Download - Statistics (cont.)
Statistics (cont.)
Psych 231: Research Methods in Psychology
Announcements
Journal Summary 2 assignment Due today
Group projects Plan to have your analyses done before Thanksgiving break,
GAs will be available during lab times to help Poster sessions are last lab sections of the semester (last
week of classes), so start thinking about your posters. I will lecture about poster presentations on Monday.
Testing 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”
Step 1: State your hypotheses
Summary to this point
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II errorXAXB
76% 80%
Is the 4% difference a “real” difference (statistically significant) or is it just sampling error?
Two sampledistributions
H0: there is no difference between Grp A and Grp B
H0: μA = μB
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 is 76%
HA: there is a difference between Grp A and Grp B
“Generic” statistical test
Tests the question: Are there differences between groups
due to a treatment?
One population
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
H0 is true (no treatment effect)
XAXB
76% 80%
Two possibilities in the “real world”
Two sampledistributions
“Generic” statistical test
XAXB XAXB
H0 is true (no treatment effect) H0 is false (is a 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
76% 80% 76% 80%
People who get the treatment change, they form a new population (the “treatment population)
People who get the treatment change, they form a new population (the “treatment population)
Tests the question: Are there differences between groups
due to a treatment?
Two possibilities in the “real world”
Two populations
Two sampledistributions
“Generic” statistical test
XBXA
Why might the samples be different?(What is the source of the variability between groups)?
ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment
“Generic” statistical test
The generic test statistic - is a ratio of sources of variability
Observed difference
Difference from chance=
TR + ID + ER
ID + ER=
Computed test statistic
XBXA
ER: Random sampling error ID: Individual differences (if between subjects factor) TR: The effect of a treatment
Difference from chance
n = 1
Population mean
x
Sampling error
(Pop mean - sample mean)
Population
Distribution
Difference from chance
n = 2
Population mean
x
Population
Distribution
x
Sampling error
(Pop mean - sample mean)
Sample mean
Difference from chance
n = 10
Population meanPopulation
Distribution
Sampling error
(Pop mean - sample mean)
Sample mean
xxx
x
x
xx
xx x
Generally, as the sample size increases, the sampling error decreases
Generally, as the sample size increases, the sampling error decreases
Difference from chance
Typically the narrower the population distribution, the narrower the range of possible samples, and the smaller the “chance”
Small population variability Large population variability
Difference from 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
XA XB XC XD
Population
Samples of size = n
Distribution of sample means
Avg. Sampling error
“chance”
“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
α-level determines where these boundaries go
Distribution of sample means
Test statistic
TR + ID + ER
ID + ER
“Generic” statistical test
Distribution of the test statistic Reject H0
Fail to reject H0
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
“Generic” statistical test
Distribution of the test statistic Reject H0
Fail to reject H0
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
“One tailed test”: sometimes you know to expect a particular difference (e.g., “improve memory performance”)
“Generic” statistical test
Things that affect the computed test statistic Size of the treatment effect
• The bigger the effect, the bigger the computed test statistic
Difference expected by chance (sample error)• Sample size• Variability in the population
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
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 (1-β)
• Sample size is too small• Effects that you’re looking for are really small
• Check your controls, maybe too much variability
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
T-test
Design 2 separate experimental conditions Degrees of freedom
• Based on the size of the sample and the kind of t-test
Formula:
T = X1 - X2
Diff by chance
Based on sample error
Observed difference
Computation differs for between and within t-tests
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.”
Analysis of Variance
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
XBXA XC
Analysis of Variance
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 chanceF-ratio =
XBXA XC
1 factor ANOVA
1 Factor, with more than two levels Now we can’t just compute a simple difference score since
there are more than one difference• A - B, B - C, & A - C
XBXA XC
1 factor ANOVA
Null hypothesis: H0: all the groups are equal
XA = XB = XC
Alternative hypotheses
HA: not all the groups are equal
XA ≠ XB ≠ XC XA ≠ XB = XC
XA = XB ≠ XC XA = XC ≠ XB
The ANOVA tests this one!!
Do further tests to pick between these
XBXA XC
1 factor ANOVA
Planned contrasts and post-hoc tests:
- Further tests used to rule out the different Alternative
hypothesesXA ≠ XB ≠ XC
XA ≠ XB = XC
XA = XB ≠ XC
XA = XC ≠ XB
Test 1: A ≠ B
Test 2: A ≠ C
Test 3: B = C
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 < 0.05. 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
Factorial ANOVAs
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
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