Biomedical Engineering Reference
In-Depth Information
6
5
4
3
2
1
0
0
20
40
60
80
100
x (Rotameter Readings, %)
FIGURE 7.1 Scatter diagram of flow rate values versus rotameter reading from Table 7.1 .
where the slope and intercept of the line are called model parameters regression coefficients.
While the mean of Y or the model for y is a linear function of x , the actual observed value y i
does not fall exactly on a straight line. The deviation of the data from a straight line is due to
the error in experimental observation. The appropriate way to generalize this trend indicated
by the experimental data to a linear model is to assume that the expected value of Y is a linear
function of x , but that for a fixed value of x the actual value of Y is determined by the mean
value function (the linear model) plus a random error term, say,
Y ¼ y þ ε ¼ a 0 þ a 1 x þ ε
(7.2)
where
is the random error term. We will call this model the simple linear regression model,
because it has only one independent variable or regressor, Y . While we have reasonable confi-
dence that Eqn (7.1) is correct qualitatively, we also know for fact that the experimental data
contain error. 1
To gain more insight into this model, suppose that we can fix the value of x (assuming x is
error-free) and observe the value of the random variable Y . Now if x is fixed, the random
component
ε
on the right-hand side of the model in Eqn (7.2) determines the properties
of Y . Suppose that the mean and variance of
ε
2 , respectively. Then
are 0 and
s
ε
EðYjxÞ¼Eða 0 þ a 1 x þ ε Þ¼a 0 þ a 1 x þ Eð ε Þ¼a 0 þ a 1 x
(7.3)
1 Well, the rotameter readings are set visually by manually adjusting the control valve that controls the flow
rate each time. In practice, one often generate a calibration curve based on the experimental data. Assuming
the same person will be performing the flow rate control, one can therefore “effectively” ignore the error for
the rotameter readings. The flow rates were measured by collecting and weighing the out flow water
from the tube using a beaker, and the collection time is measured by a stopwatch.
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