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)
What does a small R^2 mean?
The variability in the test is not really explained by the factor levels
If R^2 is close to one, then…
even small differences among the factor levels may be statistically significant (even if it’s inconsequential, or functionally different)
In the affects model, what does it mean if /alpha_i is positive?
That the ith factor on average has a greater affect than the typical average
