Introductory Biostatistics

2. Whereas ifX¼xþ1,

my¼b 0 þb 1 ðxþ 1 Þ

It can be seen that by taking the di¤erence,b 1 represents theincrease (or
decrease,ifb 1 is negative) in the mean ofYassociated with a 1-unit increase in
the value ofX,X¼xþ1 versusX¼x. For anm-unit increase in the value of
X, sayX¼xþmversusX¼x, the corresponding increase (or decrease) in the
mean ofYismb 1.

8.1.4 Estimation of Parameters

To findgoodestimates of the unknown parametersb 0 andb 1 , statisticians use a
method calledleast squares, which is described as follows. For each subject or
pair of values (Xi;Yi), we consider the deviation from the observed valueYito
itsexpected value, the meanb 0 þb 1 Xi,

Yi
ðb 0 þb 1 XiÞ¼ei

In particular, the method of least squares requires that we consider thesum of
squared deviations:

S¼

Xn

i¼ 1

ðYi
b 0 
b 1 XiÞ^2

According to the method of least squares, the good estimates ofb 0 andb 1 are
valuesb 0 andb 1 , respectively, whichminimizethe sumS. The results are

b 1 ¼

P

xy
ð

P

xÞð

P

yÞ=n

P

x^2 
ð

P

xÞ^2 =n

b 0 ¼y
b 1 x

Given the estimatesb 0 andb 1 obtained from the sample, we estimate the mean
response by

YY^¼b 0 þb 1 X

This is ourpredicted valuefor (the mean of)Yat a given level or value ofX.

Example 8.1 In Table 8.1, the first two columns give the values for the birth
weight (x, in ounces) and the increase in weight between days 70 and 100 of
life, expressed as a percentage of the birth weight (y) for 12 infants. We first let

SIMPLE REGRESSION ANALYSIS 285