Introductory Biostatistics

rl¼

expð
 4 : 892 Þ
 1

expð
 4 : 892 Þþ 1

¼
 0 : 985

ru¼

expð
 2 : 278 Þ
 1

expð
 2 : 278 Þþ 1

¼
 0 : 814

or a 95% confidence interval for the population coe‰cient of correlation of
(
0 : 985 ;
0 :814), indicating a very strong negative association between a
baby’s birth weight and his or her increase in weight between days 70 and 100
of life; that is, smaller babies are likely to grow faster during that period (that
may be why, at three months, most babies look the same size!).

4.6 BRIEF NOTES ON THE FUNDAMENTALS

Problems in the biological and health sciences are formulated mathematically
by considering the data that are to be used for making a decision as the
observed values of a certain random variableX. The distribution ofX is
assumed to belong to a certain family of distributions specified by one or sev-
eral parameters; examples include the normal distribution, the binomial dis-
tribution, and the Poisson distribution, among others. The magnitude of a
parameter often represents the e¤ect of a risk or environmental factor and
knowing its value, even approximately, would shed some light on the impact of
such a factor. The problem for decision makers is to decide on the basis of the
data which members of the family could represent the distribution ofX, that is,
to predict or estimate the value of the primary parametery.

Maximum Likelihood Estimation The likelihood functionLðx;yÞfor random
samplefxigof sizenfrom the probability density function (pdf)fðx;yÞis

Lðx;yÞ¼

Yn

i¼ 1

fðxi;yÞ

The maximum likelihood estimator (MLE) ofyis the valueyy^for whichLðx;yÞ
is maximized. Calculus suggests setting the derivative ofLwith respect toy
equal to zero and solving the resulting equation. We can obtain, for example:

1. For a binomial distribution,

Lðx;pÞ¼

n

x



pxð 1 
pÞn
x

BRIEF NOTES ON THE FUNDAMENTALS 171