CRD (Complete Randomized Design)

Question 1

What are characteristics of a CRD?

Answer 1

• A single experimental factor with g≥2 levels
• Experimental units are randomly assigned to factor levels
• One measurement of response variable is made on each experimental unit
• (not necessary, but preferred) # of experimental units is the same for each factor level (aka: BALANCED)

Question 2

What is CRD?

Answer 2

Complete Randomized Design

Question 3

In the Means Model, what is Y_ij?

Answer 3

Value of response variable.
Where i = 1 , …, g (the factor level)
and j = 1,…,n (the number of experimental units)

Y_ij is the jth exp. unit in factor level i

Question 4

What is N in the means model?

Answer 4

N = g*n

The overall sample size!

Question 5

What is µ_i in the means model?

Answer 5

Example of a parameter (fixed, but unknown value).

Specifically, it’s the mean for factor level i

Question 6

What is epsilon_ij in the means model?

Answer 6

The residual error!

Question 7

What is the primary interest for the means model?

Answer 7

H_0: µ_1 = … = µ_g
H_A: µ_i ≠ µ_j, for some i and j

Question 8

We can estimate µ_i with…

Answer 8

µ_i-hat, which is equal to Y_i•-bar (the sample mean. 1/n•sum from j=1 to n of Y_ij )

Question 9

Means model pros and cons:

Answer 9

• simple and intuitive
• easy to formulate hypotheses
• obvious parameter estimates for µ_is
• hard to generalize to more than one factor

Question 10

What is the µ and /alpha_i in the effects model?

Answer 10

µ + /alpha_i are the overall or grand mean, and the effect of factor level i (respectively)

Question 11

Number of parameters in effects model?

Answer 11

µ, and /alpha_i, for i = 1,…,g
so there are g+1
But /alpha_g is set equal to 0, which makes g parameters (to be equivalent to the means model)

Question 12

How many groups of data in the effects model?

Answer 12

g (the number of factor levels)

Question 13

What constraint to be place on the effects model?

Answer 13

µ_g = Y_g -bar, and for each i-1,…,g-1, /alpha-hat = Y_i-bar minus Y_g-bar

Question 14

What is the sum-to-zero constraint

Answer 14

sum over all the factor levels of /alpha_i = 0

Question 15

In both the means model and effects model, the parameter estimates are obtained by minimizing

Answer 15

The sum of (Y_ij - Y_ij-hat)^2

aka the square of the residuals

Question 16

What is the principle of least squares, and what properties does it give the estimates?

Answer 16

Principle: Minimizing the sum of the square of the residuals,
Properties: gives minimum bias and variance!

Question 17

What does “SS” stand for?

Answer 17

Sum of Squares

Question 18

SS_(total) = ?

Answer 18

SS_(treatment) + SS_(error)

Question 19

The SS_(total) is proportional to…

Answer 19

the variance of the data

Question 20

SS_(trt) means?

Answer 20

sum of squares due to treatment

or factors in the model

Question 21

SS_E means?

Answer 21

Sum of squares due to error

or residual sum of squares

Question 22

With ANOVA, we have N-1 total degrees of freedom because…

Answer 22

We have to estimate µ! That loses us 1 degree of freedom

Question 23

What is R^2’s calculation?

What does R^2 mean intuitively?

Answer 23

R^2 = SS_(trt)/SS_T
It’s the proportion of variation (in observations) that is explained by differences in factor level (treatment) effects

(compared to the total variation. (variation due to error + explained by model))

Question 24

What does a large R^2 mean?

Answer 24

factor levels CAUSE most of the variability (because this is a CRD)

Question 25

What does a small R^2 mean?

Answer 25

The variability in the test is not really explained by the factor levels

Question 26

If R^2 is close to one, then…

Answer 26

even small differences among the factor levels may be statistically significant (even if it’s inconsequential, or functionally different)

Question 27

In the affects model, what does it mean if /alpha_i is positive?

Answer 27

That the ith factor on average has a greater affect than the typical average