Differences between blocks and treatments

For randomised block data, the response measurement depends on both the treatment and block membership. There are major differences between the concepts of blocks and treatments:

Similarities between blocks and treatments

However from a model-building perspective, blocks and treatments are modelled in the same way. We assume that the response is affected by both the block and the treatment in an additive way:

y = (overall mean) + (effect depending on block)
         + (effect depending on treatment) + error

As in earlier models, the error is assumed to have a normal distribution with mean zero and constant standard deviation.

So we assume that changing the treatment simply adds or subtracts a constant to the response. And differences between block membership also result in a constant change to the mean response.

Three-dimensional scatterplot

The relationship between the response, blocks and treatments can be displayed in a three-dimensional scatterplot with these three variables on the three axes. This emphasises the similarity in the way that blocks and treatments are modelled.

Codeine and acupuncture for dental pain relief

The diagram below shows the pain relief score for the 32 subjects who were each given one of four treatments (combinations of codeine and acupuncture). They were grouped into blocks of four according to an initial assessment of their pain tolerance and the treatments were randomly allocated to the four subjects in each block. (I.e. a randomised block design was used.)

The diagram is 3-dimensional. Position the mouse in the middle of the diagram and drag towards the bottom right to rotate the diagram. Rotate a bit until you get a feel for how pain relief depends on both the block and treatment.

Click Colour different blocks to join the crosses with coloured lines. This may help to understand the effects of the blocks and treatments on the response.