OFFICE OF MEDICAL HISTORY AMEDD REGIMENT AMEDD MUSEUM  
HISTORY OF THE OFFICE OF MEDICAL HISTORY 
Part 4 

EXPLANATORY NOTES SECTION II* IV. EXPLANATORY NOTES AND DEVELOPMENT OF FORMULAE The basic data and material used in this study are from the sources stated in the reference list. Much of the material is from the unpublished statistical tables and sheets on file in the office of the Surgeon General, which are referred to in the list. New tabulations were made from the statistical cards to show how many cases left sick report during the first seven days, the second seven, etc., during the World War, as follows: 1. All cases wounded by gunshot missiles, and all deaths and disability discharges resulting therefrom. 2. All cases wounded by poisonous gases, and all deaths and disability discharges resulting therefrom. 3. (a) A sample of approximately 150,000 out of about 450,000 disease and nonbattle injury cases in the American Expeditionary Forces during the last six months of 1918. The sample included the. same proportion of each diagnosis. (b) All of the deaths from diseases and nonbattle injuries in the American Expeditionary Forces during 1918. 4. All deaths and discharges for disability among troops in the United States during 1918. The task of tabulating a sufficiently large sample of the duty cases in the United States during 1918 was too great a one to be undertaken with the time and personnel available. Similar data were available, however, for the troops in the United States during 19251927, and it was assumed that there was no material difference in the duration of treatment of such cases and of similar ones during 1918. Consequently those data available for the 19251927 cases were used as 1918 experience with the necessary slight modifications as indicated in Figs. 80 and 82. In the absence of more exact information, estimates were used in some instances, as noted in the appropriate places. Whenever this was done all available data were checked to test the estimated results. *Section II is intended for use by only such as are interested in the technical details in the development of the material in Section I. 134 The sources of information, the method of arriving at such estimates were used, and the changes which were made in the basic data are noted in connection with the various Figures. 53. Patients leaving sick report.  The data for Figs. 2164 are largely from formulae. Figs. 7998 inclusive. show the method of developing them. The basic material as tabulated, shows how many patients of the duty, death, or disability groups left sick report during the first second, third week, etc. The numbers leaving sick report each week were divided by the total in thousands to reduce each weeks cases to proportional parts per 1000 of the total number. 54. Graduation of material.  In some instances attempts were made to fit this basic material to second or third order parabolas and also to Pearsonian curves, but the results were not satisfactory. It soon became apparent that there was a very definite geometrical progression relationship between the groups of cases leaving sick report by time intervals or remaining on it, and that all of the curves were exponential in character. Inspection showed that this type of curve fitted the observed points quite well with only a few exceptions. The apparent failure to fit was due usually to imperfections in, or incompleteness of, the basic material; but in some instances, the necessity of relating the graduated data to those from other curves resulted in the use of a poorer fit than could have been obtained otherwise. The exponential curve not only best fitted the observed points, but it graduated also satisfactorily the grouped experience beyond the 14th or 20th week; i.e., the limits of the tabulations by weeks. 55. Patients remaining on sick report.  During the early stages of this work, the data for leaving hospital, or sick report, were first smoothed by fitting them to an exponential curve; but after a short time it was found that as satisfactory results, could be obtained by omitting that step and passing directly from the ungraduated, Leaving to the ungraduated Remaining. To find the number of patients from an original 1000 remaining sick at the end of the first week, either the graduated, or ungraduated, number leaving was deducted from 1000. From the remainder so obtained, the number leaving during the second week was subtracted to find the number remaining at the end of that time. This process was continued through the 14th, and in some instances through the 20th week. As stated, the number remaining beyond that time was ultimately distributed by the formulae obtained. The following table shows the general methods of handling the material and the results obtained in this instance. 135 Table 29.  Number of patients among white enlisted men in the United States. 19251927, inclusive, leaving sick report during each week, and also the number remaining on sick report at end of each week.
56. Exponential curve.  a. One section curve.  The basic formula of the exponential curve used was y = e^{a+bx} In graduating the Leaving material, Y_{1} designates the number of patients leaving during each week (X_{7}) ; and likewise in graduating the Remaining data, Y_{r} equals the number of patients remaining at the end of each week (X_{7}). Where X represents a period of one week, or seven days, it is written X_{7}; and where it represents a five day period, X_{5}. The basic ungraduated data were first plotted on arithmetic logarithmic paper, as is shown by Fig. 79. In some instances where the material was homogenous, as with cases wounded by poisonous gases or by gunshot missiles (See Figs. 87 and 88), a one section curve was sufficient. b. Two section curve.  In another class of cases, such as hospital cases of disease and nonbattle injury patients, certain ones like compound fractures of the femur and pulmonary tuberculosis which required prolonged treatment, were not in the proper proportion to those needing only a few days in hospital to permit of a fit by a one section curve. In such instances there were relatively more short than long duration cases, and a two section curve was required with the general formula 136 y_{r}=e^{a}1^{+b}1^{x}7+e^{a}2^{+b}2^{x}7 The sub 1 and sub 2 appended to the a and b designate the first (1) and second (2) section of the curve (See Fig. 82). c. Three section curve.  In a third class of cases; such as those of patients treated in hospital and quarters in the United States in 19251927, the short duration treatment ones, such as the majority of those in quarters, were relatively so much more numerous than the medium and long duration cases, that a three section curve was required with the general formula (See Fig. 80) y_{r}=e^{a}1^{+b}1^{x}7+e^{a}2^{+b}2^{x}7+e^{a}3^{+b}3^{x}7 d. Plus and minus section curve.  In a few instances where there was too large a proportion of long duration cases for the short duration ones, it was found necessary to use a two section curve with the second, a minus one. Here the formula was y_{r}=e^{a}1^{+b}1^{x}7e^{a}2^{+b}2^{x}7 e. Method of fitting.  In fitting the two or three section curves, the second or third section, as the case might be, with the long duration cases and the proper proportion of medium and short duration ones, was fitted first. For this purpose, the observed data were plotted on, arithlog paper, and a straight line was drawn through the lower end of the material. The angle of this line was determined by its fit of the observed points in that section, and also by the apparent accuracy of the graduation of the grouped material beyond the end of the available tabulation. This latter point was tested by comparing the total of the graduated and ungraduated material beyond the end of the available tabulation; and also by a careful check against the detail of the basic data, which although scattered and irregular, showed quite definitely the general trend and time limits of treatment. The material so graduated was next subtracted from the ungraduated total. The remainder was then plotted on the same sheet of arithmetic log paper, and the second or first sections fitted. In every instance, a number of trials was necessary before a satisfactory fit was obtained. The sum of the different sections as graduated was summated by additions or subtractions, as required, and the graduated total compared with the ungraduated one. In order to hold the higher points fixed, the fitting was by selected points rather than by the least square method. The points used for fitting 137 were usually two observed ones for each section; but in some instances where the graph or trial showed that it was necessary, the points used were selected rather than observed. f. Normal equation.  The normal equation used was log to base e of b of the equation y = e^{a+bx} or log_{e }y=a+bx The value of each x chosen with the log_{e} of its corresponding y was substituted in two equations, which were then solved for a and b. This gave a formula of the general form y = e^{a+bx} or y=e^{a}(e^{bx}) Here e^{a} represents the original height of the curve; and e^{b} the numerical value of the ratio between the height at any time (x) and that of the next following it, or in other words, the slope of the curve. The y then is a function of x, and varies as x varies. g. Illustration of use of formula.  In illustrating the method of developing the specific formulae, in the interest of simplicity the formula of a one section curve will be used. Thus the formula in Fig. 87, showing the number of gas cases remaining in hospital at the end of each week, is y_{r}=1000.00 (.842250)^{x}7 The 1000.00 is the e^{a}; and the .842250, the e^{b}. The curve then starts with 1000.00 patients of whom .842250 × 1000, or 842.25, are remaining in hospital at the end of one week (X_{7}) ; and then of this remaining group (842.25) the same fraction (.842250) or 842.25 × .842250 = 709.39 are remaining at the end of the two weeks, etc. h. Change in size of basic group and time interval.  To convert the formula into terms of 1 patient, simply divide by 1000 and then y_{r}=1.00 (.842250)^{x}7 The e^{bx} is in terms of seven day periods. To convert it into terms of one day (e^{bx}_{1}) take the seventh root of it; and then to change the result into terms of any other period, as, for example, a five day one (e^{bx}_{5}) as used here, raise it to the fifth power. 138 Thus from the above. x_{5}=antilog of 5(log .842250/7)=.884593 Then the formula by 5day periods is y_{r}=1.00 (.884593)^{x}5 Y_{r}, then is the number of patients remaining in hospital at the end of successive five day periods. If, however, it is necessary to find directly the fraction of a patient remaining at the end of any other period, as for example one year (365 days), multiply as above by that number instead of by 5. Thus x_{365}=antilog of 365 (log .842250/7)=.000130 If the e^{a} is 1.00, on the 365th day, .000130 of 1 patient will, still be in hospital; but if the e^{a} is 1000.00, there will be .13 (See Fig. 87). To convert the e^{a} into 100.00 patients, so that those remaining at any time will be in terms of percentages, divide the formula showing 1000.00 by 10. Then the formula will be y_{r}=100.00 (.884593)^{x}5 2. Percentage of patients who have left hospital.  To find the percentage of patients who have left the hospital, subtract the number remaining in hospital from the original 100.00. Patients who have left hospital = 100.00  [100.00(.884593)^{x}5] Expressed in general terms, the formula is dx=e^{a}e^{a+bx} i. Graduated increase in the number of patients on sick report or in hospital.  Cases of sickness and injury occur from day to day. Patients who are admitted to sick report remain under treatment for, varying periods of time, depending chiefly upon the severity of illness or, injury, but to a certain extent upon the proximity of the hospital to the troop 139 areas, the facility for returning the men to the organization, etc. Of those admitted to sick report, some will have returned to duty by the end of the first period (5 days in this case), others will go out during the second one, while some will remain a longer time, and a few even a year or more. During the second period, in addition to the cases coming in, a certain number of patients will be remaining from those admitted during the first one. Also during the third period, in addition to those admitted, cases will be remaining from the first and second ones. And so, the increase in the number of patients will continue as time advances, until all of the first group of cases admitted have left sick report, which, for all practical purposes, will be at the end of one year. After that time the number, of patients leaving sick report during each period by return to duty, deaths, or discharges for disability, will equal the number admitted. j. Integration of basic formula.  To find then the number of patients on sick report, or in hospital as the case may be, at any time, add to the number admitted during the period, the number remaining from all preceding ones. For this purpose the basic formula is integrated between the beginning (M day) and the constantly advancing time, X_{5}. Using the basic formula with one section, the integrated formula is ydx=(e^{a+bx}/log_{e}b)(e^{a}/log_{e}b) Substituting the values already found, and using one seventh of the log_{e} b, the formula for gas patients shows the number of gas patients in hospital at the end of each 5 day periods ydx=P=40.774720[40.774720(.884593)^{x}5] To find the total patients in hospital on any day, the day's basic admissions must be added to the P found from this or any similar formula. The patients (P) include all who will ultimately return to duty, die, or be discharged as physically disabled. The number who are disposed of in other ways is so small that it can be omitted from consideration. k. Formulae for deaths and disability discharges, which have, occurred. The total number of patients who have left hospital at any time by death, disability, discharge, or duty equals those who have been admitted less the number sick in hospital. This is also true of death and disability cases when considered separately. The integration of the basic formula shows how many cases, which will ultimately result in death or discharge, are in hospital. Then the difference between that group in hospital and the total deaths or disabilities expected among the cases which have been admitted, constitute the number of fatalities or disability discharges which have already occurred. 140 All of the special formulae relating to death or disability cases are expressed here as parts of the total cases [See Fig. 97, (1) and (2)]. Thus, for every patient wounded by gas who is admitted to hospital, .017306 will die; and if there are 5 cases admitted there will be 5 times .017306, or .086532, fatalities. The basic formula for the eventually fatal gas cases remaining in hospital from 1000.00 of them admitted on any day is (Fig. 97) y_{r}=863.02(.420133)^{x}7+136.98(.870099)^{x}7 Changing each e^{bx}_{7} to e^{bx}_{5}; reducing the sum of the two e^{a}, from 1000.00 to 1.00 by dividing each by 1000.00; multiplying each resulting e^{a} by .017306 to reduce the deaths to parts per total cases; and then integrating ydx=.239820[.120563(.538258)^{x}5+.119257(.905389)^{x}5] This formula shows the gas cases in hospital, who will eventually die. At the end of 30 days the number of such cases is .239820  (.002932 + .065689) = .171199 As above, the total patients who have died or who will die out of 5 gas cases, is .086532. Consequently, the formula for those who have died is .086532x_{5}  the above formula At the end of 30 days, or after 6 five day periods, it will be 6 (.086532)  .171199 = .347993 Then at the end of 30 days, if 30 gas cases have been admitted at the rate of 1.00 per day, .35 have died, and .17 patients, who are in hospital, will die. .35 + .17 = .017306 × 30 = .519180 l. Multiple section curves.  When two or three section exponential curves are used, the method of developing the formulae from them is the same as that used for a one section curve; and the only additional step which is required is the summation of the different sections. S. COMPUTATION OF DATA FOR SHORT DURATION CASES 57. Percentage leaving hospital each day by return to duty or death.  Calculate from the formulae on Figs. 90, 91, and 92 for duty cases; and from the last ones on Figs. 96, 97, and 98 for death cases, the percentage of each group who return to duty or die on each day up to and including the tenth. These data are shown by Tables 26 and 30, respectively. 141 The figures are cumulated and show the percentage of duty or death cases which occur during any period from the beginning of the first day the end of the tenth. The percentage leaving hospital on any one day by duty, death, etc., can. be found easily by subtraction. Table 30.  Percentage of the total deaths in hospital which occurred during the first ten days from the three classes of cases in the American Expeditionary Forces.
* See Fig. 96. ? See Fig. 97. ? See Fig. 98. 58. Average days of treatment.  As an illustration of the method, let us find the average duration of treatment for gassed cases who die in ten days or less. a. Divide the percentage of death cases leaving hospital (dying) during each day from the first day to the tenth, by the total leaving (dying) during the ten days. The results expressed in parts per 100 show the percentage of those dying in ten days or less who died on each day from the first to the tenth. (see column 4, Table 31). b. Beginning with 100; that is, the total percentage to die in 10 days', subtract successively the number leaving, hospital (dying) during each day from the first to the tenth. The results show the percentage of the 10 day fatal cases remaining in hospital each day (see column 5, Table 31). c. Reduce the results to the basis of one case instead of 100% (see column 6, Table 31) and summate. The total shows the eventual death cases in hospital at the end of ten days from the group remaining under treatment from one to ten days, when the daily admission rate of gassed cases who die during period is 1.00. 142 Table 31.  Method of finding the eventually fatal gassed cases in hospital at the end of ten days among those that die in ten days or less, when the daily admission rate from the group is 1.00.
*From Table 30. ?See Par. 58a. ?See Par. 58b §See. Par. 58c. With an admission rate of 1.00, the eventual death cases under treatment (4.55) is the same as the average days treatment for each one. Table 32 shows the average duration of treatment of duty and death cases admitted as sick, gassed, and gunshot wounded, and whose maximal days of treatment are from one to ten. Thus of the fatal gassed cases which survive five days or less, the average days was 2.77; and for the group surviving ten days or less, it was 4.55. In Table 32, the data for "Duty cases" are computed from those for "Duty cases" in Table 26 in same way as Col. 6, Table 32 (gassed) are from those in Table 31. The data for "Death cases" in Table 32, are calculated similarly from those in Table 30. Table 32.  Average duration of treatment of cases in hospital who will return to duty or die, at any time between one day or less, and ten days or less.
*From Table 26 ?From Table 30 143 59. Patients in hospital on the maximal day.  As stated above, among gassed cases who eventually die and for whom the duration. of treatment is from one to ten days, there are 4.55 cases in hospital on the tenth day, when the daily admission rate from such cases is 1.00 per day. a. In the first place, however, only 63.77% of the fatal gassed cases die within ten days. Consequently to convert the data into terms of total gassed cases which are fatal, multiply the 4.56 by 63.77%. The results show that when the daily admission rate for fatal gassed cases of all durations is 1.00, that among, the cases surviving from one to ten days, there are 2.90 (4.55 × 63.77%) such patients in hospital on the tenth day. b. In the second place, only 1.73% of all gassed cases treated in hospital are fatal. Consequently to translate the data to the basis of 1.00 gas admissions per day, the 2.90 must be multiplied by 1.73%. The result shows that there are .05 (2.90 × 1.73%) gassed cases in hospital on the tenth day in a group of such cases that will die within ten days, when the daily admission rate from war gases is 1.00 (see Table 27). The data for. the cases who are treated five days or less, six days or less, etc., are found by the same method. 60. Patients in hospital on any day, when the maximal one is the tenth, etc.  The problem here is to find how many cases there will be in hospital on any one day, such as the fifth or sixth, in a group of cases whose treatment is for a longer period; for example, ten days or less. To illustrate the method let us again use the data for the gassed cases who eventually die. The process is the same as in Table 31 until the data in the last column are obtained. Then to find the patients in hospital on any day among those surviving ten days or less, proceed as shown by Fig. 22. The following table shows the method in detail. Table 33.  Fatal gassed cases in hospital on any day in a group surviving ten days or less.
* Summation from above downward of data, last
column, Table 31, 144 Fig. 77.  Histogram and fitted skew curve for daily admission rates from diseases and nonbattle injuries to hospital and quarters per 1000 strength in 30 large camps in the United States during 1918, excluding September and October. 145 Fig. 78.  Histogram and fitted skew curve for daily admission rates from diseases and nonbattle injuries to hospital and quarters per 1000 strength in 30 large camps in the United States during 1918, excluding September and October. 146 Fig. 79.  Duration of treatment (leaving sick report) of diseases and nonbattle injury cases among white enlisted men in hospital and quarters in the United States in 1925  1927. ^{1 2} 147 Fig. 80. 
Duration of treatment (remaining on sick report) of disease and nonbattle
injury cases in hospital and quarters as they occur in the United States 148 Fig. 81. 
Duration of treatment (leaving hospital) of disease and nonbattle injury
cases in hospital as they occur in the United States. 149 Fig. 82. 
Duration of treatment (remaining in hospital) of disease and nonbattle
injury cases in hospital as they occur in the United States. 150 Fig. 83. 
Duration of treatment (leaving hospital) of disease and nonbattle injury
cases which occurred in the American Expeditionary Forces in 1918 while
in the A.E.F. and later in the U.S. 151 Fig. 84. 
Duration of treatment (remaining in hospital) in the American Expeditionary
Forces and later in the U.S. of disease and nonbattle injury cases which
occurred in the A.E.F. in 1918. 152 Fig. 85. 
Duration of treatment (leaving hospital) in the A.E.F. only of disease
and nonbattle injury cases which occurred there in 1918. 153 Fig. 86.  Duration of treatment (remaining in hospital) in the A.E.F. only of disease and nonbattle injury cases which occurred there in 1918.^{1 NOTE: Observed points from data as graduated by Fig. 85. (1) From integration of formula of curve. (2) It is estimated that the average days lost per case in the A.E.F. only was 23.75 (22.75 + 1.00). Multiply (1) by 96.575% to reduce it to (2).} 154 Fig. 87.  Duration of treatment of GAS cases which occurred in the A.E.F. in 1918.^{1 }NOTE: The observed points include practically only the time lost by cases while in the A.E.F. The lower curve (b) is sloped so as to include only that much time; while the upper one (a) cover all the time lost including that after transfer from the A.E.F. The upper curve is obviously a poor fit of the observed points, but it is as good a one as is practicable with the available data. 155 Fig. 88.  Duration of treatment of GUNSHOT cases which occurred in the A.E.F. in 1918.^{1 NOTE: See Note on Fig. 87.} 156 Fig. 89. 
Duration of treatment of disease and nonbattle injury cases who were admitted
to hospital in the United States in 1918 and returned to duty. 157 Fig. 90. 
Duration of treatment of disease and nonbattle injury cases which returned
to duty while in the A.E.F. in 1918. 158 Fig. 91. 
Duration of treatment of GAS cases which returned to duty while in the
A.E.F. in 1918. 159 Fig. 92. 
Duration of treatment of GUNSHOT cases which returned to duty while in
the A.E.F. in 1918. 160 Fig. 93. 
Duration of treatment of disease and nonbattle injury patients admitted
to hospital in the United States in 1918, who ultimately died. 161 Fig. 94. 
Duration of treatment of disease and nonbattle injury patients admitted
to hospital in the United States in 1918, who were ultimately discharged
for disability. 162 Fig. 95. 
Duration of treatment of disease and nonbattle injury patients admitted
to hospital in the United States in 1918, who ultimately died or were discharged
for disability. 163 Fig. 96. 
Duration of treatment of disease and nonbattle injury patients admitted
to hospital in the A.E.F. in 1918, who ultimately died. 164 Fig. 97. 
Duration of treatment of gas patients admitted to hospital in the A.E.F.
in 1918, who ultimately died. 165 Fig. 98. 
Duration of treatment of GUNSHOT patients admitted to hospital in the A.E.F.
in 1918, who ultimately died. 166 Fig. 99.  Relationship of the number wounded by poisonous gases to those wounded by gunshot missiles.^{3 }NOTE: The same infantry regiment battle days were used as per Fig. 100. As shown by the graph the number of men wounded by war gases increased at a slower rate than those wounded by gunshot missiles, actually causing a gradual decrease in the number of gas cases per one gunshot case as the latter increases in number. Thus the number wounded by poisonous gases to each one wounded by gunshot missiles was 1.14 when there were 3 of the latter, but only .18 to each one when there was 153. 167 Fig. 100.
 Relationship of the number killed by all causes to those wounded by gunshot
missiles or by poisonous gases in 6,022 battle days for infantry regiments
from Jan. 1  Nov. 11, 1918.^{3
}NOTE: The infantry regiments used were those of the following Divisions:
1st, 2nd, 3rd, rth, 26th, 27th, 35th, 42nd, 78th, 79th, and 80th. Return to the Table of Contents

