Burns, 1,135-l 41 135

Discriminant function analysis of 570 consecutive burns patients admitted to the Yorkshire Regional Burns Centre between 1966 and 1973 B. Moores and M. M. Rahman Department of Management Sciences, University of Manchester Institute of Science and Technology

F. S. C. Browning and J. A. D. Settle Regional Burns Centre, Pinderfields Hospital, Wakefield

Summary As part of a wider study into the management information needs of a burns unit an analysis was undertaken of the characteristics of those patients who lived and those who died over a 7-year period. Data on all 570 patients were subjected to a dis- criminant function analysis which reveals that the patient’s age, sex, extent of burns and whether or not he is pyo-prone significantly affect his prognosis. From this analysis a mortality function was produced which enables the probability of a particular patient surviving to be computed. A certain amount of discussion is directed toward the possible manipula- tion of the confusion matrix which results from such an analysis and a suggestion at the end focuses on the value of a centralized data processing facility for all burns patients.

INTRODUCTION DURING the past twenty years a number of researchers have sought to identify those characteristics of burns patients which signifi- cantly effect their prognosis, perhaps the most comprehensive such analysis being that con- ducted by Rittenbury et al. (1965). Once the relevant factors have been identified it is possible to incorporate measures of these factors into techniques which allow predictions to be made of the probability that a patient will or will not survive. The earliest of these approaches was that pursued by Bull and his co-workers (Bull and Squire, 1949; Bull and Fisher, 1954) which

utilized probit analysis and concentrated on the patient’s age and the amount of body surface area burnt. The data related to 2807 patients treated over a lo-year period up to 1952. A similar analysis was also performed by Barnes (1957) on burns patients at Massachusetts General Hospital over the 15-year period 1939- 1954. The two sets of results differ significantly, as can be seen in Fig. 1 in which are depicted the 20 per cent and 70 per cent mortality contours taken from the two studies. The difference in shape does imply a quite fundamental difference in the postulated relationship between age and mortality. An examination of the raw data underlying the U.S. figures reveals that thesuperior performance of the younger age group is essen- tially based upon the results of 37 patients out of the total population of 949. Bull’s most recent figures (Bull, 1971) also do not support the hypothesis that for the same degree of burns a patient’s prognosis improves up to the age of 25 and then progressively declines.

MATERIAL This present analysis is based on data pertaining to 570 patients* admitted to the 16bedded Yorkshire Regional Burns Centre situated at Pindertields Hospital. When a patient with burns

*This is coincidently identical to the number of patients forming the data base for a similar study carried out at the University of Texas (McCoy, 1960.

136 Burns Vol. 1 /No. 2

Fig. l.-Iso-mortality contours for various ages and percentage area burned. - - - - Barnes’ data; - Bull’s data. (Taken from Barnes (1957) and Bull and Fisher (1954).)

arrives at any hospital accident department in the Yorkshire region, the medical staff of that department can request his transfer to the Regional Burns Centre. In general, only those patients whose burns are sufficiently extensive to require intravenous resuscitation will be accepted for transfer which for an adult usually implies that more than 15 per cent of the body surface is burned. In addition, almost all patients with burns who are brought to the accident department of Pinderfields General Hospital and who require hospitalization will be admitted to the Regional Burns Centre. Thus, two forms of service are provided, a selective service for the whole of the region and a comprehensive service for the Wakefield district.

A brief summary of the performance of the unit is set out in Tables I, ZZ and ZZZ, These reveal that age, percentage burn and sex appear to significantly effect mortality although it should be stressed that any one of these tables is in- sufficient in itself to indicate that a particular factor influences mortality to the extent revealed therein. It could be, for example, that the higher mortality amongst females is attributable to the fact that the male admissions are less severely

burnt and younger than their female counter- parts. Indeed if we group those 50 patients who are particularly vulnerable either because of age or burns the resultant data set out in Table IV lends a little support to this conjecture.

METHODOLOGY Discriminant analysis essentially represents an attempt to determine from which of two distinct populations an observation or event has origin- ated. Suppose, for example, it is proposed to

Table /.-Breakdown of admissions by agegroup and the survival characteristics of these groups

Age o-2 3-12 13-40 41-60 >60

No. died 4 7 13 19 39 Total 107 142 149 94 78 Mortality (per cent) 3.7 4.9 8.7 20.2 50

characterize burn patients by the two variables we have encountered already, namely age (a) and percentage burns (b). The result of the analysis will be an index which will be some

Moores et al.: Discriminant Function Analysis 137

Table //.-Breakdown of admissions by the extent of body surface burned and the survival characteristics of these groups

Table V.-Sex breakdown of patients with more than 30 per cent body surface burns

Per cent of body surface burned O-9 1 O-l 9 20-29 30-49 50-59 >60

No. died 3 8 9 10 7 25 Total 183 164 83 90 15 35 Mortality (percent) 1.6 4.9 IO.8 33.3 46.7 71.4

Table I//.-Mortality of male and female admissions

Sex Male Female

No died 31 51 Total 339 231 Mortality (per cent) 9.1 22.1

combination of these two variables say c1 . a+ cz . b. If upon inserting a particular patient’s ‘a’ and ‘6’ values into this expression, the resultant number is greater than some predetermined

90 r + +

80 -t t

0 +

++o +

Age 41-60 >60

Male 12 8 Female 13 17

value say ‘ v ‘, then we assign the patient to population zl, whereas if the value is less than ‘ v ’ the patient is designated as belonging to a second population x2. The values c1 and cz are determined statistically so that as many of those designated as having characteristics which would cause them to be labelled as origina- ting from the first population are in fact from that source. We can imagine an observation as a point in multi-dimensional space R where the co-ordinates of the point are x values. We divide this space into two regions R, and R, by some suitable optimum method and lay down a rule of procedure. The two populations may be thought of as two clusters of points which are

+ +

+

+ t

0 IO 20 30 40 50 60 70 ----t--“--” 90 IO0

Age

Fig. 2.-Age and burn characteristics of 82 fatal (+) and 488 non-fatal (0) cases.

138 Burns Vol. 1 /No. 2

separate but to some extent overlapping. We wish to set up a boundary in the space such that as many as possible of population x, lie on one side of the boundary and as many of the popula- tion x2 on the other. The two populations could correspond to those who survive and those who don’t but they could just as readily relate to those patients who contract pseudomonas and those who don’t. If we examine Fig. 2 which sets out the age and burns data separately for those patients who died and those who lived

Table V.-Confusion matrices when the cost of mis- classifying a patient who actually died is rated X per cent higher than that of misclassifying an eventual survivor

Actual outcome

Table Va X=0

Prediction Die Survive

Died 62 20 Survived 31 457

Prediction Die Survive

Actual outcome Died 68 14 Survived 52 436

Table Vb X=22 per cent

440 1

400 :

360

320;

I ; 28Or

f 1

= 240

1

200 1

Fig. 3.-Smooth cumulative distributions of 82 fatal and 488 non-fatal cases.

then a discriminant function would be any line between the points. The objective is to find that line which minimizes the number of mis- designated points. Once having produced a discriminant (index, it need not necessarily be utilized in a strictly dichotomous sense. We could, for example, examine whether any relation- ship exists between the index and some con- tinous variable such as the probability of survival or even the length of the patient’s stay.

It is extremely unlikely that a discriminant index will provide a perfect predictor in the sense that if we contrast our predicted results _. . . _

Prediction Die Survive

Actual outcome Died 69 13 Survived 56 432

Table Vc X=51 per cent

Prediction Die Survive

Actual outcome Died 70 12 Survived 59 429

Tab/e Vd X=235 per cent

complete agreement. There will, invariably, be false positives and false negatives. In this present analysis there will be a number of patients predicted to survive who will actually die and others predicted to die who do in fact survive. Fig. 3 shows the smoothed cumulative distributions of the discriminant indices for those 488 patients who live and the 82 who die. For any specified critical value of the discriminant index say ‘ V ’ there will be n, patients incorrectly designated as survivors and n, wrongly labelled as fatalities. It is, of course, possible to vary these numbers by switching the location of the critical values. The device for accomplishing this is equivalent to assigning a ‘ cost ’ value to the two forms of mis-classification.* To illustrate this consider Table V in which are set out the confusion matrices when incorrectly designating a potential survivor is deemed to be X per cent higher than that of incorrectly predicting a death. t

*For a description of the method see for example Kendal and Stuart (1968). tThese are all based upon the first of the three dis-

with what actually transpires there will be criminant functions presented later.

Moores et al. : Discriminant Function Analysis 139

For each of the 570 patients a punched card variables were included as a consequence of record was produced on which the following examining the raw data and the emerging results particulars were coded : which appeared to indicate that age and burns

XI Sex. interacted significantly. The mean and standard X, Age. deviations of each of the measures together X, Per cent burn. with the corresponding ‘ t ’ values for the fatal X,0 Per cent full-thickness burn. and the non-fatal cases are set out in Table

X, Whether or not patient pyo-prone.: VI.

Tab/e V/.-Mean and standard deviations of fatal and non-fatal cases

Fatal Non-fatal Standard Standard ‘t’

Mean deviation Mean deviation value ______--

Sex* 1.62 0.49 1.37 0.48 4.4 Age 51.60 26.40 22.50 22.80 10.4 Per cent total burn 43.20 21.70 16.20 13.30 15.3 Per cent full-thickness burn 34.40 22.40 7.20 11.20 17.0 Pyo-pronet 0.82 0.39 0.39 0.49 7.5 Burn. Age 2036 1489 339 458 20.2 Age2 3348 2353 1025 1596 11.3 Full-thickness burns” 1675 1891 177 466 15.0 Age*. Burn 64 67 275 167 20.0 Burn. Age* 81 86 311 178 18.4

P value always less than 0.0001 *A value of 1 is assigned to a male and a value of 2 to a female tA value of 1 is assigned to a patient who is pyo-prone. 0 otherwise

In addition other variables were computed from these basic five. These were:

X, Burn . Age. X, Age”. X, Burnr. X,, (Full-thickness burn)2. X* Age*. Burn. X, Age . Burn*.

It should be stressed that these compounded

Table V/la.-Confusion matrices for first discriminant function

Prediction Die Survive

62 20 Died

70 12 Actual outcome

Survived

EEI

31 457

59 429

*The term ‘ pyo-prone ’ was originally coined by Sachs (1968).

RESULTS From the myriad of discriminant indices produced the following are considered the best three:

D,- -6.927+0*906 X,-O.0665 X,+ OGO136 X,+OGIlO X,+-O.002200 X3+ 0.416 X,+0.000708 X,

D,- -6.508-tO.970 X,+0.0204 X, + 0.0629 X4+0.00193 X,+0.235 X,

D,=7.737+-0.984 X,+0.0630 X,+0.130 X,1- 0.102 x,

The corresponding confusion matrices are set out in Tables Vlla, VIIb, and VIIc.

Tab/e V/lb.--Confusion matrrces tor second drs- criminant function

Prediction Die Survive

Died

Actual outcome

Survived

140

Tab/e V/k-Confusion matrices for third dis- criminant function

Prediction Die Survive

Actual

Died

outcome

Survived

key

‘a’ values correspond to equal weightings being applied to both forms of misclassification ’ b ’ values relate to ’ optimum ’ unequal weighting situation

From amongst these it is suggested that the index including only the patient’s sex, age, percentage burn and whether or not he is pyo- prone is probably the most appropriate and is certainly the one most readily computed. It is perhaps surprising that although the average full-thickness burn showed up as being signi- ficantly different between the fatal and non- fatal groups, it did not improve the power of any of the discriminant functions in which it was introduced. McCoy did, in fact, postulate that such an improvement would materialize and we can only observe that despite its inherent attraction as a contributory factor, the difficulty in quantifying it leads to this somewhat surprising conclusion. It is also worth noting that the correlation coefficient between full-thickness burn and total burn of 0.71 indicates the lack of independence between these two measures.

From the calculated discriminant scores it is possible to produce the cumulative distribution shown in Fig. 3 from which the relationship between the discriminant index and the probabil- ity of survival shown in Fig. 4 can be determined.

DISCUSSION At first sight there might appear to be little value in securing estimates of the probability that a particular patient will or will not survive.

Burns Vol. l/No. 2

Certainly it is not thought, nor even suggested for one minute, that the production of such a figure ought to influence the management of his treatment. It does, however, have the advantage that it would enable the mortality performance of different units to be compared on a standardized base. If, for example, one computed the discrimi- nant indices for a group of say 500 patients these indices could in turn be translated into a corresponding mortality probability. The sum of these probability figures would then obviously be the number of deaths to be expected within the unit and any deviation from this would provide some insight into the performance of that unit.

Another possible advantage offered is that stemming from an examination of the character- istics of those patients whose fate differs from their prognosis as determined from the dis- criminant function. It could be, for example, that those incorrectly designated as survivors might, on average, be more obese than the normal population and in this hypothetical case the incorporation of a measure of obesity might lead to a more sensitive discriminant

Dcrcrim,nanr Index

Fig. 4.-Probability of a patient surviving as function of his computed discriminant index (D3.

a

analysis. Alternatively, it might be the case that a significant proportion of those who defied the mathematics and lived had some medical characteristic which set them apart from the others. In this case not only could the discrimin- ant function be revised but it could lead to constructive changes in patient management. Did they, for example, have a lower incidence of pseudomonas?

Why, though, should discriminant analysis be utilized for this rather than probit analysis? Perhaps the most powerful argument is that the latter method requires the clustering of data if the

Moores et al. : Discriminant Function Analysis

analysis is to be meaningful. We have already seen in the case of Barnes’ data how this can lead to distortions due to particular clusters containing such a small number of entries that random effects are exaggerated. If one wishes to incorporate other patient characteristics such as sex and whether or not the patient is pyo-prone the number of entries in any one cell is obviously going to be much reduced and this problem of size will be exacerbated. The discriminant function approach on the other hand treats the data in a continuous manner without recourse to this initial grouping.

It is not suggested that the function used in the production of Fig. 4 is the final word. On the contrary it is to be expected that more refined functions would be forthcoming from analysis of data originating from a number of sources. Indeed, for this and other reasons it is proposed that considerable value would result from the institution of a national data retrieval system. This would initially require a group of specialists to meet and determine which characteristics of burns patients ought to be measured and routinely recorded. Some of these would be straightforward as for example age and sex, but others would be less obvious and reflect one individual’s ‘ hunch ’ that it was an important attribute. These are all factors which would be collected prospectively but another set might be retrospective and relate to incidents occurring during the patient’s stay or the type of treatment carried out. Once these had been determined a suitable data collection form could be designed. One such completed form would be submitted for each admission to some central processing facility which would be responsible for producing appropriate statistics in a format to be determined by this original group. Should such an idea appeal to a sufficient number of readers they might perhaps care to write to either of the two main authors. Certainly no computational difficulties are foreseen.

With regard to the particular results obtained it is difficult to compare them with earlier studies due to the inclusion of the two additional patient characteristics. It is perhaps interesting that sex should be so important in view of the com- ment made earlier that it could in fact be merely a statistical illusion. However, in all the functions computed the patient’s sex did appear as a contributory factor. Why a female of a particular age and with a particular extent of burns should

141

be more likely to die than her male counterpart is itself an interesting question and typifies others that might emerge from the more comprehensive survey alluded to above.

Finally a word is in order regarding the original motivation for this work and the intended development. Basically it stemmed from an examination of how the nursing service could be better controlled so as to reduce the considerable fluctuations inherent in a burns unit between the availability of nursing staff and the needs of patients. It became obvious that the ward sister requires better information in order to secure a better balance between the two sides of this equation. On the assumption that the nursing needs of a patient reflected to some extent the severity of his condition a measure of this was required. As it also seemed obvious that the severity of a patient’s condition upon admission would influence his chances of surviving, the idea of a discriminant index was pursued. It is hoped to report on the wider study within twelve months by which time it is visualized that we will, in conjunction with the ward sister on the unit, have developed a tool for determining, within limits, the nursing work load imposed by a particular patient not just today but into the future. The implications for nurse management of having such a tool and a method of predicting arrivals are, of course, considerable.

REFERENCES BARNES B. A. (1957) Mortality of burns at the

Massachusetts General Hospital, 1939-1954. Ann. Surg. 145, 21G221.

BULL J. P. (1971) Revised analysis of mortality due to burns. Lancet 2, 1133-l 134.

BULL J. P. and FISHER A. J. (1954) A study of mortal- ity in a burns unit: A revised estimate. Ann. Surg. 139, 269-274.

BULL J. P. and SQUIRE J. R. (1949) A study of mortal- ity in a burns unit. Ann. Surg. 130, 160-173

KENDAL M. G. and STUART A. (1968) The Advanced Theory ofStatistics, Vol. 3. London, Charles Griffin.

MCCOY J. A. (1968) Discriminant function probability model for predicting survival in burns patients. JAMA 203,128-130.

RITTENBURY M. S., SCHMIDT F. H. and MADDOX R. W. (1965) Factors significantly affecting mort- ality in the burned patient. J. Trauma 5, 587-599.

SACHS A. (1968) Proceedings of Meeting at U.K. Section International Society of Burns Injuries, Roussel Laboratories Ltd., Wembley Park, Middle- sex.

Rryuests for reprints or further information should be addressed to either: Dr. B. Moore, MSc Tech, PhD, Senior Lecturer, Depart- ment of Management Sciences. UMIST, PO Box 88, Manchester, M60 IQD, or Dr. John A. D. Settle, MPhil, MRCS, LRCP, DA, Burns Consultant. Regional Burns Centre, Pinderfields Hospital, Wakefield, WFl 4DG.