1. Experimental Statistics - week 3
Chapter 8: Inferences about More Than 2 Population Central Values

2. PC SAS on Campus Library BIC Student Center SAS Learning Edition $125

3. Hypothetical Sample Data
Scenario A
Scenario B
Pop Pop 2
Pop Pop 2
For one scenario, | t | = 1.17 For the other scenario, | t | = 3.35

4. In general, for 2-sample t-tests:
To show significance, we want the difference between groups to be ___________ compared to the variability within groups

5. Completely Randomized Design 1-Factor Analysis of Variance (ANOVA)
Setting (Assumptions):
- t populations
- populations are normal
denote the mean and variance
of the ith population
- mutually independent random samples are taken from the populations
- the sample sizes to not have to all be equal

6. 1-Factor ANOVA m1 s m2 s mk s
. . .

7. Question: Notes: - not directional i.e. no “1-sided / 2-sided” issues
- alternative doesn’t say that all means are distinct

8. Completely Randomized Design 1-Factor Analysis of Variance
Example data setup where t = 5 and n = 4

9. Notation:

10. A Sum-of-Squares Identity
Note: This is for the case in which all sample sizes are equal ( n )
The 3 sums of squares measure:
- variability between samples
- variability within samples
- total variability
Question: Which measures what?

11. In words: Total SS = SS between samples + within sample SS where
TSS(total SS) = total sample variability
SSB(SS between samples) = variability due to factor effects
SSW(within sample SS) = variability due to uncontrolled error
Note: Formula for unequal sample sizes given on page 388

12. Pop
Pop
Pop

13. Pop
Pop
Pop

14. Recall: For 2-sample t-test, we tested using
To show significance, we want the difference between groups
compared to the variability within groups

15. Note:
Our test statistic for testing
will be of the form
This has an F distribution
Question: What type of F values lead you to believe the null is NOT TRUE?

16. Analysis of Variance Table
Note:

17. Note:

18. CAR DATA Example
For this analysis, 5 gasoline types (A - E) were to be tested. Twenty cars were selected for testing and were assigned randomly to the groups (i.e. the gasoline types). Thus, in the analysis, each gasoline type was tested on 4 cars. A performance-based octane reading was obtained for each car, and the question is whether the gasolines differ with respect to this octane reading.  
  A
91.7
91.2
90.9
90.6 B
91.7
91.9
90.9 C
92.4
91.2
91.6
91.0 D
91.8
92.2
92.0
91.4 E
93.1
92.9
92.4
means

19. ANOVA Table Output - car data
Source SS df MS F p-value
Between
  samples
Within
Totals

20. F-table -- p.1106

21. Extracted from From Ex. 8.2, page 390-391
3 Methods for Reducing Hostility
12 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded.
Method
Method
Method
Test:

22. ANOVA Table Output - hostility data
Source SS df MS F p-value
Between
  samples
Within
Totals

23. SPSS ANOVA Table for Hostility Data
    
SPSS ANOVA Table for Hostility Data

24. ANOVA Models Note: Example: Population has mean m = 5.
Consider the random sample

25. For 1-factor ANOVA

26. Alternative form of the 1-Factor ANOVA Model
General Form of Model:
Alternative form of the 1-Factor ANOVA Model
(pages )
- random errors follow a Normal distribution, are independently distributed, and have zero mean and constant variance
-- i.e. variability does not change from group to group

28. Analysis of Variance Table
Recall:
Note:
- if no factor effects, we expect F _____
- if factor effects, we expect F _____

29. The CAR data set as SAS needs to see it:   A A A A B B B C C C C D D D D E E E

30. SAS file for CAR data Case 1: Data within SAS FILE : DATA one;
DATA one;
INPUT gas$ octane;
DATALINES; A A . E ;
PROC GLM;
CLASS gas;
MODEL octane=gas;
TITLE 'Gasoline Example - Completely Randomized Design';
MEANS gas;
RUN;
PROC MEANS mean var;
class gas;

31. The SAS Output for CAR data:
Gasoline Example - Completely Randomized Design
General Linear Models Procedure
Dependent Variable: OCTANE
Sum of Mean
Source DF Squares Square F Value Pr > F
Model
Error
Corrected Total
R-Square C.V Root MSE OCTANE Mean   
Source DF Type I SS Mean Square F Value Pr > F
GAS
Textbook Format for ANOVA Table Output - car data
Source SS df MS F p-value
Between
  samples
Within
Totals

32. Problem 1. Descriptive Statistics for CAR Data
The MEANS Procedure
Analysis Variable : octane
Mean Std Dev Minimum Maximum
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33. Problem 3. Descriptive Statistics by Gasoline  
gas=A
  The MEANS Procedure
  Analysis Variable : octane
  Mean Std Dev Minimum Maximum
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   gas=B
   gas=C
Mean Std Dev Minimum Maximum
  gas=D
Analysis Variable : octane
gas=E

35. Question 1: Which gasolines are different?
Question 2: Why didn’t we just do t-tests to compare all combinations of gasolines?
i.e. compare
A vs B
A vs C
D vs E

36. Simulation:
i.e. using computer to generate data under certain known conditions and observing the outcomes

37. Setting: Simulation Experiment: Question:
Normal population with: m = 20 and s = 5
Simulation Experiment:
Generate 2 samples of size n = 10 from this population and run t-test to compare sample means.
i.e test:
Question:
What do we expect to happen?

38. (which is what we expected)
Simulation Results:
t-test procedure: (a = .05)
Reject H0 if | t | > 2.101
t = .235 so we do not reject H0
(which is what we expected)

39. Simulation results: Now - suppose we obtain 10 samples and test
Note: Comparing means 4 vs 5 we get t = 2.33
-- i.e. we reject the null
(but it’s true!!)

40. Suppose we run all possible t-tests at significance level a =
Suppose we run all possible t-tests at significance level a = .05 to compare 10 sample means of size n = 10 from this population
- it can be shown that there is a 63% chance that at least one pair of means will be declared significantly different from each other
F-test in ANOVA controls overall significance level.

41. Probability of finding at least 2 of k means significantly different using multiple t-tests at the a = .05 level when all means are actually equal.
k Prob.

42. Fisher’s Least Significant Difference (LSD)
Protected LSD: Preceded by an F-test for overall significance.
Only use the LSD if F is significant. X
Unprotected: Not preceded by an F-test (like individual t-tests).

43. Gasoline Example - Completely Randomized Design -- All 5 Gasolines
The GLM Procedure
Dependent Variable: octane
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE octane Mean
Source DF Type I SS Mean Square F Value Pr > F
gas