Daniel W.W. Biostatistics: A Foundation for Analysis in the Health Sciences

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EXAMPLE 12.8.2

Let us refer again to the data on primary malignant tumors of the sternum presented

in Example 12.8.1. Examination of the data reveals that there are 20 time periods

(strata). For each of these a table following the pattern of Table 12.8.3 must be

constructed. The ﬁrst of these tables is shown as Table 12.8.4. By Equations 12.7.5 and

12.7.6 we compute and as follows:

The data for Table 12.8.4 and similar data for the other 19 time periods are shown in

Table 12.8.5. Using data from Table 12.8.5, we compute the log-rank statistic by Equa-

tion 12.7.7 as follows:

Reference to Appendix Table F reveals that since the p value for this

test is We, therefore, reject the null hypothesis that the survival experience is

the same for patients with low-grade tumors and high-grade tumors and conclude that

they are different.

There are alternative procedures for testing the null hypothesis that two survival

curves are identical. They include the Breslow test (also called the generalized Wilcoxon

test) and the Tarone–Ware test. Both tests, as well as the log-rank test, are discussed in

Parmar and Machin (26). Like the log-rank test, the Breslow test and the Tarone–Ware test

are based on the weighted differences between actual and expected numbers of deaths at

6 .005.

24.704 7 7.879,

19 - 17.8112

3.140

= 24.704

10 + 12125 + 13210 + 25211 + 132

1382

= .230

10 + 1210 + 252

= .641

2 * 2

12.8 SURVIVAL ANALYSIS 657

TABLE 12.8.3 Contingency Table for Stratum (Time)

for Calculating the

Log-Rank Test

Group A Group B Total

Number of deaths observed

Number of patients alive

Number of patients “at risk” n

= a

+ b

+ c

+ d

+ c

+ d

+ b

TABLE 12.8.4 Contingency Table for First Stratum (Time

Period) for Calculating the Log-Rank Test, Example 12.8.2

Low-Grade High-Grade Total

Deaths 0 1 1

Patients alive 25 13 38

Patients at risk 25 13 39

the observed time points. Whereas the log-rank test ranks all deaths equally, the Breslow

and Tarone–Ware tests give more weight to early deaths. For Example 12.8.1, SPSS com-

putes a value of for the Breslow test and a value of

for the Tarone–Ware test. Kleinbaum (27) discusses another test called the Peto test. For-

mulas for this test are found in Parmar and Machin (26). The Peto test also gives more

weight to the early part of the survival curve, where we ﬁnd the larger numbers of sub-

jects at risk. When choosing a test, then, researchers who want to give more weight to the

earlier part of the survival curve will select either the Breslow, the Tarone–Ware, or the

Peto test. Otherwise, the log-rank test is appropriate.

We have covered only the basic concepts of survival analysis in this section. The

reader wishing to pursue the subject in more detail may consult one or more of several

books devoted to the topic, such as those by Kleinbaum (27), Lee (28), Marubini and

Valsecchi (29), and Parmar and Machin (26).

Computer Analysis

Several of the available statistical software packages, such as SPSS, are capable of per-

forming survival analysis and constructing supporting graphs as described in this section.

A standard SPSS analysis of the data discussed in Examples 12.8.1 and 12.8.2 is

shown in Figure 12.8.3.

■

25.22 1p = .0000224.93 1p = .00002

658 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES

TABLE 12.8.5 Intermediate Calculations for the Log-Rank Test, Example 12.8.2

Time,

0 0 25 25 1 13 14 39 0.641 0.230

2 0 25 25 1 12 13 38 0.658 0.225

3 0 25 25 1 11 12 37 0.676 0.219

4 0 25 25 1 10 11 36 0.694 0.212

6 0 25 25 1 9 10 35 0.714 0.204

7 0 24 24 1 8 9 33 0.727 0.198

9 0 23 23 2 6 8 31 1.484 0.370

11 1 22 23 0 6 6 29 0.793 0.164

12 0 22 22 1 5 6 28 0.786 0.168

15 2 20 22 0 5 5 27 1.630 0.290

16 0 20 20 1 4 5 25 0.800 0.160

17 0 20 20 1 3 4 24 0.833 0.139

21 1 19 20 0 3 3 23 0.870 0.113

23 0 19 19 1 2 3 22 0.864 0.118

27 0 18 18 1 1 2 20 0.900 0.090

29 1 17 18 0 1 1 19 0.947 0.050

33 1 16 17 0 1 1 18 0.944 0.052

79 1 13 14 0 1 1 15 0.933 0.062

102 1 10 11 0 1 1 12 0.917 0.076

212 1 1 2 0 0 0 2 1.000 0.000

Totals 9 17.811 3.140

 d

+ c

The Proportional Hazards or Cox Regression Model In previous

chapters, we saw that regression models can be used for continuous outcome measures

and for binary outcome measures (logistic regression). Additional regression techniques

are available when the dependent measures may consist of a mixture of either time-until-

event data or censored time observations. Returning to our example of a clinical trial of

the effectiveness of two different medications to prevent a second myocardial infarction,

we may wish to control for additional characteristics of the subjects enrolled in the study.

12.8 SURVIVAL ANALYSIS 659

Means and Medians for Survival Time

Mean

Median

95% Conﬁdence Interval 95% Conﬁdence Interval

tumor_ Std. Lower Upper Std. Lower Upper

grade Estimate Error Bound Bound Estimate Error Bound Bound

H 18.357 8.251 2.186 34.528 9.000 1.852 5.371 12.629

L 88.040 15.258 58.134 117.946 82.000 16.653 49.359 114.641

Overall 63.026 11.490 40.505 85.546 27.000 7.492 12.317 41.683

Estimation is limited to the largest survival time if it is censored.

Overall Comparisons

Chi-Square df Sig.

Log Rank (Mantel-Cox) 24.704 1 .000

Breslow (Generalized

Wilcoxon) 24.927 1 .000

Tarone-Ware 25.217 1 .000

Test of equality of survival distributions for the different levels of tumor_grade.

FIGURE 12.8.3 SPSS output for Examples 12.8.1 and 12.8.2.

For example, we would expect subjects to be different in their baseline systolic blood

pressure measurements, family history of heart disease, weight, body mass, and other

characteristics. Because all of these factors may inﬂuence the length of the time interval

until a second myocardial infarction, we would like to account for these factors in deter-

mining the effectiveness of the medications. The regression method known as Cox regres-

sion (after D. R. Cox, who ﬁrst proposed the method) or proportional hazard regression

can be used to account for the effects of continuous and discrete covariate (independent

variable) measurements when the dependent variable is possibly censored time-until-

event data.

We describe this technique by ﬁrst introducing the hazard function, which describes

the conditional probability that an event will occur at a time just larger than condi-

tional on having survived event-free until time This conditional probability is also

known as the instantaneous failure rate at time and is often written as the function

The regression model requires that we assume the covariates have the effect of

either increasing or decreasing the hazard for a particular individual compared to some

baseline value for the function. In our clinical trial example we might measure k covari-

ates on each of the subjects where there are subjects and is the base-

line hazard function. We describe the regression model as

(12.8.3)

The regression coefficients represent the change in the hazard that results from

the risk factor, that we have measured. Rearranging the above equation shows that

the exponentiated coefficient represents the hazard ratio or the ratio of the conditional

probabilities of an event. This is the basis for naming this method proportional haz-

ards regression. You may recall that this is the same way we obtained the estimate of

the odds ratio from the estimated coefficient when we discussed logistic regression in

Chapter 11.

(12.8.4)

Estimating the covariate effects, requires the use of a statistical software package

because there is no straightforward single equation that will provide the estimates

for this regression model. Computer output usually includes estimates of the regres-

sion coefficients, standard error estimates, hazard ratio estimates, and confidence

intervals. In addition, computer output may also provide graphs of the hazard func-

tions and survival functions for subjects with different covariate values that are use-

ful to compare the effects of covariates on survival. In summary, Cox regression is a

useful technique for determining the effects of covariates with survival data. Addi-

tional information can be found in the texts by Kleinbaum (27), Lee (28), Kalbfleisch

and Prentice (30), Elandt-Johnson and Johnson (31), Cox and Oakes (32), and Fleming

and Harrington (33).

h1t

= exp1b

+ b

h1t

2= h

2exp1b

+ b

2I = 1,...,n

h1t

660 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES

EXERCISES

12.8.1 Fifty-three patients with medullary thyroid cancer (MTC) were the subjects of a study by Dot-

torini et al. (A-24), who evaluated the impact of different clinical and pathological factors and the

type of treatment on their survival. Thirty-two of the patients were females, and the mean age of

all patients was 46.11 years with a standard deviation of 14.04 (range 18–35 years). The follow-

ing table shows the status of each patient at various periods of time following surgery. Analyze

the data using the techniques discussed in this section.

Subject Time

(Years) Status

Subject Time

(Years) Status

1 0 doc 28 6 alive

2 1 mtc 29 6 alive

3 1 mtc 30 6 alive

4 1 mtc 31 6 alive

5 1 mtc 32 7 mtc

6 1 mtc 33 8 alive

7 1 mtc 34 8 alive

8 1 mtc 35 8 alive

9 1 alive 36 8 alive

10 2 mtc 37 8 alive

11 2 mtc 38 9 alive

12 2 mtc 39 10 alive

13 2 alive 40 11 mtc

14 2 alive 41 11 doc

15 3 mtc 42 12 mtc

16 3 mtc 43 12 doc

17 3 alive 44 13 mtc

18 4 mtc 45 14 alive

19 4 alive 46 15 alive

20 4 alive 47 16 mtc

21 4 alive 48 16 alive

22 5 alive 49 16 alive

23 5 alive 50 16 alive

24 5 alive 51 17 doc

25 5 alive 52 18 mtc

26 6 alive 53 19 alive

27 6 alive

Time is number of years after surgery.

doc dead of other causes; mtc dead of medullary thyroid cancer.

Source: Dr. Massimo E. Dottorini. Used with permission.

12.8.2 Banerji et al. (A-25) followed non–insulin-dependent diabetes mellitus (NIDDM) patients from

onset of their original hyperglycemia and the inception of their near–normoglycemic remission fol-

lowing treatment. Subjects were black men and women with a mean age of 45.4 years and a stan-

dard deviation of 10.4. The following table shows the relapse/remission experience of 62 subjects.

Use the techniques covered in this section to analyze these data.

EXERCISES 661

Total Total Total

Duration of Duration of Duration of

Remission Remission Remission Remission Remission Remission

(Months) Status

3 1 82261

3 2 92271

3 1101282

3 1101291

3 1112312

4 1131311

4 1161332

4 1162392

5 1172411

5 1182441

5 1201461

5 1221462

5 1222481

5 1222482

5 1231481

6 1242491

6 1252501

6 1252531

7 1261702

8 2261941

1  yes (the patient is still in remission); 2  no (the patient has relapsed).

Source: Dr. Mary Ann Banerji. Used with permission.

12.8.3 If available in your library, read the article, “Impact of Obesity on Allogeneic Stem Cell Transplant

Patients: A Matched Case-Controlled Study,” by Donald R. Fleming et al. [American Journal of

Medicine, 102 (1997), 265–268] and answer the following questions:

(a) How was survival time determined?

(b) Why do you think the authors used the Wilcoxon test (Breslow test) for comparing the sur-

vival curves?

(d) What speciﬁc statistical results allow the authors to arrive at their stated conclusion?

12.8.4 If available in your library, read the article, “Improved Survival in Patients with Locally Advanced

Prostate Cancer Treated with Radiotherapy and Goserelin,” by Michel Bolla et al. [New England

Journal of Medicine, 337 (1997), 295–300], and answer the following questions:

(a) How was survival time determined?

(b) Why do you think the authors used the log-rank test for comparing the survival curves?

662 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES

(d) What speciﬁc statistical results allow the authors to arrive at their stated conclusion?

12.8.5 Fifty subjects who completed a weight-reduction program at a ﬁtness center were divided into

two equal groups. Subjects in group 1 were immediately assigned to a support group that met

weekly. Subjects in group 2 did not participate in support group activities. All subjects were fol-

lowed for a period of 60 weeks. They reported weekly to the ﬁtness center, where they were

weighed and a determination was made as to whether they were within goal. Subjects were con-

sidered to be within goal if their weekly weight was within 5 pounds of their weight at time of

completion of the weight-reduction program. Survival was measured from the date of completion

of the weight-reduction program to the termination of follow-up or the point at which the sub-

ject exceeded goal. The following results were observed:

Status Status

(G  Within Goal (G  Within Goal

Time GExceeded Goal Time GExceeded Goal

Subject (Weeks) L  Lost to Follow-Up) Subject (Weeks) L  Lost to Follow-Up)

Group 1 Group 2

160 G 120 G

232 L 226 G

360 G 310 G

422 L 42 G

56 G+ 536 G

660 G 610 G

760 G 720 G

8 20 G+ 8 18 L

9 32 G+ 9 15 G

10 60 G 10 22 G

11 60 G 11 4 G

12 8 G+ 12 12 G

13 60 G 13 24 G

14 60 G 14 6 G

15 60 G 15 18 G

16 14 L 16 3 G

17 16 G+ 17 27 G

18 24 L 18 22 G

19 34 L 19 8 G

20 60 G 20 10 L

21 40 L 21 32 G

22 26 L 22 7 G

23 60 G 23 8 G

24 60 G 24 28 G

25 52 L 25 7 G

Analyze these data using the methods discussed in this section.

EXERCISES 663

12.9 SUMMARY

In this chapter some uses of the versatile chi-square distribution are discussed. Chi-square

goodness-of-ﬁt tests applied to the normal, binomial, and Poisson distributions are pre-

sented. We see that the procedure consists of computing a statistic

that measures the discrepancy between the observed and expected frequencies of

occurrence of values in certain discrete categories. When the appropriate null hypothesis is

true, this quantity is distributed approximately as When is greater than or equal to

the tabulated value of for some the null hypothesis is rejected at the level of sig-

niﬁcance.

Tests of independence and tests of homogeneity are also discussed in this chapter.

The tests are mathematically equivalent but conceptually different. Again, these tests

essentially test the goodness-of-ﬁt of observed data to expectation under hypotheses,

respectively, of independence of two criteria of classifying the data and the homogene-

ity of proportions among two or more groups.

In addition, we discussed and illustrated in this chapter four other techniques for

analyzing frequency data that can be presented in the form of a contingency table:

the Fisher exact test, the odds ratio, relative risk, and the Mantel–Haenszel procedure.

Finally, we discussed the basic concepts of survival analysis and illustrated the compu-

tational procedures by means of two examples.

SUMMARY OF FORMULAS FOR CHAPTER 12

Formula

Number Name Formula

12.2.1 Standard normal random variable

12.2.2 Chi-square distribution with n

degrees of freedom

12.2.3 Chi-square probability density

function

12.2.4 Chi-square test statistic

12.4.1 Chi-square calculation formula

for a 2  2 contingency table x

n1ad - bc2

1a + c21b + d21a + b21c + d2

- E

f1u2=

- 1b!

k>2

1k>22-1

-1u>22

1n2

= z

+ z

- m

2 * 2

aa,x

21O

- E

664 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES

12.4.2 Yates’s corrected

chi-square calcu-

lation for a 2  2

contingency table

12.6.1–12.6.2 Large-sample

approximation

to the chi-square

12.7.1 Relative risk

estimate

12.7.2 Conﬁdence

interval for the

relative risk

estimate

12.7.3 Odds ratio

estimate

12.7.4 Conﬁdence

interval for the

odds ratio

estimate

12.7.5 Expected

frequency in

the Mantel–

Haenszel statistic

12.7.6 Stratum expected

frequency in the

Mantel–Haenszel

statistic

12.7.7 Mantel–Haenszel

test statistic

12.7.8 Mantel–Haenszel

estimator of the

common odds ratio

i = 1

+ b

21c

+ d

21a

+ c

21b

+ d

- 12

+ b

21a

+ c

10011 - a2%CI = OR

1; 1z

>1x

10011 - a2%CI = RR

1; 1z

>1x

corrected

ad - bc

-.5n2

1a + c21b + d21a + b21c + d2

SUMMARY OF FORMULAS FOR CHAPTER 12 665

where

= 1a + b2>1A + B2

z =

1a>A2- 1b>B2

11 - p

211>A + 1>B2

(Continued)

RR =

a>1a + b2

c>1c + d2

OR =

a>b

c>d

12.8.1 Survival number of subjects surviving at least (t  1) time period

probability

12.8.2 Estimated

survival

function

12.8.3 Hazard

regression

model

12.8.4 Proportional

hazard model

Symbol • a, b, c, d  cell frequencies in a 2  2 contingency table

Key • A, B  row totals in the 2  2 contingency table

•  regression coefﬁcient

• (or X

)  chi-square

• e

 expected frequency in the Mantel–Haenszel statistic

• E

 expected frequency

•  expected value of y at x

• k  degrees of freedom in the chi-square distribution

•  mean

• O

 observed frequency

• OR  odds ratio estimate

•  standard deviation

• RR  relative risk estimate

• v

 stratum expected frequency in the Mantel–Haenszel statistic

• y

 data value at point i

• z  normal variate

REVIEW QUESTIONS AND EXERCISES

1. Explain how the chi-square distribution may be derived.

2. What are the mean and variance of the chi-square distribution?

3. Explain how the degrees of freedom are computed for the chi-square goodness-of-ﬁt tests.

4. State Cochran’s rule for small expected frequencies in goodness-of-ﬁt tests.

5. How does one adjust for small expected frequencies?

6. What is a contingency table?

7. How are the degrees of freedom computed when an value is computed from a contingency

table?

h1t

= exp1b

+ b

h1t

2= h

2exp1b

+ b

1t2= p

* p

666 CHAPTER 12 THE CHI-SQUARE DISTRIBUTION AND THE ANALYSIS OF FREQUENCIES

who also survive t

period

number of subjects alive at end of time period 1t - 12