1. Inferential Statistics
Psych 231: Research Methods in Psychology

2. Inferential Statistics
Purpose: To make claims about populations based on data collected from samples
What’s the big deal?
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

3. Testing Hypotheses Step 1: State your hypotheses
Null hypothesis
Alternative hypothesis
This is the hypothesis that you are testing
Step 2: Set your decision criteria
Alpha level (prob of Type I error)
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
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 “statistically significant differences”
Fail to reject H0 “not statistically significant differences”
Testing Hypotheses

4. Testing Hypotheses XA XB About populations 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 average memory score is 76%
H0: μA = μB
H0: there is no difference between Grp A and Grp B
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Is the 4% difference a “real” difference (statistically
significant) or is it just sampling error?
Two sample
distributions XA XB
76%
80%
Testing Hypotheses

5. “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%
“Generic” statistical test

6. “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)
“Generic” statistical test

7. “Generic” statistical testXA XB
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)?
“Generic” statistical test

8. “Generic” statistical testXA XB
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
“Generic” statistical test

9. Estimating Sampling Error
Observed difference
Difference from chance
“Statistically significant differences”
When you “reject your null hypothesis”
Essentially this means that the observed difference is above what you’d expect by chance
“Chance” is determined by estimating how much sampling error there is
Factors affecting “chance” (sampling error)
Sample size
Population variability
Estimating Sampling Error

10. Estimating Sampling Error: Sample size
Population mean
Population
Distribution x
Sampling error
(Pop mean - sample mean)
n = 1
Estimating Sampling Error: Sample size

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

12. Estimating 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)
n = 10
Estimating Sampling Error: Sample size

13. Estimating Sampling Error: Pop. Variance
Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance”
Large population variability
Small population variability
Estimating Sampling Error: Pop. Variance

14. Estimating Sampling Error: Standard Error
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 XA XD
Avg. Sampling error XB
“chance”
Estimating Sampling Error: Standard Error

15. “Generic” statistical test
The generic test statistic distribution
A transformation of the distribution of sample means into a standardized distribution (e.g., z-distribution, t-distribution, F-distribution)
Observed difference
Difference from chance
Distribution of the test statistic
Test statistic
Distribution of sample means
“Generic” statistical test

16. Testing Hypotheses Step 1: State your hypotheses
Null hypothesis
Alternative hypothesis
Step 2: Set your decision criteria
Alpha level (prob of Type I error)
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 “statistically significant differences”
Fail to reject H0 “not statistically significant differences”
Testing Hypotheses

17. “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 to reject H0? 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

18. “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 to reject H0? The alpha level gives us the decision criterion
Distribution of the test statistic
Reject H0
Fail to reject H0
“Generic” statistical test

19. “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 to reject H0? 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”)
Fail to reject H0
“Generic” statistical test

20. “Generic” statistical testNR
exp
Observed difference
TR + ID + ER
ID + ER
Computed
test statistic = =
Difference from chance
Things that affect the computed test statistic
Difference expected by chance (sample error)
Sample size
Variability in the population
Size of the treatment effect
The bigger the effect, the bigger the computed test statistic
“Generic” statistical test

21. Significance “A statistically significant difference” means:
the researcher is concluding that there is a difference above and beyond chance
with the probability of making a type I error at 5% (assuming an alpha level = 0.05)
Note “statistical significance” is not the same thing as theoretical significance.
Only means that there is a statistical difference
Doesn’t mean that it is an important difference
Significance

22. Non-Significance Failing to reject the null hypothesis
Generally, not interested in “accepting the null hypothesis” (remember we can’t prove things only disprove them)
Usually check to see if you made a Type II error (failed to detect a difference that is really there)
Check the statistical power of your test
Sample size is too small
Effects that you’re looking for are really small
Check your controls, maybe too much variability
Non-Significance

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. Relationships between variables
Example: Suppose that you notice that the more you study for an exam, the better your score typically is.
This suggests that there is a relationship between study time and test performance.
We call this relationship a correlation.
Relationships between variables

35. Relationships between variables
Properties of a correlation
Form (linear or non-linear)
Direction (positive or negative)
Strength (none, weak, strong, perfect)
To examine this relationship you should:
Make a scatterplot
Compute the Correlation Coefficient
Relationships between variables

36. Scatterplot Plots one variable against the other
Useful for “seeing” the relationship
Form, Direction, and Strength
Each point corresponds to a different individual
Imagine a line through the data points
Scatterplot

37. Y X 1 2 3 4 5 6
Hours study X
Exam perf. Y 6 1 2 5 3 4
Scatterplot

38. Correlation Coefficient
A numerical description of the relationship between two variables
For relationship between two continuous variables we use Pearson’s r
It basically tells us how much our two variables vary together
As X goes up, what does Y typically do
X, Y
X, Y
X, Y
Correlation Coefficient

39. Linear
Non-linear
Form

40. Direction Positive Negative Y X Y X As X goes up, Y goes up
X & Y vary in the same direction
Positive Pearson’s r
As X goes up, Y goes down
X & Y vary in opposite directions
Negative Pearson’s r
Direction

41. Strength Zero means “no relationship”.
The farther the r is from zero, the stronger the relationship
The strength of the relationship
Spread around the line (note the axis scales)
Strength

42. r = 0.0
“no relationship”
r = 1.0
“perfect positive corr.”
r = -1.0
“perfect negative corr.”
-1.0
0.0
+1.0
The farther from zero, the stronger the relationship
Strength

43. Rel A
Rel B
r = -0.8
r = 0.5
-.8 .5
-1.0
0.0
+1.0
Which relationship is stronger?
Rel A, -0.8 is stronger than +0.5
Strength

44. Compute the equation for the line that best fits the data pointsY X 1 2 3 4 5 6
Y = (X)(slope) + (intercept)
0.5
2.0
Change in Y
Change in X
= slope
Regression

45. Regression Can make specific predictions about Y based on X X = 5
Y = (X)(.5) + (2.0) Y X 1 2 3 4 5 6
Y = (5)(.5) + (2.0)
Y = = 4.5
4.5
Regression

46. Regression Also need a measure of error Y = X(.5) + (2.0) + error
Same line, but different relationships (strength difference) Y X 1 2 3 4 5 6 Y X 1 2 3 4 5 6
Regression

47. Cautions with correlation & regression
Don’t make causal claims
Don’t extrapolate
Extreme scores (outliers) can strongly influence the calculated relationship
Cautions with correlation & regression