Jump to ContentJump to Main Navigation
Statistical Analysis of Epidemiologic Data$

Steve Selvin

Print publication date: 2004

Print ISBN-13: 9780195172805

Published to Oxford Scholarship Online: September 2009

DOI: 10.1093/acprof:oso/9780195172805.001.0001

The Analysis of Contingency Table Data: Logistic Model I

Chapter:
(p.190) 7 THE ANALYSIS OF CONTINGENCY TABLE DATA: LOGISTIC MODEL I
Source:
Statistical Analysis of Epidemiologic Data
Author(s):

Steve Selvin

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780195172805.003.07

Abstract and Keywords

This chapter discusses the use of a logistic regression model to analyze data classified into a multiway table. Topics covered include the simplest model, the 2 x 2 x 2 table, the 2 x k table, the 2 x 2 x k table, the multiway table, and goodness-of-fit.

Keywords:   logistic regression model, epidemiologic data analysis, 2 x 2 x 2 table, 2 x k table

Epidemiologic data frequently consist of cases of disease or deaths tabulated by categorical variables or tabulated by continuous variables made into categories. Converting continuous variables into categorical variables simplifies the presentation and is usually based on traditional definitions (at a cost of statistical power, as already noted). For example, individuals are considered hypertensive or not depending primarily on whether their systolic blood pressure exceeds 140 mm Hg, and smokers are placed into exposure categories based on the reported numbers of cigarettes smoked per day. The end result is a table of counts. A table is certainly a useful summary of collected data, but, in addition, it is often important to explore the sometimes complex relationships found within a table. Such a table (Table 7–1 ) describes three risk factors and a disease outcome from the Western Collaborative Study (WCGS; Appendix A in this volume).

Table 7–1 contains the counts of coronary heart disease (CHD) events and the number of individuals at risk classified by behavior type, systolic blood pressure, and smoking exposure. Clearly, increased risks are associated with increasing amounts smoked, high blood pressure, and type-A behavior pattern. A number of questions, however, are not easily answered:

Does the risk of smoking have a threshold influence, or does risk increase more or less consistently as the number of cigarettes smoked increases?

(p.191)

What is the influence of blood pressure and smoking on the behavior/CHD relationship?

Do these three risk factors have joint or separate influences on the probability of a CHD event?

What is the relative influence of each risk factor on the likelihood of a coronary event?

Frequently, the inability to answer complex or even elementary questions about risk/disease relationships in a satisfactory way results, to a large extent, from lack of data. If hundreds of thousands of individuals were classified into a table with numerous categories, most questions about the role of the risk factors in the occurrence of a disease could be directly answered. When more than a few variables are investigated or large amounts of data are not available, a table often fails to effectively describe the relationships among the categorical variables. Additionally, human disease is, almost always, a relatively rare event, and it causes low cell frequencies in most tabulated data. For example, only three coronary events occur in the low blood pressure, heavy smoking, type-B category among 34 observations when a relatively large sample of 3154 men was collected. Therefore, it becomes necessary for analytic procedures to account for the impact of sampling variation. A statistical analysis based on a logistic regression model is one way to describe simply and efficiently the relationships between risk factors and a binary disease outcome classified into a multiway table.

The use of a logistic regression model to analyze data classified into a multiway table is the topic of this chapter. The extension to include continuous risk variables is the topic of the next chapter. Logistic regression is by no means the only approach to the analysis of risk/disease relationships, but it is frequently used and demonstrates both the strengths and weaknesses of multivariable techniques applied to epidemiologic data.

THE SIMPLEST MODEL: DISCRETE CASE

A 2 × 2 table provides the simplest illustration of a logistic model applied to the relationships among contingency table variables. To start, the variables to be studied are the presence (D = 1) and absence (D = 0) of a disease investigated at two levels of a risk factor (F = 1, risk factor present; F = 0, risk factor absent). The general notation (repeated) for a contingency table applies (Table 7–2 ).

A chi-square analysis is the most common approach for assessing an association within a 2 × 2 table. Other methods based on conditional probabilities such as P(D | F) and  The Analysis of Contingency Table Data: Logistic Model I are also used to describe the risk/disease relationship. (p.192)

Table 7–1. Coronary heart disease classified by behavior type, systolic blood pressure, and smoking exposure: a summary table of probabilities of CHD

Blood pressure (mm Hg)

Behavior type

Smoking frequency (cigarettes per day)

0

1–20

21–30

> 30

≥ 140

A

29/184 = 0.158

21/97 = 0.216

7/52 = 0.135

12/55 = 0.218

≥ 140

B

8/179 = 0.045

9/71 = 0.127

3/34 = 0.088

7/21 = 0.333

< 140

A

41/600 = 0.068

24/301 = 0.080

27/167 = 0.162

17/133 = 0.128

< 140

B

20/689 = 0.029

16/336 = 0.048

13/152 = 0.086

3/83 = 0.036

(p.193)

Table 7–2. Notation for a 2 × 2 contingency table

Factor present

Disease

D = 1 (present)

D = 0 (not present)

Total

F = 1

n 11

n 12

n 1.

F = 0

n 21

n 22

n 2.

   Total

n .1

n .2

n

Another choice is to compare the odds as a measure of association between a risk factor and a disease calculated from data recorded in a 2 × 2 table. The odds of disease among individuals without the risk factor are the ratio of the probabilities P(D | F = 0) to  The Analysis of Contingency Table Data: Logistic Model I , and the odds among individuals with the risk factor are the ratio of the probabilities P(D | F = 1) to  The Analysis of Contingency Table Data: Logistic Model I . A statistically strategic measure is the logarithm of the odds, called the log-odds, or logit. Taking the logarithms of an odds changes a ratio measure into a difference that is a linear measure of risk. Linear measures are far simpler to understand and to analyze statistically.

A baseline measure of disease of risk is the estimated log-odds when the risk factor is absent, or

 The Analysis of Contingency Table Data: Logistic Model I
so that â is an estimate of a where
 The Analysis of Contingency Table Data: Logistic Model I
A measure of particular interest is an estimate of the increase or decrease in the log-odds due to the presence of the risk factor measured relative to the baseline log-odds (a difference in log-odds), or
 The Analysis of Contingency Table Data: Logistic Model I
so that  The Analysis of Contingency Table Data: Logistic Model I is an estimate of b where
 The Analysis of Contingency Table Data: Logistic Model I
(p.194) and, furthermore,
 The Analysis of Contingency Table Data: Logistic Model I
This is the odds ratio measure of association from a 2 × 2 table (Appendix C ).

The two quantities (a, b) form the simplest possible linear logistic model. That is, log-odds = a + bF, where F = 0 or F = 1, so that when F = 0, a is estimated by log-odds = log(n 21/n 22) = â and when F = 1, a + b is estimated by  The Analysis of Contingency Table Data: Logistic Model I .

The value  The Analysis of Contingency Table Data: Logistic Model I estimates, on the log-odds scale, the change in risk of disease associated with the presence of the risk factor (F) relative to the absence of the same factor  The Analysis of Contingency Table Data: Logistic Model I . The risk factor in the context of a linear model has other names—it is sometimes called the predictor variable, the explanatory variable, or the independent variable. A difference in log-odds is not a particularly intuitive way to assess a risk factor/disease association, but it has tractable mathematical properties and, of most importance, relates directly to the odds ratio measure of association, which is the primary reason for using a logistic regression model. In symbols,  The Analysis of Contingency Table Data: Logistic Model I .

Consider a 2 × 2 table (Table 7–3 ) from the WCGS data describing the relationship between behavior type (risk factor) and a coronary event (disease outcome). From these data, â = log(79/1486) = – 2.934, and  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . The estimated change in the log-odds of CHD associated with type-A individuals (risk factor present) relative to type-B individuals (risk factor absent—baseline) is  The Analysis of Contingency Table Data: Logistic Model I (Fig. 7–1 ). Since  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , then  The Analysis of Contingency Table Data: Logistic Model I and is the usual estimate of the odds ratio calculated from a 2 × 2 table (Appendix C ). For the behavior type data (Table 7–3 ), the odds ratio estimate is  The Analysis of Contingency Table Data: Logistic Model I . The odds of a coronary event is, therefore, 2.373 times greater for a type-A individual than for a type-B individual. This same odds ratio can also be calculated directly from the tabled data where  The Analysis of Contingency Table Data: Logistic Model I .

When the number of model parameters estimated equals the number of observed log-odds values calculated from a table, the results of using a model to describe associations and making direct calculations from the data are always identical. Such a model is called saturated. The two-parameter logistic model

Table 7–3. CHD by A/B behavior type

Behavior type

CHD

No CHD

Total

A

178

1411

1589

B

79

1486

1565

   Total

257

2897

3154

(p.195)
 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–1. Log-odds associated with CHD risk by behavior type (A/B).

applied to the two log-odds values generated by a 2 × 2 table is an example of a saturated model.

The two primary ways to evaluate the influence of this sampling variation on the estimated log-odds  The Analysis of Contingency Table Data: Logistic Model I are a significance test and a confidence interval.

Aside: The relative merits of a confidence interval versus a significance test have been occasionally discussed, even debated ([ 1 ], [ 2 ], and [ 3 ]). The confidence interval and the two-sided significance test rarely give contradictory results. The confidence interval conveys more information in the sense that the interval width gives some idea of the range of parameter possibilities (power). A confidence interval is expressed in the same units as the estimated quantity which is sometimes helpful. A significance test reported in isolation tends to reduce conclusions to two choices (“significant” or “not significant”), which, perhaps, emphasizes the role of chance variation too simply. For example, estimates can be “significant” (chance not a plausible explanation) because a large number of observations are involved, but the results do not reflect an important biological or physical measurement. However, a (p.196) confidence interval approach is complicated when more than one estimate is involved. In these situations several estimates can be assessed simultaneously with a significance test where the analogous confidence region is difficult to construct and complex to interpret. Of course, a significance test and a confidence interval can both be presented. However, if one approach must be chosen, a confidence interval usually allows a direct and more comprehensive description of the influence of the impact of sampling variation on a single estimated quantity.

Lack of precision, incurred because the collected data represent a fraction of the population sampled, must be taken into account to realistically assess the influence of a risk factor on the likelihood of a disease. To evaluate the impact of this sampling variation associated with the estimate  The Analysis of Contingency Table Data: Logistic Model I , both a significance test and a confidence interval require an estimate of the variance of the distribution of the estimate  The Analysis of Contingency Table Data: Logistic Model I , which is

 The Analysis of Contingency Table Data: Logistic Model I
For the behavior type data (Table 7–3 ), the estimated variance associated with the estimate  The Analysis of Contingency Table Data: Logistic Model I is
 The Analysis of Contingency Table Data: Logistic Model I
To test the null hypothesis that a risk factor is unrelated to the disease outcome (H 0: b = 0), the test-statistic
 The Analysis of Contingency Table Data: Logistic Model I
has an approximate chi-square distribution with one degree of freedom when the null hypothesis is true. A test-statistic consisting of a squared estimate divided by its estimated variance is frequently called Wald’s test. A statistical test of the hypothesis of b = 0 is equivalent to testing a number of other hypotheses, including odds ratio = 1.0,  The Analysis of Contingency Table Data: Logistic Model I , or  The Analysis of Contingency Table Data: Logistic Model I . These expressions all indicate the same property—namely, the disease outcome and the risk factor are unrelated. More technically, the disease outcome and risk factor are statistically independent, or P(D | F) = P(D). That is, the probability of disease is unaffected by the risk factor.

For the WCGS data, the Wald’s chi-square test-statistic is X 2 = (0.864)2/0.020 = 37.985, yielding a p-value of P(X 2 ≥ 37.985 | b = 0) < 0.001 and providing substantial evidence that A/B-behavior type is associated with CHD incidence. (p.197) An approximate (1 – α)% confidence interval based on the estimate  The Analysis of Contingency Table Data: Logistic Model I has  The Analysis of Contingency Table Data: Logistic Model I and  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I because  The Analysis of Contingency Table Data: Logistic Model I has an approximate normal distribution for moderate or large samples of observations. The value z 1–α/2 is the (1 – α/2)th-percentile of a standard normal distribution. A (1 – α)% confidence interval for the odds ratio or = eb is then (e lower, e upper). The approximate 95% confidence interval based on the estimate  The Analysis of Contingency Table Data: Logistic Model I from the behavior-type data is (0.589, 1.139). The approximate 95% confidence interval for the odds ratio (or) is then (e 0.589 = 1.803, e 1.139 = 3.123), indicating a likely range of the possible values for the underlying “true” odds ratio, estimated by or = e 0.864 = 2.373.

Using a linear model to represent the log-odds is related to describing the probabilities from a 2 × 2 table with a logistic function—thus the name “logistic regression.” The logistic function, which has historically been used in a variety of contexts to study biological phenomena, is formally

 The Analysis of Contingency Table Data: Logistic Model I
The function f(x) is an “s-shaped” curve in which large negative values of x yield values of f(x) near zero, large positive values of x yield values of f(x) near one, and a value of x = 0 produces f(0) near one-half. Because the logistic function is simple and produces values that are always greater than zero and less than one, it is ideal for representing probabilities, which are also bounded between zero and one.

In symbols, the logistic probabilities applied to a 2 × 2 table are

 The Analysis of Contingency Table Data: Logistic Model I
where a and b are the quantities previously defined in terms of log-odds. In other words, the same parameters (a, b) describe the relationships within a contingency table using either log-odds or logistic probabilities to measure risk.

Logistic probabilities from a 2 × 2 table imply a linear relationship among the log-odds, and vice versa. For example, the log-odds measures risk when the risk factor is present and

 The Analysis of Contingency Table Data: Logistic Model I
(p.198) and when the risk factor is absent
 The Analysis of Contingency Table Data: Logistic Model I
In general, the relationship between the log-odds and the logistic probability of disease is
 The Analysis of Contingency Table Data: Logistic Model I
where the log-odds are typically a function of the risk factors.

An odds ratio is most easily interpreted when the frequency of the disease is low (less than 0.1 in both risk-factor groups, F and  The Analysis of Contingency Table Data: Logistic Model I ) since, in this case, the estimated odds ratio is approximately equal to a simpler measure of association between risk factor and disease called the relative risk. In symbols, the  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . Relative risk is a natural measure of risk to compare the probability of disease between one group that possesses the risk factor (F) to another that does not the possess the risk factor  The Analysis of Contingency Table Data: Logistic Model I and is discussed in detail elsewhere (for example, [ 4 ] and [ 5 ]). Nevertheless, the odds ratio measures association regardless of the frequency of the outcome variable.

A summary of the logistic regression model applied to estimate the association between behavior type and coronary heart disease using the WCGS data is given in Table 7–4 . The logistic probabilities are estimated as  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I and  The Analysis of Contingency Table Data: Logistic Model I , whether calculated from the logistic function using estimates â and  The Analysis of Contingency Table Data: Logistic Model I or directly from the data, again indicating that the logistic model is saturated when it is applied to a 2 × 2 table. Specifically, for the type-A individuals (F = 1)

 The Analysis of Contingency Table Data: Logistic Model I

Table 7–4. CHD by A/B: the two-parameter logistic model

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 2.934

0.115

A/B

 The Analysis of Contingency Table Data: Logistic Model I

0.864

0.140

< 0.001

2.373

− 2 log-likelihood = 1740.334; number of model parameters = 2.

(p.199) and for the type-B individuals (F = 0)
 The Analysis of Contingency Table Data: Logistic Model I
and again
 The Analysis of Contingency Table Data: Logistic Model I

THE 2 × 2 × 2 TABLE

When an association is detected in a 2 × 2 table, a natural question arises: How is this association influenced by other variables? The WCGS data show a strong association between behavior type and coronary disease, but the question remains: Could this association be, at least in part, a result of the influences of other risk factors? For example, systolic blood pressure level is related to both the risk of coronary disease and behavior type and could substantially influence the observed association. It is entirely possible that type-A individuals have on average higher blood pressure levels than type-B individuals, and it is this increased level of blood pressure that produces the higher risk of CHD and not the difference in behavior types. A 2 × 2 × 2 contingency table sheds some light on the influence of a third variable on the relationship under investigation (Chapter 6 ). To illustrate, the role of systolic blood pressure as a possible influence on the relationship between behavior type and coronary disease is explored from a 2 × 2 × 2 table of WCGS data (Table 7–5 ). Another description of these data is possible in terms of the log-odds (Table 7–6 ).

The influence of behavior type is measured by the difference in log-odds, or  The Analysis of Contingency Table Data: Logistic Model I for individuals with blood

Table 7–5. CHD by A/B and systolic blood pressure

Behavior type

Blood pressure ≥ 140

Blood pressure < 140

CHD

No CHD

Total

CHD

No CHD

Total

A

69

319

388

109

1092

1201

B

27

278

305

52

1208

1260

   Total

96

597

693

161

2300

2461

 The Analysis of Contingency Table Data: Logistic Model I

 The Analysis of Contingency Table Data: Logistic Model I

(p.200)

Table 7–6. CHD by A/B and systolic blood pressure: a summary

Blood pressure

Behavior type

Log-odds

Notation

At-risk

CHD

No CHD

≥ 140

A

− 1.531

l 11

388

69

319

≥ 140

B

− 2.332

l 12

305

27

278

< 140

A

− 2.304

l 21

1201

109

1092

< 140

B

− 3.145

l 22

1260

52

1208

pressure ≥ 140 and by the difference in log-odds, or  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , for individuals with blood pressure < 140. These two quantities do not differ from the log-odds measure previously described for a 2 × 2 table but are calculated twice, once within each blood pressure 2 × 2 subtable. In the case of a 2 × 2 × 2 table, measures of CHD risk associated with blood pressure can also be estimated for each of the two behavior categories. The two log-odds measures of the effect on risk from blood pressure levels are  The Analysis of Contingency Table Data: Logistic Model I for type-A individuals and  The Analysis of Contingency Table Data: Logistic Model I for type-B individuals.

The failure of these two measures of association ( The Analysis of Contingency Table Data: Logistic Model I ’s or  The Analysis of Contingency Table Data: Logistic Model I ’s) to be equal in each subtable brings up a fundamental issue. One of two possible situations exists:

  1. 1. The influence of the risk factor under study is the same in both subtables and the observed differences arise only because of random variation (interaction absent).

  2. 2. The influence of the risk factor under study is different in each subtable (interaction present).

The first possibility suggests that the two estimates should be combined to produce a single summary measure of the association between risk factor and disease. The second suggests quite the opposite. As noted before, if a variable behaves differently in each subtable, then a combined estimate possibly conceals or exaggerates the difference and will occasionally be completely misleading. As always, the presence of interactions restricts summarization.

In symbolic terms, for a 2 × 2 × 2 table, an interaction measured by the log-odds is interaction = b 1b 2 = (l 11l 12) – (l 21l 22) = l 11l 12l 21 + l 22 or, similarly, interaction = c 1c 2 = (l 11l 21) – (l 12l 22 = ) = l 11l 12l 21 + l 22. The expression l 11l 12l 21 + l 22 is the difference between two differences and quantifies the amount of interaction associated with two risk factors on a log-odds scale. An estimate of the magnitude of the interaction associated with behavior type and blood pressure is  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I .

(p.201) A key issue is whether the data provide evidence of a nonrandom (“real”) interaction. Testing the null hypothesis that l 11l 12l 21 + l 22 = 0 is equivalent to testing the hypothesis that b 1 = b 2 = b or c 1 = c 2 = c (are the differences different?). A statistical test to evaluate the magnitude of the estimated interaction effect requires an expression for the variance. The variance associated with the distribution of the estimated interaction effect in a 2 × 2 × 2 table is estimated by

 The Analysis of Contingency Table Data: Logistic Model I
where nijk represents one of eight cell frequencies in the 2 × 2 × 2 table (notation displayed in Table 6–14 ). For example, the estimated interaction effect associated with blood pressure and behavior type is – 0.040, and the estimated variance associated with the distribution of this estimate is
 The Analysis of Contingency Table Data: Logistic Model I
The test-statistic (another version of the Wald test)
 The Analysis of Contingency Table Data: Logistic Model I
has an approximate chi-square distribution with one degree of freedom when no interaction exists and provides a statistical assessment of the magnitude of the estimated interaction effect. Continuing the WCGS example, the chi-square statistic of X 2 = (– 0.040)2/0.088 = 0.018 yields a p-value of 0.892 that formally indicates no evidence of an interaction between the risk factors of blood pressure and behavior type in the analysis of CHD risk.

The linear model underlying the log-odds (denoted lij ) approach to describing relationships in a 2 × 2 × 2 table is

 The Analysis of Contingency Table Data: Logistic Model I
(p.202)
 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–2. Log-odds associated with CHD risk by behavior type (A/B) and systolic blood pressure (< 140 and ≥ 140).

where d represents the magnitude of the interaction effect (d = l 11l 21l 21 + l 22). The parameters b 1 and c 1 are redundant since b 1 = b 2 + d and c 1 = c 2 + d (four parameters uniquely establish four log-odds values, Figure 7–2 ). The coefficient a represents a log-odds measure of the baseline level of risk. The logistic model in the log-odds form is log-odds = a + b 2 F + c 2 C + dFC, where F = 1 (type A) or F = 0 (type B), and C = 1 (blood pressure ≥ 140) or C = 0 (blood pressure < 140).

Another view of the parameter d is d = b 1b 2 = c 1c 2, or

 The Analysis of Contingency Table Data: Logistic Model I
where or 1 is the odds ratio from the first 2 × 2 subtable and or 2 is the odds ratio from the second 2 × 2 subtable. For the blood pressure and behavior type (p.203) data,  The Analysis of Contingency Table Data: Logistic Model I (blood pressure ≥ 140) and  The Analysis of Contingency Table Data: Logistic Model I (blood pressure < 140), yielding  The Analysis of Contingency Table Data: Logistic Model I and again  The Analysis of Contingency Table Data: Logistic Model I . When d = 0 (e 0 = 1.0), the odds ratios are identical in both subtables; otherwise, the magnitude of ed measures the degree of interaction in terms of the ratio of odds ratios.

Table 7–7 summarizes the estimates of these four components (a, b 2, c 2, and d) that describe the relationship of the risk variables, blood pressure, and behavior type to coronary disease in terms of the logistic model.

The application of the logistic model to a 2 × 2 × 2 table can be geometrically interpreted as two straight lines on a log-odds scale, each depicting the influence of the risk factor at each level of the third variable. The parameter d indicates the degree to which these two lines are not parallel. Figure 7–2 shows two lines representing the change in CHD risk associated with behavior type for each level of blood pressure estimated from the WCGS data. The model estimated odds ratios (last column in Table 7–7 ) could have been calculated directly from the tabled data, indicating that the four-parameter model is saturated. For example,  The Analysis of Contingency Table Data: Logistic Model I , or  The Analysis of Contingency Table Data: Logistic Model I , and is identical to  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . Four log-odds values means a four-parameter logistic model is saturated.

When no interaction exists (d = 0; that is, or 1 = or 2), an additive logistic model (not saturated) represents the relationships in a 2 × 2 × 2 table, again in terms of the log-odds:

 The Analysis of Contingency Table Data: Logistic Model I
A compact form of the same model is represented by
 The Analysis of Contingency Table Data: Logistic Model I

Table 7–7. CHD by A/B by systolic blood pressure: the four-parameter logistic model

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.145

0.142

A/B

 The Analysis of Contingency Table Data: Logistic Model I

0.841

0.174

< 0.001

2.319

Blood pressure

ĉ 2

0.813

0.246

< 0.001

2.256

Interaction

 The Analysis of Contingency Table Data: Logistic Model I

− 0.040

0.297

0.892

0.960

− 2 log-likelihood = 1709.934; number of model parameters = 4.

(p.204) where again F = 1 (type A) or F = 0 (type B), and C = 1 (blood pressure ≥ 140) or C = 0 (blood pressure < 140).

This statistical model is additive because the log-odds associated with possessing both risk factors (l 11) relative to possessing neither risk factors (l 22) is exactly the sum of the log-odds influences from each risk factor—namely, b and c. In symbols, l 11l 22 = a + b + ca = b + c. A direct result of an additive model is that d = 0, or d = l 11l 12l 21 + l 22 = (a + b + c) – (a + c) – (a + b) + a = 0. The values b and c take the place of the previous values b 1, b 2, c 1, and c 2. As required, b 1 = b 2 = b and c 1 = c 2 = c because d = 0. An iterative procedure or a computer program is necessary to estimate the model parameter values of a, b, and c from a 2 × 2 × 2 table.

The maximum likelihood estimates (Appendix E ) from the WCGS data are â = – 3.136,  The Analysis of Contingency Table Data: Logistic Model I , and ĉ = 0.786. That is, the estimated model is log-odds = – 3.136 + 0.827F + 0.786C. From the additive model and the estimated coefficients (Table 7–8 ), the model-generated log-odds, and probabilities can be estimated (Table 7–9 ). For example,  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , giving

 The Analysis of Contingency Table Data: Logistic Model I
Using this model, the estimate  The Analysis of Contingency Table Data: Logistic Model I is the number of CHD events expected among n = 388 individuals who are type A with high blood pressure. The number of type-A study participants with high blood pressure (≥ 140) who are expected to be free of a coronary event is directly 388 – 69.46 = 318.54.

The additive (no-interaction) model requires that the influence of the risk factor be exactly the same in both subtables, or b 1 = b 2 = b and c 1 = c 2 = c. The estimated odds ratio associated with behavior type is  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , which contrasts the CHD risk associated with behavior type adjusted for the influence of blood pressure. A type-A individual has odds of a coronary event

Table 7–8. CHD by A/B by systolic blood pressure: the three-parameter additive logistic model

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.136

0.124

A/B

 The Analysis of Contingency Table Data: Logistic Model I

0.827

0.141

< 0.001

2.286

Blood pressure

ĉ

0.786

0.138

< 0.001

2.195

− 2 log-likelihood = 1709.954; number of model parameters = 3.

(p.205)

Table 7–9 CHD by A/B and systolic blood pressure: expected values based on the additive model parameter estimates (Table 7–8 )

Blood pressure

Behavior type

Log-odds

Notation

At risk

CHD

No CHD

≥ 140

A

− 1.523

l 11

388

69.46

318.54

≥ 140

B

− 2.350

l 12

305

26.54

278.46

< 140

A

− 2.309

l 21

1201

108.54

1092.46

< 140

B

− 3.136

l 22

1260

52.46

1207.54

2.286 times greater than a type-B individual regardless of blood pressure group ( ≥ 140 or < 140). Similarly,  The Analysis of Contingency Table Data: Logistic Model I is the estimated odds ratio reflecting CHD risk of individuals with blood pressure equal to or exceeding 140 relative to individuals with values less than 140 regardless of behavior type. Also the product of these odds ratios— The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I —measures the CHD risk associated with an individual simultaneously possessing both risk factors (type A with blood pressure ≥ 140) relative to an individual possessing neither risk factor (type B with blood pressure < 140). In terms of the additive model, the estimated difference in log-odds associated with possessing both risk factors relative to possessing neither factor is  The Analysis of Contingency Table Data: Logistic Model I , and the  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . In general, the odds ratio estimated by a logistic regression coefficient, when no interaction is present (additive model), measures the separate contribution of a specific factor to the risk of disease, adjusted for the influences of other risk factors in the model.

In the absence of an interaction, the factors influencing the outcome act additively on the log-odds scale and multiplicatively on the odds ratio scale. No interaction, therefore, requires each factor to contribute to the risk of disease separately, and the overall odds ratio becomes a product of a series of specific odds ratios, each associated with a single risk factor (in symbols, or = Πori ). Under these additive conditions, the risk factors are said to have an “independent” influence on the risk of disease.

A comparison of the model-generated values from the additive (no-interaction) model in Table 7–9 with the corresponding values in Table 7–6 (observed values) shows no important differences occur when the interaction term is deleted from the model (interaction present versus interaction absent). The influence of an interaction is formally evaluated by contrasting the log-likelihood statistics (Appendix E ) associated with each model. The logarithm of the likelihood value multiplied by – 2 is called the log-likelihood statistic. A log-likelihood statistic is a relative measure of the goodness-of-fit of a specific model and the difference between two log-likelihood statistics produces a test-statistic with an approximate chi-square distribution. A log-likelihood statistic increases with each (p.206) parameter deleted from a model. This increase is likely to be small when the deleted parameter is unrelated to the disease under study and substantial if the deleted parameter influences the risk of disease. The difference between two log-likelihood statistics, therefore, reflects the comparative strengths of two models to “explain” the data.

For the WCGS data, the two log-likelihood statistics are L d=0 = 1709.954 from the no interaction model (the parameter d deleted from the model) and L d≠0 = 1709.934 from the saturated model (the parameter d maintained in the model), showing that the descriptive power of the logistic model is essentially unchanged by assuming that no interaction exists (d = 0). The difference between two log-likelihood statistics has an approximate chi-square distribution when the observed increase arises from strictly capitalizing on random variation. The degrees of freedom are equal to the difference in the number of parameters necessary to describe each model. Using this log-likelihood chi-square test verifies that setting d = 0 in the logistic model has essentially no impact. Formally, the chi-square statistic is X 2 = L d=0L d≠0 = 1709.954 – 1709.934 = 0.020 with one degree of freedom and produces a p-value of P(X 2 ≥ 0.020 | d = 0) = 0.888. Contrasting log-likelihood statistics to evaluate the descriptive worth of two competing models is a fundamental statistical tool and is used repeatedly in the following discussion.

To evaluate the role of systolic blood pressure, the logistic model with blood pressure deleted from the model is the key to contrasting log-likelihood statistics. The additive model with the parameter c set to zero is summarized in Table 7–4 . The difference in the log-likelihood statistics for the model including blood pressure and the model excluding blood pressure indicates the contribution of the blood pressure variable to the additive model describing the risk of coronary disease. Thus, L c=0 = 1740.344 (blood pressure deleted) and L c≠0 = 1709.954 (blood pressure included) gives X 2 = L c=0L c≠0 = 30.390. The test-statistic X 2 has a chi-square distribution with one degree of freedom when c = 0. The associated p-value is P(X 2 ≥ 30.390 | c = 0) < 0.001, showing that systolic blood pressure is a vital part of the description of CHD risk. The comparison is not influenced by behavior type because this risk variable is maintained in both additive models.

A log-likelihood approach to evaluating a single variable is not basically different from the statistical test of the null hypothesis stating that a single parameter is zero. As indicated earlier, a squared estimated coefficient divided by its variance has an approximate chi-square distribution with one degree of freedom when the expected value of the coefficient is zero (Wald’s test). Specifically, the blood pressure test-statistic is X 2 = ĉ 2/variance(ĉ) = (0.786/0.138)2 = 32.4 with a p-value < 0.001. The usefulness of employing log-likelihood statistics to evaluate hypotheses stems from the fact that combinations of several risk (p.207) factors can be assessed simultaneously by comparing a model containing these risk factors to a model excluding these factors, resulting in a rigorous statistical assessment.

We can add here a note on the power to detect interaction effects. The power of a statistical test to detect an interaction can be relatively low, compared to tests of direct effects. In terms of the WCGS data, the probability of identifying an interaction between blood pressure and behavior type is less than the probability that such variables as blood pressure and behavior type will be identified as significant risk factors.

In a 2 × 2 × 2 contingency table, the reason for a relatively less powerful test is directly seen. Recall that the variance of the estimated measure of interaction is

 The Analysis of Contingency Table Data: Logistic Model I
where nijk represents a cell frequency in the 2 × 2 × 2 table. From the previous example, the estimated measure of an interaction effect associated with blood pressure and behavior type is – 0.040, and the estimated variance of the estimated coefficient  The Analysis of Contingency Table Data: Logistic Model I is
 The Analysis of Contingency Table Data: Logistic Model I
This variance involves all eight cells of the contingency table, and its magnitude is primarily determined by the cell with the lowest frequency—in this case, 27. Therefore, the variance of the interaction cannot be smaller than 1/27 = 0.037. Because the variance of an interaction is largely determined by the cell with the fewest observations, estimates of the interaction effects tend to have large variances causing statistical tests that tend to have an accompanying low statistical power. Estimates of the direct effects are less likely to involve low-frequency cells and thus produce more reliable estimates (smaller variances) and increased power. In more complex situations, the investigation of interactions also suffers from problems of low power for much the same reason as illustrated in the 2 × 2 × 2 case. Failure of a statistical test to reject the null hypothesis of no interaction provides some justification for employing an additive model, but the power of this test is frequently low. An analysis based on the proposition that the risk factors are additive simplifies the interpretation at the cost of potentially incurring biased estimates.

It is essential to detect interaction effects when they exist. It is not as critical to eliminate interaction terms when the data can support an additive model. (p.208) In terms of assessing the null hypothesis of no interaction (H 0), a type I error (rejecting H 0 when H 0 is true) is not as important as a type II error (accepting H 0 when H 0 is not true) when it comes to testing for an interaction. For a test of the impact of an interaction, it is a good idea to increase the level of significance (say, α = 0.2) to increase the power (Chapter 3 ). Increasing the type I error α to attain more statistical power (decreasing the type II error) is a conservative strategy in the sense that relatively minor losses occur, such as some loss of efficiency and increased complexity of the model, if interaction terms are unnecessarily included in the model (type I error). Mistaken elimination of interaction effects (type II error), however, substantially disrupts the validity of any conclusions (“wrong model bias”). To repeat one more time, ignoring an interaction leads to potentially misleading results.

THE 2 × k TABLE

A 2 × k table is effectively analyzed using a log-odds measure of risk. Underlying the log-odds approach is the assumption that the levels of the risk factor act as series of multiplicative effects. The previous approach to a 2 × k table used a straight line to summarize a series of proportions (Chapter 6 ), implying a linear relationship between the probability of disease and the levels of the risk factor. The WCGS data (Table 7–10 ) provide an example of a 2 × 4 table relating coronary heart disease to the amount smoked (k = 4 levels).

A logistic model that exactly duplicates (i.e., is a saturated model) the information contained in a 2 × k contingency table in terms of log-odds (denoted li ;  The Analysis of Contingency Table Data: Logistic Model I ) is, for k = 4, l 0 = a, l 1 = a + b 1, l 2 = a + b 2, and l 3 = a + b 3, or log-odds = a + b 1 x 1 + b 2 x 2 + b 3 x 3, where x 1, x 2, and x 3 are three binary variables (0, 1) that identify each level of the categorical risk factor. For example, the values x 1 = 0, x 2 = 1, and x 3 = 0 produce a model value log-odds = l 2 = a + b 2, which represents the risk from smoking 21 to 30 cigarette per day. The values x 1 = x 2 = x 3 = 0 establish a baseline or referent group (log-odds = l 0 = a) that reflects the CHD risk among nonsmokers.

Table 7–10. CHD by smoking: a 2 × k table

Cigarettes per day

0

1–20

21–30

> 30

Total

CHD

98

70

50

39

257

No CHD

1554

735

355

253

2897

Total

1652

805

405

292

3154

(p.209)

Aside: In Chapter 6 it was noted that the values of the k level variable in a 2 × k table can have at least three forms: numeric, ordered, and nominal. Each variable type suggests a specific analytic approach: regression, ridit, and chi-square, respectively. In using a logistic model to analyze categorical data, similar issues arise. Categories that can be characterized numerically are typically used directly in the regression analysis, thereby implying a specific relationship among the values. However, nominal categorical values have no specific ordering, and no logical numeric ordering is possible. A nominal variable is incorporated into a logistic regression equation using a binary indicator variable. For example, if two races are being considered, say white and African-American, one is coded 0 and the other 1. The logistic regression coefficients and the odds ratios are then relative to the variable coded zero. If the odds ratio is 2 and whites are coded as 0 and African-Americans as 1, then the odds are two times greater among African-Americans than among whites.

The principle of using an indicator variable can be extended. When more than two categories are present (for example, whites, African-Americans, Hispanics, and Asians), a series of indicator variables is used. For k categories, k – 1 indicator variables, each taking on the values 0 or 1, allow the analysis to include unordered categorical variables in a regression model. Such a variable is frequently called a design variable. Again, a baseline category is established by setting the k – 1 components of the design variable equal to zero. The members of other categories are identified by a single component of the design variable that takes on the value one while the remaining components are zero. For example, if African-Americans, Hispanics, and Asians are to be compared relative to the whites, then for k = 4 categories,

Whites: x 1 = 0, x 2 = 0, and x 3 = 0

Blacks: x 1 = 1, x 2 = 0, and x 3 = 0

Hispanics: x 1 = 0, x 2 = 1, and x 3 = 0

Asians: x 1 = 0, x 2 = 0, and x 3 = 1.

Like the binary case, the resulting logistic regression coefficients measure the role of each category relative to the category with all components set equal to zero. Any number of categories can be identified with this scheme, which allows the assessment of risk by using a logistic model without requiring a numeric value or even requiring that the categories be ordered. Many software analysis systems set up design variables automatically.

Values illustrating a design variable to identify four race/ethnic categories are given in Table 7–11 . Twelve individuals (three whites, four African-Americans, two Hispanics, and three Asians) make up the “data” set.

The logistic model describing four categories of smoking exposure and CHD risk is not restricted in any way because this saturated model can produce any pattern of log-odds values. A model that does not imply a particular pattern is said to be unconstrained. Estimates of the unconstrained model parameters are (notation is given in Table 6.2 )

 The Analysis of Contingency Table Data: Logistic Model I
(p.210)

Table 7–11. Illustration of the use of a design variable to establish membership in four race/ethnic categories

Individual

Race/ethnicity

x 1

x 2

x 3

1

African-American

1

0

0

2

White

0

0

0

3

Asian

0

0

1

4

White

0

0

0

5

African-American

1

0

0

6

African-American

1

0

0

7

Hispanic

0

1

0

8

Asian

0

0

1

9

White

0

0

0

10

Asian

0

0

1

11

African-American

1

0

0

12

Hispanic

0

1

0

The estimates from the WCGS smoking exposure data are displayed in Table 7–12 .

The odds ratios  The Analysis of Contingency Table Data: Logistic Model I measure the multiplicative risk for each smoking category relative to a nonsmokers (baseline = nonsmokers). For example, smoking more then 30 cigarettes a day produces an odds 2.444 (e 0.894) times greater than the odds among nonsmokers. Like all saturated models, the same odds ratio can be directly calculated from the data:  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . The odds ratios associated with increasing levels of smoking show an increasing risk of CHD (increasing from 1.0 to 1.5 to 2.2 to 2.4; Table 7–12 and Fig. 7–3 ). This increasing pattern is a property of the data and not the model. Any pattern, as mentioned, can emerge from an unconstrained model.

One reason to construct a 2 × k table is to explore the question of whether a risk factor has a specific pattern of influence on the probability of disease. For example, does the likelihood of a coronary event increase in a specific pattern

Table 7–12. CHD logistic model: smoking exposure (unconstrained and saturated model)

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 2.764

0.104

Smoking (1–20)

 The Analysis of Contingency Table Data: Logistic Model I

0.412

0.163

< 0.001

1.510

Smoking (21–30)

 The Analysis of Contingency Table Data: Logistic Model I

0.804

0.183

< 0.001

2.233

Smoking (> 30)

 The Analysis of Contingency Table Data: Logistic Model I

0.894

0.201

< 0.001

2.444

− 2 log-likelihood = 1751.695; number of model parameters = 4.

(p.211)
 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–3. Log-odds associated with CHD risk by amount smoked (unconstrained and constrained models).

with increasing levels of smoking? In terms of a logistic model, the log-odds described by the straight line is
 The Analysis of Contingency Table Data: Logistic Model I
where Fi represents one of k numeric levels of the risk factor and li represents the log-odds resulting from the ith level of the risk factor. The choices of the Fi - values brings a degree of subjectivity to the analysis that influences the results (Chapter 6 ). For the WCGS example, the values F 0 = 0, F 1 = 1, F 2 = 2, and F 3 = 3 are chosen to reflect the four smoking levels. Other equally spaced Fi -values reflecting each category give the same analysis (same p-values and log-likelihood statistics, but different values of the model parameters a and b). For example, using the levels of 0, 10, 20, and 30 cigarettes per day does not change (p.212) the analytic results (Chapter 6 ). For this straight-line model, the log-odds must follow a linearly increasing or decreasing pattern, making the model constrained. Such a constrained model no longer “fits” the data perfectly but represents the underlying relationships in a simple and frequently useful way.

Computer-generated maximum likelihood estimates for the parameters of the linear model from the smoking data are â = – 2.725 and  The Analysis of Contingency Table Data: Logistic Model I (Fig. 7–3 ) and  The Analysis of Contingency Table Data: Logistic Model I . The log-odds increases by an amount 0.325 between each of the four smoking categories. The log-likelihood statistic that best reflects the relative fit of the two-parameter constrained model is L 0 = 1753.045 (number of number parameters = 2). The previous unconstrained saturated model yields a log-likelihood statistic of L 1 = 1751.695 (number of number parameters = 4). The difference in log-likelihood statistics has an approximate chi-square distribution with two degrees of freedom when the models differ strictly because of sampling variation (X 2 = L 0L 1 = 1.350, p-value = P(X 2 ≥ 1.350 | no systematic difference) = 0.509). The comparison indicates that using a single straight line to represent the smoking/CHD relationship differs little from the saturated, unconstrained model (again Fig. 7–3 ). The straight-line model adequately and simply describes the relationship between smoking and CHD risk using two instead of four parameters. Consequently, the effect of the smoking categories can be effectively represented as increasing the risk of a CHD event in a parsimonously multiplicative pattern. In symbols,  The Analysis of Contingency Table Data: Logistic Model I implies  The Analysis of Contingency Table Data: Logistic Model I and  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , where  The Analysis of Contingency Table Data: Logistic Model I represents the odds ratio associated with the ith level of the risk factor relative to a baseline. For the smoking data, then,  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I , where the baseline is the nonsmoking group (Fi = 0) and the estimated odds ratios increase multiplicatively (Table 7–13 ).

The linear constrained model translates into estimated logistic probabilities since

 The Analysis of Contingency Table Data: Logistic Model I
providing an alternative expression of the relationship between smoking exposure and the likelihood of CHD (Table 7–13 ). For example, based on the estimated

Table 7–13. CHD risk: smoking exposure (straight-line model)

Levels

F = 0

F = 1

F = 2

F = 3

Odds ratio (or 0i )

(1.384)0

(1.384)1

(1.384)2

(1.384)2

Odds ratio (or 0i )

1.000

1.384

1.916

2.651

 The Analysis of Contingency Table Data: Logistic Model I

0.062

0.083

0.112

0.147

(p.213) linear model, smoking 30 or more cigarettes a day produces an odds of a CHD event 2.651 times greater than the odds among nonsmokers and a probability of a CHD event of 0.147. Necessarily, these two estimated values are related. That is,
 The Analysis of Contingency Table Data: Logistic Model I

THE 2 × 2 × k TABLE

When a logistic model is used to explore the relationships within the sampled data, there are two typical ways to begin; the simplest model or the most complex model. In the first case, variables are added to the model until a satisfactory description is achieved (the “forward” method). In the second case, variables are removed from the most complex model until a simpler but satisfactory statistical structure is found (the “backward” method). In this section the most complicated (most parameters) model serves as a starting point for analyzing the simultaneous influences of k levels of one categorical risk factor and two levels of another, a 2 × 2 × k table. The most complicated model (i.e., the saturated model) is log-odds = a + b 1 x 1 + b 2 x 2 + b 3 x 3 + cC + d 1 Cx 2 + d 2 Cx 2 + d 3 Cx 3 where the xi -values are, as before, components of a design variable indicating the categories of the four-level risk factor. The variable C = 1 identifies that another risk factor is present, and C = 0 identifies that it is absent. The three terms  The Analysis of Contingency Table Data: Logistic Model I incorporate into the model an interaction between the k-level and the two-level risk factor variables. In this context, an interaction is the failure of the four-level categorical variable to have the same relationship with the outcome variable at both levels of the variable C. All direct influences and all possible interactions are included in this statistical model, yielding an eight-parameter and, therefore, a saturated model. To be concrete, suppose x 0, x 1, x 2, and x 3 again indicate four smoking categories and C indicates the two categories of behavior type (A type coded C – 1 and B type coded C = 0). The WCGS data for the four smoking and the two behavior categories form a 2 × 2 × 4 table (Table 7–14 ). A logistic model applied to these data produces eight estimated parameters (Table 7–15 ).

Summary values calculated from a saturated model, as before, are the same as those calculated directly from the data themselves. The contingency table produces eight log-odds values, and the model employs eight parameters. Although a saturated model is not a summary of the sample of data, it provides a baseline against which to compare the utility of reduced models (models with fewer parameters) that are simpler representations of the relationships (p.214)

Table 7–14. CHD logistic model: smoking by behavior types (A/B) data

Cigs/day

0

1–20

21–30

> 30

Total

Type-A behavior

 CHD

70

45

34

29

178

 No CHD

714

353

185

159

1411

  Total

784

398

219

188

1589

Type-B behavior

 CHD

28

25

16

10

79

 No CHD

840

382

170

94

1486

  Total

868

407

186

104

1565

under study. One such reduced model postulates that no interaction exists between the k-level factor (smoking exposure) and the dichotomous risk factor (behavior type).

When the two risk factors do not interact, the log-odds model becomes

 The Analysis of Contingency Table Data: Logistic Model I
This special case of the previous saturated model is formed by setting the interaction coefficients to zero (d 1 = d 2 = d 3 = 0), making the relationship of the log-odds to the four-level categorical variable and the two-level variable C additive. The maximum likelihood estimates of the five parameters of the additive (no-interaction) model are given in Table 7–16 .

Geometrically, the additive model is represented by two parallel lines (Fig. 7–4 ). The pattern is not constrained to either increase or decrease with increasing levels of smoking, but CHD risk associated with type-A and type-B individuals differs by a constant amount (measured by the estimated coefficient ĉ = 0.815 from the model) on a log-odds scale for all four levels of smoking exposures.

Table 7–15. CHD logistic model: smoking and behavior type (saturated)

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.401

0.192

Smoking (1–20)

 The Analysis of Contingency Table Data: Logistic Model I

0.675

0.282

0.017

1.963

Smoking (21–30)

 The Analysis of Contingency Table Data: Logistic Model I

1.038

0.324

0.001

2.824

Smoking (> 30)

 The Analysis of Contingency Table Data: Logistic Model I

1.160

0.384

0.003

3.191

A/B

ĉ

1.079

0.229

< 0.001

2.941

Interaction

 The Analysis of Contingency Table Data: Logistic Model I

− 0.412

0.347

0.235

0.662

Interaction

 The Analysis of Contingency Table Data: Logistic Model I

− 0.410

0.395

0.299

0.664

Interaction

 The Analysis of Contingency Table Data: Logistic Model I

− 0.540

0.452

0.232

0.583

− 2 log-likelihood = 1713.694; number of model parameters = 8.

(p.215)

Table 7–16. CHD logistic model: smoking by behavior type (no interaction)

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.223

0.139

Smoking (1–20)

 The Analysis of Contingency Table Data: Logistic Model I

0.401

0.164

0.014

1.493

Smoking (21–30)

 The Analysis of Contingency Table Data: Logistic Model I

0.761

0.185

< 0.001

2.141

Smoking (> 30)

 The Analysis of Contingency Table Data: Logistic Model I

0.775

0.203

< 0.001

2.170

A/B

ĉ

0.815

0.141

< 0.001

2.259

− 2 log-likelihood = 1716.069; number of model parameters = 5.

The additive model dictates that the unconstrained pattern of log-odds associated with smoking exposure be exactly the same for both type-A and type-B individuals. The adjusted odds ratio measuring the impact of behavior type on CHD risk is eĉ = e 0.815 = 2.259 and is independent of smoking exposure (same at all four levels).

 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–4. Log-odds associated with CHD risk by amount smoked and behavior type (unconstrained and additive models).

(p.216) The increase in the log-likelihood value incurred by deleting the interaction terms from the model is evaluated by comparing the saturated model and the no-interaction model log-likelihood statistics (X 2 = L 0L 1 = 1716.069 – 1713.694 = 2.375 with three degrees of freedom producing the p-value = P(X 2 ≥ 2.375 | d 1 = d 2 = d 3 = 0) = 0.498). The comparison shows no persuasive evidence that smoking exposure and behavior type interact, thereby implying that the two risk factors are usefully represented as additive influences on CHD risk measured on a log-odds scale (multiplicatively on an odds scale). For example, the type-A (C = 1) individuals who smoke more than 30 cigarettes per day (x 1 = 0, x 2 = 0, x 3 = 1) have an estimated odds ratio of  The Analysis of Contingency Table Data: Logistic Model I relative to type-B individuals (C = 0) who are nonsmokers (x 1 = x 2 = x 3 = 0). Figure 7–4 contrasts the no-interaction model with the data (saturated model).

A further step in building a description of the relationships of the three variables in a 2 × 2 × k contingency table is to postulate a model that is linearly constrained. When the influence from the k-level factor is linearly related to the risk of disease, the log-odds model becomes

 The Analysis of Contingency Table Data: Logistic Model I
where Fi is a numeric value representing the ith level of a risk factor. To start, the interaction term FiC is included. Continuing the analysis of the smoking/behavior type data, as before, Fi = 0, 1, 2, or 3 represents levels of smoking exposure, and C is again a binary variable (1, 0) representing the dichotomous risk factor behavior type (A and B behavior types).

Geometrically, this model requires the log-odds values from a 2 × 2 × k contingency table to be random deviations from two straight lines with different slopes and intercepts—one line within each level of the dichotomous variable C. In terms of the example, a straight line for type-A individuals and a straight line for type-B individuals reflects the log-odds measure of CHD risk associated with the four levels of smoking exposure. Specifically, this statistical model expressed as two straight lines is

 The Analysis of Contingency Table Data: Logistic Model I
and
 The Analysis of Contingency Table Data: Logistic Model I

These two straight lines do not differ in principle from the straight-line logistic model used to analyze the 2 × k contingency table (previous section) since this constrained, interaction model can be viewed as summarizing two separate (p.217)

Table 7–17. CHD logistic model: smoking by behavior type (constrained with interaction)

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.305

0.164

Smoking

 The Analysis of Contingency Table Data: Logistic Model I

0.423

0.108

< 0.001

1.527

A/B

ĉ

1.005

0.198

< 0.001

2.733

Interaction

 The Analysis of Contingency Table Data: Logistic Model I

− 0.190

0.130

0.144

0.827

− 2 log-likelihood = 1715.859; number of model parameters = 4.

2 × k contingency tables. The estimates of the four parameters necessary to define this model are given in Table 7–17 .

Using the estimated parameters, expressions for the two estimated straight lines are

 The Analysis of Contingency Table Data: Logistic Model I
and
 The Analysis of Contingency Table Data: Logistic Model I
Contrasting the differences in log-likelihood statistics of this constrained model with the unconstrained saturated model (X 2 = L 0L 1 = 1715.859 – 1713.694 = 2.165 with four degrees of freedom, producing a p-value of 0.705) shows that using two different straight lines to describe the response in CHD risk to increasing levels of smoking exposure within each behavior type is an excellent summary (Fig. 7–5 ).

A further refinement of this constrained model postulates that the response to the k-level risk factor is the same for both levels of the dichotomous risk factor. This no-interaction model is achieved by setting d = 0 in the previous model, yielding an additive, three-parameter expression where

 The Analysis of Contingency Table Data: Logistic Model I

Again, the data are described by two straight lines, one within each category of the dichotomous risk factor, but the lines have identical slopes—namely, b. Geometrically, once again, “parallel” is synonymous with “no interaction.” Using this simple additive model, estimates of the three logistic model parameters are given in Table 7–18 .

The comparison of the log-likelihood statistics from the two constrained models indicates that describing the WCGS data (Table 7–14 ) with two parallel straight lines is not misleading and provides an adequate description of CHD risk (p.218)

 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–5. Log-odds associated with CHD risk by amount smoked and behavior type (constrained model).

(X 2 = L 0 = L 1 = 1717.972 – 1715.859 = 2.113 with one degree of freedom, producing p-value = P(X 2 ≥ 2.113 | d = 0) = 0.146). This parsimonious three-parameter description of the relationships among smoking exposure, behavior type, and CHD risk allows a simple description of the multiplicative roles of these two factors in the likelihood of coronary disease (Fig. 7–6 , dashed line).

Table 7–18. CHD logistic model: smoking exposure by behavior type (constrained with no interaction)

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.171

0.129

Smoking

 The Analysis of Contingency Table Data: Logistic Model I

0.290

0.060

< 0.001

1.336

A/B

ĉ

0.808

0.141

< 0.001

2.244

− 2 log-likelihood = 1717.972; number of model parameters = 3.

(p.219)

 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–6. Log-odds associated with CHD risk by amount smoked and behavior type (constrained and additive models).

The expressions for the two estimated parallel lines (same slope with different intercepts) are

 The Analysis of Contingency Table Data: Logistic Model I
and
 The Analysis of Contingency Table Data: Logistic Model I
The additive constrained model translates into estimated odds ratios:
 The Analysis of Contingency Table Data: Logistic Model I
and
 The Analysis of Contingency Table Data: Logistic Model I
(p.220)

Table 7–19. CHD risk by smoking exposure and behavior type (parallel straight-line model)

Levels

F = 0

F = 1

F = 2

F = 3

Type A  The Analysis of Contingency Table Data: Logistic Model I

2.244

2.998

4.008

5.356

Type B (or 0i )

1.000

1.336

1.786

2.387

where  The Analysis of Contingency Table Data: Logistic Model I and  The Analysis of Contingency Table Data: Logistic Model I are again the odds ratios associated with the ith level of smoking relative to the nonsmokers for type-B and type-A behavior types. The three-parameter additive and constrained model generates seven odds ratio estimates (Table 7–19 ).

These estimated odds ratios succinctly summarize the separate roles of behavior type and smoking in the risk of a coronary event incurring a modest lack of fit to achieve a simple and easily interpreted three-parameter model.

To a large extent, the analysis of trend is contained in the previous discussion, but it is worth focusing specifically on this aspect of a logistic regression model. Data on smoking exposure and CHD frequencies along with a series of odds ratios estimated under different conditions are given in Table 7–20 .

The directly calculated odds ratios and the odds ratios estimated from the logistic model do not substantially differ. The model

 The Analysis of Contingency Table Data: Logistic Model I
summarizes, as before, the linear trend in the log-odds and produces the model estimated odds ratio based on the regression coefficient  The Analysis of Contingency Table Data: Logistic Model I . Both the direct and model estimates show increasing odds ratios associated with increasing levels of smoking but fail to take into account influences from other possibly confounding variables. For example, a variable such as blood pressure may have an effect on the observed linear trend in the log-odds attributed to increased smoking. To account for the influence of blood pressure, a term is added to the logistic model so that the model becomes

Table 7–20. Odds ratios: smoking and CHD

Fi

0

1

2

3

CHD

98

70

50

39

No CHD

1554

735

355

253

 The Analysis of Contingency Table Data: Logistic Model I direct

1.0

1.510

2.233

2.444

 The Analysis of Contingency Table Data: Logistic Model I model

1.0

1.384

1.916

2.651

 The Analysis of Contingency Table Data: Logistic Model I adjusted

1.0

1.378

1.900

2.620

(p.221)
 The Analysis of Contingency Table Data: Logistic Model I
where D = 0 (blood pressure < 140) and D = 1 (blood pressure ≥ 140). The extended model yields an estimate of the coefficient describing the linear trend in CHD risk (log-odds) associated with smoking exposure taking into account the influence of blood pressure using an additive model, producing an adjusted regression coefficient  The Analysis of Contingency Table Data: Logistic Model I . Comparison of the estimated regression coefficients shows that systolic blood pressure measured as a binary variable has little influence on the increasing risk of a coronary event associated with the amount smoked ( The Analysis of Contingency Table Data: Logistic Model I and  The Analysis of Contingency Table Data: Logistic Model I , adjusted). The lack of difference in these coefficients is, of course, reflected in the similarity of the estimates of the odds ratios (Table 7–20 ; last two lines).

Adjustment for the influence of any number of variables follows the same pattern. The effect of each variable is accounted for by including a term in an additive logistic model. More exactly, the model

 The Analysis of Contingency Table Data: Logistic Model I
yields an estimate of the coefficient b adjusted for the influence of the other variables in the model and produces an unconfounded description of the linear trend in the log-odds associated with the variable represented by Fi . Wald’s test of the coefficient  The Analysis of Contingency Table Data: Logistic Model I provides a formal test of linear trend in log-odds. A chi-square statistic to evaluate linear trend is, then  The Analysis of Contingency Table Data: Logistic Model I and X 2 has a chi-square distribution with one degree of freedom when no linear trend exists (H 0: b = 0). The effectiveness of this description of the log-odds, however, depends on the effectiveness of the additive model to represent accurately the relationships among the variables in the model.

THE MULTIWAY TABLE

A logistic model directly extends to summarizing relationships among three categorical risk variables and a binary disease outcome. Interest is again focused on evaluating the impact of behavior type (two levels) on the risk of a coronary event while accounting for the influences from smoking (k = four levels) and blood pressure (two levels). As before, the blood pressure variable is defined as a binary variable based on systolic blood pressure measurements (< 140 and ≥ 140). These three risk variables produce a 2 × 4 × 4 contingency table of WCGS data (Table 7–21 ). The same data, in a different format, are given at the beginning of the chapter (Table 7–1 ).

The 16-parameter saturated model estimated from the WCGS data (Table 7–21 ) serves as a point of comparison for exploring reduced models. The log-likelihood (p.222)

Table 7–21. CHD by smoking by A/B by blood pressure

Cigs/day

0

1–20

21–30

> 30

Total

Type A and blood pressure ≥ 140

 CHD

29

21

7

12

69

 No CHD

155

76

45

43

319

  Total

184

97

52

55

388

Type A and blood pressure < 140

 CHD

41

24

27

17

109

 No CHD

559

277

140

116

1092

  Total

600

301

167

133

1201

Type B and blood pressure ≥ 140

 CHD

8

9

3

7

27

 No CHD

171

62

31

14

278

  Total

179

71

34

21

305

Type B and blood pressure < 140

 CHD

20

16

13

3

52

 No CHD

669

320

139

80

1208

  Total

689

336

152

83

1260

statistic is the minimum possible since a saturated model produces the minimum possible log-likelihood value. No other model could possible “fit” the data better. Incidentally, a chi-square goodness-of-fit test-statistic would be exactly zero. This minimum log-likelihood statistic is L 1 = 1667.138.

A basic question to be addressed arises: To what extent can the 16-parameter saturated model be reduced (fewer parameters) and still maintain a faithful but simpler representation of the relationships among the three risk variables (type-A/B behavior, smoking, blood pressure) and the probability of a CHD event? A “minimum” model that accounts for influences from the three risk factors involves four parameters, no interactions among the risk factors, and smoking exposure constrained to have the same linear impact on CHD risk within the four levels of the other two risk variables. This four-parameter logistic model is

 The Analysis of Contingency Table Data: Logistic Model I
where again Fi indicates one of the four levels of smoking (coded again, Fi = 0, 1, 2, and 3), the variable C = 0 or 1 indicates the behavior type (B, A), and the variable D = 0 or 1 indicates the level of blood pressure (< 140, ≥ 140). The estimates of the four parameters of this reduced model (Table 7–22 ) produces a log-likelihood statistic of L 0 = 1688.422.

Geometrically the model represents the relationships within the 2 × 4 × 4 table as four parallel straight lines on a log-odds scale (one for each of four blood pressure/behavior type categories), each describing the same linear increase in (p.223)

Table 7–22. CHD logistic model: A/B, smoking exposure, and blood pressure (no interaction and linearly constrained model)

Variable

Term

Estimate

SE

p-value

 The Analysis of Contingency Table Data: Logistic Model I

Constant

â

− 3.365

0.136

Smoking

 The Analysis of Contingency Table Data: Logistic Model I

0.286

0.060

< 0.001

1.331

A/B

ĉ

0.769

0.142

< 0.001

2.157

Blood pressure

 The Analysis of Contingency Table Data: Logistic Model I

0.779

0.139

< 0.001

2.179

− 2 log-likelihood = 1688.422; number of model parameters = 4.

CHD risk from increased levels of smoking. The distance between these straight lines measures the differing influences of blood pressure and behavior type on the risk of a coronary event. The four parallel lines along with the data (saturated model) are displayed in Figure 7.7 .

 The Analysis of Contingency Table Data: Logistic Model I

Figure 7–7. Log-odds associated with CHD risk by systolic blood pressure (< 140 and ≥ 140), amount smoked, and behavior type (constrained and additive four-parameter model).

(p.224) The utility of this four-parameter logistic model is evaluated by contrasting log-likelihood statistics—reduced (L 0) versus saturated (L 1)—where X 2 = L 0L 1 = 1688.422 – 1667.138 = 21.284 with 12 degrees of freedom, producing a p-value = P(X 2 ≥ 21.284 | model fits) = 0.046. The phrase “model fits” means that the differences between the observed values and the model-generated values are due to chance alone. This considerably simpler logistic model is not an extremely accurate reflection of the relationships among the risk factors and the log-odds associated with a coronary event. The increase in the log-likelihood statistic illustrates the typical trade-off between lack of fit and simplicity of the model. Simpler models fit less well. Although the simpler model is not ideal, it gives an approximate measure of the influences of the three risk factors on CHD risk as if they had independent effects, producing an extremely parsimonious description (Table 7–23 ).

The odds ratios in Table 7–23 are derived from combinations of the parameter estimates  The Analysis of Contingency Table Data: Logistic Model I , ĉ, and  The Analysis of Contingency Table Data: Logistic Model I using the relationship  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I . The odds ratios associated with the 15 different levels of the risk factors are relative to the baseline category—nonsmokers (F = 0), type B (C = 0), and blood pressure < 140 (D = 0) and or = 1.

GOODNESS-OF-FIT: DISCRETE CASE

For contingency table data, the goodness-of-fit of a logistic model is assessed in typical fashion by generating a series of expected values based on the model (denoted ek ) and comparing these values to the observed counts (denoted ok ) using a Pearson chi-square statistic. The estimated model parameters generate a logistic probability that produces the estimated number of individuals with and without CHD events (Table 7–24 ).

The model-generated frequencies for each of the 32 cells in the 2 × 4 × 4 table based on the four-parameter logistic model (Table 7–22 ) produce 32 comparisons  The Analysis of Contingency Table Data: Logistic Model I . The logistic probabilities are estimated using the relationship

 The Analysis of Contingency Table Data: Logistic Model I
and ek is either  The Analysis of Contingency Table Data: Logistic Model I individuals with CHD events or  The Analysis of Contingency Table Data: Logistic Model I individuals without CHD events where n is the total number of individuals in the category, or the sum of the CHD and non-CHD individuals. For example, when (p.225)

Table 7–23. CHD by A/B by systolic blood pressure: odds ratios

Blood pressure

Behavior type

F = 0

F = 1

F = 2

F = 3

≥ 140

A

4.700

6.256

8.327

11.083

≥ 140

B

2.179

2.900

3.860

5.138

< 140

A

2.157

2.871

3.821

5.086

< 140

B

1.000

1.331

1.772

2.358

Table 7–24. Goodness-of-fit: various summaries

Outcome

Blood pressure

Fi

ok

ek

ok ek

 The Analysis of Contingency Table Data: Logistic Model I

(ok ek )2/ek

Behavior type A

 CHD

≥ 140

0

29

25.722

3.278

0.646

0.418

 CHD

≥ 140

1

21

17.251

3.749

0.903

0.815

 CHD

≥ 140

2

7

11.625

− 4.625

− 1.357

1.840

 CHD

≥ 140

3

12

15.239

− 3.239

− 0.830

0.689

 No CHD

≥ 140

0

155

158.278

− 3.278

− 0.261

0.068

 No CHD

≥ 140

1

76

79.749

− 3.749

− 0.420

0.176

 No CHD

≥ 140

2

45

40.375

4.625

0.728

0.530

 No CHD

≥ 140

3

43

39.761

3.239

0.514

0.264

 CHD

< 140

0

41

42.027

− 1.027

− 0.158

0.025

 CHD

< 140

1

24

27.428

− 3.428

− 0.655

0.428

 CHD

< 140

2

27

19.663

7.337

1.655

2.738

 CHD

< 140

3

17

20.062

− 3.062

− 0.684

0.467

 No CHD

< 140

0

559

557.973

1.027

0.043

0.002

 No CHD

< 140

1

277

273.572

3.428

0.207

0.043

 No CHD

< 140

2

140

147.337

− 7.337

− 0.604

0.365

 No CHD

< 140

3

116

112.938

3.062

0.288

0.083

Behavior type B

 CHD

≥ 140

0

8

12.422

− 4.422

− 1.255

1.574

 CHD

≥ 140

1

9

6.411

2.589

1.023

1.046

 CHD

≥ 140

2

3

3.968

− 0.968

− 0.486

0.236

 CHD

≥ 140

3

7

3.141

3.859

2.177

4.741

 No CHD

≥ 140

0

171

166.578

4.422

0.343

0.117

 No CHD

≥ 140

1

62

64.589

− 2.589

− 0.322

0.104

 No CHD

≥ 140

2

31

30.032

0.968

0.177

0.031

 No CHD

≥ 140

3

14

17.859

− 3.859

− 0.913

0.834

 CHD

< 140

0

20

23.018

− 3.018

− 0.629

0.396

 CHD

< 140

1

16

14.778

1.222

0.318

0.101

 CHD

< 140

2

13

8.771

4.229

1.428

2.039

 CHD

< 140

3

3

6.256

− 3.256

− 1.302

1.694

 No CHD

< 140

0

669

665.982

3.018

0.117

0.014

 No CHD

< 140

1

320

321.222

− 1.222

− 0.068

0.005

  No CHD

< 140

2

139

143.229

− 4.229

− 0.353

0.125

 No CHD

< 140

3

80

76.744

3.256

0.372

0.138

  Total

3154

3154

0.0

0.0

22.126

(p.226) Fi = 3, C = 1, and D = 1, then  The Analysis of Contingency Table Data: Logistic Model I is the number of expected CHD events and e 8 = 55 – 15.239 = 39.761 is the number of expected individuals without CHD events among n = 55 study subjects who are type-A individuals with blood pressures greater than or equal to 140 who smoke more than 30 cigarettes per day. The corresponding observed values are o 4 = 12 and o 8 = 43. The remaining 31 model-generated and observed values are similarly calculated. A chi-square goodness-of-fit statistic (the sum of the last column) yields X 2 = Σ(ok ek )2/ek = 22.126 with 12 degrees of freedom. The p-value is 0.036. The similarity between the Pearson goodness-of-fit chi-square approach and the comparison of the log-likelihood statistics is typical (22.126 versus 21.284, each with 12 degrees of freedom). The two methods are likely to be similar for large sample sizes and usually do not substantially differ even when they are applied to moderately small samples of data.

The four-parameter logistic model, as seen in Table 7–24 (columns 7 and 8), is a reasonable representation of the observed frequencies in most categories. That is, the contributions to the chi-square statistic are small. Only a few categories are seriously misrepresented by the additive, constrained four-parameter logistic model. For example, individuals with high blood pressure who are type B and smoke more than 30 cigarettes per day are poorly predicted by the model (e 20 = 3.141, o 20 = 7 and (o 20e 20)2/e 20 = 4.741).

Although the four-parameter logistic model is not an ideal representation of the relationships among the risk variables, it unambiguously addresses the questions suggested by the table presented at the beginning of this chapter (Table 7–1 ). Blood pressure, behavior type, and smoking are represented as additive influences on the likelihood of a CHD event, allowing the influences of these risk factors to be evaluated separately (no interaction). Statistical tests demonstrate that these odds ratios represent substantial influences (not likely due to chance alone; p-values < 0.001). Therefore, the relative impacts on the likelihood of CHD estimated by odds ratios are  The Analysis of Contingency Table Data: Logistic Model I (behavior type), orbp = 2.157 (blood pressure type), and  The Analysis of Contingency Table Data: Logistic Model I (smoking more than 30 cigarettes per day). The three variables show about equal influence. Individuals with high blood pressure (greater than 140), with type-A behavior and who smoke more than 30 cigarettes per day have a combined risk (measured in terms of odds) 11 times the risk of individuals who have low blood pressure, are type B, and are nonsmokers. Specifically, the joint effect of these three risk factors produces an estimated odds ratio of (2.179)(2.157)(2.358) = 11.08. The log-odds measure of risk of CHD increases consistently (linearly) and independently as the level of smoking increases. The description of the risk factors and inferences based on the estimated additive model are unequivocally a function of the model parameters. However, as illustrated (p.227) by the CHD data, the correspondence between the model and the data is frequently a difficult issue.

SUMMARIZING A SERIES OF 2 × 2 TABLES

As already noted, one way to control for the confounding influence of a variable is to stratify the data into a series of more or less homogeneous groups based on values of the confounding variable (discussed in Chapter 2 and again in Chapter 10 ). One result of this process is a series of 2 × 2 tables (one table per stratum). For example, a series of strata formed on the basis of age might each contain individuals classified by the presence or absence of a coronary event, as well as by behavior type (A or B). To summarize the information contained in a series of 2 × 2 tables, three issues arise:

  1. 1. Interaction (homogeneity): Is the association between risk factor and disease the same for all tables (all strata)?

  2. 2. Association (independence): If the association is the same, then is it substantial and not likely due to random variation?

  3. 3. Estimation: If the association is the same and it is not likely due to random variation, then what is the magnitude of the risk factor/disease association?

Model-free methods answer these questions and are briefly reviewed. In addition, the parallel answers from a multivariable logistic model analysis are presented.

Test of Homogeneity

A single value summarizing the relationships within a series of 2 × 2 tables is most useful when the relationships summarized are the same for all tables (no interaction). In terms of an odds ratio, if a series of 2 × 2 tables reflect the same degree of association between risk factor and disease outcome (random fluctuations from a common overall value), then a single summary odds ratio serves as an accurate measure of that association. Otherwise, as with all interactions, a single summary odds ratio is not easily interpreted and can be entirely misleading.

Woolf [ 6 ] proposed a model-free approach to assessing the observed differences among a series of independent odds ratios (homogeneity). Again using the notation described earlier (Chapter 3 and Appendix C ), the null hypothesis (p.228) states that the odds ratio estimates calculated from each of k tables differ only because of random variation, or  The Analysis of Contingency Table Data: Logistic Model I , where ori is the odds ratio associated with the ith-specific 2 × 2 table. To develop a test-statistic to assess this hypothesis of no interaction, instead of the odds ratios themselves, the logarithms of the odds ratios are used. This transformation produces estimates with approximately normal distributions (Appendix C ).

A stratum-specific (ith stratum) odds ratio is estimated by

 The Analysis of Contingency Table Data: Logistic Model I
where, as before, the 2 × 2 table for the ith-stratum is as shown in Table 7–25 . A summary value of the logarithms of the odds ratio [denoted  The Analysis of Contingency Table Data: Logistic Model I ], derived from a weighted average of the logarithms of the k stratum–specific odds ratios  The Analysis of Contingency Table Data: Logistic Model I , is then
 The Analysis of Contingency Table Data: Logistic Model I
(W for Woolf who first presented the estimate) with the weights wi given by
 The Analysis of Contingency Table Data: Logistic Model I

The intuitive rationale for these weights is that, in determining the overall summary value, reliable estimates (small variance) should have relatively large weight, and unreliable estimates (large variance) should have relatively little weight.

Table 7–25. Notation for a specific 2 × 2 table: coronary disease and behavior type counts

Behavior type

CHD

No CHD

Total

A

ai

bi

ai + bi

B

ci

di

ci + di

  Total

ai + ci

bi + di

n

(p.229)

Table 7–26. WCGS data: age by behavior type by CHD outcome

Age

Type A

Type B

Total (ni )

CHD (ai )

No CHD (bi )

CHD (ci )

No CHD (di )

< 40

20

241

11

271

543

40–44

34

462

21

574

1091

45–49

49

337

21

343

750

50–54

38

209

17

184

448

> 54

37

162

9

114

322

   Total

178

1411

79

1486

3154

The question of whether the individual odds ratios (ori ) systematically differ from the overall odds ratio (or) is assessed by the Woolf test for homogeneity using the test-statistic

 The Analysis of Contingency Table Data: Logistic Model I
The test-statistic  The Analysis of Contingency Table Data: Logistic Model I has an approximate chi-square distribution with k – 1 degrees of freedom when the 2 × 2 tables are independent and homogeneous with respect to the k odds ratios (H 0 is true). A large chi-square value indicates that the odds ratios are not likely the same in all or some of the k strata.

The WCGS data once again illustrate. Study participants divided into five age groups to examine the relationship between behavior type and CHD risk allows the estimation of a summary odds ratio “removing” the influence of age (Table 7–26 ). These five separate 2 × 2 tables each generate an independent odds ratio to measure the association between behavior type and CHD events within each of the five strata (Table 7–27 , column 2). These odds ratios are minimally affected by the age of the study participants because the individuals within each stratum are essentially the same age.

Table 7–27. WCGS data: odds ratios and log-odds ratios by age category

Age

 The Analysis of Contingency Table Data: Logistic Model I

 The Analysis of Contingency Table Data: Logistic Model I

 The Analysis of Contingency Table Data: Logistic Model I

wi

< 40

2.045

0.715

0.149

6.723

40–45

2.012

0.699

0.081

12.355

45–49

2.375

0.865

0.074

13.530

50–54

1.968

0.677

0.095

10.487

> 54

2.893

1.062

0.153

6.532

   Total

2.373

0.864

49.627

(p.230) Using the example data,  The Analysis of Contingency Table Data: Logistic Model I , and  The Analysis of Contingency Table Data: Logistic Model I . The test of homogeneity ( The Analysis of Contingency Table Data: Logistic Model I with four degrees of freedom yields a p-value = 0.934) gives no reason to believe that the odds ratios systematically differ among the five age categories. That is, no evidence exists of an interaction.

Test of Association

A second step in summarizing a series of k-separate 2 × 2 tables is to assess the association between the risk factor and disease, using the data from all k tables. One such test is called the Mantel-Haenszel chi-square test [ 7 ]. William Cochran suggested essentially the same test in an earlier paper [ 8 ]. The approach generates cell frequencies estimated within each 2 × 2 table as if the risk factor and disease are independent (i.e., the null hypothesis). Again (see Chapter 6 ), the estimated cell frequency is

 The Analysis of Contingency Table Data: Logistic Model I
The variance of the distribution of ai is estimated by
 The Analysis of Contingency Table Data: Logistic Model I
The Mantel-Haenszel chi-square test-statistic compares the observed sum Σai to the same sum generated under the hypothesis that risk factor and disease are independent  The Analysis of Contingency Table Data: Logistic Model I and is
 The Analysis of Contingency Table Data: Logistic Model I
(MH for Mantel-Haenszel; Appendix D ).

The Mantel-Haenszel chi-square statistic combines information from each table, resulting in a summary test-statistic that reflects the association between risk factor and disease outcome as long as the odds ratios are homogeneous (i.e., have no interaction). The value  The Analysis of Contingency Table Data: Logistic Model I has an approximate chi-square distribution with one degree of freedom when risk factor and disease are unrelated in all k strata. Continuing the WCGS example, since Σai = 178 (observed) and  The Analysis of Contingency Table Data: Logistic Model I  The Analysis of Contingency Table Data: Logistic Model I (p.231) (expected), then  The Analysis of Contingency Table Data: Logistic Model I (p-value = P(X 2 ≥ 32.663 | independence) < 0.001), producing strong evidence that behavior type and CHD remain associated after accounting for influence from age.

Estimation of a Common Odds Ratio

The third step in summarizing a series of 2 × 2 tables is the estimation of a common measure of association. A popular estimate that provides an overall measure of association is the Mantel-Haenszel summary odds ratio [ 7 ] given by

 The Analysis of Contingency Table Data: Logistic Model I
When a series of odds ratios is homogeneous, the summary odds ratio  The Analysis of Contingency Table Data: Logistic Model I estimates the common value. For the age and behavior type data, the Mantel-Haenszel summary odds ratio is
 The Analysis of Contingency Table Data: Logistic Model I
The estimated odds ratio 2.214 summarizes, combining information from each age stratum, the association between behavior type and the occurrence of CHD and is “free” of influences from age.

Aside: The Mantel-Haenszel estimate combines a series of ratios to produce a single summary ratio. A summary ratio is typically constructed from a weighted average of each of a series of k ratios where

 The Analysis of Contingency Table Data: Logistic Model I
The choice of the weights wi determines the properties of the resulting summary ratio. The simplest choice of weights is wi = 1.0, giving
 The Analysis of Contingency Table Data: Logistic Model I
where k ratios are directly averaged to form a single estimate.

(p.232)

A more sophisticated estimate is based on the weights wi = xi . That is, the “worth” of each ratio is proportional to the denominator xi , giving

 The Analysis of Contingency Table Data: Logistic Model I
This form is the most common way ratios are combined from k sources of data.

If the weights are chosen so the  The Analysis of Contingency Table Data: Logistic Model I , then

 The Analysis of Contingency Table Data: Logistic Model I
which is an estimate of the slope of a straight line (least squares estimate) through the origin.

The weights that yield the Mantel-Haenszel estimate of the summary odds ratio are wi = b i c i/ni from each of the k individual odds ratios and

 The Analysis of Contingency Table Data: Logistic Model I
This choice of weights produces an efficient and effective summary estimate of the common odds ratio. Other choices are possible and have other properties.

The Woolf summary odds ratio  The Analysis of Contingency Table Data: Logistic Model I is similarly an estimate of the odds ratio common to the k = 5 homogeneous 2 × 2 tables. In addition, the estimated variance of the Woolf’s estimate of the logarithm of the summary odds ratio is particularly simple. It is the reciprocal of the sum of the weights wi (Table 7–27 ). In symbols, the estimated variance of  The Analysis of Contingency Table Data: Logistic Model I is variance = 1/Σwi . For the age-stratified data,

 The Analysis of Contingency Table Data: Logistic Model I
Approximate 95% confidence intervals follow the usual pattern. The confidence interval based on  The Analysis of Contingency Table Data: Logistic Model I is (0.511, 1.068), and the confidence interval based on the estimate  The Analysis of Contingency Table Data: Logistic Model I is (e 0.511, e 1.068), or (1.668, 2.909).

Logistic Regression Approach

The Woolf test for interaction, the Mantel-Haenszel test for association, and the Mantel-Haenszel summary odds ratio have analogous measures estimated from (p.233) a logistic regression model. In most situations, the results are similar from a seemingly different approach.

Four logistic models relating CHD outcome to the risk factors age (five categories) and behavior type are necessary:

  1. 1. Saturated model: log-odds = a + bF + c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 + d 1 Fx 1 + d 2 Fx 2 + d 3 Fx 3 + d 4 Fx 4.

    The variable F represents type-A (F = 1) and type-B (F = 0) behavior, and x 1, x 2, x 3, and x 4 are the components of a design variable that accounts for the five age categories. The log-likelihood statistic associated with this saturated 10-parameter model is L 1 = – 2 log-likelihood = 1702.156.

  2. 2. Additive model (behavior type and ageno interaction): log-odds = a + bF + c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 (see Table 7–28 , top).

  3. 3. Additive model (age onlyb = 0, no influence from behavior type): log-odds = a + c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 4 (see Table 7–28 , middle).

  4. 4. Additive model (behavior type onlyc 1 = c 2 = c 3 = c 4 = 0, no influence from age): log-odds = a + bF (see Table 7–28 , bottom).

Table 7–28. Three additive models

Variable

Term

Estimate

SE

p-value

Odds ratio

Behavior type and age—no interaction

 Constant

â

− 2.461

0.192

 A/B

 The Analysis of Contingency Table Data: Logistic Model I

0.793

0.141

< 0.001

2.210

 Age 40–44

ĉ 1

− 0.112

0.232

0.629

0.894

 Age 45–49

ĉ 2

0.510

0.225

0.023

1.665

 Age 50–54

ĉ 3

0.793

0.236

< 0.001

2.211

 Age > 54

ĉ 4

0.921

0.246

< 0.001

2.512

L 2 = − 2 log-likelihood = 1703.010; number of model parameters = 6.

Age only; b = 0—no influence from behavior type

 Constant

â

− 2.894

0.192

 Age 40–44

ĉ 1

− 0.132

0.231

0.569

0.877

 Age 45–49

ĉ 2

0.531

0.224

0.018

1.700

 Age 50–54

ĉ 3

0.838

0.234

< 0.001

2.311

 Age > 54

ĉ 4

1.013

0.246

< 0.001

2.753

L 3 = − 2 log-likelihood = 1736.578; number of model parameters = 5.

Behavior type only; c 1 = c 2 = c 3 = c 4 = 0—no influence from age

 Constant

â

− 2.934

0.115

 A/B

 The Analysis of Contingency Table Data: Logistic Model I

0.864

0.140

< 0.001

2.373

L 4 = − 2 log-likelihood = 1740.344; number of model parameters = 2.

(p.234)

Table 7–29. Comparison of four logistic models

Model

Li

Likelihood

 The Analysis of Contingency Table Data: Logistic Model I

SE

 The Analysis of Contingency Table Data: Logistic Model I

Parameters

1.

A/B + age + interactions

L 1

1702.156

0.715

0.386

2.045

10

2.

A/B + age

L 2

1703.010

0.793

0.141

2.210

6

3.

Age only

L 3

1736.578

5

4.

A/B only

L 4

1740.344

0.864

0.140

2.373

2

Summary results from these four models applied to the WCGS data allow a multivariable logistic analysis summarizing k separate 2 × 2 tables that parallels the three steps of the model-free approach (Table 7–29 ).

To test for possible interaction effects (that is, do the odds ratios systematically differ among some or all of the five age categories?), two log-likelihood statistics are compared. The log-likelihood statistic calculated from the saturated model with the interaction terms included (model 1: L 1 = 1702.156) is compared to the log-likelihood statistic from the model with the interaction terms ignored (model 2: L 2 = 1703.010), producing an increase of X 2 = L 2L 1 = 0.854, which reflects the lack of homogeneity among the five odds ratios. The contrast of the log-likelihood values from these two logistic models shows no evidence of an interaction (p-value = 0.931). This result is similar to the Woolf homogeneity chi-square value (p-value = 0.834). These two approaches are likely to be similar in general, particularly when each stratum contains a moderate or a large number of observations.

A Mantel-Haenszel-like test of the association between behavior type and CHD can be conducted in the context of an additive logistic model. The difference between the additive model containing the risk factors behavior type and age (model 2) and the model containing only age (model 3) indicates the degree of association between behavior type and CHD risk while accounting for the influence of age. These two additive (no-interaction) models produce a difference in log-likelihood statistics analogous to the Mantel-Haenszel test-statistic  The Analysis of Contingency Table Data: Logistic Model I . The increase in log-likelihood values X 2 = L 3L 2 has a chi-square distribution with one degree of freedom when the risk variable is unrelated to the outcome (b = 0). Both the Mantel-Haenszel chi-square and the logistic model approaches require that no interactions exist among the k tables. For the age and behavior type data, the difference in log-likelihood values is X 2 = L 3L 2 = 1736.578 – 1703.010 = 33.568 (p-value < 0.001), which is not very different from the Mantel-Haenszel chi-square value calculated to assess the same association  The Analysis of Contingency Table Data: Logistic Model I . As with the tests to assess interaction effects, these two tests of association will usually be similar.

(p.235)

Table 7–30. Two approaches to summarizing k independent 2 × 2 tables (strata)

No model

Logistic model

Homogeneity

 The Analysis of Contingency Table Data: Logistic Model I

X 2 = 0.854

Association

 The Analysis of Contingency Table Data: Logistic Model I

X 2 = 33.568

Odds ratio

 The Analysis of Contingency Table Data: Logistic Model I

 The Analysis of Contingency Table Data: Logistic Model I

Odds ratio

 The Analysis of Contingency Table Data: Logistic Model I

When a series of 2 × 2 tables shows no likely interaction effects, the strictly additive logistic model (model 2) produces an estimate of the strength of the association between risk factor and disease adjusted for confounding influences from the strata-variable. For the WCGS data, the behavior type coefficient  The Analysis of Contingency Table Data: Logistic Model I is 0.793 (model 2) and the estimated adjusted odds ratio is  The Analysis of Contingency Table Data: Logistic Model I , which is almost identical to  The Analysis of Contingency Table Data: Logistic Model I . The 95% confidence interval based on the estimate  The Analysis of Contingency Table Data: Logistic Model I is (0.517, 1.069), and the corresponding confidence for the odds ratio is (1.676, 2.914). Both estimates measure the influence of A/B behavior type in determining the risk of CHD, accounting for the influence of age. In general, the odds ratio estimated from the additive logistic model, the Mantel-Haenszel and the Woolf summary odds ratio estimates will be similar. To repeat, these three estimates require homogeneity of the odds ratios (no interactions) among the k-independent tables to summarize the association between behavior type and CHD. The maximum likelihood estimate  The Analysis of Contingency Table Data: Logistic Model I from the additive logistic model is very slightly more efficient (smaller variance) than either  The Analysis of Contingency Table Data: Logistic Model I or  The Analysis of Contingency Table Data: Logistic Model I . Table 7–30 provides a summary comparison of the model-free and logistic model approaches to summarizing an association from a series of k-independent 2 × 2 tables (Table 7–30 ).

The confounding influence of the age variable is identified by comparing values of the estimate  The Analysis of Contingency Table Data: Logistic Model I from two specific additive logistic models. The value of  The Analysis of Contingency Table Data: Logistic Model I estimated from the logistic model accounting for age is compared to the estimate from the model excluding age (model 2 versus model 4). When age in included,  The Analysis of Contingency Table Data: Logistic Model I ; when age is excluded,  The Analysis of Contingency Table Data: Logistic Model I . In terms of odds ratios,  The Analysis of Contingency Table Data: Logistic Model I , and  The Analysis of Contingency Table Data: Logistic Model I . In both cases, the difference measures the confounding influence of age on a measure of the relationship between behavior and CHD.

The Mantel-Haenszel summary odds ratio calculated from the five age strata is 2.214. The same odds ratio calculated ignoring the five age strata is 2.373. The difference also measures the confounding influence of age (bias) on the odds ratio.