Interaction
It must be stressed again that the no-interaction model does not always fit factorial data. Sometimes the effect of one factor is different for different levels of the others. Interactions in models with 3 or more factors are however difficult to understand, so we only briefly mention their existence here.
Shrimp culture
Commercial cultivation of the Californian brown shrimp was being planned, so scientists conducted a factorial experiment to assess how shrimp growth was affected by temperature, salinity and the density of shrimps in the tanks. The table below shows the average 4-week gain in weight per shrimp (mg) from the post-larval stage for each combination of factor levels.
80 shrimps/litre | 160 shrimps/litre | ||||
---|---|---|---|---|---|
Salinity | 25°C | 35°C | 25°C | 35°C | |
10% | 73 | 349 | 86 | 364 | |
25% | 482 | 330 | 208 | 316 | |
40% | 397 | 205 | 243 | 281 |
The data are displayed in the diagram below
Click the top three checkboxes to fit the main effects for the factors. Weight gain appears to be highest at 25% salinity, density 80 shrimps per litre and 35°C.
This however does not tell the whole story. Several data points are far from the corresponding fitted value for the model — they have large residuals (the vertical red lines). In particular, at 10% salinity, the weight gains are under-estimated if the temperature is 35°C and over-estimated if the temperature is 25°C.
Click the checkbox for a Temperature-by-salinity interaction. This allows the effect of salinity to be different at the different temperatures. Observe that there is a very different pattern for the two temperatures.