1. Psych 200 Methods & Analysis
Dr. Kulstad for Dr. Lindgren
Fall 2008

2. Outline
1. Turn in homework 2. Stats in the news 3. ANOVA con’t 4. Bring to next class (Tuesday) COMPLETED ungraded pop quiz (available on Blackboard) as of end of class List of statistical tests for group project

3. 3. Components of the F-statistic
Example A researcher is interested in testing the effectiveness of medication on reducing pain. She is interested in examining the effectiveness of Aspirin, Tylenol, and a placebo. She recruits subjects and randomly assigns them to one of the three groups. Participants perceptions on how much their pain is reduced is measured. What kind of design is this? What is the IV? DV? How many levels or conditions of the IV are there? What are the null & alternative hypotheses?
Aspirin
Tylenol
Placebo 3 2 5 1 4
Xbar
Used to describe & communicate how different scores are from each other

4. 3. Components of the F-statistic
MSwn: Basic formula
Add the squared deviations from the
mean for each condition & divide by df
Aspirin: (3-4)2 + (5-4)2+ (3-4)2 + (5-4)2 = 4
Tylenol: (2-3)2 + (4-3)2+ (2-3)2 + (4-3)2 = 4
Placebo: (2-2)2 + (1-2)2+ (3-2)2 + (2-2)2 = 2
Sum of all of the squared deviations in
each condition: = 10
df: N –k = = 9
SSwn = = 1.111 df
Aspirin
Tylenol
Placebo 3 2 5 1 4
Xbar
Used to describe & communicate how different scores are from each other
Sum of Squares within (SSwn)
Mean Squares within (MSwn)
df within (dfwn)

5. 3. Components of the F-statistic
Components of ANOVAs are often arrayed in tables
Let’s fill in what we know so far with our sample data…
Source
Sum of Squares df
Mean square F
Between
Within 10 9
1.111
Total
Used to describe & communicate how different scores are from each other

6. 3. Components of the F-statistic
Mean square between groups (MSbn)
Estimate of variability of scores that occurs between levels/conditions in a factor
How much does each level mean deviate from overall mean of the experiment
Way to measure how much the levels means differ from one another
Takes into account error AND effect of treatment
If null is true, MSbn is only estimating one population -- only estimating σ2error like MSwn
If null is not true, MSbn is estimating error variance and treatment variance
Used to describe & communicate how different scores are from each other

7. 3. Components of the F-statistic
Mean square between groups (MSbn)
Basic formula
Σn(Xbar - GM)2
Calculate the deviation of a condition mean from the grand mean & multiply by the number in that condition
Repeat for each condition
Add those values
Divide by df: k-1 (k = # conditions)
Used to describe & communicate how different scores are from each other
Grand mean (GM) = mean of all scores in the entire experiment

8. 3. Components of the F-statistic
Aspirin
Tylenol
Placebo 3 2 5 1 4
Xbar
GM (Grand mean)
MSbw: Basic formula
Calculate the deviation of a condition mean from the grand mean (mean of ALL scores) & multiply by the number in that condition
Repeat for each condition
Add those values
Aspirin: 4* (4-3)2 = 4
Tylenol: 4 * (3-3)2 = 0
Placebo: 4 * (2-3)2 = 4
Sum for all conditions: = 8
df: k-1 = 3 -1 = 2
SSbt = 8 = 4 df
Used to describe & communicate how different scores are from each other
Sum of Squares between (SSbt)
Mean Squares between (MSbt)
df between (dfwn)

9. 3. Components of the F-statistic
Sum of Squares total (SStot) =
SSbt + SSwn =
Σ(X- GM)2
Back to our ANOVA table
Now, we can complete it
Source
Sum of Squares df
Mean square F
Between 8 2 4
Within 10 9
1.111
Total
df total (dftot) =
dfbt + dfwn =
N-1
Fobt = MSbt /MSwn
Used to describe & communicate how different scores are from each other
Source
Sum of Squares df
Mean square F
Between 8 2 4
3.60
Within 10 9
1.111
Total 18 11

10. 3. Components of the F-statistic
Logic of the F ratio (Fobt)
To conduct an ANOVA, we compare MSbn / MSwn
If H0 is true
MSbn should = MSwn
NO treatment + Error = Error
Fobt = 1
If H0 is false
MSbn should be > MSwn
Treatment + Error vs. Error
Fobt > 1
Does not mean we reject H0 every time Fobt > 1
Effect has to be strong enough to rule out differences due to chance alone
Used to describe & communicate how different scores are from each other

11. 3. Components of the F-statistic
F-Distribution
Sample distribution that shows values of F that occur when H0 is true
Positively skewed
Like t-distribution, F is a group of curves
Dependent on df
Unlike t-distribution, F, depends on 2 dfs
dfbw
dfwn
Need both to determine Fcrit
(By hand) use chart in back of book
(SPSS) if Fobt < α, reject H0,
Used to describe & communicate how different scores are from each other

12. 3. Components of the F-statistic
Source
Sum of Squares df
Mean square F
Between 8 2 4
3.60
Within 10 9
1.111
Total 18 11
Back to our data
α = .05
Fcrit = 4.26
The exact p-value for Fobt = .071
What do we decide?
Fail to reject H0
How do we write that up?
A one-way analysis of variance was conducted to examine the effect of three drugs on pain relief. There were no significant differences in pain relief, F(2,9) = 3.60, p > .05. Therefore, there was no evidence that Tylenol or Aspirin worked better than a placebo.
Fobt > 1 & not rejecting H0?Why? Due to sample size (which determines df), can’t rule out that differences observed are simply due to chance – i.e., treatment effects aren’t “extreme”enough to rule out chance.
Used to describe & communicate how different scores are from each other

13. 3. Components of the F-statistic
What if data and results were different?
α = .05
Fcrit = 4.26
The exact p-value for Fobt = .031
What do we decide?
What trouble do we run into?
Aspirin
Tylenol
Placebo 3 2 5 1 4
Xbar
1.75
Source
Sum of Squares df
Mean square F
Between
10.167 2
5.083
5.229
Within
8.750 9
.972
Total 18 11
Used to describe & communicate how different scores are from each other

14. 4. Performing Post hoc Comparisons
Procedures to identify significant differences in condition/level/group means
Variety of procedures
Some are more/less conservative
Book covers 2
Fisher’s protected t-test
Used if groups have unequal n’s
Tukey’s HSD multiple comparisons test
Used if groups have equal n’s
Used to describe & communicate how different scores are from each other

15. 4. Performing Post hoc Comparisons
Fisher’s protected t-test
tobt = (Xbar1 – Xbar2)
[ MSwn (1/n1 + 1/n2)]1/2
Same as independent samples t-test BUT use MSwn in lieu of s2pool
Use this formula for every pair of means in experiment
Use t table in book & find tcrit (use two-tailed values)
df is dfwn
Remember not used for equal n’s
Used to describe & communicate how different scores are from each other

16. 4. Performing Post hoc Comparisons
Tukey’s HSD (Honestly Significant Difference)
Used with equal n’s
Computes the minimum difference between means that is required for them to differ significantly
Steps
Identify qk (table in book)
Need k (α, # levels, and dfwn)
Multiply qk by (MSwn/n)½
Determine the differences between all the means
Compare each difference to the HSD difference
If difference is > than HSD, means differ significantly
If difference is < HSD, means do not differ significantly
Used to describe & communicate how different scores are from each other

17. 4. Performing Post hoc Comparisons
Which post hoc test would we run?
α = .05
Determine which means differ significantly & write up results using APA style & GST
qk =3.95
Aspirin & Placebo differ significantly
Aspirin
Tylenol
Placebo 3 2 5 1 4
Xbar
1.75
Source
Sum of Squares df
Mean square F
Between
10.167 2
5.083
5.229
Within
8.750 9
.972
Total 18 11
Used to describe & communicate how different scores are from each other

18. Tukey’s HSD (SPSS) Multiple Comparisons pain Tukey HSD (I) drug
(J) drug
Mean Difference (I-J)
Std. Error
Sig.
95% Confidence Interval
Lower Bound
Upper Bound 1 2
.69722
.365
-.9466
2.9466 3 *
.025
.3034
4.1966
.9466
.226
-.6966
3.1966 *
-.3034
.6966
*. The mean difference is significant at the 0.05 level.

19. 5. Summary of Steps Initial steps Computations Identify Fcrit
Determine null & alternative hypotheses, choose alpha, check assumptions, collect data
Computations
Sums of squares
dfs
Mean squares
Identify Fcrit
If you reject the null, perform the appropriate post hoc tests
Interpret & write up your results
Used to describe & communicate how different scores are from each other

20. 6. Describing the relationship
Can also calculate confidence intervals
Upper bound: [( MSwn /n)1/2 * +tcrit]+ Xbar
Lower bound: [( MSwn /n)1/2 * -tcrit]+ Xbar
Calculated for every significant condition/mean/group
Can also calculate the effect size (eta squared or η2)
New correlation coefficient
Akin to a squared correlation coefficient
Remember squared correlation coefficient tells you proportion of variance of accounted for
Formula: SSbn / SStot
Used to describe & communicate how different scores are from each other

21. 7. Power F statistic (Fobt) : MSbn / MSwn Maximize power through
Good design
Strong manipulation
Increase numerator
Added control (lessen variability and error)
Decreases denominator
Larger n’s
Increases dfwn
Minimizes MSwn
Decreases size of Fcrit
All of the above increase Fobt
Used to describe & communicate how different scores are from each other