Model allows us to estimate response distribution at any X

Bivariate data sets contain response measurements corresponding to a few specific values of X, whereas a normal linear model provides a response distribution for all X. By fitting a normal linear model to the data, it is therefore possible to estimate the response distribution at x-values for which we do not have data.

Oxygen intake and heart rate

In athletes, oxygen intake (VO2) and heart rate both increase when exercise is increased. Sports scientists are extremely interested in monitoring VO2, but it is a much harder quantity to measure than heart rate, so an experiment was conducted to examine the relationship between heart rate and VO2 for a single individual. Both variables were recorded for the individual at various work loads.

Oxygen intake Heart rate   Oxygen intake Heart rate
0.473
0.753
0.929
0.939
0.832
0.983
1.049
1.178
1.176
1.292
94
96
95
95
94
95
94
104
104
106
 
1.403
1.499
1.529
1.599
1.749
1.746
1.897
2.040
2.231
 
108
110
113
113
118
115
121
127
131
 

The relationship between oxygen intake and heart rate seems to be reasonably linear, so we will try to fit a normal linear model to the data. The least squares line is shown in the diagram below with the grey band representing ± twice the estimate of σ.

Drag the slider to display the estimated normal distribution of VO2 at each value of X, the heart rate.

The mean of this estimated distribution (i.e. the least squares line) provides a prediction of oxygen intake, Y at any heart rate, X.