1. Experimental Statistics - week 5
Chapters 8, 9: Miscellaneous topics Chapter 14: Experimental design concepts Chapter 15: Randomized Complete Block Design (15.3)

2. yij = mi + eij yij = m + ai + eij 1-Factor ANOVA Model or
observed data
unexplained part
mean for ith treatment

3. were rewritten as:

4. In words:
TSS(total SS) = total sample variability among yij values
SSB(SS “between”) = variability explained by differences in group means
SSW(SS “within”) = unexplained variability (within groups)

5. Analysis of Variance Table
Note: unequal sample sizes allowed

6. 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:

7. ANOVA Table Output – extracted hostility data - calculations done in class
Source SS df MS F p-value
Between <.001
  samples
Within
Totals

8. 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).

9. Hostility Data - Completely Randomized Design The GLM Procedure
t Tests (LSD) for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of t
Least Significant Difference
Means with the same letter are not significantly different.
t Grouping Mean N method A B B B

10. Notice unequal sample sizes
Ex. 8.2, page
3 Methods for Reducing Hostility
24 students displaying similar hostility were randomly assigned to 3 treatment methods. Scores (HLT) at end of study recorded.
Method
Method
Method
Notice unequal sample sizes
Test:

11. ANOVA Table Output – full hostility data
Source SS df MS F p-value
Between <.0001
  samples
Within
Totals

12. The GLM Procedure
t Tests (LSD) for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of t
Comparisons significant at the 0.05 level are indicated by ***.
Difference
method Between % Confidence
Comparison Means Limits
***
***
***
***
***
***
Notice the different format since there is not one LSD value with which to make all pairwise comparisons.

13. Duncan's Multiple Range Test for score
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
Error Degrees of Freedom
Error Mean Square
Harmonic Mean of Cell Sizes
NOTE: Cell sizes are not equal.
Number of Means
Critical Range
Means with the same letter are not significantly different.
Duncan Grouping Mean N method A B C
Note: Duncan’s test (another multiple comparison test) avoids the issue of different sample sizes by using the harmonic mean of the ni’s.

14. Some Multiple Comparison Techniques in SAS
FISHER’S LSD (LSD)
BONFERONNI (BON)
DUNCAN
STUDENT-NEWMAN-KEULS (SNK)
DUNNETT   
RYAN-EINOT-GABRIEL-WELCH (REGWQ)
SCHEFFE
TUKEY

15. Balloon Data Col. 1-2 - observation number
Col observation number
Col color (1=pink, 2=yellow, 3=orange, 4=blue)
Col inflation time in seconds
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2

16. Balloon Data Col. 1-2 - observation number
Col observation number
Col color (1=pink, 2=yellow, 3=orange, 4=blue)
Col inflation time in seconds
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2

17. ANOVA --- Balloon Data
General Linear Models Procedure
Dependent Variable: TIME
Sum of Mean
Source DF Squares Square F Value Pr > F
Model
Error
Corrected Total
R-Square C.V Root MSE TIME Mean
Mean
Source DF Type I SS Square F Value Pr > F
Color

18. ANOVA --- Balloon Data The GLM Procedure t Tests (LSD) for time
NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate.
Alpha
Error Degrees of Freedom
Error Mean Square
Critical Value of t
Least Significant Difference
Means with the same letter are not significantly different.
t Grouping Mean N color A A A B B B

19. Experimental Design: Concepts and Terminology
Designed Experiment
- an investigation in which a specified framework is used to compare groups or treatments
Factors
- any feature of the experiment that can be varied from trial to trial
- up to this point we’ve only looked at experiments with a single factor

20. - these are called replicates
Treatments
- conditions constructed from the factors (levels of the factor considered, etc.)
Experimental Units
- subjects, material, etc. to which treatment factors are randomly assigned
- there is inherent variability among these units irrespective of the treatment imposed
Replication
- we usually assign each treatment to several experimental units
- these are called replicates

21. Examples: 1. factor 2. treatments 3. experimental units 4. replicates
Car Data
Hostility Data
Balloon Data

22. Question: Balloon Data
1122.4
2324.6
3120.3
4419.8
5324.3
6222.2
7228.5
8225.7
9320.2
Balloon Data
Col observation number (run order)
Col color (1=pink, 2=yellow, 3=orange, 4=blue)
Col inflation time in seconds
Question:
Why randomize run order? i.e. why not blow up all the pink balloons first, blue balloons next, etc?

23. Scatterplot Using GPLOT What do we learn from this plot?
Time
Run Order
What do we learn from this plot?

24. RECALL: 1-Factor ANOVA Model
- random errors follow a Normal (N) distribution, are independently distributed (ID), and have zero mean and constant variance
-- i.e. variability does not change from group to group

25. Checking Validity of Assumptions
Model Assumptions:
- equal variances
- normality
Checking Validity of Assumptions
Equal Variances
1. F-test similar to 2-sample case
- Hartley’s test (p.366 text)
- not recommended
2. Graphical
- side-by-side box plots

26. Graphical Assessment of Equal Variance Assumption

27. Note: Optional approaches if equal variance assumption is violated:
1. Use Kruskal Wallis nonparametric procedure – Section 8.6
2. Transform the data to induce more nearly equal variances – Section 8.5
-- log
-- square root
Note: These transformations may also help induce normality

28. Assessing Normality of Errors The e ij’s are called residuals.
yij = m + ai + eij so
eij = yij - (m + ai)
= yij - mi
eij is estimated by
The e ij’s are called residuals.

29. SAS Code for Balloon Data
proc glm;
class color;
model time=color;
title 'ANOVA --- Balloon Data';
output out=new r=resball;
means color/lsd;
run;
proc sort;
by color;
proc boxplot;
plot time*color;
title 'Side-by-Side Box Plots for Balloon Data';
proc univariate;
var resball;
histogram resball/normal;
title 'Histogram of Residuals -- Balloon Data';
proc univariate normal plot;
title 'Normal Probability Plot for Residuals - Balloon Data';
proc gplot;
plot time*id;
title 'Scatterplot of Time vs ID (Run Order)';

30. Normal Probability Plot*+
| * *+++
| *+++
| *+
| ***
| ****
***+
| **
| ***
| *****
| *+
| *+*+*
* ++++

31. Caution: Chapter 15 introduces some new notation
- i.e. changes notation already defined

32. Recall: Sum-of-Squares Identity 1-Factor ANOVA
In words:
Total SS = SS between samples + within sample SS

33. - new notation for Chapter 15
Recall: Sum-of-Squares Identity Factor ANOVA
- new notation for Chapter 15

34. - new notation for Chapter 15
Recall: Sum-of-Squares Identity Factor ANOVA
- new notation for Chapter 15

35. - new notation for Chapter 15
Recall: Sum-of-Squares Identity Factor ANOVA
- new notation for Chapter 15
In words:
Total SS = SS for “treatments” + SS for “error”

36. Revised ANOVA Table for 1-Factor ANOVA (Ch. 15 terminology - p.857)
Source SS df MS F
Treatments SST t - 1
Error SSE N - t
  Total TSS N - 1

37. yij = mi + eij yij = m + ai + eij
Recall 1-factor ANOVA (CRD) Model for Gasoline Octane Data
yij = mi + eij or
yij = m + ai + eij
observed octane
mean for ith gasoline
unexplained part
-- car-to-car differences
-- temperature
-- etc.

38. Similar to diet t-test example:
Gasoline Octane Data
Question:
What if car differences are obscuring gasoline differences?
Similar to diet t-test example:
Recall: person-to-person differences obscured effect of diet

39. Possible Alternative Design for Octane Study:
Test all 5 gasolines on the same car
- in essence we test the gasoline effect directly and remove effect of car-to-car variation
Question:
How would you randomize an experiment with 4 cars?

40. Blocking an Experiment
- dividing the observations into groups (called blocks) where the observations in each block are collected under relatively similar conditions
- comparisons can many times be made more precisely this way

41. Terminology is based on Agricultural Experiments
Consider the problem of testing fertilizers on a crop
- t fertilizers
- n observations on each

42. Completely Randomized DesignB A A C B B A C A C C B C A
t = 3 fertilizers
n = 5 replications B
- randomly select 15 plots
- randomly assign fertilizers to the 15 plots

43. Randomized Complete Block Strategy
A | C | B
B | A | C
C | B | A
A | B | C
t = 3 fertilizers
C | A | B
- select 5 “blocks”
- randomly assign the 3 treatments to each block
Note: The 3 “plots” within each block are similar
- similar soil type, sun, water, etc

44. Randomized Complete Block Design
Randomly assign each treatment once to every block
Car Example
Car 1: randomly assign each gas to this car
Car 2:
etc.
Agricultural Example
Randomly assign each fertilizer to one of the 3 plots within each block

45. Model For Randomized Complete Block (RCB) Design
yij = m + ai + bj + eij
effect of ith treatment
effect of jth block
unexplained error
(gasoline)
(car)
-- temperature
-- etc.

46. Previous Data Table from Chapter 8 for 1-factor ANOVA
column averages don’t make any sense

47. Back to Octane data: “Restructured” Data Car Old Data Format Gas Gas
Suppose that instead of 20 cars, there were only 4 cars, and we tested each gasoline on each car.
“Restructured” Data
Car
Old Data Format A B C D E A B C D E
Gas
Gas

48. - using new notation for Chapter 15
Recall: Sum-of-Squares Identity Factor ANOVA
- using new notation for Chapter 15
In words:
Total SS = SS for “treatments” + SS for “error”

49. A New Sum-of-Squares Identity
In words:
Total SS = SS for treatments + SS for blocks + SS for error

50. Hypotheses: To test for treatment effects - i.e. gas differences
we test
To test for block effects
- i.e. car differences
(not usually the research hypothesis)
we test

51. Randomized Complete Block Design ANOVA Table
Source SS df MS F
Treatments SST t - 1
Blocks SSB
Error SSE
  Total TSS bt - 1
See page 866

52. Test for Treatment Effects
Note:

53. Test for Block Effects

54. “Restructured” CAR Data - SAS Format
A B
A B
A B
A B
B B
B B
B B
B B
C B
C B
C B
C B
D B
D B
D B
D B
E B
E B
E B
E B
The first variable (A - E) indicates gas as it did with the Completely
Randomized Design. The second variable (B1 - B4) indicates car.

55. SAS file - Randomized Complete Block Design for CAR Data
INPUT gas$ block$ octane;
PROC GLM;
CLASS gas block;
MODEL octane=gas block;
TITLE 'Gasoline Example -Randomized Complete Block Design';
MEANS gas/LSD;
RUN;

56. 1-Factor ANOVA Table Output - octane data
Source SS df MS F p-value
Gas
 (treatments)
Error  
Totals

57. 1-Factor ANOVA Table Output - car data
Source SS df MS F p-value
Gas
 (treatments)
Cars (blocks)
Error  
Totals

58. SAS Output -- RCB CAR Data
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 Anova SS Mean Square F Value Pr > F
GAS
BLOCK

59. Multiple Comparisons in RCB Analysis

60. CAR Data -- LSD Results CRD Analysis RCB Analysis
t Grouping Mean N gas
A E
B D B
C B C
C B
C B B C
C A
RCB Analysis
t Grouping Mean N gas
A E
B D B
C B C C
C B
C A 

61. CAR Data -- Bonferroni Results
CRD Analysis
  Bon Grouping Mean N gas
A E A
B A D B
B C
B B
B A
RCB Analysis
Bon Grouping Mean N gas
A E
B D B
B C
B B
B A