Response distribution at each X
In experiments, the values of the explanatory variable, X, are controlled by the experimenter. Several response measurements are often made at each distinct value of X.
At any single value of X, the repeated response measurements can be considered as a univariate data set and can be modelled as a random sample from some distribution — commonly a normal distribution. The characteristics of the distribution will often depend on the value of X.
The collection of distributions of Y at different values of X comprise a model for the complete bivariate data set called a regression model.
Antibiotic effectiveness
An experiment was conducted to assess how well an antibiotic, polymyxin B, killed the bacterium, Brucella bronchiseptica. Petri dishes containing the bacterium (grown in agar) were used and different doses of the antibiotic were added. The diameter of the area cleared around each addition point was recorded. Each of 6 different doses of antibiotic was used 10 times.
The diagram below shows the resulting data. The crosses have been jittered a little (randomly moved) to separate them in the scatterplot.
This diagram is 3-dimensional. Position the mouse in the middle of the diagram and drag towards the top left of the screen to rotate the plot (or click the 3D rotation button). The histogram at each x-value describes the distribution of cleared diameters at that dose of antibiotic.
Possible model for antibiotic effectiveness
The next diagram shows a possible model for the data above— a normal distribution for each dose of antibiotic (X).
You may use the mouse (or the buttons at the top right) to rotate the 3-dimensional diagram. Click Take sample to show a random sample of 10 values from each of these normal distributions. Our model claims that the observed data are a data set of this form.