Clinical Trials

Clinical Trials: A Practical Guide ■❚❙❘If a variable X follows N(μ,σ 2 ) then it can be mathematically transformed into anew variable Z with a standard normal distribution N(0,1) by the following formula:Z = X – μσZ is sometimes called the standard normal deviate [2]. Take the SBP data asan example:as SBP follows N(130,15 2 ), then Z = SBP – 130 follows the N(0,1)15The transformation does not alter the shape of the distribution. All that happens is:• The standardized mean takes the value 0 instead of 130 mm Hg,with lower values to the left and higher values to the right.• The horizontal units are now standard deviations: +1 means onestandard deviation away from the mean on the right.Calculating the area under the curveWe can see from the last section that any normal distribution is linked to thestandard normal distribution through a proper transformation. We can use thisrelationship to calculate some very useful statistics through the standard normaldistribution table. Table 3 gives areas in the tail of the standard normaldistribution for some selected Z-values: the rows of the table refer to Z to onedecimal place, and the columns to the second decimal place. The table shows theproportion of the area lying on the right (or upper tail) for each Z-value of thestandard normal distribution. We will now use the SBP data (SBP followsN[130,15 2 ]) to demonstrate four common calculations performed on such data.Area under the curve in the upper tailQuestion 1: What proportion of subjects have an SBP above 160 mm Hg?The above statistic can be computed in two steps:Step 1: Obtain the standardized Z-value: Z = (160 – 130) / 15 = 2.00.Step 2:Obtain a value of area corresponding to a Z-value of 2.00, by referringto Table 3, which states that the proportional area to the right of thatZ-value is 0.0228.Converting the proportion to a percentage, we can answer that about 2.28% ofsubjects will have an SBP above 160 mm Hg.179