an introduction to generalized linear models - GDM@FUDAN ...

Central F
Non-central F
Figure 2.5 Central and non-central F distributions.
tion. However, as σ 2 is unknown, we cannot compare ( � S0 − � S1)/σ 2 directly
with the χ 2 (J − 1) distribution. Instead we eliminate σ 2 by using the ratio
of( � S0 − � S1)/σ 2 and the random variable � S1/σ 2 with a central chi-squared
distribution, each divided by the relevant degrees offreedom,
F = ( � S0 − � S1)/σ 2
(J − 1)
/
�S1/σ 2
(JK − 2J) = ( � S0 − � S1)/(J − 1)
�S1/(JK − 2J)
IfH0 is correct, from Section 1.4.4, F has the central distribution F (J −
1,JK − 2J). IfH0 is not correct, F has a non-central F -distribution and
the calculated value of F will be larger than expected from the central F -
distribution (see Figure 2.5).
For the example on birthweight and gestational age, the value of F is
(658770.8 − 652424.5)/1
652424.5/20
=0.19
This value is certainly not statistically significant when compared with the
F (1, 20) distribution. Thus the data do not provide evidence against the hypothesis
H0 : β0 = β1, and on the grounds ofsimplicity model (2.6), which
specifies the same slopes but different intercepts, is preferable.
These two examples illustrate the main ideas and methods ofstatistical
modelling which are now discussed more generally.
2.3 Some principles of statistical modelling
2.3.1 Exploratory data analysis
Any analysis ofdata should begin with a consideration ofeach variable separately,
both to check on data quality (for example, are the values plausible?)
and to help with model formulation.
1. What is the scale ofmeasurement? Is it continuous or categorical? Ifit
© 2002 by Chapman & Hall/CRC
.