# 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:*

`y`

= outcome score for the_{i}`i`

^{th}unit`β`

= coefficient for the_{0}*intercept*`β`

= mean difference for treatment_{1}`β`

= blocking coefficient for block 2_{2}`β`

= blocking coefficient for block 3_{3}`β`

= blocking coefficient for block 4_{4}`Z`

= dummy variable for treatment (_{1i}`0`

=control,`1`

=treatment)`Z`

=_{2i}`1`

if block 2,`0`

otherwise`Z`

=_{3i}`1`

if block 3,`0`

otherwise`Z`

=_{4i}`1`

if block 4,`0`

otherwise`e`

= residual for the_{i}`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 `Z`

, _{2}`Z`

and _{3}`Z`

to indicate blocks 2, 3 and 4 respectively. Analogously, the beta values (_{4}`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 (`β`

) is the estimate for the Block 1 control group. For the Block 1 treatment group, _{0}`Z`

= 1 and the estimate is equal to _{1}`β`

+ _{0}`β`

. By substituting the appropriate dummy variable “switches” you should be able to figure out the equation for any block or treatment group._{1}

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.