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Shortcut method to calculate the sample size in trials of screening for chronic disease
J Clii E&l
Vol.43,No. 11,pp. 1261-1266,1990
0895-4356/90 $3.00+ 0.00 Copyright0 1990PergamonPres8plc
F’rinted in Grcat Britain. AU rights rcscrvcd
SHORTCUT METHOD TO CALCULATE THE SAMPLE SIZE IN TRIALS OF SCREENING FOR CHRONIC DISEASE HUUB STRAATMAN and ANDRÉ. L. M. VERBEEK Department of Epidemiology, Institute of Social Medicine, Nijmegen University, Verlengde Groenestraat 75, 6525 EJ Nijmegen, The Netherlands (Received in revised farm 2 November 1989; received for publication 26 April 1990)
Ahstract-One of the first questions arising in the planning of a randomized trial to evaluate mortality reduction by screening concerns the sample size of the trial required to detect an expected mortality reduction in the study group for given significante leve1 a, and power 1 - 8. If estimates exist of the underlying average annual incidence rate of the disease r, and the annual mortality rate 6, or survival data for patients in the population under consideration before screening started, then a simple formula for the probability of dying from the disease within Tyears after entry into the trial can be given for the control group. Standard formulas may then be used for sample size calculations in randomized trials, which compare the risk of death from the disease in the control and the study group accrued at T years after entry. A simple correction for loss of follow-up, due to mortality from other causes or, for instance, migration is possible. Power calculation
Trial
Mass screening
INTRODUCTION
The ultimate goal of screening for chronic disease (e.g. a life-threatening disease like cancer) is to reduce mortality from that disease among the people screened by early treatment of the cases detected [ll. In studies of efficacy of screening, randomization is an important means of controlling confounders. With a view to randomization it is decided in advance how many people per trial group ought to be enrolled to reach a predetermined power. Power and size calculations for randomized controlled trials performed to assess the mortality reduction due to screening programmes are rather cumbersome, and very often hampered by a lack of detailed baseline data. These calculations are carried out on the basis of expected risks of mortality from the disease. According to the statistical null-hypothesis this expected risk is
Mortality reduction
the same in both groups (yes or no screening offered). This hypothesis could be tested using a one-sided test for the differente [2] or the ratio of the two risks. Mass-screening for disease is defined as the examination of asymptomatic people, i.e. people who have no history of the disease and who are not in the process of evaluating signs and symptoms which may lead to confirmation of the disease. People who are not asymptomatic at the moment of entry into the trial are excluded from the study. Massscreening is aimed at detecting patients with no obvious signs or symptoms caused by the disease, but who are nevertheless in a preclinical state of the disease. In sample size and power calculations two parameters have to be incorporated: the incidence rate of the disease; and the case fatality rate. Here we would like to present a shortcut method for sample size and power calculations.
1261
HWB
1262
STRAATMAN and
METHOD
The risk of death from the disease
In Fig. 1 the natura1 history of disease for a healthy individual entering the control group at time 0 is given. The disease state (healthy or disease manifestation) has to be checked at the moment of entry in the trial for al1 members of the cohort defined. Those with clinical disease now or in the past are excluded from the trial. Some time, say P years, wil1 have elapsed before this information has been gathered for al1 individuals. To obtain follow-up evidente of T years for al1 individuals, the duration of the trial is to be P + T years. Of course it is a waste not to use the additional follow-up for individuals entering the trial before OSP years after the start of the study. Instead of waiting until al1 individuals have a follow-up period of T years, one could use the mean follow-up time reached at the end of the study as the period of risk. If the entry into the trial is uniform, then T years after the start of the study the mean follow-up wil1 be of the order of T - 0.5P years. In breast cancer screening P is often the period between successive screenings and varies from 1 to 3 years. In the example discussed later in this paper the use of the mean follow-up at the end of the study is shown to lead to satisfactory results. In Fig. 1 Xis the time in years from entry into the trial until occurrence of the disease. Z is the time in years after X until death from the disease. X + Z is the time in years from entry into the trial until death from the disease. We assume that X and Z are independent and both follow an exponential distribution with parameters r and 6, respectively. If r and 6 are smal1 the average annual incidence and mortality rate can be used as estimates for r and 6. An expression for the risk of death from the disease within T years (Rr) after entry into the trial is given by (see the Appendix): rT - f [l - exp( -6T)].
(1)
Survival estimates for patients as given by actuarial life tables or the Kaplan Meier method can be seen as risk estimates corrected for loss Entry into the study
ANDRÉL. M.
VERBEEK
of follow-up due to death from other causes than the disease screened for, migration or the termination of the survival study. From this estimate the instantaneous rate and the annual mortality rate for patients can be derived directly). Suppose the 5-year survival for patients is 0.7. The survival after diagnosis was assumed to follow an exponential distribution with instantaneous rate 6. So, Pr(Z > 5) = exp( - 6 * 5) = 0.7. Solving 6 in this equation leads to 6 = - ln(0.7)/5 = 0.071. And the annual mortality rate 6, per person year is 1 - exp( -6) = 1 exp( - 0.071) = 0.069. Annual incidence rates are most often available from cancer registries and survival risks from clinical patient files. Size calculations
If screening the study group in fact reduces mortality caused by the disease by, say, 30%, the risk of death from the disease in the study group is given by 0.70 * R,. If the control group has N, individuals and the study group f *iV,( f > 0) individuals and a one-sided statistical test is performed, Schwarz et al. [2] give the following relation between N, (sample size required in the control group), f (f*N, sample size in the control group), a (significante level), p (probability error of kind 11) and R,: N, =
(f + 4f [sin-’ (ar)
l)(Cl+ q* - sin- ’(Jmr)12’
If R, approaches 0: N, x
(f + 1) (% + q2 4f (Jg-
JíGözQ’
(2)
where ca and cB, respectively, are the 1 - a and the 1 - p percentiles of the standard normal distribution. Then the statistical power is 1 - fl. If a two-sided test is performed a should be replaced by a/2. Power calculations
If the risk of death from the disease within T years after entry (R,), c, and the sizes of the study and the control group are known, the power to detect a mortality reduction of 30% in Death from the disease
Disease occurrence
x
XL ??
Time in years (t)
Fig. 1. Natura1 history of disease.
Sample Size Calculation in Trials
the study group follows from formula Rewriting formula (2) gives
(2).
6p=(JRi-JOT7*Rrjr
;$-C a. (3) $_ By means of formula (3) I+ is calculated. The corresponding j?-value, and thus the statistical power 1 - /?, can be found in tables of the cumulative standard normal distribution. Loss of fellow -up Let L be the time in years from entry into the study until loss of follow-up due to migration or death from other causes. Assume that L is exponentially distributed with annual rate cc,per individual. Then the instantaneous rate p of loss of’follow-up is -ln( 1 - u). The risk of dying from the disease within T years after entrance into the study is (see the Appendix): R +1
- exp( --
-Pil r
a+j?
-exp(
-(6
+jf)T)].
Substituting this expression for RT in formulas (2) and (3) leads to size and power calculations corrected for a constant rate of loss of follow-up during the trial. Linearly increasing incidence It is not very realistic to assume that the
incidence rate of the disease under consideration is constant. Generally, cancer incidence increases with age. If the relation between incidence and time after start of the study is linear, then a simple correction can be given for the risk of death from the disease. Let r, denote the annual incidence rate in year [i - 1, i], for i=l,2,..., T, in the control group and let the linear relation between incidence and time be given by ri = u + wi(w > 0). With (weighted) regression analysis the linear relation between the annual incidence rate and time can be found. The average annual incidence rate ra over the period [O, í”J is ra =v+w(T+1)/2.IfR,isthe risk of death from the disease in the control group based on the average annual incidence rate and RP is the risk of death based on the linear relation between the incidence and time, and 6T c 0.50, the following approximate expression can be derived for R7 (see the Appendix): Riin_6~+w(2~+3)R ’ -62,+w(3T+3)
”
(4)
1263
From formula (4) it follows immediately that 2/3 RTG RP < RT. It is not surprising that Rr is less than RT, because the total follow-up after diagnosis of the disease in the case of increasing incidence is less than the follow-up compared with constant incidence rate. In the following example the effect of a linear increase of 50% in the incidence in a 11-year period is calculated. Suppose ri = 0.0019 + O.OOOli,i = 1,2, . . . , ll. Then the average annual incidence rate is 0.0025 and with formula (4): RP=0.93*R
T’
It is also possible to derive expressions for the risk of dying if the relation between the annual incidence and time is a higher degree polynomial, but the calculations become laborious. EXAMPLE
In the last 20 years a number of trials have been performed to evaluate the breast cancer mortality reduction achieved by screening for breast cancer. Examples are the HIP-trial [4], the W&E [5] and Malmö-trials [6] in Sweden, and a multicentre trial in the U.K. [7l. Incidence rates for breast cancer in these studies vary from 0.0015 to 0.0025 per person-year. The 5-year survival estimates (breast cancer death) for patients in the control group vary from 0.6 to 0.8. After 7-10 years the breast cancer mortality reduction in the study group is of the order of 30%. The data of the Malmö-trial [6] in Sweden were presented in a way that makes direct verification of the risk formula presented here possible. This trial started in October 1976 and the predetermined end of the trial was 31 December 1986. Al1 women bom in 1908-1932 were identifìed from the population registry of Malmö. Each birth year was randomized separately into the control group and the study group. The planned interval between the screenings in the study group was 18-24 months. The control group consisted of 21,195 women and the study group of 21,088 women. Both groups entered the trial in a period of 2 years. The number of woman-years involved in the control group was 187,016, which corresponds with a mean follow-up of 8.82 years. Out of these women 447 proved to have breast cancer. These 447 women experienced 1869 breast cancer years until the end of the follow-up period. So, the average annual incidence rate per womanyear is 447/(187,016 - 1869) = 0.0024. Out of the 447 women with breast cancer, 66 died from
HWB STRAATMAN and ANDRÉ L. M. VERBEEK
1264
the disease, yielding an average annual mortality rate per woman-year for patients of 6611869 = 0.035. Then the instantaneous mortality rate is -ln(l - 0.035) = 0.036. The expected number of breast cancer deaths at the end of the study in the control group is given by [see formula (l)]: 21195*R, =21195*r*(T-
1/6*[1 -exp(-62))
=21195*0.0024*{8.82
- 1/0.036
*[l -exp(-0.036*8.82)]} = 21195 *0.00303 = 64.3. The expected number of breast cancer deaths (64.3) is very close to the observed number of breast cancer deaths (66) in the control group, which supports the use of this paper’s formula for calculating the risk of death from the disease. The study group was almost equal in size (21,088 individuals) and experienced also a mean follow-up of 8.82 years at the end of the study. The researchers in the Malmö-trial hoped to demonstrate a mortality reduction of 35% in the study group with a power of 0.90 at the end of the study with a = 0.05 (one sided). Then Q is given by [see formula (3)]: C&q = (JGEG
- JóZXöW3) *dm-
1.645 = 0.55.
The value Q = 0.55 corresponds to fl = 0.29, and hence a power of 1 - p = 0.7 1. This power of 0.71 is considerably smaller than the power of 0.9 initially required. The researchers of the Malmö-trial did not have any precise estimates for r and 6 in the control group at the start of the study, which may explain the differente. One might ask what size control and study group should have been to reach the predetermined power of 0.9. Formula (2) with c1= 0.05 (E.= 1.645) and /? =O.lO (ca= 1.282) andf= 1 gives: (1.645 + 1.282)’
N, z 2*($öGG
- JO_)2
x 38000. This is almost double the size used in the Malmö-trial. Other questions arising are the following. How long would it take to reach the predetermined power of 0.9 with the given sample sizes
of the control and the study group? From formulas (1) and (3) it can be derived that a power of 0.90 is reached after the study and control group have experienced a mean followup of 12-13 years, which is about 4 years after the actual termination of the Malmö-trial. In Table 1 size calculations for equal control and study groups are given for several values of the annual incidence and 5-year survival for patients, where a = 0.05, j = 0.1 and the expected mortality reduction in the study group is 30%. The values for the incidence and survival are in agreement with epidemiologic data. Until now, no clear effect of screening for women under age 50 years has been observed. One should note, however, that none of the relevant studies met the required size (al1 less than 20,000 screenees). In young women the incidence of breast cancer is the range of 0.0015-0.0020 per woman-year, 5-year survival from breast cancer is about 0.8. Assuming no loss of follow-up and a study period of 7 years, the required size of both control and screened group is the order Table 1.Sample size control group and study group required to detect a mortality reduction of 30% in the study group with a power of 0.9 S-Year survival for patients Years after entry
0.6 0.7 0.8 Annual rate of loss of follow-up Incidence
0
0.02
0
0.02
0
0.02
Size of control group and study group r =0.0015 99 105 135 144 207 220 155 165 0.0020 74 79 101 108 81 86 124 132 0.0025 59 63 r =0.0015 71 0.0020 53 0.0025 43
76 57 46
96 72 58
103 78 62
146 109 88
157 118 94
r =0.0015
54 0.0020 40 0.0025 32
59 44 35
72 54 43
79 59 47
109 82 65
119 89 71
r =0.0015 42 0.0020 32 0.0025 26
47 35 28
56 42 34
62 47 37
84 63 51
93 70 56
r =0.0015 0.0020 0.0025
35 26 21
39 29 23
46 34 27
51 38 31
68 51 41
76 57 45
r =0.0015 29 0.0020 22 0.0025 17
33 24 20
38 28 23
43 32 26
56 42 33
63 47 38
One-sided test, a = 0.05. The size required is given for several values of the incidence per woman-year of the disease (r) and the 5-year survival from the disease for patients. Sample sizes are specified for “no loss of follow-up” and for “0.02 loss of follow-up per person year”. Entries for the sample size must be multiplied by 1000.
Sample Size Cakulation in Trials
of 80,000-110,000. Assuming an annual rate of loss of follow-up of 0.02, the quired size increases to 90,000-120,000 (see Table 1). DISCUSSION
If age-specific registers for the annual incidence rate of the disease under study exist, it is possible to estimate the average annual incidence rate in the control group during the trial. If, furthermore, data are available on the surviva1 of patients and the annual rate of loss of follow-up caused by mortality from other causes and migration, the method presented here is a quick method to calculate the required sample size of the trial. However, some restrictions should be mentioned. As a consequente of the assumption of the exponential distribution for X and 2 the incidence rate and the mortality rate are supposed to be constant over time. The study cohorts grow older during the trial and the incidence of most diseases increases with age. If the relation between incidence and time after entry is linear, a correction for the risk of death from the disease is possible. Also, the mortality from other causes increases with age. The annual disease mortality rate for patients may increase or decrease as time-since-diagnosis passes. Therefore the method presented here should not be used in those cases where the information available violates the assumptions in a serious way. Methods described by Moss et al. [8] or Carlsson et al. [9] should be used if one wishes to take changing incidence (not linear) and mortality from other causes into account. The example presented in this paper indicates the method is reliable if breast cancer screening is the subject and the duration of the trial is of the order of 10 years. REFERENCFS Morrison AS. Scre&ng ln Chronlc Disease. New York Oxford University Press; 1985. Schwarz D, Flamant R, Lellouch J. Cllnical TriaIs. Londen: Academie Press; 1980. Kalbfleish JD, Prentie RL. l%e Statlstlcal Ar&& of Failure Time Data. New York: Wiley: 1980. Shaoiro S. Venet W. Strax P et al. Ten to fourteen vear effe&s of breast cancer screening on mortality. J NatI Caricer Inst 1982; 69: 349-355. 5. Tabár L, Gad A, Holmberg LH et al. Reduction in mortality from breast cancer after mass screening with mammography. Lancet 1985; i: 829-832. 6. Andersson 1, Aspegren K, Janzon L er al. Mammographic screening and mortality from breast cancer: the Malmö screening trial. Br Med J 1988; 297: 943-948. 1. Chamberlain J, Coleman D, Ellman R er al. First
1265
results on mortality reduction in the UK trial of early detectíon of breast cancer. Laneet 1988; ii: 411-416. 8. Moss S, Draper GJ, Hardcastle JD et al. Calculation of sample size in trials of screening for early diagnosis of disease. ZnrJ. Epidemial. 1987; 16e 105-110. 9. Carlsson U, Gören E, Eriksson R et al. Evaluation of possibilities for mass screening for, colorectal cancer with Hemoccult fecal blood test. Dis Colon Rectmn 1986; % 553-557.
APPENDM Risk Calculation For an exponential distribution it can be shown that the hazard or instantaneous rate is the parameter of the distribution [3]. This instantaneous rate is deflned as limPr(rdXdr+drIX,Q
=limh(KW+dr)=
dr
drl0
drl0
Pr(X,
t)+dt
r’
independently of t (years after entry). In words, the risk of disease manifestation in the interval [t, t + dr] under the condition that there is no disease at time t, divided by the length of the interval (dr), quals r, when the length (dt) approaches 0. The risk of death from the disease within T years (R,) after entry into the trial is
. =1-exp(-rT)-k[exp(-rT)-exp(dT)1, where 0 < r # 6. If r «S, which is true when the disease is cancer, this formula can be approximated by rT-i[l
-exp(-dn].
The annual incidence rate r, per individual is r,=Pr(r
dying from the disease within T years after the study and without loss of follow-up is + Z < L, X + Z < T). The approximate den[sec formula (1) for the distribution] is
1
$[ rt-i[l-exp(-bf)]
=r-rexp(-bt).
The densities for X + Z and L are known. Furthermore we assume that X + Z and L are independent. Standard rules in probability calculus give the following results. Pr(X+Z
r
+1
-
r exp( -at)
- exp( -PI
p exp(-$)dl -&
[l -exp(-(6
Note that ~tgWX+Z
1 dr
+r)T)].
1266
*1
HUUB STRMTMANand ANDRÉL. M. VERBEEK
If p approaches 0, which is equivalent to a negligible loss of follow-up, formula (1) applies again. Linearly Increasing Incidence Let the linear relation between the annual incidence and the time be given by ri = u + wi, where r, is the annual incidence rate in year [i - 1,i] after entry for i = 1,2, . . . , T. The average annual incidence rate is then given by r, = v + w(T+ 1)/2. The risk of developing the disease within T (positive integer) years after entry is r,T and the risk of death from the disease based on the average annual incidence rate is R, = r;c, where c = T- l/S[l -exp(-U)]. If we choose the density f(x) = u + w(x + 1/2) for X it fellows that the annual incidence in year [i - 1, i] is
,.
u + wi and the number of new patients in a period of T years after entry is the same for the constant incidence rate r, as for the linearly increasing one. The risk of death from the disease with f(x) as above for the density of X is given by R~=~~ru+w(x+f)[~~r-‘dexp(-6~)di]dx
1 1 = u+jw c+2WP-% C5 ( > A third order Taylor expansion of exp(-GT) 3,
6u + w(2T + 3)
Rr N 6u +w(3T+3)’
in c gives:
Vol.43,No. 11,pp. 1261-1266,1990
0895-4356/90 $3.00+ 0.00 Copyright0 1990PergamonPres8plc
F’rinted in Grcat Britain. AU rights rcscrvcd
SHORTCUT METHOD TO CALCULATE THE SAMPLE SIZE IN TRIALS OF SCREENING FOR CHRONIC DISEASE HUUB STRAATMAN and ANDRÉ. L. M. VERBEEK Department of Epidemiology, Institute of Social Medicine, Nijmegen University, Verlengde Groenestraat 75, 6525 EJ Nijmegen, The Netherlands (Received in revised farm 2 November 1989; received for publication 26 April 1990)
Ahstract-One of the first questions arising in the planning of a randomized trial to evaluate mortality reduction by screening concerns the sample size of the trial required to detect an expected mortality reduction in the study group for given significante leve1 a, and power 1 - 8. If estimates exist of the underlying average annual incidence rate of the disease r, and the annual mortality rate 6, or survival data for patients in the population under consideration before screening started, then a simple formula for the probability of dying from the disease within Tyears after entry into the trial can be given for the control group. Standard formulas may then be used for sample size calculations in randomized trials, which compare the risk of death from the disease in the control and the study group accrued at T years after entry. A simple correction for loss of follow-up, due to mortality from other causes or, for instance, migration is possible. Power calculation
Trial
Mass screening
INTRODUCTION
The ultimate goal of screening for chronic disease (e.g. a life-threatening disease like cancer) is to reduce mortality from that disease among the people screened by early treatment of the cases detected [ll. In studies of efficacy of screening, randomization is an important means of controlling confounders. With a view to randomization it is decided in advance how many people per trial group ought to be enrolled to reach a predetermined power. Power and size calculations for randomized controlled trials performed to assess the mortality reduction due to screening programmes are rather cumbersome, and very often hampered by a lack of detailed baseline data. These calculations are carried out on the basis of expected risks of mortality from the disease. According to the statistical null-hypothesis this expected risk is
Mortality reduction
the same in both groups (yes or no screening offered). This hypothesis could be tested using a one-sided test for the differente [2] or the ratio of the two risks. Mass-screening for disease is defined as the examination of asymptomatic people, i.e. people who have no history of the disease and who are not in the process of evaluating signs and symptoms which may lead to confirmation of the disease. People who are not asymptomatic at the moment of entry into the trial are excluded from the study. Massscreening is aimed at detecting patients with no obvious signs or symptoms caused by the disease, but who are nevertheless in a preclinical state of the disease. In sample size and power calculations two parameters have to be incorporated: the incidence rate of the disease; and the case fatality rate. Here we would like to present a shortcut method for sample size and power calculations.
1261
HWB
1262
STRAATMAN and
METHOD
The risk of death from the disease
In Fig. 1 the natura1 history of disease for a healthy individual entering the control group at time 0 is given. The disease state (healthy or disease manifestation) has to be checked at the moment of entry in the trial for al1 members of the cohort defined. Those with clinical disease now or in the past are excluded from the trial. Some time, say P years, wil1 have elapsed before this information has been gathered for al1 individuals. To obtain follow-up evidente of T years for al1 individuals, the duration of the trial is to be P + T years. Of course it is a waste not to use the additional follow-up for individuals entering the trial before OSP years after the start of the study. Instead of waiting until al1 individuals have a follow-up period of T years, one could use the mean follow-up time reached at the end of the study as the period of risk. If the entry into the trial is uniform, then T years after the start of the study the mean follow-up wil1 be of the order of T - 0.5P years. In breast cancer screening P is often the period between successive screenings and varies from 1 to 3 years. In the example discussed later in this paper the use of the mean follow-up at the end of the study is shown to lead to satisfactory results. In Fig. 1 Xis the time in years from entry into the trial until occurrence of the disease. Z is the time in years after X until death from the disease. X + Z is the time in years from entry into the trial until death from the disease. We assume that X and Z are independent and both follow an exponential distribution with parameters r and 6, respectively. If r and 6 are smal1 the average annual incidence and mortality rate can be used as estimates for r and 6. An expression for the risk of death from the disease within T years (Rr) after entry into the trial is given by (see the Appendix): rT - f [l - exp( -6T)].
(1)
Survival estimates for patients as given by actuarial life tables or the Kaplan Meier method can be seen as risk estimates corrected for loss Entry into the study
ANDRÉL. M.
VERBEEK
of follow-up due to death from other causes than the disease screened for, migration or the termination of the survival study. From this estimate the instantaneous rate and the annual mortality rate for patients can be derived directly). Suppose the 5-year survival for patients is 0.7. The survival after diagnosis was assumed to follow an exponential distribution with instantaneous rate 6. So, Pr(Z > 5) = exp( - 6 * 5) = 0.7. Solving 6 in this equation leads to 6 = - ln(0.7)/5 = 0.071. And the annual mortality rate 6, per person year is 1 - exp( -6) = 1 exp( - 0.071) = 0.069. Annual incidence rates are most often available from cancer registries and survival risks from clinical patient files. Size calculations
If screening the study group in fact reduces mortality caused by the disease by, say, 30%, the risk of death from the disease in the study group is given by 0.70 * R,. If the control group has N, individuals and the study group f *iV,( f > 0) individuals and a one-sided statistical test is performed, Schwarz et al. [2] give the following relation between N, (sample size required in the control group), f (f*N, sample size in the control group), a (significante level), p (probability error of kind 11) and R,: N, =
(f + 4f [sin-’ (ar)
l)(Cl+ q* - sin- ’(Jmr)12’
If R, approaches 0: N, x
(f + 1) (% + q2 4f (Jg-
JíGözQ’
(2)
where ca and cB, respectively, are the 1 - a and the 1 - p percentiles of the standard normal distribution. Then the statistical power is 1 - fl. If a two-sided test is performed a should be replaced by a/2. Power calculations
If the risk of death from the disease within T years after entry (R,), c, and the sizes of the study and the control group are known, the power to detect a mortality reduction of 30% in Death from the disease
Disease occurrence
x
XL ??
Time in years (t)
Fig. 1. Natura1 history of disease.
Sample Size Calculation in Trials
the study group follows from formula Rewriting formula (2) gives
(2).
6p=(JRi-JOT7*Rrjr
;$-C a. (3) $_ By means of formula (3) I+ is calculated. The corresponding j?-value, and thus the statistical power 1 - /?, can be found in tables of the cumulative standard normal distribution. Loss of fellow -up Let L be the time in years from entry into the study until loss of follow-up due to migration or death from other causes. Assume that L is exponentially distributed with annual rate cc,per individual. Then the instantaneous rate p of loss of’follow-up is -ln( 1 - u). The risk of dying from the disease within T years after entrance into the study is (see the Appendix): R +1
- exp( --
-Pil r
a+j?
-exp(
-(6
+jf)T)].
Substituting this expression for RT in formulas (2) and (3) leads to size and power calculations corrected for a constant rate of loss of follow-up during the trial. Linearly increasing incidence It is not very realistic to assume that the
incidence rate of the disease under consideration is constant. Generally, cancer incidence increases with age. If the relation between incidence and time after start of the study is linear, then a simple correction can be given for the risk of death from the disease. Let r, denote the annual incidence rate in year [i - 1, i], for i=l,2,..., T, in the control group and let the linear relation between incidence and time be given by ri = u + wi(w > 0). With (weighted) regression analysis the linear relation between the annual incidence rate and time can be found. The average annual incidence rate ra over the period [O, í”J is ra =v+w(T+1)/2.IfR,isthe risk of death from the disease in the control group based on the average annual incidence rate and RP is the risk of death based on the linear relation between the incidence and time, and 6T c 0.50, the following approximate expression can be derived for R7 (see the Appendix): Riin_6~+w(2~+3)R ’ -62,+w(3T+3)
”
(4)
1263
From formula (4) it follows immediately that 2/3 RTG RP < RT. It is not surprising that Rr is less than RT, because the total follow-up after diagnosis of the disease in the case of increasing incidence is less than the follow-up compared with constant incidence rate. In the following example the effect of a linear increase of 50% in the incidence in a 11-year period is calculated. Suppose ri = 0.0019 + O.OOOli,i = 1,2, . . . , ll. Then the average annual incidence rate is 0.0025 and with formula (4): RP=0.93*R
T’
It is also possible to derive expressions for the risk of dying if the relation between the annual incidence and time is a higher degree polynomial, but the calculations become laborious. EXAMPLE
In the last 20 years a number of trials have been performed to evaluate the breast cancer mortality reduction achieved by screening for breast cancer. Examples are the HIP-trial [4], the W&E [5] and Malmö-trials [6] in Sweden, and a multicentre trial in the U.K. [7l. Incidence rates for breast cancer in these studies vary from 0.0015 to 0.0025 per person-year. The 5-year survival estimates (breast cancer death) for patients in the control group vary from 0.6 to 0.8. After 7-10 years the breast cancer mortality reduction in the study group is of the order of 30%. The data of the Malmö-trial [6] in Sweden were presented in a way that makes direct verification of the risk formula presented here possible. This trial started in October 1976 and the predetermined end of the trial was 31 December 1986. Al1 women bom in 1908-1932 were identifìed from the population registry of Malmö. Each birth year was randomized separately into the control group and the study group. The planned interval between the screenings in the study group was 18-24 months. The control group consisted of 21,195 women and the study group of 21,088 women. Both groups entered the trial in a period of 2 years. The number of woman-years involved in the control group was 187,016, which corresponds with a mean follow-up of 8.82 years. Out of these women 447 proved to have breast cancer. These 447 women experienced 1869 breast cancer years until the end of the follow-up period. So, the average annual incidence rate per womanyear is 447/(187,016 - 1869) = 0.0024. Out of the 447 women with breast cancer, 66 died from
HWB STRAATMAN and ANDRÉ L. M. VERBEEK
1264
the disease, yielding an average annual mortality rate per woman-year for patients of 6611869 = 0.035. Then the instantaneous mortality rate is -ln(l - 0.035) = 0.036. The expected number of breast cancer deaths at the end of the study in the control group is given by [see formula (l)]: 21195*R, =21195*r*(T-
1/6*[1 -exp(-62))
=21195*0.0024*{8.82
- 1/0.036
*[l -exp(-0.036*8.82)]} = 21195 *0.00303 = 64.3. The expected number of breast cancer deaths (64.3) is very close to the observed number of breast cancer deaths (66) in the control group, which supports the use of this paper’s formula for calculating the risk of death from the disease. The study group was almost equal in size (21,088 individuals) and experienced also a mean follow-up of 8.82 years at the end of the study. The researchers in the Malmö-trial hoped to demonstrate a mortality reduction of 35% in the study group with a power of 0.90 at the end of the study with a = 0.05 (one sided). Then Q is given by [see formula (3)]: C&q = (JGEG
- JóZXöW3) *dm-
1.645 = 0.55.
The value Q = 0.55 corresponds to fl = 0.29, and hence a power of 1 - p = 0.7 1. This power of 0.71 is considerably smaller than the power of 0.9 initially required. The researchers of the Malmö-trial did not have any precise estimates for r and 6 in the control group at the start of the study, which may explain the differente. One might ask what size control and study group should have been to reach the predetermined power of 0.9. Formula (2) with c1= 0.05 (E.= 1.645) and /? =O.lO (ca= 1.282) andf= 1 gives: (1.645 + 1.282)’
N, z 2*($öGG
- JO_)2
x 38000. This is almost double the size used in the Malmö-trial. Other questions arising are the following. How long would it take to reach the predetermined power of 0.9 with the given sample sizes
of the control and the study group? From formulas (1) and (3) it can be derived that a power of 0.90 is reached after the study and control group have experienced a mean followup of 12-13 years, which is about 4 years after the actual termination of the Malmö-trial. In Table 1 size calculations for equal control and study groups are given for several values of the annual incidence and 5-year survival for patients, where a = 0.05, j = 0.1 and the expected mortality reduction in the study group is 30%. The values for the incidence and survival are in agreement with epidemiologic data. Until now, no clear effect of screening for women under age 50 years has been observed. One should note, however, that none of the relevant studies met the required size (al1 less than 20,000 screenees). In young women the incidence of breast cancer is the range of 0.0015-0.0020 per woman-year, 5-year survival from breast cancer is about 0.8. Assuming no loss of follow-up and a study period of 7 years, the required size of both control and screened group is the order Table 1.Sample size control group and study group required to detect a mortality reduction of 30% in the study group with a power of 0.9 S-Year survival for patients Years after entry
0.6 0.7 0.8 Annual rate of loss of follow-up Incidence
0
0.02
0
0.02
0
0.02
Size of control group and study group r =0.0015 99 105 135 144 207 220 155 165 0.0020 74 79 101 108 81 86 124 132 0.0025 59 63 r =0.0015 71 0.0020 53 0.0025 43
76 57 46
96 72 58
103 78 62
146 109 88
157 118 94
r =0.0015
54 0.0020 40 0.0025 32
59 44 35
72 54 43
79 59 47
109 82 65
119 89 71
r =0.0015 42 0.0020 32 0.0025 26
47 35 28
56 42 34
62 47 37
84 63 51
93 70 56
r =0.0015 0.0020 0.0025
35 26 21
39 29 23
46 34 27
51 38 31
68 51 41
76 57 45
r =0.0015 29 0.0020 22 0.0025 17
33 24 20
38 28 23
43 32 26
56 42 33
63 47 38
One-sided test, a = 0.05. The size required is given for several values of the incidence per woman-year of the disease (r) and the 5-year survival from the disease for patients. Sample sizes are specified for “no loss of follow-up” and for “0.02 loss of follow-up per person year”. Entries for the sample size must be multiplied by 1000.
Sample Size Cakulation in Trials
of 80,000-110,000. Assuming an annual rate of loss of follow-up of 0.02, the quired size increases to 90,000-120,000 (see Table 1). DISCUSSION
If age-specific registers for the annual incidence rate of the disease under study exist, it is possible to estimate the average annual incidence rate in the control group during the trial. If, furthermore, data are available on the surviva1 of patients and the annual rate of loss of follow-up caused by mortality from other causes and migration, the method presented here is a quick method to calculate the required sample size of the trial. However, some restrictions should be mentioned. As a consequente of the assumption of the exponential distribution for X and 2 the incidence rate and the mortality rate are supposed to be constant over time. The study cohorts grow older during the trial and the incidence of most diseases increases with age. If the relation between incidence and time after entry is linear, a correction for the risk of death from the disease is possible. Also, the mortality from other causes increases with age. The annual disease mortality rate for patients may increase or decrease as time-since-diagnosis passes. Therefore the method presented here should not be used in those cases where the information available violates the assumptions in a serious way. Methods described by Moss et al. [8] or Carlsson et al. [9] should be used if one wishes to take changing incidence (not linear) and mortality from other causes into account. The example presented in this paper indicates the method is reliable if breast cancer screening is the subject and the duration of the trial is of the order of 10 years. REFERENCFS Morrison AS. Scre&ng ln Chronlc Disease. New York Oxford University Press; 1985. Schwarz D, Flamant R, Lellouch J. Cllnical TriaIs. Londen: Academie Press; 1980. Kalbfleish JD, Prentie RL. l%e Statlstlcal Ar&& of Failure Time Data. New York: Wiley: 1980. Shaoiro S. Venet W. Strax P et al. Ten to fourteen vear effe&s of breast cancer screening on mortality. J NatI Caricer Inst 1982; 69: 349-355. 5. Tabár L, Gad A, Holmberg LH et al. Reduction in mortality from breast cancer after mass screening with mammography. Lancet 1985; i: 829-832. 6. Andersson 1, Aspegren K, Janzon L er al. Mammographic screening and mortality from breast cancer: the Malmö screening trial. Br Med J 1988; 297: 943-948. 1. Chamberlain J, Coleman D, Ellman R er al. First
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results on mortality reduction in the UK trial of early detectíon of breast cancer. Laneet 1988; ii: 411-416. 8. Moss S, Draper GJ, Hardcastle JD et al. Calculation of sample size in trials of screening for early diagnosis of disease. ZnrJ. Epidemial. 1987; 16e 105-110. 9. Carlsson U, Gören E, Eriksson R et al. Evaluation of possibilities for mass screening for, colorectal cancer with Hemoccult fecal blood test. Dis Colon Rectmn 1986; % 553-557.
APPENDM Risk Calculation For an exponential distribution it can be shown that the hazard or instantaneous rate is the parameter of the distribution [3]. This instantaneous rate is deflned as limPr(rdXdr+drIX,Q
=limh(KW+dr)=
dr
drl0
drl0
Pr(X,
t)+dt
r’
independently of t (years after entry). In words, the risk of disease manifestation in the interval [t, t + dr] under the condition that there is no disease at time t, divided by the length of the interval (dr), quals r, when the length (dt) approaches 0. The risk of death from the disease within T years (R,) after entry into the trial is
. =1-exp(-rT)-k[exp(-rT)-exp(dT)1, where 0 < r # 6. If r «S, which is true when the disease is cancer, this formula can be approximated by rT-i[l
-exp(-dn].
The annual incidence rate r, per individual is r,=Pr(r
dying from the disease within T years after the study and without loss of follow-up is + Z < L, X + Z < T). The approximate den[sec formula (1) for the distribution] is
1
$[ rt-i[l-exp(-bf)]
=r-rexp(-bt).
The densities for X + Z and L are known. Furthermore we assume that X + Z and L are independent. Standard rules in probability calculus give the following results. Pr(X+Z
r
+1
-
r exp( -at)
- exp( -PI
p exp(-$)dl -&
[l -exp(-(6
Note that ~tgWX+Z
1 dr
+r)T)].
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*1
HUUB STRMTMANand ANDRÉL. M. VERBEEK
If p approaches 0, which is equivalent to a negligible loss of follow-up, formula (1) applies again. Linearly Increasing Incidence Let the linear relation between the annual incidence and the time be given by ri = u + wi, where r, is the annual incidence rate in year [i - 1,i] after entry for i = 1,2, . . . , T. The average annual incidence rate is then given by r, = v + w(T+ 1)/2. The risk of developing the disease within T (positive integer) years after entry is r,T and the risk of death from the disease based on the average annual incidence rate is R, = r;c, where c = T- l/S[l -exp(-U)]. If we choose the density f(x) = u + w(x + 1/2) for X it fellows that the annual incidence in year [i - 1, i] is
,.
u + wi and the number of new patients in a period of T years after entry is the same for the constant incidence rate r, as for the linearly increasing one. The risk of death from the disease with f(x) as above for the density of X is given by R~=~~ru+w(x+f)[~~r-‘dexp(-6~)di]dx
1 1 = u+jw c+2WP-% C5 ( > A third order Taylor expansion of exp(-GT) 3,
6u + w(2T + 3)
Rr N 6u +w(3T+3)’
in c gives: