Ординатура / Офтальмология / Английские материалы / Using and Understanding Medical Statistics_Matthews, Farewell_2007

repeatedly in chapters 8 and 9. To indicate that Z is a standardized normal random variable we simply write Z N(0, 1).

An important fact which we will often use is the following:

if X ~ N( , 2 ), then X N(0, 1).

Therefore, the probability distribution of (X – )/ is exactly the same as the probability distribution of Z, and so we write Z = (X – )/ . The formula asserts that Z is the separation, measured in units of , between X and ; the sign of Z indicates whether X lies to the left (Z ! 0) or right (Z 1 0) of . This relationship is best appreciated by comparing the probability curves for X N( , 2) and Z N(0, 1). Notice that the shape of the curve for X, shown in figure 8.2b, corresponds exactly with the curve for Z shown in figure 8.1b, except that the center of symmetry is located at rather than at zero, and each major division on the horizontal axis is one standard deviation, , rather than one unit. In figure 8.2b, the equal probabilities which are a result of the symmetry about (see the shaded areas) correspond to the probabilities Pr(X 1 + a ) on the right and Pr(X ! – a ) on the left. And by comparing figures 8.1b and 8.2b we can also see that

Pr(Z 1 a) = Pr(X 1 + a ) and Pr(Z ! –a) = Pr(X ! – a ).

This equality between probabilities for X N( , 2) and probabilities for Z N(0, 1) means that probabilities for X can be calculated by evaluating the equivalent probabilities for the standardized normal variable Z. In fact, since Pr(Z 1 a) = Pr(X 1 + a ), it follows that

Pr(Z > a) = Pr(X >µ+a )

= Pr(X µ>a ) [subtract from both sides of the inequality] = Pr X µ >a . [divide both sides of the inequality by ]

This illustrates, once more, the fact that

X µ =Z N(0, 1),

i.e., the random variable X– has a standardized normal distribution. This relationship, Z = X– , is usually called the standardizing transformation because it links the distribution of X N( , 2) to the distribution of Z N(0, 1). In fact, the concept of the standardizing transformation is fundamental to the calculation of normal probabilities and to the use of the normal distribution in the advanced methodology we intend to discuss in chapters 10 through 15.

In §8.1, we indicated that the nature of medical data often precludes the use of methods which assume that the observations, e.g., survival times, are

In chapter 13, we discuss a method for estimating the ratio of the separate mortality rates for patients presenting with A and B symptoms. This method leads to an estimate that the mortality rate for patients with B symptoms is 3.0 times that of patients with A symptoms. The actual calculations which are involved concern the logarithm of this ratio of mortality rates, which we repre-

ˆ

sent by b, and lead to an estimate of b which we denote by b = log 3.0 = 1.10. Simultaneously, the calculations also yield ˆ, the estimated standard deviation

ˆ standard error (b) interchangeably with ˆ.

Much of the methodology in chapters 10–15 is concerned with quantities like b. The usual question of interest is whether or not b equals zero, since this value frequently represents a lack of association between two variables of interest in a study. For example, in the lymphoma data, the value b = 0 corresponds to a ratio of mortality rates which is equal to e0 = 1.0. Thus, b = 0 represents the hypothesis that there is no relationship between presenting symptoms and

survival. In this section, we will consider a test of the hypothesis H: b = 0 which

ˆ is based on the quantities b and ˆ.

Because of the central limit theorem, which we discussed briefly at the end

ˆ

of §8.2, we can state that b, which is usually a complicated function of the data, has an approximate normal distribution with mean b and variance ˆ2, i.e.,

b N(b, ˆ ). We can remark here, without any attempt at justification, that

one reason for using the logarithm of the ratio of mortality rates, rather than the ratio itself, is that the logarithmic transformation generally improves the

ˆ

In general, both positive and negative values of b can constitute evidence against the hypothesis that b is zero. This prompts us to use the test statistic

value is negative. Let to be the observed value of T which corresponds to bo, i.e., to = bo ˆ. Since T is always non-negative, values which exceed to are less consistent with the null hypothesis than the observed data. Therefore, the signifi-

8 An Introduction to the Normal Distribution 82

This equation specifies that statistical tables for the 21 distribution can also be used to evaluate the significance level of an observed Z-statistic, i.e., an observed value of T. To use the 21 table, we first need to square to. Occasionally, this relationship between T and the 21 distribution is used to present the results of an analysis. In general, however, significance levels are calculated from observed Z-statistics by referring to critical values for the distribution of T =Z , cf., table 8.1.

8.4. The Normal Distribution and Confidence Intervals

In all the statistical methods which we have thus far considered, the use of significance tests as a means of interpreting data has been emphasized. The concept of statistical estimation was introduced in chapter 6, but has otherwise been absent from our discussion. Although significance tests predominate in the medical literature, we can discern a shift towards the greater use of estimation procedures, the most useful of which incorporate the calculation of confidence intervals. From the statistical point of view, this is a development which is long overdue.

The perspective which we intend to adopt in this section is a narrow one; a definitive exposition of the topic of confidence intervals goes beyond the purpose of this book. There are well-defined methods for calculating confidence intervals in all of the situations that we have previously described; we shall not consider any of these techniques, although the calculations are generally not difficult. In chapter 9, we will introduce the calculation of a confidence interval for the mean of a population when the data are normally distributed. This particular interval arises naturally in our discussion of specialized procedures for data of this kind. However, in this section we will consider only those confidence intervals which pertain to the role of the normal distribution in the advanced statistical methods which are the subject of chapters 10 through 15. In this way, our discussion will parallel that of the preceding section.

In chapter 2, we defined a significance test to be a statistical procedure which determines the degree to which observed data are consistent with a specific hypothesis about a population. Frequently, the hypothesis concerns the value of a specific parameter which, in some sense, characterizes the population, e.g., or 2. By suitably revising the wording of this definition, we can accurately describe one method of obtaining confidence intervals. Basically, a confidence interval can be regarded as the result of a statistical procedure which identifies the set of all plausible values of a specific parameter. These will be values of the parameter which are consistent with the observed data. In fact,

values for the 95 and 99% intervals are 3.92ˆ and 5.152ˆ, respectively. Therefore, if we wish to increase our confidence that the interval includes the true value of b, the interval will necessarily be longer.

For the lymphoma data which we discussed in §8.3, we have stated that the method of chapter 13 leads to an estimate for b, the logarithm of the ratio of

ˆ ˆ ˆ mortality rates, which is b = 1.10; the estimated standard error of b is = 0.41.

Therefore, a 95% confidence interval for the true value of b is the set of values

{1.10 – 1.96(0.41), 1.10 + 1.96(0.41)} = (0.30, 1.90).

But what information is conveyed by this interval of values for b? In particular, does this 95% confidence interval provide information concerning the survival of lymphoma patients which we could not deduce from the significance test which was discussed in §8.3?

The test of significance prompted us to conclude that symptom classification does influence survival. Given this conclusion, a token examination of the data would indicate that the mortality rate is higher for those patients who present with B symptoms. Notice that this same information can also be deduced from the confidence interval. The value b = 0, which corresponds to equal survival experience in the two groups, is excluded from the 95% confidence interval for b; therefore, in light of the data, b = 0 is not a plausible value. Moreover, all values in the interval are positive, corresponding to a higher estimated mortality rate in the patients with B symptoms.

But there is additional practical information in the 95% confidence inter-

ˆ

val for b. The data generate an estimate of exp(b) = 3.0 for the ratio of the mortality rates. However, the width of the confidence interval for b characterizes the precision of this estimate. Although 3.0 is, in some sense, our best estimate of the ratio of the mortality rates, we know that the data are also consistent with a ratio as low as exp(0.30) = 1.35, or as high as exp(1.90) = 6.69. If knowledge that this ratio is at least as high as 1.35 is sufficient grounds to justify a change in treatment patterns, then such an alteration might be implemented. On the other hand, if a ratio of 2.0, for example, was necessary to justify the change, then the researchers would know that additional data were needed in order to make the estimation of b more precise. In general, a narrow confidence interval reflects the fact that we have fairly precise knowledge concerning the value of b, whereas a wide confidence interval indicates that our knowledge of b, and the effect it represents, is rather imprecise and therefore less informative.

In succeeding chapters, we will use significance tests, parameter estimates and confidence intervals to present the analyses of medical studies which demonstrate the use of more advanced statistical methodology. By emphasizing this combined approach to the analysis of data, we hope to make confidence intervals more attractive, and more familiar, to medical researchers.

Table 8.2. Values of the cumulative probability function, (z) = Pr(Z ^ z), for the standardized normal distribution

Adapted from: Pearson ES, Hartley HO: Biometrika Tables for Statisticians. London, Biometrika Trustees, Cambridge University Press, 1954, vol 1, pp 110–116. It appears here with the kind permission of the publishers.