CRD (Complete Randomized Design) Flashcards
What are characteristics of a CRD?
- 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)
What is CRD?
Complete Randomized Design
In the Means Model, what is Y_ij?
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
What is N in the means model?
N = g*n
The overall sample size!
What is µ_i in the means model?
Example of a parameter (fixed, but unknown value).
Specifically, it’s the mean for factor level i
What is epsilon_ij in the means model?
The residual error!
What is the primary interest for the means model?
H_0: µ_1 = … = µ_g
H_A: µ_i ≠ µ_j, for some i and j
We can estimate µ_i with…
µ_i-hat, which is equal to Y_i•-bar (the sample mean. 1/n•sum from j=1 to n of Y_ij )
Means model pros and cons:
- simple and intuitive
- easy to formulate hypotheses
- obvious parameter estimates for µ_is
- hard to generalize to more than one factor
What is the µ and /alpha_i in the effects model?
µ + /alpha_i are the overall or grand mean, and the effect of factor level i (respectively)
Number of parameters in effects model?
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)
How many groups of data in the effects model?
g (the number of factor levels)
What constraint to be place on the effects model?
µ_g = Y_g -bar, and for each i-1,…,g-1, /alpha-hat = Y_i-bar minus Y_g-bar
What is the sum-to-zero constraint
sum over all the factor levels of /alpha_i = 0
In both the means model and effects model, the parameter estimates are obtained by minimizing
The sum of (Y_ij - Y_ij-hat)^2
aka the square of the residuals
What is the principle of least squares, and what properties does it give the estimates?
Principle: Minimizing the sum of the square of the residuals,
Properties: gives minimum bias and variance!
What does “SS” stand for?
Sum of Squares
SS_(total) = ?
SS_(treatment) + SS_(error)
The SS_(total) is proportional to…
the variance of the data
SS_(trt) means?
sum of squares due to treatment
or factors in the model
SS_E means?
Sum of squares due to error
or residual sum of squares
With ANOVA, we have N-1 total degrees of freedom because…
We have to estimate µ! That loses us 1 degree of freedom
What is R^2’s calculation?
What does R^2 mean intuitively?
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))
What does a large R^2 mean?
factor levels CAUSE most of the variability (because this is a CRD)
