A TEST OF SEVERAL PARAMETRIC STATISTICAL MODELS FOR ESTIMATING SUCCESS RATE IN THE TREATMENT OF CARCINOMA CERVIX UTERI

-The parametric statistical models discussed include all those which have previously been described in the literature (Boag, 1948-lognormal; Berkson and Gage, 1952-negative exponential; Haybittle, 1959-extrapolated actuarial) and the basic data used to test the models comprised some 3000 case histories of patients treated between 1945 and 1962. The histories were followed up during the period 1969-71 and thus provided adequate information to validate long-term survival fractions predicted using short-term follow-up data. The results with the lognormal model showed that for series of staged carcinoma cervix patients treated during a 5-year period, satisfactory estimates of long-term survival fractions could be predicted after a minimum waiting period of 3 years for stages I and II, and 2 years for stage III. The model should be used with a value assumed for the lognormal paramater S in the range S = 0-35 to S = 0 40. Although alternative models often gave adequate predictions, the lognormal proved to be the most consistent model. This model may therefore now be used with more confidence for prospective studies on carcinoma cervix series and can provide good estimates of long-term survival fractions several years earlier than would otherwise be possible.

ALTHOUGH the 5-year survival rate or, for us and if the logical framework of a in more general terms, the m-year survival model can be shown to be valid, the rate, determined from an m-year follow-up evaluation of the various parameters is of all the surviving patients, is widely now easy. Such models do provide a used as a criterion of success in cancer way of bringing to bear a great deal of therapy, it is too crude and too long valuable past experience upon the assessdelayed a statistic to be a satisfactory ment of new short-term results. Indeed, way of comparing alternative treatments they often allow a useful prediction of during the working life of a surgeon or longer term results to be made from the radiotherapist. Even if this rate is as-available short-term data. Moreover, the sessed by the actuarial (i.e. life table) detailed classification they demand can method, it still requires that a consider-be of help in assessing whether an improveable proportion of all cases shall have ment in m-year survival rate is due to survived the full m-year term. Statistical long-term cures or merely to protracted models which attempt to allow for the survival with cancer. Confidence in any delayed mortality during the follow-up such model must, however, be built up period have rarely been used, partly by its successful use on actual follow-up perhaps because when they were first data. This can be done retrospectively put forward (Boag, 1949; Berkson and by using records of cases treated many Gage, 1952) the tedious computation inyears ago and followed up at intervals volved had to be done by hand. The until death with or without cancer or digital computer has solved that problem long-term symptom-free survival had been proved. However, detailed case histories several possible statistical models which are necessary and these are not readily have been suggested, and some new available in sufficient numbers or over ones. long enough periods certainly not in a These tests have been made in 2 single cancer centre. The Regional Can-stages-firstly, the actual survival time cer Registries which provide data for distribution for each group of patients the Office of Population Censuses and examined has been compared, for each Surveys do, indeed, have data in model, with the postulated analytical bulk but not in sufficient detail for form, choosing the model parameters to testing a parametric model, and since give the best fit, and assessing the good-1970 they are no longer required to ness of fit achieved by a x2 test. Secondrecord the disease stage (O.P.C.S., 1970). ly, accepting only the limited survival Also, there is no uniformity of data data which would have been available collection, storage and retrieval within a few years (2, 3 or 4 years) after the end the medical records departments of difof the 5-year period under review, the ferent hospitals. The only accurate models were used to predict the 7-year, method of obtaining the essential treat-10-year or 15-year survival fractions as ment and follow-up information is to well as the proportion of long-term consult the original hospital case records cures " C ". These predicted values were at a number of centres.
then compared with the observed 7-, 10-, For the present study on a single or 15-year results, taking account of the site carcinoma cervix uteri-material standard errors of both predicted and has had to be gathered from 6 large observed results. The rationale of this cancer centres, covering a 25-year period. " prediction " and "proof" test is illus-We have used this material to test trated in Fig. 1 assumed that data before 1945 are fragmentary inasmuch as many of the early records have either been The stage IV group was also small and lost or destroyed. In view of this uncer-wasgathered onlyfromtheLondonhospitals tainty, only post-1945 records have been Stage IV is not of any value for testing used to test the various statistical models predictive models but we have tested its and the post-1945 era has been subdivided conformity with the survival time distribuinto three 5-year treatment periods-1945tion of the unsuccessfully treated cases. 49, 1950-54 and 1955-59. Since the records The letters C, H, U, M, Z and N refer to the were examined in the period 1969-72 there 6 hospital centres of Table I. was a minimum follow-up period of 20 years for the 1945-49 group, of 15 years for the 1960-54 group and of 10 years for the Methods 1955-59 group.
(a) Construction of a statistical model.-The stage I groups from the 4 London When a large group of patients is treated for hospitals were much smaller than the stage II cancer, a temporary remission is achieved in or stage III groups, and therefore additional many cases and in some there is no return of data for stage I was obtained from Man-the disease before the death of the patient chester and from Oslo for the period 1945-59. from some other cause many years later. Table II shows the grouping of cases avail-Although one cannot claim a certain " cure " able to test the validity of the different in any individual case, in view of the residual statistical models. For stages I-Ill there risk of recurrence, it is surely not unduly are data from at least 2 different single or optimistic to attempt to distinguish and grouped centres for each 5-year treatment estimate a " proportion cured " by approperiod, except for stage III during the priate statistical techniques applied to any period 1945-49 where only a single group large group of patients. Two kind, it is necessary to find an appropriate formula for the distribution of survival times~~~~whc ocu wihi thsfato Table III lists the sources of these several proposals and the methods of analysis (1/C). The general shape of the curve is used. skew, the mortality from persistent or The second type of model (Fig. 3), was recurrent cancer reaching a peak during the first put forward by Haybittle (1959) and first one or 2 years after treatment and declining gradually thereafter. Several ana-was called by him the " extrapolated lytical forms for this curve have been pro-carial" mode postulateslan anayi pose, amng tem he lgnoral crvecal form for the gradually declining cancer posed, among them the lognormal curve mortality which affects the whole group of (equation 1), the negative exponential (equa-patients subsequent to treatment. Although tion 2) and the skew exponential (equation the " cured " group was not explicitly 3). The latter is a particular example of postulated, it is implicit in this model also, a family of skew curves with the general since   Haybittle (1965) ft ( Various expressions have been tried for (b) Data storate and retrievaa . This surthe function +(t). Haybittle (1959Haybittle ( , 1965 vey was undertaken at a time when digital chose 0(t) =b . exp (-agt) and called this computers were readily available for calthe " extrapolated actuarial " model. We culation but much less available for data shall test this model in various ways in later storage and retrieval for these functions paragraphs using our carcinoma cervix data. make quite different demands on the machine. This form for 0(t) implies that cancer mor-The number of cases we had to examine was tality in the treated group will be a raximum not too largeq some 6000 to be dealt with at t = 0, that is, immediately aftertreatment, manually on an edge-punched card system whereas all clinical experience indicates that and we chose this for our data base, extracting mortality is low at t -0 and rises to a peak all the relevant information for each patient which occurs at anything from a few months onto a single 8 X 5 inch card of the design to a few years after treatment, depending illustrated in Fig. 4 which shows the Formica on the site and stage of the disease. The template used to assist punching the data. various skew curves tried out in the Type I All the information for each patient was thus models to fit the distribution of survival in an immediately visible form, making times may be tested again as hypotheses checking easy, and the cards could be sorted for +(t). In an attempt to find a simple quickly into their various groupings by means single parameter representation for 0(t), of the edge-punched holes and slots. Surwe have tested the form 0(t) = E 2texp (-,Et), vival time data derived from these card calling, this the skewed extrapolated actu-sorting operations were punched onto paper arial. CDtape as required and entered into the com-puter in this form for the necessary statistical other models uses only 2. This extra estimation procedures.
parameter makes the distribution curve more (c) Estimation of the parameters of the flexible and thus facilitates a good fit with statistical models being studied.-In the first the observations, but another consequence type of test referred to in the introduction, is that the standard errors of the paranamely, testing the " goodness of fit " of a meter values increase so that the estimate completed histogram of survival times with of any one parameter-such as C -is less some postulated analytical distribution, the stable. A 2-parameter model is clearly best values of a single parameter of the simpler than a 3-parameter one and it is distribution could be estimated directly by a shown below that the parameter S in the standard " least squares " method.
lognormal can often be treated as a constant, When 2 or 3 parameters have to be esti-thus converting this model also to a 2mated simultaneously from the incomplete parameter one. In the present survey of data of a treatment series-incomplete be-ca. cervix uteri, S = 0 40 fits practically all cause further deaths with cancer will still be our data. added to the histogram of survival times-(d) Extrapolated survival fractions.-The more general estimation methods must be various models may be used simply as a adopted and we have chosen the " method of framework for extrapolation instead of maximum likelihood " (Lea, 1945; Fisher, attaching absolute significance to the quan-1922). tity, C, as " proportion cured ". Thus the The logic of this method is to take as " m-year survival fraction " may be cal-"best " values of the parameters those culated from the model (Fig. 2) as: which would yield the highest chance of obtaining a sample of the type actually observed, when the calculation of probability even when the parameter estimates are is based on the chosen statistical model. based on survival data for less than m The detailed algebra involved in applying years. This is the " prediction " indicated maximum likelihood to the several models in in Fig. 1. The " proof" is then the actual Table III has been given elsewhere (Mould survival fraction observed after m years 1973). The iterative computations involved follow-up when causes of death other than in solving the equations have been carried cancer are excluded, this fraction being out by writing programmes either in BASIC evaluated by the actuarial method as deor in FORTRAN IV for each of the models. scribed by Greenwood (1926), Merrell and Four mutually exclusive follow-up groups Shulman (1955) and Cutler and Ederer can be seen in the top right-hand area of (1958). Fig. 4 with codes numbered 1, 2, 3 and 4 respectively. Groups 5 and 6 occur when RESULTS follow-up data in the patients' notes are (a) Testing the analytical form of the incomplete: further supplementary informasurvival time distribution tion, if eventually available, may require the transfer of a patient from these groups Agreement between the observed surto one of the Groups 1, 2, 3 or 4. If no vival time distributions and the proposed additional information is forthcoming, a analytical formulae was tested by groupdecision on this transfer must be taken on ing survival times into equal logarithmic the basis of the last detailed follow-up report. intervals* and comparing observed with The small Group 9 may be combined with theoretical numbers in each interval by Group 1 and the even smaller Group 10 means of a X-squared test for the 27 combined with Group 2, of which it is a .
. iT special case. Thus we can allocate all the hospital series in Table II. The theorcases to one or other of the first 4 mutually etical parameters were varied stepwise exclusive follow-up groups. in the programme until a minimum x- The lognormal model employs 3 inde-squared value was found and the computer pendent parameters, whereas each of the then printed out this value together * Basically the groups were 0-6, 6-9, 9-13-5, 13-5-20-25, 20-25-30-5, 30 5-45 5, 45-5-68-5, 68-5-102-5, 102-5-153 -5, etc. but for small sample series these groups were sometimes combined in pairs.  Table II) ?=1 00 C=0-67 C=0-50 C=0-40 1=0-33 C=0-29 C=0-25 of series for which a good fit 4 6 8 5 5 5 2 to the data is obtained, P > 0 * 05 to the data is obtained, P> 0 * 05 with the corresponding values of the are tested against the data from the 4 parameters-M and S for the lognormal, London hospitals, Manchester and Oslo, ,8 for the negative exponential and y for the results are those shown in Table IV. each member of the family of skew The data in this table are for patients curves given by equation 4. We tried treated in the 5-year periods 1945-49, 7 members of this family with C defined 1950-54, 1955-59 and followed up until by the formula: 1969 so that the minimum follow-up 2/(Ir) period was 10 years, which gives some assurance that the tail of the distribution where r is integral and 1 : r 7. This of recurrences is adequately represented. restriction ensured that integration of The C value which fits the largest proporequation 4 would lead to a complete tion of the individual stage groups is gamma function and would therefore be C = 0 5. C-067 and ( = 0 40 also proeasily evaluated. vide reasonable fits but curves C 1 When the skew exponential curves and -= 0-25 provide poor fits to the data. We have therefore concluded from complete London hospital series for 1945- Table IV that for carcinoma cervix, 59 are combined, the data are not fitted by C-0 5 is the best choice of exponent any skew exponential curve, nor indeed for the skew exponential model of the by any lognormal or negative exponential survival time distribution in follow-up curve either. For other sites also, if the Group 1 (see Fig. 4). We have noticed data comprise a mixture of different that if a skew exponential distribution stages, it is not usually possible to is chosen, many published observational obtain a good fit to any of these disdata including sites other than the cervix, tributions. are also best fitted by putting C 0 5 In Table V the lognormal and negative (Boag, 1948(Boag, , 1949; Wood and Boag, exponential curves are fitted to the same 1950; Smithers et al., 1952;Haybittle, observational data, again using minimum 1959;Ronnike, 1968;Sorensen, 1958). x2 to fix the best values of the parameters.
It is noticeable that when all the It is seen that the 2-parameter lognormal  In each case the figure in the Table gives the number of series for which a good fit to the data was obtained, P> 0 05, for the stage on that horizontal level and the distribution at the head of the vertical column.
curve provides a good fit to all but one dictions for long-term survival fractions of the 26 samples of data grouped indi-for many carcinoma cervix series. vidually by stage while the negative exponential fits only 15 of them satisfactorily, Table VI.

(b) Estimation of the long-termp survivors
When the lognormal is reduced to when a 10-year mnimum follow-up interval a single variable curve by fixing S equal is available to 0 40, it still provides an adequate fit With follow-up data available in for 20 of the 27 series of data. When 1969-71 the observation periods ranged S is fixed and equal to 0*35, the lognormal from 10 years to 25 years and the actuarial fits 12 series and when S is fixed and method of calculating long-term survival equal to 0 45, it fits 24 series. Moreover, should, and does, converge towards an when the model is used for prediction, estimate of " cure rate ". We have taken as we shall see later, the predicted the value at 20 years subsequent to treatvalue changes little in the range S equals ment as this asymptotic value, with 0 30-0A40.
which the estimates of "cure rate" In testing the distribution of survival based on each of the parametric models times given by " extrapolated actuarial " can be compared. and similar models, one has to determine In addition to this comparison of first the best values of the 2 parameters "cure rates " our computer programme by fitting the model to the whole of the calculated for each of the 22 groups of data and then, using these parameter cases in Table II, the expected survival values, to calculate the expected number fractions at times 5, 6, 7, 8, 9, 10 and of cancer deaths in each interval along 15 years after treatment using both the the time scale for comparison with the actuarial method and each of the 5 paranumbers observed. This we have done metric models of Table III. A detailed for the original Haybittle model and for listing of all these results (except the our modification of it but the results skewed extrapolated actuarial) is given by of a X2 test showed that the original Mould (1973).
Haybittle model provided an adequate Table VII compares "cure rate" fit for only 12/27 series and the skewed estimates for stages I, II and III carcinoma extrapolated actuarial model an adequate cervix based on each model with that fit for only 9/27 series. Nevertheless, as from the actuarial calculation. The value will be seen later, both these type II of one standard error of the actuarial models (Table III) give adequate preestimate is included in Table VII and it   can be seen that the " cure rate " estimates of some 100-150 cases. The subdivision derived by the other methods nearly all of the data into stage groups is highly lie within one standard error of this desirable in any carefully planned clinical actuarial estimate. Thus, with longtrial and 5 years is a reasonable period term follow-up available it is clear that for a trial if clinical interest and continuity all these statistical models will give an of plan are to be maintained. Any acceptable estimate of C. The 3 para-suggested modifications in treatment techmeter lognormal model requires for sta-nique can then be applied without too bility a larger number of cases than are long a delay. Standard errors of this available in these separate quinquennial magnitude must therefore be regarded as groups, but the 2-parameter lognormal typical in most stratified clinical trials. is satisfactory for any fixed value of S To reduce the error by a factor of /2 between 0-25 and 0-50 (only values for would involve doubling the sample size 0-3-0-4 are quoted in Tables). The and in this survey we have reviewed standard errors in " C " were usually some 2000 case histories of carcinoma close to 0-05 for the values of C en-cervix from the 4 London centres alone. countered and the small sample sizes Clinical trials in cancer therapy are very  seldom as comprehensive as that and it actuarial and parametric estimates to is evident that small treatment differences within one standard error of the actuarial of the order of 5% will rarely be found to estimate. The skewed extrapolated actube significant. arial model gives consistently lower esti-Using a similar format to Table VII, mates for the survival fraction than a comparison of the observed 10-year those given by the other models. The survival fractions with those calculated low values given by the skewed extrafrom the parametric models for stage polated actuarial model for stages II and groups I and II, and of the 7-year survival III are due to the fact that this distribution fraction for stage group III, is given in has a very broad peak. The 7-year Table VIII. For the lognormal, skew survival fraction was chosen as the exponential ( = 0.5), negative exponen-criterion for stage III as almost all cancer tial and extrapolated actuarial models, deaths among patients first seen in this there is nearly always agreement between stage will have occurred before 10 years have elapsed, so that the 10-year survival The format of Tables IX-XI is similar fraction is virtually identical with the to that of Tables VII and VIII. estimate of C. Close agreement was Tables VII and VIII give the results observed between the extrapolated actu-calculated from the long-term follow-up arial and negative exponential. These data and a single column of figures 2 models, and the skew exponential appears beneath the heading for each model, gave predictions for 10-year and model. Tables IX-XI give results based 7-year survival fractions which agreed on short-term follow-up information and fairly well with those given by the logthe date at which the predictions were normal model, taking a fixed value of made is defined as " n years after the S in the range 0 30-0 45. series closed " (see notation in Fig. 1). Hence for Tables IX and X (for carcinoma cervix stages I and II) there are 2 columns (c) Eotimation of the long-term survival Of figures beneath the heading for each fraction when only relatively short-term model. They correspond to predictions follow-up data are available made at 4 years or 3 years after the series The data already presented confirm closed (n = 4 and n = 3). In Table XI that several of the statistical models for stage III carcinoma cervix, the preexamined can provide an accurate repredictions were made at 2 years or 1 year sentation of the life experience of carafter the series closed (n = 2 and n = 1). cinoma cervix patient groups when long- Figures 5 and 6 show the results for term follow-up data are used to estimate treatment series A, B and C which are the parameters of the model. It is quoted in Table IX, and in addition therefore of great interest to determine results for the lognormal model with with what accuracy the subsequent life fixed values of S ranging from 0-25 to experience can be predicted when only 0 50 and also for the same analysis shorter term follow-up data are used, carried out for n 2 years. Series A, as would normally be the case in a planned B and C represent the combined data clinical trial some 5-8 years from its for stage I of the 4 London teaching commencement. To do this, the para-hospitals for the three 5-year treatment meters of the statistical model to be periods 1945-49, 1950-54 and 1955-59. tested were first estimated by the method A similar combination of data for stage II of maximum likelihood from the incom-has been annotated W, X and Y, see plete follow-up data which would have Table XII. been available in our series after only a limited follow-up period and these esti-DISCUSSION mated parameters were used to calculate TiscUlm ION the expected 10-, 15-or 20-year survival Type I 8tatistical models fractions. These extrapolated survival The lognormal model.-The lognormal fractions were then compared with the model with 3 floating parameters, M, S actual survival fraction calculated by and C, requires for its stability a larger the actuarial method from the long-term number of cases than are available in follow-up data on the same group of most of our quinquennial stage groups cases (see Fig. 1). The results for the even when long-term follow-up is availseveral models, both type I and type II able. This was evident in the study of (Table III), are set out in Tables IX, X " information content " in the original and XI, for disease stages I, II and III publication (Boag, 1949) and has been respectively. For stages I and II, the confirmed in other practical examples 10-year and 15-year survival fractions (Wood and Boag, 1950; Smithers et are shown but for stage III the 7-year and al., 1952;Mould, 1973   long-term follow-up data are used, the the actuarial value. This is due to the predicted 10-year survival fractions for fact that, in this series, 7 patients died the various values of S in this range do from carcinoma cervix 12-20 years subsenot usually differ by more than 0 03. quent to treatment and this frequency With short-term follow-up data, however, of later recurrences is unusual. extending over only 3 or 4 years subse-For stage II carcinoma cervix, 10-year quent to treatment, the long-term extra-and 15-year survival fractions, there is polated survival fractions depend more good agreement between actuarial calstrongly on the value of S adopted and culation and lognormal prediction for in Tables VII-XI we (Table X). Results for range of S. stages I and II carcinoma cervix have not For stage I carcinoma cervix, 10-year been included for the shortest follow-up survival fraction, there is good agreement (n 2 years) since good agreement could between actuarial calculation (" proof ", not be expected after only 2 years in see Fig. 1) and lognormal prediction these early stages where recurrence tends (" prediction ", see Fig. 1) for fixed values to be longer delayed.
of S equal to 0 30, 0-35 or 0 40, and for For stage III carcinoma cervix, 7-year both n = 4 years and n = 3 years short-and 10-year survival fractions, there is term follow-up information (Table IX). reasonable agreement between actuarial The largest discrepancy occurs for series calculation and lognormal prediction for F, when n = 3 years and S _ 0 40. For n = 2 years (Table XI). this series (Table IX) no results were A summary of these conclusions is obtained using the skew exponential shown in Table XIII. The choice of S model since the iterative procedure did equal to 0 30 is not recommended because not converge, while the standard errors when testing the analytical form of the of the parameters in all the other models survival time distribution of patients were very large indeed. Evidently this known to have died with carcinoma series had a somewhat abnormal time cervix present, this particular value of S distribution. in the lognormal curve did not provide There is also a good general agreement an adequate fit to most of the data between actuarial calculation and log-under review (see Results, (a)). The normal prediction of the 15-year survival lognormal curve with S = 0 40 provided fractions for stage I carcinoma cervix. a fit to more data than the S = 0 35 Discrepancies occur again for series F, curve, but the data of Tables IX-XI and also for series B with S 0 40 andc indicate that either value is suitable for n = 4 years (but not for n 3 years!). the purpose of predicting long-term sur-For series A, the predicted 15-year vival fractions. survival fractions are always higher than The skew exponential model.-Although a maximum likelihood solution was always estimates are of little practical value and found for the skew exponential model this model cannot be regarded as suitable for stage 1 carcinoma cervix when long-for stage I series with sample sizes term follow-up data were used, the equa-similar to those available for this study. tions did not always yield a solution For stage II carcinoma cervix, there when only short-term data were available. is better agreement when n = 4 years This failure of the iterative procedure to than when n = 3 years, and for n = 3 converge in 3 of 7 series when n -4 years the model is satisfactory for only and n = 3 years, indicates that this some half of the series studied (Table X). model is unsuitable for predictive esti-For stage III carcinoma cervix, the mates on stage I series. It is perhaps negative exponential model is unsatissurprising that in those cases where a factory when n 1 year, but when solution did exist good agreement was n -2 years the results are comparable found between observation and prediction with those obtained using the other type I (Table IX). models (Table XI). For stage II carcinoma cervix, the results using the skew exponential model Type II statistical models were inferior to those obtained with the The extrapolated actuarial model.-The lognormal model. This is particularly extrapolated actuarial model was intronoticeable for short-term follow-up when duced by Haybittle (1959) mainly for n = 3 years. Of the 9 stage II series carcinoma breast data but has also been in Table X, only series P showed a large used by him for 2 series of carcinoma proportion of the cancer deaths occurring cervix patients obtained from follow-up before the analysis time n = 4 years. information reported by Sorensen (1958) Also, most of the remaining patients and by University College Hospital (1958). who would eventually die with cancer However, only estimates of C were depresent were then already showing a rived and the efficiency of the model for recurrence. (This may reflect some dif-predicting 10-year and 15-year survival ferences in staging.) This high propor-fractions from short-term data was not tion of early cancer deaths has a more discussed (Haybittle, 1960). marked influence on the skew exponential For stage I carcinoma cervix, it is model than on the other models, since seen from Table IX that the predicted the area under the " tail " of the skew values of the 10-year survival fractions exponential curve is larger than that of using the type I negative exponential the similar curves in the other models. model and the type II extrapolated This explains the low survival fractions actuarial model are very similar. Howpredicted for series P using this model. ever, each of the model parameters a For stage III carcinoma cervix, the and , is often subject to a standard skew exponential model is unsatisfactory error of some 50% of its value, so these for n -1 year, but for n = 2 years the models are unsuitable for use with carresults are comparable with those obtained cinoma cervix stage I series. using the other type I statistical models For stage II carcinoma cervix series, (Table XI). the extrapolated actuarial model does The negative exponential model.-For not always give good agreement with stage I carcinoma cervix, short-term actuarial estimates of long-term survival follow-up when n = 3 years, the standard rates (Table X). error in the negative exponential para-For stage III carcinoma cervix, the meter ac, was greater than 0-5a in 3 of model is unsatisfactory for n = 1 year, the 7 series (Table IX). Thus although but for n = 2 years the results are there is generally good agreement between comparable with those obtained using actuarial calculation and prediction, the type I statistical models (Table XI).
The skewed extrapolated actuarial term results can be made, within calmodel. Only one type II statistical model culable error limits. has previously been suggested, namely, Three parametric statistical models the extrapolated actuarial model, and this have previously been described, the logmodel postulated an exponential mortality normal (Boag, 1949), the negative expocurve with maximum at time zero. A nential (Berkson and Gage, 1952) and skew curve rising to a peak within the the extrapolated actuarial (Haybittle, first year or two might be expected to 1959). Each of these models makes a represent the mortality curve with greater different assumption about the analytical accuracy and the skewed extrapolated form of the distribution of survival actuarial model was devised as a possible times of the unsuccessful cases. improvement. The form of this curve is In the present study, the validity of M(t) -(log C62t e-4 these several survival time distributions has been assessed, using the x2 test, with but the peak proved to be too broad and reference to 27 different series of carcinoma generally too far from the origin to of the cervix patients, drawn from several provide a good fit for the survival time hospitals. The patients had all been at distribution (Results, (a)) and its use risk for at least 10 years, having been in a predictive model is therefore some-treated during the period 1945-59 and what artificial. For carcinoma cervix followed up until 1969-71. Two further stages I and II the predicted values were survival time distributions were introfound to be inferior to those derived duced and tested-the skew exponential from the ordinary extrapolated actuarial and the skewed extrapolated actuarial. model, and for stage III they were similar A summary of the results of the tests to those of the other models tests (Tables for goodness of fit is given in Table VI. IX, X and XI). The lognormal and the skew exponential No doubt single-parameter skew curves with C-05 give the best fit to the could be found, possibly from the family observed data. given by Equation 4, which would provide Previous tests of these parametric a better fit but since the lognormal, models have generally been limited to with S fixed, has now been shown to checking the goodness of fit of the survival be of rather wide application (see logtime distribution with the proposed fornormal model, Discussion) there are little mula, but the extrapolated actuarial incentive to seek alternatives which are model has also been tested by comparing likely to be analytically much less con-predicted long-term survival rates with venient.
the observed values for carcinoma of the breast (Haybittle, 1965).
In the present study, all 5 models CONCLUSIONS referred to above have been tested as Parametric models seem to provide predictive models for carcinoma of the a useful alternative to the actuarial cervix with the results shown in Fig. 1, method of calculating survival percentages 5 and 6 and Tables IX, X and XI. When even when follow-up data are sufficiently all these models are tested on stage I extensive to allow the latter method to cases, the lognormal is consistently the be used (Mould, 1976). They certainly most accurate in its prediction of longer extract more information from the clinical term results; the other 4 models sometimes data than the crude m year survival fail to give any satisfactory solution. figures which are still the common form For stage II cases, the lognormal is still of reporting treatment results in clinical the best model but the disparity between journals. They offer the unique ad-this and the other models is not so vantage that an early prediction of longer marked. For stage III cases, where the number of long-term survivors is inevitably comparatively small, there are understandably no great differences between the predictions from the several models.
In summary, the lognormal model, with S fixed at an appropriate value (Table XIII), has been shown to be of wider validity than any of the other models tested and to give reliable extrapolated estimates of long-term survival rate for the separate stage groups in carcinoma of the cervix.