Randomized Block Analysis

I’ve decided to present the statistical model for the Randomized Block Design in regression analysis notation. Here is the model for a case where there are four blocks or homogeneous subgroups.

$$ y_{i}=\beta_{0}+\beta_{1} Z_{1 i}+\beta_{2} Z_{2 i}+\beta_{3} Z_{3 i}+\beta_{4} Z_{4 i}+e_{i} $$

Where:

• yi = outcome score for the i^th unit

• β0 = coefficient for the intercept

• β1 = mean difference for treatment

• β2 = blocking coefficient for block 2

• β3 = blocking coefficient for block 3

• β4 = blocking coefficient for block 4

• Z1i = dummy variable for treatment (0=control, 1=treatment)

• Z2i = 1 if block 2, 0 otherwise

• Z3i = 1 if block 3, 0 otherwise

• Z4i = 1 if block 4, 0 otherwise

• ei = residual for the i^th unit

Notice that we use a number of dummy variables in specifying this model. We use the dummy variable Z1 to represent the treatment group. We use the dummy variables Z2, Z3 and Z4 to indicate blocks 2, 3 and 4 respectively. Analogously, the beta values (b’s) reflect the treatment and blocks 2, 3 and 4. What happened to Block 1 in this model? To see what the equation for the Block 1 comparison group is, fill in your dummy variables and multiply through. In this case, all four Zs are equal to 0 and you should see that the intercept (β0) is the estimate for the Block 1 control group. For the Block 1 treatment group, Z1 = 1 and the estimate is equal to β0 + β1. By substituting the appropriate dummy variable “switches” you should be able to figure out the equation for any block or treatment group.

The data matrix that is entered into this analysis would consist of five columns and as many rows as you have participants: the posttest data, and one column of 0’s or 1’s for each of the four dummy variables.