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.