Kallen A. Understanding Biostatistics

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130 LEARNING ABOUT PARAMETERS, AND SOME NOTES ON PLANNING

Instead we will build on the observation that the difference of two Gaussian stochastic

variables also has a Gaussian distribution and therefore that

ˆp

− ˆp

∈ N



− p

(1 − p

)

(1 − p

)



, (5.1)

at least approximately for large n

and n

. In order to make inference about the risk differ-

ence  = p

− p

, we rewrite equation (5.1) in  and one further parameter, for which we

choose p

 = ˆp

− ˆp

∈ N



, p

(1 − p

)





(1 − 2p

− )



The problem when we use this to derive a conﬁdence function for  is that we need to do

something about the nuisance parameter p

. One suggestion is to estimate it from data. This

is an approximation, but it is only the variance that depends on p

, and only mildly so, so we

do not really need a high-precision estimate of it. We will illustrate this in the next section.

5.4.2 Analysis of the group difference

To see how we can do the statistical analysis of the difference  we use the data of

the ﬁrst Hodgkin’s lymphoma example in Section 4.5.1, with p

∗

= 67/101 = 0.66 and

∗

= 43/107 = 0.40. The difference is therefore estimated to be 

∗

= 0.26. Based on the

discussion in the previous section, we can derive knowledge about the true difference  from

the conﬁdence function

C() = 



 − 

∗

σ(, p

∗

)



where we have inserted the estimate p

∗

for p

in the variance expression

(, p) = p(1 − p)





(1 − 2p − )

. (5.2)

The function C() is graphed as the solid curve in Figure 5.5, from which we can deduce,

for example, that the symmetric 90% conﬁdence interval around 

∗

is (0.15, 0.37). What

is not shown is the one-sided p-value for the null hypothesis of  ≤ 0. This is given by

C(0) = (−

∗

/σ(0,p

∗

)) = 0.00008 and is too small to show up in the graph. We note

instead that if we test the hypothesis  ≤ 0.15 we get a p-value of 0.05; this is part of the

information found in the (symmetric) 90% conﬁdence interval above.

This is actually not the standard p-value computed for the null hypothesis  ≤ 0. What is

usually done is to compute the standard deviation using ¯p = (67 + 43)/(101 + 107) = 0.53

instead of p

∗

. This is because when  = 0, this should be a better estimate of the parameter

than p

∗

. Similarly, it is common practice to replace  in the expression for the variance

with its estimate 

∗

(cf. Box 4.7). The conﬁdence function obtained with these substitutions

is shown as the dashed curve in Figure 5.5. We see that there is a difference, but it is very

small and of no consequence in this particular case.

We may also note the following. A short calculation shows that



∗

σ(0, ¯p)

− E

)

√

)

STATISTICAL ANALYSIS OF TWO PROPORTIONS 131

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Conﬁdence function C (Δ)

0.1 0.2 0.3 0.4

Risk difference Δ = p

– p

Figure 5.5 Conﬁdence functions for the two-group binomial test. The two functions differ

in how the variance is estimated. For the solid curve we vary  and use the frequency in the

second group to estimate p

, whereas the dashed curve uses the probability estimates in the

variance estimate.

where the mean and variance on the right are computed under the assumption of a hyperge-

ometric distribution for x

(as part of a 2 × 2 table with ﬁxed margins). Actually this is not

quite true, there is a factor

√

n/(n − 1) in front of the right-hand side, but we can eliminate

it if we bias-correct the estimator σ

(0, ¯p) (see page 158). This means that, when we use the

large-sample approximations, the binomial test and Fisher’s exact test give identical p-values

for the hypothesis of no difference. The difference between the tests is in their small-sample

properties, which is when the different assumptions play a role.

We next turn our attention to the relative risk θ = p

. We can obtain a conﬁdence

function for this ratio in a way very similar to what we did for the difference by using a trick

due to Fieller (see Box 7.7), which uses the observation that

ˆp

− θ ˆp

∈ AsN



θ(1 − p

θ)

+ θ

(1 − p

)



As was the case when we analyzed  above, it is only the variance that depends on p

, and

it does so only mildly. Carrying through the analysis as we did for  above, we can deduce

that the estimated relative risk is 1.65 with 90% conﬁdence interval (1.32, 2.08).

In order to get a reasonable estimate of the odds ratio θ we proceed along a different

route, building on the observation made at the end of Section 4.6.1 about the large-sample

approximation to the logarithm of the odds x/(n − x). Since the logarithm of the odds ratio

is the difference of two independent log-odds, we see that, for the empirical odds ratio

θ = x

− x

)/x

− x

θ ∈ AsN



ln(θ),

(1 − p

)

(1 − p

)



132 LEARNING ABOUT PARAMETERS, AND SOME NOTES ON PLANNING

If we insert estimates into this variance expression, this becomes the same analysis as we did

in the previous section, though at that time we assumed a hypergeometric distribution, and

we found there that the estimate and 90% conﬁdence interval for the odds ratio are 2.93 and

(1.82, 4.71), respectively.

We conclude this section with a comment about how a Bayesian statistician might approach

this problem. Theoretically he would do the obvious follow-up on what he did for a single

proportion, possibly using different priors for the two proportions. Once the priors are decided,

the data will give him a two-dimensional distribution for the two probabilities, and with some

mathematical tricks he can use standard probability theory to make any kind of probability

statements he wants for any particular function of these parameters (with no adjustments made

for multiplicity issues).

To be more speciﬁc, let us use the same data as above and assume that we believe, a priori,

that there is no difference between the two groups and that the two proportions have indepen-

dent and identical a priori distributions. If the common prior has a β(x; a, b) distribution, we

get an a posteriori distribution for the pair (p

) as the product of two beta distributions:

β(x; x

+ a, n

− x

+ b) and β(x; x

+ a, n

− x

+ b). From this we can compute the dis-

tribution of any function of (p

) we want to, which is illustrated in Figure 5.6 for the

difference  = p

− p

. There are two curves. The black one to the left uses independent

informed priors β(x;4, 6) for the two parameters, whereas the gray one to the right uses unin-

formed priors. The median for the distribution of  with uninformed priors is 0.25 with 90%

credibility interval (0.14, 0.36), which is similar to what the conﬁdence function produced

in terms of point estimate and 90% conﬁdence interval. The corresponding estimate with

informed priors is smaller, 0.23 with 90% credibility interval (0.13, 0.34). The reason why

it is smaller is that our prior belief was that there was no difference between p

and p

,so

that  was symmetrically distributed around zero, and old beliefs are hard to change. But the

evidence was somewhat overwhelming, and we therefore ended up relatively close to what

the uninformed priors provide.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Posterior probability CDF

0.1 0.2 0.3 0.4

Risk difference Δ = p

– p

Informed prior

Uninformed prior

Figure 5.6 Bayesian analysis of the difference of two binomial proportions.

ADJUSTING FOR CONFOUNDERS IN THE ANALYSIS 133

5.5 Adjusting for confounders in the analysis

If we want to investigate an exposure–response relationship deﬁning a 2 × 2 table and we have

known and measurable confounders, one way to adjust for them in the analysis is to divide the

population into smaller groups for which we have similar values for these confounders. This

gives us a number N of 2 × 2 tables within each of which we can apply Fisher’s exact test,

as outlined in Section 4.5.1. The assumption is that within each table there is homogeneity to

such an extent that a common odds ratio within that group is a reasonable description. The

problem is to summarize this information into a common odds ratio applicable to all groups

or, if that is not possible, a simple description of the variability of the odds ratios between

groups. As already mentioned, these groups are called strata, and the process of dividing the

population up into strata is called stratiﬁcation.

For example, if we want to see whether there is an effect of a particular exposure on the

occurrence of Down’s syndrome, we know from previous examples that we should stratify

with respect to the age of the mother. This means that we deﬁne a sequence of age classes,

and do the comparison between exposed and non-exposed pregnancies within each age class.

In order to deﬁne the age spans of the different strata, it may be helpful to look at the risk

function for the event as a function of age, as shown in Figure 2.2(a). We can also deﬁne strata

by combining a set of different confounders, such as age, smoking habit and social class.

A particular example is when we want to do a meta-analysis and try to combine the

information from a number of different studies, possibly of different designs, each provid-

inga2× 2 table from which we want a collated summary. The strata in this case are the

individual studies.

Example 5.4 In Section 4.5.1 we discussed two studies investigating tonsillectomy as a risk

factor for Hodgkin’s lymphoma. The second of these had much more data than was used in the

analysis, because the investigators ﬁrst ascertained the tonsillectomy history of 174 patients

with Hodgkin’s lymphoma and all of their 472 siblings. They then matched the data in a way

that actually ignored more than 70% of these data. We want to use all 646 patients in this

study and combine them with those of the ﬁrst study in a meta-analysis. The data are given

in the following table:

Patients Controls

+−+− OR

Study 1 67 34 43 64 2.93

Study 2 90 84 165 307 1.99

Sum 157 118 208 371 2.37 (1.83, 3.00)

The last row contains the marginal table, in which we have ignored which study the data

come from. It also contains the corresponding empirical odds ratio and its 90% conﬁdence

interval. In this table we ignore the familial relations in the second study. To do a traditional

meta-analysis we need a way to combine the information about a common odds ratio from

those of the individual tables. Some ideas about how to obtain knowledge about the odds ratio

within a single table were addressed in Section 5.3, and our objective is to extend one of these

ideas to a series of tables.

134 LEARNING ABOUT PARAMETERS, AND SOME NOTES ON PLANNING

The method to generalize in Section 4.5.1 is the one that equated the observation of x

with its mean. There are two ways to extend this to two tables. Both methods are based on

the idea of looking at the squared difference between what we observe and what a model

predicts, normalized to what we expect this squared difference to be. The difference between

the methods is whether you sum the observations ﬁrst, or sum the normalized squares:

1. If we ﬁrst sum the observations from the two tables, and then normalize with the

variance for the sum, we get the function

(θ) =

+ y

− E

) − E

))

) + V

)

We will call this the Mantel–Haenszel quadratic form (it is a quadratic form in the

observations, not the parameter, but we study it as a function of the parameter).

2. If we ﬁrst normalize each squared difference with its variance and then compute the

sum, we get the function

(θ) =

− E

))

)

− E

)

which we call the Breslow–Day quadratic form.

With large numbers, x

+ y

is approximately Gaussian in distribution, and we should have

that Q

(θ) is distributed approximately as χ

(1). We can use this to deﬁne a pooled odds

ratio for the two tables by ﬁnding the θ for which the quadratic form is zero, and we can

also derive a two-sided conﬁdence function for θ, which is shown as the solid function in

Figure 5.7 for our data. From this we get the estimate of the pooled odds ratio as 2.22, and

the symmetric 90% conﬁdence interval is given by (1.73, 2.85).

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Two-sided conﬁdence function

321

Odds Ratio

Figure 5.7 The two curves in this graph are the χ

(1) maps of the quadratic forms Q

(θ)

and Q

(θ) discussed in the text. The solid curve is the one for the Mantel–Haenszel form,

and deﬁnes a two-sided conﬁdence function for θ. The dashed one is for the Breslow–Day

form, and is used to test for homogeneity as indicated by the dashed lines.

ADJUSTING FOR CONFOUNDERS IN THE ANALYSIS 135

The Breslow–Day quadratic form Q

(θ) is used differently, namely to assess how

reasonable it is that a particular pooled odds ratio θ is a good description of the data. If a

particular value θ applies to both tables, this quadratic form should also be zero. The amount

it differs from zero is therefore a measure of the heterogeneity of the two tables. To translate

this to a probability we need to compare its value to the χ

(1) distribution. The corresponding

function is shown in Figure 5.7 as the dashed curve and by taking one minus the conﬁdence

in the estimate for the pooled odds ratio, we get a p-value for the hypothesis of homogeneity

of the odds ratios in the two studies. In this case it turns out to be 0.26.

However, this example does not really describe how most

epidemiologists work with pooled odds ratios. In order to un-

derstand their preferred method we ﬁrst generalize the present

discussion to a general series of 2 × 2 tables. In doing this we

change the notation so that we have N different strata, and in

each a 2 × 2 table with data as shown in the table shown on the right. We can then deﬁne the

following two quadratic forms:

1. The Mantel–Haenszel quadratic form is deﬁned as the function

(θ) =

(



k=1

− E

))



k=1

)

A pooled estimate θ

∗

is obtained as the point where this is zero, and a conﬁdence

function for θ is obtained as C(θ) = χ

(θ)).

2. The Breslow–Day quadratic form is deﬁned as the function

(θ) =



k=1

− E

))

)

It is used to test for homogeneity among strata of the individual odds ratios by com-

paring Q

(θ

∗

)toaχ

(N − 1) distribution.

Example 5.5 Before we continue, let us see what the quadratic form Q

(θ) means in the

case where we have a one-to-one matched case–control study. In such a study, the investigator

selects for each case a control with a speciﬁed set of characteristics that are the same as those

of the case, and then determines, for both case and control, whether there has been an exposure

or not. Each such pair produces a small 2 × 2 table, in which the total number in each row is

one. In this case, what is the quadratic form Q

(θ)?

Each table has row sums n

= n

= 1, and for the sum of the elements in the ﬁrst column,

r, we can have only three possible values, r = 0, 1, 2. However, when r = 0 neither case nor

control was exposed, and the expected value of a

must be zero, and when r = 2 both case

and control were exposed, so the expected value of a

is necessarily one. In both cases we

have a

− E

) = 0, so the terms involved in Q

(θ) are only non-zero if r = 1; only those

pairs contribute to the analysis for which either the case or the control is exposed, but not

both. Let the number of such tables be n.

For a table with r = 1 we have (see Appendix 5.A.1) that E

) = p = θ/(1 + θ)

and that V

) = θ/(1 + θ)

= p(1 − p). It follows that if z =



is the total

136 LEARNING ABOUT PARAMETERS, AND SOME NOTES ON PLANNING

number of cases that were exposed, and if we express Q

as a function of p,

(p) = (z − np)

/np(1 − p). This is the same test as the large-sample approximation

for a Bin(n, p) distribution, and the test for the null hypothesis θ = 1 is therefore McNemar’s

test mentioned in Section 4.5.1.

Using Q

(θ) for the estimation of θ is, however, somewhat complex and it is not clear

how the pooled estimate relates to the empirical odds ratio of the individual tables, given by

∗

= a

. Epidemiologists have therefore largely chosen to use a different pooled esti-

mate, one that is explicitly derived from the individual θ

∗

s. When a

follows a hypergeometric

distribution we have that E

− θb

) = 0 and if we specify stratum weights w

,wecan

therefore deﬁne a pooled odds ratio from the equation





− θb

)



= 0.

The corresponding pooled odds ratio estimate is obtained by inserting observed table data,

from which we derive the pooled estimate

pool

= (



k=1

)/(



k=1

). Each

choice of weights gives its own pooled odds ratio. The particular choice w

= 1/N

, where

= a

+ b

+ c

+ d

is the total number of observations in stratum k, provides the Mantel–

Haenszel pooled odds ratio



k=1



k=1

This is a weighted sum of the individual empirical odds ratios,



k=1

∗

, where

= S



k=1

. We know that the variances of the individual empirical odds ratios are

approximately given by

+ d

)θ

+ b

+ c

≈

where the last approximation is valid when θ

≈ 1, which means that the Mantel–Haenszel

pooled odds ratio is close to being the most efﬁcient weighting of the individual table odds

ratios. It is so if the individual odds ratios are one.

As an example, consider the one-to-one matched case–control study. In each cell we have

that a

= 1 if and only if case is exposed and control is not, whereas b

= 1 if the reverse

is true. In all other cases these products are zero, so the Mantel–Haenszel odds ratio becomes

the ratio (



k=1

)/(



k=1

) of the off-diagonal elements in the overall summary table

(cf. page 98).

In order to obtain a conﬁdence function for the Mantel–Haenszel estimator we need its

variance. As we will see in Example 13.3, we should do the analysis on the log scale, with

the variance of ln

well approximated by



)



k=1

−2

+ b

)(a

+ d

+ (b

+ c

)θ

). (5.3)

ADJUSTING FOR CONFOUNDERS IN THE ANALYSIS 137

(There are other approximations for the variance of

, on the original scale and on the log

scale, but experience and simulation studies indicate that the variance given by equation (5.3)

is to be preferred over others.) If we apply these results to the data in Example 5.4, we ﬁnd the

pooled odds ratio to be 2.23 with 90% conﬁdence interval given by (1.73, 2.86), very close

(but not identical) to what was obtained with the ﬁrst method.

We noted above, and it is worth repeating, that the Mantel–Haenszel pooled estimate of

the odds ratio is obtained as a weighted average of the individual empirical odds ratios. The

weights were so determined that they were essentially optimal from a statistical perspective.

They had no other, independent, meaning. From a practical point of view this means that the

estimate obtained is really only meaningful if all individual odds ratios (the true ones, not the

empirical ones) are equal. In such a case the pooled estimate will estimate this number in an

efﬁcient way.

There are alternative ways to estimate a pooled odds ratio from a stratiﬁed analysis.

A common alternative to the Mantel–Haenszel method is derived from the problem

of estimating a pooled odds ratio from the equation Q

(θ) = 0, which was men-

tioned in Example 5.4. The equation to solve is G(θ) =



− E

)) = 0, which

is a nonlinear equation, so a solution must be obtained by iterative methods. If we

write θ = e

and consider only one iteration (starting with β = 0), we get the equation



− E

) − βV

)) = 0, and if we solve this we arrive at Peto’s suggestion for a

pooled odds ratio,

Peto

= exp





− E

))/V

)



How to obtain conﬁdence intervals, etc., is left to the reader. Even though the Peto estimate

is expected to behave well when the true odds ratio is close to one, it is not known for good

properties when this is not the case.

Pooling the odds ratios is the only option for case–control studies, but in a cohort study

we might prefer to estimate pooled proportions for exposed and non-exposed groups, re-

spectively, and derive a pooled relative risk instead. Alternatively, we might consider rates.

The corresponding Mantel–Haenszel methodology is outlined in Box 5.3. Again the weights

are obtained for optimal statistical efﬁciency, and the value of the pooled group estimates

depends on the relative contribution of the individual strata. If the stratum-speciﬁc probabil-

ities/rates differ, the pooled ones may be a rather meaningless estimate of some population

probability/rate.

To be more speciﬁc, consider rates and assume that we have stratiﬁed on age, so that we

have a rate λ =



, where λ

denotes the rate in age stratum a.Ifwewanttheλ to

represent the overall risk in the population, we should take weights w

corresponding to the

size of the age class a in the population. However, that may not be the age distribution in

the study, unless we have representative sampling. The weights of the pooled analysis may

therefore provide an estimate for a population with a different age structure than the one we

want to relate to. In order to give meaning to pooled measures, we therefore need to deﬁne

the weights properly, a process called standardization. Note that the standardized rate will

typically be estimated to a lower precision than the pooled one. Epidemiologists often talk

about the standardized mortality ratio, which is the rate ratio obtained when both nominator

and denominator use the weights of the exposed group. It represents the ratio of the number

of observed cases, divided by the number of expected cases.

138 LEARNING ABOUT PARAMETERS, AND SOME NOTES ON PLANNING

Box 5.3 Mantel–Haenszel type estimate of a pooled relative risk

For a cohort study we can adjust the Mantel–Haenszel method of obtaining a pooled

estimate to obtain one for a relative risk. This can be applied equally well to proportions

and rates. Suppose that we have observed x

events in group j in stratum i and assume

that one of the following two cases applies:

1. x

∈ Bin(T

), where T

is total number of observations;

2. x

∈ Po(λ

), where T

is observation time.

Let the relative risk in stratum i be denoted θ

, so that θ

= p

in the ﬁrst case

and θ

= λ

/λ

in the second case. We wish to deﬁne a pooled relative risk θ

from

these these.

The relative risk in stratum i satisﬁes the equation E

− θ

) = 0soa

pooled relative risk could be deﬁned from an equation of the form





− θx

)



= 0.

If, for each i, we take as weights the harmonic mean of the T

, that is, w

= 2T

where T

= T

+ T

, we arrive at the Mantel–Haenszel relative risk estimate



)



)

This corresponds to the pooled probability estimates p

∗





∗

for group j, where



= w



, and similarly for rates. Much as in Example 13.3 the variance of ln

is approximately given by



)





(

+ T

)



We are appealing to Gaussian approximations here, so we can alternatively use the

Fieller approach (see Box 7.7) to obtain a conﬁdence function for θ

5.6 The power curve of an experiment

In the planning phase for an experiment it is important to ﬁgure out how large it should

be – how many patients to include in a clinical study. If the experiment is too small, so

that the statistical analysis has no realistic chance of detecting a difference, it may be a

waste of money to make the experiment at all (or unethical, when it comes to clinical

trials). On the other hand, if it is too large, providing more evidence than necessary,

there is also some concern about money spent unnecessarily (and possibly patients put

at risk). To ﬁnd the right size of an experiment we need to investigate the statistical

properties of a proposed size. To do that, we ﬁrst need to be explicit about the objective of

the experiment.

THE POWER CURVE OF AN EXPERIMENT 139

The standard protocol for this is as follows. We want to reject the null hypothesis θ = 0at

a speciﬁed signiﬁcance level α. That means that we consider the experiment a success if we

get a p-value which is less than α. That part is the actual statistical test, and controls the Type

I error. As noted at the end of Section 5.2, this deﬁnes a particular value, x

crit

, such that if the

value of the test statistic is equal to this, the p-value is α. (We continue to consider one-sided

tests where a large value of the test statistic is evidence against the null hypothesis.)

We also want to understand the Type II error. This involves the true distribution

G(x) = F (x, θ

true

) for the test statistic. We want to know what our chances are of a suc-

cessful outcome of the experiment. However, we do not know θ

true

, so we investigate

the probability

β(θ) = 1 − F(x

crit

,θ)

for a range of θ where we expect θ

true

to be. This function is called the power function of the

experiment and is such that β(0) = α by construction. Actually the notation is unconventional

here; usually β(θ) is used to denote the Type II error, so the power function should be β

(θ).

If we vary x

crit

we get a family of different power curves. Each such curve corresponds to

a particular choice of signiﬁcance level, which can be read off from its value at θ = 0. Some

further insight into the nature of power curves is obtained in the following example. (This

is the slightly confusing case when we estimate a location parameter, so that the experiment

outcome x works on the same scale as the parameter θ, and we therefore label values on the

x-axis in two different ways.)

Example 5.6 Assume that the parameter θ is a location parameter (such as a mean) and that the

statistical model is a shift model, which means that F(x, θ) takes on the form F (x − θ) for some

CDF F (x), assumed to be symmetric around x = 0. This special case has some geometrical

properties which are illustrated in Figure 5.8. For a symmetric and continuous distribution we

have that F (−x) = 1 − F (x), which implies not only that F (0) = 0.5, but also that

β(θ) = 1 − F(x

crit

− θ) = F (θ − x

crit

so the power function is obtained from a horizontal shift of magnitude x

crit

of the null

distribution F(x). If we introduce the alternative notation θ

for the critical value x

crit

we can

write β(θ) = F (θ − θ

), and we have that β(θ

) = 0.5. In other words, the value of θ which

gives 50% power deﬁnes the critical value x

crit

for the test statistic.

The dashed curve on the left in Figure 5.8 is a part of the graph of F (θ). The solid curve

is this function shifted by the amount θ

. This is an increasing function which starts with the

value α at θ =0, passes through the value 0.5 at θ = θ

(= x

crit

), and reaches the level 1 − α

for θ = 2θ

, after which it increases asymptotically to one. To be successful, the experiment

must produce an observation of the test statistic which is larger than x

crit

= θ

. When θ is less

than θ

, this probability is less than 50%, and when it is greater than θ

, the probability is

greater than 50%.

So far we have discussed the power curve for a particular experiment. Most often we

want to use power considerations to ﬁnd the appropriate size of the experiment. We therefore

introduce the study size n into the notation, so that the CDF for the test statistic becomes

(x, θ). To determine n we must hold something ﬁxed, and this can be done in different

ways. We can, for example, decide that we want the test on signiﬁcance level α and specify