1. The Completely Randomized Design (§8.3)
Introduction to the simplest experimental design - the Completely Randomized Design.
Introduce a statistical model for the observations in a completely randomized design.

2. Completely Randomized Design
Two different Names for the Same Design:
Experimental Study - Completely randomized design (CRD)
Sampling Study One-way classification design
Randomization: The t treatments are randomly allocated to the experimental units in such a way that n1 units receive treatment 1, n2 receive treatment 2, etc.
Assumptions:
Independent random samples (response from one experimental unit does not affect responses from other experimental units).
Responses follow a normal distribution.
Common true variance, s2, across all groups/treatments.
True mean for population i is mi.
Interest is in comparing means.

3. AOV Model of Responses/Effects
random error ~ N(0,s2)
overall mean
effect due to population i
Requirement for  to
be the overall mean:
Expected response
Estimate
All ai = 0 implies all groups have the same mean (m)

4. Example
A manufacturer of concrete bridge supports is interested in determining the effect of varying the sand content on the strength of the supports. Five supports are made for each of five different amounts of sand in the concrete mix and each is tested for compression resistance.
Percent Sand 15 20 25 30 35 7 17 14 12 18 24 10 11 22 19 9 23

5. Basic Statistics and AOV Effects
Percent Sand 15 20 25 30 35 7 17 14 12 18 24 10 11 22 19 9 23
9.6
15.4
17.6
21.6
10.8
-5.4
0.4
2.6
6.6
-4.2
Overall Mean
MEAN
EFFECT
Sum of Effects

6. Decomposing the Data
m = overall mean
ai = i – m = group i effect
ij = yij – m – ai = residual
(Note that sum of residuals for each treatment is zero)
Sum of squares

7. Decomposing Sums of Squares
SSB
SSW
6275.0
=650.0
-486.4
=163.6
-163.6
=0.0
TSS
SSB
SSW

9. Best Treatment?
Is 30% significantly better than 25%?

10. Estimation

11. Reference Group/Cell Model
random error ~ N(0,s2)
reference group
mean
effect due to population i
This is the
model SASâ
uses.
Mean for the last group (i=t) is mt.
Mean for the first group (i=1) is mt + b1
Thus, b1 is the difference between the
mean of the reference group (cell) and the
target group mean. Any group can be the
reference group.
All bi = 0 implies all groups have the same mean.

12. Basic Statistics and Reference Cell Effects
Percent Sand 15 20 25 30 35 7 17 14 12 18 24 10 11 22 19 9 23
9.6
15.4
17.6
21.6
10.8
-1.2
4.6
6.8 21
Reference Cell Mean
MEAN
EFFECT
Sum of Effects

13. Reference Cell Decomposition
Note: Sums of squares don’t quite add up.
Due to fact that sum of bi is not zero.
6275.0
=3369.0
-927.4
=2441.6
-163.6
=2278.0

14. Decomposing Sums of Squares
6275 =

15. Reference Cell Model

16. SAS Program options ls=78 ps=49 nodate; data stress;
input sand resistance
datalines; ;
proc glm data=stress;
class sand;
model resistance = sand / solution;
title2 'Compression resistance in concrete beams as';
title2 ' a function of percent sand in the mix';
run;

17. SAS Output(1) Compression resistance in concrete beams as
a function of percent sand in the mix
The GLM Procedure
Dependent Variable: resistance
Sum of
Source DF Squares Mean Square F Value Pr > F
Model <.0001
Error
Corrected Total
R-Square Coeff Var Root MSE resistance Mean

18. SAS Output(2) Source DF Type I SS Mean Square F Value Pr > F
sand <.0001
Source DF Type III SS Mean Square F Value Pr > F
Standard
Parameter Estimate Error t Value Pr > |t|
Intercept B <.0001
sand B
sand B
sand B
sand B <.0001
sand B
NOTE: The X'X matrix has been found to be singular, and a generalized inverse
was used to solve the normal equations. Terms whose estimates are
followed by the letter 'B' are not uniquely estimable.

19. Minitab One-way ANOVA: Resist versus Sand
Analysis of Variance for Resist
Source DF SS MS F P
Sand
Error
Total
Individual 95% CIs For Mean
Based on Pooled StDev
Level N Mean StDev
(----*-----)
(-----*----)
(----*-----)
(----*-----)
(-----*----)
Pooled StDev =

20. Minitab
Stat  ANOVA  One-Way
Multiple comparisons (later)

21. Minitab Dot Plot

22. SPSS AOV Table ANOVA RESIST Sum of Squares df Mean Square F Sig.
Between Groups 4
14.866
.000
Within Groups 20
8.180
Total 24

23. SPSS Descriptives Descriptives RESIST 95% Confidence Interval for Std.
Mean N
Mean
Deviation
Std. Error
Lower Bound
Upper Bound
15.00 5
9.6000
5.5195
20.00 5
25.00 5
.92736
30.00 5
.92736
35.00 5
7.2444
Total 25
Model
Fixed Effects
.57201
Random Effects
8.8767
7.00
15.00
11.00
19.00
14.00
24.00
Minimum
Maximum
Between-
Component
Variance

24. CRD Analysis in R
> resist <- c(7,7,10,15,9,17,12,11,18,19,14, …,19,23,7,10,11,15,11)
> sand <- factor(rep(seq(15,35,5),rep(5,5)))
> myfit <- aov(resist~sand)
> summary(myfit)
Df Sum Sq Mean Sq F value Pr(>F)
sand e-06 ***
Residuals
---
Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1
> coef(myfit)
(Intercept) sand sand sand sand35
R functions aov() & lm() by default reference first cell mean!

25. Fixed Effects
Normally, the “effect” of a particular treatment is assumed to be a constant value (ai) added to the response of all units in the group receiving the treatment.
If the treatments are well defined, easily replicable and are expected to produce the same effect on average in each replicate, we have a fixed set of treatments and the AOV model is said to describe a fixed effects model.
Examples:
A scientist develops 3 new fungicides. Her interest is in these fungicides only.
The impact of 4 specific soil types on plant growth are of interest.
Three particular milling machines are being compared.
Four particular lakes are of interest in their weed biomass densities.
Three tests for assessing developmental learning are being compared.

26. Random Effects
If the treatments cannot be assumed to be from a prespecified or known set of treatments, they are assumed to be a random sample from some larger population of potential treatments. In this case, the AOV model is called a random effects model and the ai are called random effects.
Examples:
A scientist is interested in how fungicides work. Ten (10) fungicides are selected (at random) to represent the population of all fungicides in the research (plots as replicates).
Four soil sub groups are selected for examining plant growth (pots as replicates).
Three milling machines selected at random from the production line are compared (runs as replicates).
16 lakes selected at random are measured for their weed biomass densities (water samples as replicates).
A standard test for development is given to 20 middle school classes selected at random from the over 200 available among all middle schools in the county (student as replicate).
In each case, we assume the values for the effects would change if our sample had changed. Inference is directed not to answering “which treatment is different from which other treatment?” but to the issue of “is the variability among treatments significantly greater than the residual variability?”.

27. Closing Comments on CRD
Even though we have introduced several variations on the same basic model for defining “effects”, the final F-test for the hypothesis of overall equal group means is the same one developed as part of the analysis of variance. It turns out that there may be computational advantages to using the one formulation of the model over another, but this has absolutely no effect on the hypothesis test. We will see this in the next Section.
For simple one-factor designs, whether the treatment effect is considered random or fixed, the F-test is the same, the interpretation is different.