiles leads to a predictable pattern
mean
mean
mean 2 2 2 4 6 5 8 2 5 2 4 3 4 8 6 8 4 6 2 6 4 6 2 4 8 6 7 2 8 5 6 4 5 8 8 8 4 2 3 6 6 6 4 4 4 6 8 7
Distribution of sample means

50. Properties of the distribution of sample means
Shape
If population is Normal, then the dist of sample means will be Normal
If the sample size is large (n > 30), regardless of shape of the population
Distribution of sample means
Population
N > 30
Properties of the distribution of sample means

51. Properties of the distribution of sample means
Center
The mean of the dist of sample means is equal to the mean of the population
Population
Distribution of sample means
same numeric value
different conceptual values
Properties of the distribution of sample means

52. Properties of the distribution of sample means
Center
The mean of the dist of sample means is equal to the mean of the population
Consider our earlier example 2 4 6 8
Population
Distribution of sample means
means 2 3 4 5 6 7 8 1 4
μ =
= 5 16 =
= 5
Properties of the distribution of sample means

53. Properties of the distribution of sample means
Spread
Standard deviation of the population
Sample size
Putting them together we get the standard deviation of the distribution of sample means
Commonly called the standard error
Properties of the distribution of sample means

54. The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean
In other words, it is an estimate of the error that you’d expect by chance (or by sampling)
Standard error

55. All three of these properties are combined to form the Central Limit Theorem
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will approach a normal distribution with a mean of and a standard deviation of as n approaches infinity
(good approximation if n > 30).
Central Limit Theorem

56. Distribution of sample means
Keep your distributions straight by taking care with your notation
Population σ μ
Distribution of sample means
Sample s X
Distribution of sample means
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