Explained variation
As explained more fully in the introductory chapter, the concepts of explained and unexplained variation in the response measurement are the basis of analysis of experimental data.
In experiments with a single factor, the explained variation is summarised by differences between the response means at the different factor levels (treatment means).
A table of treatment means is a good summary of the effect of the factor.
Are the differences real?
Although a table of treatment means provides a good summary, it is incomplete without taking into account the unexplained variation in the data.
Could differences between the means have arisen by chance?
We address this question in later pages of the chapter.
The following experiments were described more fully in a previous page. In each case, the mean reponses for the treatments summarise their differences.
Rice yield and insecticides
Six different insecticides and a control treatment were each randomly allocated to 4 out of 28 plots and the rice yield was recorded from each plot.
The variation between the 7 treatment means (6 insecticides and control) is explained by differences between the insecticides. The variation of yields within each treatment group (e.g. between 1,077 and 1,510 for the control treatment) is unexplained by differences between the insecticides.
Lamb weight gain and hormones
An experiment was conducted to determine the effects of three different hormones, each at a single dose, on the weight gained by lambs. The three hormones and a control treatment (no hormone) were randomly allocated to 16 lambs.
In this experiment, there is considerable unexplained variation — with the hormone kept constant, the range of weight gains from lamb to lamb is about 15 lb/day. From these data, it is unclear whether the difference between the four means (explained variation) could have arisen by chance — a formal test is required.
Effect of copper on aquatic animals
An experiment was conducted to determine how copper concentrations in water affect the lifetime of Daphnia magna, a small aquatic animal. The lifetimes (days) were recorded from fourteen daphnia that were kept in separate containers containing water with different copper levels.
The overall variation in lifetimes (from the smallest value, 25 days, to the largest value, 90 days) is partly caused by differences in copper concentration. The differences between the three treatment means (39.0 to 76.5) describe the variation that is explained by the treatment. The variation of lifetimes within each treatment group (e.g. between 60 and 90 days when there is no copper) is unexplained by copper concentration.