Principles and Practice of Pharmaceutical Medicine

statistician to test the null hypothesis H 0 :
¼
pd
pp¼0 against the alternative hypothesis
H 1 :
¼pd
pp>0.
This very simple model provides sufficient
structure for the statistician to design a statistical
test to test these hypotheses. As was discussed in
Section 25.3 above, the statistical test is a device
providing a rule for decision-making associated
with possible errors. The study design must be
such that the error probabilities are properly con-
trolled. In other words, the researcher must
decide on acceptable levels ofaandb, the prob-
abilities of type I and type II errors. Typically,ais
chosen to be 0.05, or 5% andbbetween 0.05 and
0.20, depending on how serious the consequences
are of committing a type II error (‘False Posi-
tive’). As the type II error probability is calcu-
lated under the assumption that the alternative
hypothesis is true, it depends on the value of
.
The investigator must specify a value of
for
which the type II error should be calculated. This
value is thesmallest clinically important
.In
our example, the clinician might consider an
increase in the probability of response, of less
than 50% not clinically meaningful. So, if it is
known that 15% of patients treated with placebo
report the disappearance of their headache,

¼ 0 :075, or 7.5%. Using the model and this
information, the statistician can calculate the
number of subjects required in the trial to guar-
antee that the statistical test will have the desired
power, say 90%, to detect this increase if it is true,
while maintaining the type I error below a desired
low level, say 5%.
Another commonly used statistical model is the
Linear Model, which represents a family of models
of a similar structure. The most commonly
employed linear model is the analysis of variance
model (ANOVA). We shall illustrate this model
using the simplest case, the one-way ANOVA
model.
The model is used to describe continuous data
such as blood pressure. The model assumes that the
observed variable of interestY(e.g. diastolic blood
pressure) can be expressed as a sum of a number of
factors:

Y¼þtþe ð 1 Þ

Heremrepresents the overall mean diastolic blood

pressure in the population under study,trepresents

the increase (or decrease) of the blood pressure due

to treatment anderepresents a random error. The

model makes two additional assumptions:

(a)ebehaves like a Normal (Gaussian) variable

with mean zero and some (unknown) standard

deviation, and

(b) the measurements obtained from different sub-

jects are independent of each other.

The quantitiesmandtare called themodel para-

meters, sometimes referred to as theindependent

variables. There is one additional parameter in

this model which is the standard deviation of the

random errore. It is not explicitly evident from

Equation (1) above but is implicit in assumption

(a). The model parameters are unknown quantities

that must be estimated from the data. The data here

are represented by the symbol Y, sometimes

referred to as thedependent variable. The relation-

ship between the data and the model parameters is

expressed by the linear equation (1), hence the

name Linear Model.

Linear models can be quite complicated when

additional structure, parameters and assumptions

are introduced. For example, one may include

another termcin the model to account for the effect

of the investigator (center) on the measurements, or

another parametert*cto account for the interaction

between the treatment and the investigator effect.

We will discuss this important parameter in some

detail in Section 25.12 below.

There are two common features to all linear

models: The relationship between the data and

the model parameters is always assumed to be

linear, and the errors are assumed to be Normal.

It is important to remember that all the statisti-

cian’s quantitativework and calculations are model

dependent. That is, their application to real life

depends on the extent to which the model assump-

tions are satisfied in reality. Much of the work the

statistician does in planning the trial, in discussing

the nature of the efficacy and safety variables,

randomizing, blinding and so forth, is expressed

in the model. Obviously, the more complex the

326 CH25 STATISTICAL PRINCIPLES AND APPLICATION IN BIOPHARMACEUTICAL RESEARCH