A Quadratic Equation Adequately Describes the Cumulative Food Intake Curve in Man

Appetite: Journal for Intake Research, 1982,3,255-272

A Quadratic Equation Adequately Describes the Cumulative Food Intake Curve in Man HARRY R. KISSILEFF Departments of Medicine and Psychiatry, St. l.uke's-Roosevett Hospital and Columbia University of Physicians and Surgeons

JOHN THORNTON Deparment of Biomathematics, Mount Sinai School of Medicine

EMIL BECKER

A quadratic equation adequately describes the cumulative food intake curve in single-course meals in non-obese men and women. Intake of the meal and both the linear and quadratic coefficients ofthe equation were larger in absolute value in men than in women. The quadratic coefficient was negative. The coefficients have a simple physical interpretation. The linear coefficient is the initial rate of eating, and the quadratic is half the rate ofdeceleration (in absolute value). We can therefore say that men exhibit a higher initial rate of eating but also decelerate faster than women, on the test diet used in this study. There were no differences in intake or the coefficients of the cumulative intake curve owing to visual cues. It is suggested that this difference from other studies was because, in this study, the reservoir was wide, not narrow, and because visual cues operate to control intake only when they induce rapid changes in perception of the amount consumed. Finally, intake was less variable in the laboratory than it was outside the laboratory, but the sex difference in intake persisted outside the laboratory. It is suggested that the quadratic equation could be useful for characterizing factors that lead to changes in the rate of eating during the course of a meal and that such changes might have diagnositic value in assessing eating disorders.

Cumulative food intake during single-course liquid meals in man was first described by Jordan, Wieland, Zebley, Stellar and Stunkard (1966). Like previously studied cumulative intake curves in animals (Skinner, 1932; Bousefield, 1935; Stellar and Hill, 1952) these curves were described as negatively accelerated, i.e., the rate of intake gradually slowed during the meal or draught. Although verbal descriptions are useful in qualitatively distinguishing curves of different shapes, in order to make optimal use ofthe information contained in such observations, it must first be reduced

We thank Jennifer Bass for assistance in collecting the data. Supported by grant AM-17624 from the National Institutes of Health (Obesity Research Center). Portions of this work were presented at a meeting of the American Society of Clinical Nutritionists, April, 1978, in San Francisco. The analyses of variance were carried out with aid ofprogram BMDP-2V (Dixon & Brown, 1979)revised, November, 1979. Programs were developed at the Health Sciences Computing Facility UCLA, which is sponsored by NIH special research resources grant RR-3. Requests for reprints should be sent to Harry R. Kissileff, St. Luke's Hospital, I 14th St & Amsterdam Av., New York, NY 10025, U.S.A. 0195---{j663/82/030255 + 18 $03'00/0

(0 1982 Academic Press Inc. (London) Limited

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H. R. KISSILEFF ET AL.

to simple quantitative terms. Animal work, in the past, followed two approaches to this problem. First, the amount consumed by a group at each point in time was averaged and the effects of experimental manipulations were determined by comparing treatment effectsat each point in time (e.g.,see Booth, 1972).The other approach was to describe curves from individual sessions of a single animal by their best fitting equations, therby reducing the data to a few parameters (Skinner, 1932; Bousefield, 1935).However, beyond fitting equations to data, little additional work has been done. Davis & Levine (1977) combined these approaches by fitting averaged data from groups of animals to an exponential equation to evaluate a quantitative feedback model of the cumulative intake curve. They showed that the equation adequately described the effects of manipulations of taste and intestinal filling. The Davis and Levine (1977) model has two shortcomings. First it is non-linearizable (i.e., the parameters enter the equations as exponentials), which makes it more difficult to apply to data than if the curve were described by a simple linearizable model in which the parameters entered additively. Second, because animals tend to stop consuming at different times during the session, the average curves for a session of fixed length are much flatter at the end than any individual curve, and, therfore, give the impression of a gradual slowing before consumption stops, which might not by typical of the individual. Another quantitative approach to describing the cumulative intake curve was taken by Pudel (1971). He noted that curves of cumulative intake were negatively accelerated in lean but not obese subjects, and he suggested that a quadratic function could be used to fit the data. However, rather than pursuing the tedious approach of estimating parameters of individual curves, he used an index of consumption which was the percentage of total intake consumed in the first temporal half of meal. If the index was above 50%, the curve was considered negatively accelerated. This approach is less precise than fitting the data to an equation, because an infinite combination oflinear and quadratic coefficients could result in the same percentage of intake in the first temporal half of the meal. The present approach provides more information and completely specifies the curve. Our approach to describing and utilizing the cumulative intake curve is to find an optimal mathematical description of the curve and to use it to characterize intake curves from men and women, eating with or without visual cues. An optimal description must meet both descriptive and analytical criteria. However, because these two domains generally conflict in their demands for parsimony, the optimal description will usually be a compromise, McCleery (1977) has suggested four criteria for the optimal equation: (1) successful fit and prediction of observations, (2) simplicity, (3) implication of mechanical causation, and (4)consistency with function considered from an evolutionary point of view. The first two can be reduced to a single criterion, adequate description. The second two can also be reduced to one, theoretical utility. As McCleery points out, there must always be a trade-off between an equation with perfect fit, having a large number of parameters, such as an nth-order polynomial, and a simpler equation which does not fit perfectly, but which has fewerparameters and more theoretical appeal. If each parameter can be assigned a theoretical role which enables its value to be predicted under various experimental conditions, its utility as an experimental tool will be greater than a better fitting equation whose parameters have only a poor relation to theory. We must also add a third criterion, ease of application. Even though computers and software are widely available, the methods of fitting equations to data are more difficult to apply to non-linearizable equations than to equations which are linear in the parameters (linearizable, i.e., the parameters enter

257

FOOD INTAK E IN MAN

add itively as first-order con stants times functions of the independent variable). The rea son for this difficult y is that the parameter estim ate s so metimes cannot be obta ined because solutions to the itera tive algorithm fail to con verge (Dra per & Smith, 1981). For further information on the problem of solving non-Iinearizable equation s the reader should consult a text on numerical analysis (e.g., Ralston , 1965, chap. 8). Besides being easier to solve th an non-Iine ar izab le equation s, linearizable equations can be fitted to data using mo st hand -held calculators, thereby greatly extending the ir accessibilit y to users. Therefore , if descriptive and the oretic al criteria evenly suppo rt a linearizable and a non-linearizable model, ease of application would favor cho ice o f a linearizable model. On the other hand if compelling theoretical reasons favored a non-I inearizable model, it should be preferred, even if harder to apply . We sha ll, therefore, apply our three criteria successively to obtain the optimal mathemati cal description, and we sha ll begin by finding the best fitting equation or equations.

METHODS

Subje cts

Eight participants of each sex were selected from a non -smoking group of 26 men and 25 women ranging in age from 19 to 24 years and within 10% of de sirable weight (Bray, 1975). They were selected by four cr iteria after participating in a taste test and eating two meals sepa ra ted by a 3-h inter val: (1) Test meal must have been given a ratin g of 6 or higher on a 9 point sca le o f liking (Peryam and Pilgrim , 1957); (2) The y ate a t least 280 g of the test meal ; (3) The y ate all of the pretest meal ; (4) Th ey followed instructions about not tou ch ing the reser voir conta ining the food (see "E xperimen tal Setting and Test Foods"). All subjects gave informed consen t and were paid $4·00 per da y of participation . Their dem ographic inform ation is show n in Table 1. T ABLE

1

Subject characteristics (Mea n ± SO )

Height (em)

Weight (kg)

Percent desirable weight

Age (year s)

Rating of test food"

Restr aint score"

Male

179±4'7

67'5±8'2

93-1±9'2

21'7± 1·3

6'5/7'1 ± 1'2/0'8

2·96 ± 2-4

Fem ale

164± 7·3

54'5±5'3

97'1± 7-0

21'7± 1·2

7'0/7'0 ± 1·3/1'1

6-45± 3-8

Sex

a b

First number is the rat ing in the taste test, second is the rat ing after eating the food as a meal. Scored according to the meth od of Herman and Mack (1975).

Preliminary Tes ts and Instru ctions to the Subjec ts

In order to o btain a gro up of subjects who liked the diet and would eat sufficient quantities of it for sufficient durati on to model the curve, subjects were given a taste test prior to consuming the main mea l and were given the test meal. Further methodological det ails, includ ing portions of the instruction s appear in the Appendix. Th ese details are provided because o f the po ssible impac t of instructions on the

258

H. R. KISSILEFF ET AL.

subject's responses (Pudel, 1976). In order to maximize the reproducibility of each test meal, subjects were given a pretest meal three hours before the test meal. Experimental Setting and Test Foods

Eating took place in a special room containing a universal eating monitor (UEM, Kissileff et al. 1980). The pretest meal consisted of two cinnamon-flavored Breakfast Squares, containing 380 kcal, and 240 ml apple juice (Red Cheek) containing an estimated 113 kcal. The test meal was the liquified version of the yogurt and fruit diet previously described (Kissileff et al. 1980). It was served at 55°F through a straw (8f' x 0'24" dia.) from an opaque tapered cylindrical 2 qt container (15'6 cm top dia., lO-5cm bottom dia., and 17·6cm high). Sequence of Procedures

Each of the selected subjects came to the laboratory for a pretest meal (breakfast) on an empty stomach (no eating since retiring) between 0800 and 1020 hrs and returned 3 h later for the test meal. The first four men and women received eight pairs of pretest and test meals in addition to the screening test over a 2 to 3 week period on non-consecutive days. The reservoir containing the diet was covered with an opaque plastic top which had a hole in it just large enough to admit the straw, so that subjects had no visual cues about the amounts they consumed. The last four men and women received four pairs of pretest and test meals, after the screening, and the container was never covered. thus providing them with visual cues about the amounts.consumed. The reduction in the number of trials was made because results with the first eight subjects indicated no trends over time (see "Results"), after the fourth day. Food diaries, kept by the subjects for the days of, and days after, each test meal, indicated amounts and types of foods eaten along with times of eating. These diaries were subsequently used to estimate the relationship between reported breakfasts (food eaten between 0700 and 1000hrs) and luncheons (food eaten between 1100 and 1400 hrs) and amounts consumed in the laboratory. At the pretest meal subjects consumed the entire portion. At the test meal a tape recorded set ofinstructions told the subjects to "eat as much as you like". When the subject had not eaten for 15min a rating sheet was brought in. When thay had finished filling it out, they were free to leave. This approach differs from that of other previous laboratory studies (e.g. Pudel, 1971; Jordan et al. 1966) in that the duration of the meal was determined by the subject and was a dependent variable rather than a fixed variable set by the experimenter. Each subject was tested alone. Further important details of the procedure and instructions to the subjects appear in the Appendix. Data Collection and Analysis

The UEM provided an intake every 3 sec as previously described (Kissileff et al. 1980). However since the objective was to obtain the best fitting average curve, noisy components were removed by using 15 sec intervals, i.e., using only every fifth point collected, and by eliminating data points which were less than a previous point by more than 10% of the final intake or 20 g, whichever was smaller. (See Appendix for further details of the "cleaning" procedure.) These procedures objectively removed obvious spike depressions in the curves which were occasionally seen when subjects pressed on the container, e.g., by pushing the straw against the bottom or by accidentally leaning on it.

259

FOOD INTAKE IN MAN

The cleaned data from the open reser voir condition were fitted to 11 different models using the SAS GLM (Barr et al . 1976) procedure for the first eight linearizable models in Table 2, and by the SAS NONLIN (Barr et al, 1976) procedure for the remaining three non-linearizable models (see Table 2). A stepwise procedure was used to compare the linearizable models with one another. Since models 5 through 8 are all polynomials, logically model 8 should provide the best fit and model 5 the least, under an y conditions. The minimum degree pol ynomial "necessar y to adequately describe the behavior of the dependent variable" (Ostle & Mensing, 1975, p. 202) was selected first. That is no significant increase in R 2 (the coefficient of determination) would be attained by adding a higher order term to the minimum degree polynomial. The minimum degree pol ynomial which adequately described the data was then compared with models 1 to 4 on the basis of two objective criteria: the coefficient of determination, TABLE

2

Models tested for fit to the cumulati ve intak e curve No. Type Linearizable a Linearizable Linearizable Linearizable Linearizable Linearizable Linearizable Linearizable Non-linearizable" Non-linearizable Non-linearizable

1 2 3 4 5 6 7 8 9 10 11

Description

Power Linear Qu adr atic Cubic Qu artic Exponenti al Exponenti al

Reference" Formula

2 3

I = (l /a )t/(t+ b/a) I =e'eb / l I=a+bln(t+l) I =e't b I= a +bt I= a +bt+ct 2 I=a +bt+ct 2 +dt 3 I = a + bt -l-ct.' + dt 3 + et 4 I = a(l -e- b l ) I=a-be- c l I = a+ bt"

3 Linearizable models are those for which there exists a tran sform of I and t such that the coefficients enter the model linearly. For example , Jog transform at ion of model 4 yields In I = a + bin t. Ln I and In are therefore linearly related . The advantage of such models is that standa rd regression programs can be used to obtain the constants. b Models for which no linear transformation exists are denoted non-Iinearizable , Denotes previously proposed models-l = Skinner , 1932; 2 = Pudel, 1971; 3 = Bousefield, 1935; Davis & Levine, 1977. C

which indicates the percentage of variance accounted for b y the equation, and the standard deviation of the residual s. The main difference between these criteria is that R 2 measures goodness offit of data after the linearizing transformations are appl ied, as in models 1 to 4, whereas the standard deviation ofthe residuals estimates variability in the untransformed space. Next, the best fitting non-linearizable model was selected on the basis of R 2 . The final step was to compare the best fitting non-linearizable model with best fitting linearizable model. For all cases of each model residual s were examined for patterns and outliers. The reason that dat a from the open condition only were used in these comparisons was one of expediency. The cur ves from the two conditions were so similar in form, and fit the optimal model so well, that there was no point in testing the closed cond ition with all the model s. In addition to data from the UEM, the inta ke and meal duration were also mea sured (seeAppendi x for detail s), and analysis of variance (Winer , 1971; Dixon, 1979)

260

H. R. KISSILEFF ET AL.

was undertaken on the parameters of the optimal model, and on intake and meal duration to determine the effectsof time, sex,and reservoir visibility. The main analyses of variance in these comparisons used data from the first through fourth meals after screening in order to have the two reservoir conditions contribute equally, since there were eight postscreening meals in the closed condition and only four in the open.

RESULTS

Models

Of the polynomial models, 6, the quadratic, was selected as the simplest adequate model for the following reasons: (1) It was the minimum degree polynomial which adequately described the observed values, (2) The linear model was rejected because a significantly larger percentage of variation could be accounted for by adding the quadratic term; (3) Adding a cubic or higher terms did not significantly increase the percentage of variation (R 2 ) accounted for. The percentage of variation accounted for by the quadratic ranged from 96·3 for the meal which fitted worst to 99·9 for the meal which fitted best. The linear model was rejected because it accounted for variation ranging from only 88'1% for the meal which fitted worst to 99'7% for the meal which fitted best. The quadratic would, therefore, be generally more useful because a simple linear curve could be described as a quadratic with a zero quadratic coefficient, but a decelerating curve could not be adequately described without a quadratic coefficient. Comparison of the quadratic model with the first four linearizable models, showed that the quadratic received the highest percentage of first rank scores, 72% or 23 of the 32 meals, on R 2 . The other models were ordered from best to worst as follows: 4, 1,3,2 (see Table 3). Because the coefficient of determination was taken on transformed data for models 1 to 4, another method of comparison was used, based on the standard deviation of the residuals ofthe untransformed data. In this case the quadratic was also the highest ranking model and received the rank of 1 in 87% of the meals (see Table 3). TABLE

3

Selection of best model Value of R 2 Model no. 6 4 1 3 2

Minimum Maximum 0·9628 0·9104 0·7346 0·8478 0'5601

0·9986 0·9979 0·9979 0·9831 0·9758

Standard deviation of residuals

Range

Mean rank

0·0358 0,0875 0·2633 0·1353 0,4157

1-45 2·22 2·75 3-69 4·88

Minimum Maximum 5·8 7-49 7·29 20·08 31·61

29·32 93-18 509·28 110·91 240·93

Range

Mean rank

23-52 85'69 501·99 90·83 209·32

1·13 2·13 3·52 3-35 4·87

Note: Mean rank is the mean of the ranks which each model received when fitted to data of each meal and evaluated by the magnitude of R 2 or standard deviation of the residuals. Rank of I was assigned to model with the lowest value of each measure and rank 6 was assigned to the model with the highest value of each measure. The values in the table are arranged by best to worst fitting on the basis of mean rank of R 2 . Both measures give similar results except for the reversal of models 1 and 3 when SD of residuals is used.

261

FOOD INTAKE IN MAN

The best fitting non-linear model was model 10. It was found to have a slightly better fit to the data than the quadratic but the improvement was not substantial, and a cubic would usually provide better fit than model 10. It should be noted, for purposes of comparing models, that any model which provides an R 2 in excess of that obtained by the minimum degree polynomial which adequately fits the data (as described under "Methods") does not provide a significant decrease in "unexplained" or "error" variance, and, therefore, all such models can be considered equivalent in their goodness of fitting the observed data. For this reason, it is permissible for selecting the optimal model to use other criteria than strict goodness of fit. These criteria are theoretical utility and ease of application. Because of the difficulty in obtaining estimates for the parameters of non-linearizable equations (Draper & Smith, 1981), we felt that more could be gained by describing the data using a quadratic model. In addition, the parameters of the quadratic model can be given a simple theoretical interpretation (see Discussion, and Kissileff & Thornton, 1982). We, therefore, next describe the effects of trial, sex, and reservoir visibility on the coefficients of the quadratic model and the realtionships between the linear and quadratic coefficients. Ability of the Coefficients to Characterize the Curves, by Graphical Inspection

An analysis of variance with repeated measures (Table 5) revealed no significant differences across trials for the coefficients or for meal duration. Consequently, an individual's cumulative intake curve could be characterized by a curve which was drawn from the average of the coefficients for the average meal duration. When such an average curve is compared with individual curves (see Figure 1), it is readily seen 1520 1140 760

380 0

jl

5

10

15

20

15

20

2280

.E c:

Time (min)

FIGURE 1. Individual cumulative intake curves (-~) and the average cumulative intake curve (--) for the subject whose linear coefficients were least and most variable. The coefficients for the averagecurve along with standard deviations for the coefficients ofeach curve are given by the equations 1= [27 ± 19+(72 ± 15)t + (1·5 ± 0·6)t2 ] and 1= [2-9 ± 45+(222 ± 55)t +(5-3 ± 3-3)t 2 ] ; I indicates intake and t, time. The average curves were constructed by plotting intake as a function of time, using the mean coefficients from the individual curves to generate values for each point in time, and extending the average curve for the mean duration.

262

H. R. KISSILEFF ET AL.

that it provides an excellent graphical summarization of the data. Similarly average curves for each subject (shown in Figure 2) illustrate effects of differences between sexes and similarity of reservoir conditions. Curves for men appear to rise more sharply and decelerate more quickly than those for women. Furthermore, those with the highest initial rates appear to decelerate most quickly. Quantitative analysis of these graphs confirms this impression. However, it should be noted that final intakes, extrapolated from curves such as these, can be as much as 30% different from the actual intakes averaged over several meals. Graphical analysis is, therefore, fine for illustrating the shapes of curves, but for interpreting their relationships to intake, one must rely on separate analyses of intake and the coefficients of the cumulative intake curve.

1800

(a )

187'9/-3.43

(b )

183,62/-6'68 1200

74,6/+4 1311/-4'19

124'3/-5'06~

600 0'

'" 0

-'"

C

1800

(d)

(c )

1200 99,96/-3,54 562/-0'83

600

102'8/-075

496/-1'16

4

8

12

4

16

8

12

16

Time (min)

FIGURE 2. Average cumulative intake curves produced as described in Figure 1 for each subject; (a) men open; (b) men closed; (c)women open; (d) women closed. The numbers above the curves are the mean linear and quadratic coefficients for each subject. To equalize comparisons, only data from the first four postscreening days are used in constructing these curves.

Quantitative Analysis and Analysis of Variance

The differences in initial rates of eating are reflected most accurately by the linear coefficients of the cumulative intake curves. This variable had marginally significantly (F=4·25, p=0·06, 1,12 df) higher values for men (118g/min) than women (74 g/rnin). There were no other significant effects for the linear coefficient. The quadratic coefficient, whose absolute value is equivalent to half the rate of deceleration (by mathematical derivative of the quadratic function) was significantly (F = 6·28, p < 0'05, 1,11 df) higher in absolute value in men (- 3·13 g/min-) than in women (-1'22 gymm"). For it, too, there were no other significant effects than for sex. Analyses of variance and

263

FOOD IN T AKE IN MAN

summarized mea ns for these varia bles as well as for intake and duration are given in Tables 4 and 5. T he finding of higher rates of decelera tion (large r absolute valu es of the quadratic coefficients) in those ind ivid uals with higher initial rate s (linear coefficients), suggested that the two coefficients were correlated . If they were, the higher rates of deceleration in TABLE

4

M ean values of major dependent variables'

Sex

Res."

Intake (g) Screen Mean"

Clo sed Op en Closed Open

Male Male Fern. Fern.

Duration (min) Screen Mean

1004 946 486 596

697 530 390 375

10·28 8·10 8'5 8 8·15

Linea r (gjmin)

Quadratic

118·98 117-22 65·31 83-09

-2,8838 -3,7425 -0,6119 -1'8383d

10·60 12·77 8·85 10·45

(gjmin 2 )

a Values are based on four subjects per gro up, for 4 days each , following the screening meal, except as noted . Variability associa ted with all means except the screening is attributed to variation between subjects and within subjects. The best estim ates of these sources are the respective error terms (lines 4 and 11)of the analysis ofvariance (Table 5) and standard errors for the individual means or for contrasts can be computed from the error mean squares according to standard formulas (Winer, 1971). b Reservoir condition, open means the top was removed from the reser voir so that the subject cou ld see how much was being con sumed; closed means a top was on the reser voir. C The mean of4 days for four subjects . Screen refers to single screening meal and is a mean of 1 da y for four subjects . d Based on only three subjects. Because of unusual vari ability, one subject's values were dropped from consideration of this mean .

TABLE 5 Summarized analysis of variance

Intake Source

df"

MS

Sex (S) Res. (R) Sx R Error TrialjL b Error Trial T xS TxR T x R xS Error

1 1 1 12 1 12 3 3 3 3 36

301499 8 10327 11382 1 470929 201753 15109 77604 6727 24120 4723 14785

Duration

F

MS

66·2188 1 56-43761 1·29300 74·16959 13-55** 1-47153 3-88072 5'25** 1·37682 (}45 1·64682 0·05557 1-63 1·86432 0·32 3-03047 6'4* 0·02 0·24

Linear F

MS

0·89 30837-55 0·76 1027'28 0·02 1525·59 7250·38 0·38 707-42 493-88 (}45 744'13 0·54 110·93 0·02 1189'13 0·62 1353-0 1 55 1'77

Quad ra tic F 4·25 0-14 0·21

MS

64·39 134 16·05450 0'499 23 10'26150 1-43 3·24306 3-87784 1·35 4·6 1507 0·20 2·73752 2-16 8·58860 2-45 4·38083 3-71943

F 6-28* 1·56 0·05 0·84 1-24 0-74 2·31 1-18

. a Degrees of freedom for the between subject error term (first one) a re 11 for the qu adratic coefficient becau se one subject's data were elim inated owing to er raticness. Likewise, degrees of freedom for the linea r effect oftria ls is I I for the quadratic coeffi cient. Degrees offreedom for the last error term are 33 for the qu adratic coefficient. b j L indica tes the effect of linea r trend for trial s. *=p <0·05. ** = p < O·Ol.

264

H. R. KISSILEFF ET AL.

men than in women would be associated with the higher initial rates. Therefore, a separate regression analysis was undertaken. There were correlations between the quadratic and linear coefficients ranging from 0·523 to 0·996 (M =0·92±0·105 SD for men and O' 815 ± 0·184 SD for women). Two of these were significant for women and five for men. The corresponding slopes relating the quadratic coefficient to the linear ranged from 0·014 to 0'16min -1 (M =0'0772± 0·042 SD for men, 0·0697±0'044 SD for women). The correlation between the quadratic and linear coefficients, averaged from each subject, was significant (p = 0'017) in men (r =0'80, 6 df) but not in women (r=0·52). Figure 3 indicates that the regression lines and intercepts for the sexes were not significantly different (see legend of Figure 3).

a

•

-I

n ~

C1

N

c

:€

-2

1: Q)

-3

2 '0

~

-4

u

+=

e

-0 0

-5

~

a

-6

40

60

00

100

120

140

160

180

200

Linear caefficienl(g/min)

FIGURE 3. Relationship of quadratic coefficient of cumulative intake curve to linear coefficient: men closed (e); men open (0); women closed (.); women open (0). Slope of the regression line was -0,03203 ± 0·00978 SE for eight men and -0'02324±0'01707 SE for eight women. The intercept was 0·4700± 1·234SE for men and 0-4132± 1·203SE for women.

Intake and Duration

There were no significant differences in meal duration under any conditions. On the other hand, intakes were significantly higher in men (975 g) than women (541 g). The only other significant effects for intake were related to trials. There was a significant trials effect which can best be described (see Figure 4) as an increase from the first postscreening trial to the second (108 g). Further increases were only 18 and 35 g, on average, over the next two trials. A Neuman-Keuls test confirmed that intake in the first post-screening trial was less than intake in all other trials and that intakes in the other three trials were not significantly different from one another. In the subjects tested for eight meals after the screening, preliminary analyses, done before the data were pooled, also revealed no significant differences across trials among the last seven trials. Therefore, in our setting, after two trials, intakes are stabilized. There is, however, another large difference, that between the screening trial and the one that follows it (M = 166 g). It should be pointed out that variance of the above effects can be estimated from the error terms in Table 5, and, for this reason, SDs have not been included with the means.

265

FOOD INTAKE IN MAN 1200 1000

0. ~

E c:

800 600

400

/

/

/

/

/

/

",/'-------

----/

/

.

......--.-.

~.?.::r.- - 0 _ . " " ' /

__

.

200

Screen

2

3

4

5

6

7

8

Days

FIGURE 4. Mean daily intakes of all subjects: by sex and reservoir condition: men closed (--); men open (---); women closed (_.-); women open (---). Note the stabilization of intake after the second day.

Relation of Laboratory to Non-laboratory Intakes

Data from the food diaries revealed that the average amounts consumed by subjects at their non-laboratory lunches showed a significant (p=0'014) correlation (r=0'598) with amounts eaten in the laboratory. However, correlations within sexes were not significant. The overall correlation, therefore, reflected the fact that the sizes oflunches consumed outside the laboratory were significantly (p < 0'05) larger in men (850kcal ±547 SD) than in women (381 kcal± 112SD), just as they were in the laboratory. Furthermore, lunches eaten in the laboratory (888 kcal ± 387 SD for men and 492 kcal ± 143 SD for women) were larger, but not significantly so, than those eaten outside the laboratory. (Note that the S D's just given are for the means of 4 days/subject, between subjects). The slightly larger variability between subjects in the non-laboratory, compared with laboratory situation, was even greater for within subject variability (see "error" rows on Table 6). Variances were ten times higher for men and five times higher for women outside, than in, the laboratory. TABLE

6

Analysis of variance for intakes by location and sex

Source

df

Non-laboratory Mean square

Laboratory Mean square

Between S's Within S's Trials Error

7

Men 1197369·21429

601128·94588

3 21

67902·00000 197521·38095

45028-45283 16360'26815

7

Women 50563-26786

82089'58925

3 21

6604·79667 43303·07700

24806· 38415 8041·33098

Between S's Within S's Trials Error

Note: Values are all in kcal and so laboratory mean squares will differ from those calculated in Table 5.

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The size of the average non-laboratory breakfast was 357 kcal ± 221 SD in men and 292 kcal ± 96 SD for women (between subject SDs) which is considerably less than the 492 kcal fixed breakfast used as the pretest meal. It should be pointed out that the size of the average breakfast is based only on breakfasts actually consumed. One man skipped his non-laboratory breakfasts twice and another skipped on three out offour reported days. One woman skipped breakfast once, two skipped twice and a fourth woman skipped three breakfasts. If breakfasts not consumed are averaged as zeros, the sizes of the average breakfasts for men drops to 300 kcal ± 244 SD and for women to 212 kcal ± 129 SD. DISCUSSION

A quadratic model adequately accounts for 97 to 99% of the variance in the cumulative food intake curve of men and women eating a liquid diet through a straw as a single-course meal. Since curves with similar shapes have been seen in other eating situations (Pudel, 1971; Jordan et al. 1966; Adams, Ferguson, Stunkard, and Agras, 1978;Kissileff et at. 1980), it is likely that they too would fit such a model. The quadratic model could therefore be generally useful for studies of single-course meals or for single courses within meals in a variety of situations. Such modelling can serve both descriptive and theoretical purposes. As a description of the cumulative intake curve, the quadratic model reduces the amount consumed at several points in time to a few parameters. Curves from different individuals or individuals under different treatment conditions can, therefore, be readily quantified for comparison. Although other models could be used for summarizing cumulative intake data (Davis & Levine, 1977; McCleery, 1977), there are two advantages to application of the quadratic. First its parameters are easily determined by polynomial regression methods. Non-linearizable models such as those used by Davis and Levine (1977)and by McCleery (1977) require prior initializing, and algorithms for the coefficients sometimes fail to converge on a solution (Draper & Smith, 1981). Therefore, although these models may have appeal because of their relationship to feedback theory, they do not provide significantly better fit to the data and are more difficult to apply. Second, the parameters of the quadratic model ha ve a readily identifiable physical interpretation. The constant term, or intercept, represents the initial mouthful; the linear term is the initial rate ofeating; and the quadratic term is half the rate of deceleration. On the theoretical side, the quadratic model offers a simpler interpretation of control of intake than the non-linear models do. Theoretically the intercept should be zero, since intake begins at zero time. However, because it is possible that an initial acceleration could occur, the curve could have an intercept below zero. A significant positive intercept could also arise as a result of a delay in activation of the DEM. Since its sensitivity is ± 4 g, at least 8 g of initial intake would be needed to indicate that the meal had started. The first point recorded at 3 sec could be as high as 10g if the subject was sucking rapidly. However, for theoretical purposes, the positive intercept is less interesting than the negative, and since on the average, the intercept was close to zero, we will not consider it further. The initial rate of eating and rate of deceleration, on the other hand suggest that two separate processes could be at work controlling the cumulative intake curve. Two process theories, as opposed to one or multiprocess theories, have dominated discussions of intake control for decades, but the labels applied to these processes (e.g.,

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hunger/satiety, excitatory/inhibitory) have differed among investigators. For example Bousefield (1935) noted that cumulative intake curves in animals tended to be more linear and had lower asymptotes after 24 h of deprivation than after a 6-h deprivation, and he suggested that their physical capacity (which he equated with the asymptote) was reduced. On the other hand, the rate of approach to the asymptote (which he equated with voracity) was increased. Davis and Levine (1977) offer an alternative interpretation of the exponential model (model 9, Table 1). They suggest that the constant term which describes the asymptote (a) is the quotient of palatability divided by accumulation in the gut, and the exponential term (b) is the product of a "drinking mechanism" (ratio of rate of ingestion to an error signal which is the sum of palatability and gut contents accumulation) and time. Clearly in both formulations the equations are fundamentally two- or three-parameter functions, although in the Davis and Levine (1977)model, the initial rate of eating and asymptote are specified as products of several factors. Our approach (see also Kissileff & Thornton, 1982) is somewhat more empirical. We begin by describing the cumulative intake curve as a quadratic. We then ask what the parameters reflect. We dismiss the intercept as uninteresting and move on to determine the possible implication of the initial rate and rate of deceleration. We suggest that the initial rate reflects the magnitude of an excitatory neural process which initiates a meal and its cognitive correlate, hunger. If this fact is true, then experimental manipulations which stimulate eating should enhance the initial rate of eating. Two such manipulations are food deprivation (up to a point) and positively rewarding sensory qualities ofthe diet. The other process, reflected in the rate of deceleration, may reflect the magnitude of neural inhibition and its cognitive correlate, satiety, which brings the meal to an end. The quadratic model suggests that this process grows as a linear function of time. The finding that perceived satiety increases as a linear function of time (Teghtsoonian, Becker, and Edelman, 1981) supports this hypothesis. If only two processes were operative, intake would stop when the magnitudes of facilitation and inhibition balanced. However, examination ofthe curves plotted in Figure 2 shows that meals actually end abruptly. That is, meals stop several minutes before the predicted time of termination. The termination time can be estimated from the ratio of the linear coefficient to twice the quadratic. The mean values of these coefficients are shown in Figure 2. It was found that the theoretical termination times were much closer to the actual durations for men than for women, although the differences were not significant. The theoretical duration for men was a mean of7'87min±4'10SD longer than the actual duration and for women it was a mean of 13-98 min ± 11·24SD longer. There are at least three possible reasons that meals terminate sooner than the theoretically predicted duration, aside from the obvious possibility that the theory is completely wrong. First, sensory specific satiety (Rolls, Rolls, and Rowe, 1979; Rolls, Rowe, and Rolls, 1982; Le Magnen, 1967)may occur. That is, true inhibition of eating has not been achieved, but only a relative inhibition, specificto the food being ingested. It is conceivable that if another food was offered, additional intake, albeit at a slower rate, would occur. However, the amount and duration of eating would be predicted to be in accordance with the curve (corrected for palatability and diet consistency of the second course) projected from the first course curve. It seems unlikely that such an explanation is operating in the present situation because subjects indicated in response to one of the post-meal rating questions that they would eat only 11-6% (r=o-20) of their favorite lunch, ifthay had time. Second, subjects may have learned to anticipate discomfort that would eventually occur if they continued to eat. Utilizing internal cues,

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they may have learned to stop when they experienced a certain feelingof fullness which they have learned serves as a precursor of that source of inhibition. Third, cognitive factors, such as concern about eating, or thoughts about other obligations after the meal could induce the subject to terminate sooner than physiologically necessary. The slightly larger differences between actual and theoretical durations in women than in men would therefore be in accord with the finding of higher restraint scores (increased concern about eating) in women than in men, although within sexes, the correlation between restraint score and difference between theoretical and actual durations was not significant. If the quadratic coefficient reflects the growth of inhibitory processes, any experimental manipulation which accelerated the onset of such processes would result in a more rapid rate of deceleration. Three possible manipulations could serve such a function: (1) simultaneous oral ingestion and gastric loading; (2) change in dietary composition such that inhibitory factors would be activated more rapidly; (3) administration of agents that would accelerate the inhibitory process. Obviously testing of such hypotheses requires further guesses about the nature of the inhibitory factors. However, it should be clear that the use of the quadratic model in the examination ofsingle meals could be fruitful in uncovering and relating the multiplicity of factors which govern intake in people. It should also be kept in mind that measurement of changes in the quadratic coefficient must be assessed with respect to changes in the linear coefficient because of the significant relationship we have found between the two. In the previous work of Meyer and Pudel (1972; cf. their Figure 1),it is also apparent that the subject with the highest initial rate decelerated most quickly. Two other aspects of these results deserve further comment: the failure of reservoir visibility to influence intake and the sex differences. The present negative results on reservoir visibility contrast sharply with results of others (Levitz, 1975; Pudel and Oetting, 1977). Both of these studies used a liquid diet served from a long slender cylinder. Levitz reported that normal weight subjects decreased intake from 1049kcal, when they could not see the reservoir, to 826kcal, when they could see the reservoir as they consumed. Pudel and Oetting (1977)showed that manipulation of visual feedback about the amount in the reservoir could exert control over intake. A reasonable interpretation of the present experiment is that the shape of the present reservoir (see Methods) reduced cues about the amount consumed to such a large extent that they no longer had a significant influence. Therefore, the visibility of the reservoir, an external factor, is not important in the control of intake, but the subject's perception of the reservoir, a cognitive factor, is. A parallel finding is that actual calories are not important in the control of intake, but perceived calories are (Wooley, 1972). The critical variable in reservoir visibility is probably the rate of change of fluid in the reservoir. In the present situation and in a similar situation, in which a two-quart open bowl was used (Kissileff et al.; 1980),the slow change in level of the reservoir probably minimizes the influence of visual cues on intake. The second important finding is the difference between sexes in the initial rate of eating and amount consumed. Although differences in intake should be expected from energy expenditure data, they have not been reported for single meals (cf. Warner & Balagura, 1975; Wing, Carrol & Jeffery, 1978; Moon, 1979). The initial rate of eating (linear coefficient ofthe cumulative intake curve) was also higher in men than in women in our previous study (Kissileff & Thornton, 1982). The difference in initial rate of eating between sexescould be due to physiological or personality differences. The latter

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would accord with suggestions (Witherly, Pangborn, & Stern, 1980)that average rate of eating is related to personality. Caution must therefore be exerted in drawing any conclusions from studies in which both sexes are used unless groups are balanced for sex. Now that we have seen that the rate of deceleration is correlated with the initial rate of eating, and that the initial rate of eating is lower in women than men, it is important to reassess the interpretation placed on the lack of deceleration by obese subjects reported by Meyer and Pudel (1972). Pudel and Oetting (1977) point out that cumulative intake curves also become more linear in subjects who are given preloads (Walike, Jordan and Stellar, 1969).It seems unlikely that individuals whose cumulative intake curves fail to decelerate, do not experience satiety as much as those whose curves do exhibit deceleration, as proposed by Meyer and Pudel (1972) for obese subjects. It seems more likely that deceleration is an indication of a rapid decline in the desire to eat, which can only be seen when that desire is initially high as it would be in a deprived individual or one given a highly palatable diet (cf. Kissileff & Thornton, 1982). Deceleration may be more of an index of decline in excitability than increase in inhibition of eating. However, we must remain flexible in our interpretation of the deceleration until we have more information about the factors which influence it. In spite of our ignorance of the proper interpretation for the deceleration, its application to clinical problems could be helpful. The study of cumulative intake curves could potentially provide a clinical test for the magnitudes of hunger and rate of satiation in individuals with eating disorders, and hence enable possible classification and perhaps etiology of obese individuals suspected of having an eating problem. Ifwe could establish that the coefficients of the cumulative intake curve reflected internal processes that control food intake, we could then determine whether these processes differed between non-obese and obese individuals. Furthermore, by employing certain psychological and physiological correlates that could be related to the control of eating, we should be able to develop an eating control profile that could be diagnostic. Such profiles could be useful in developing differential treatment of individuals having eating disturbances characterized by polarized etiologies that have been suggested for classifying locus of eating control, e.g., push-pull (Van Itallie and Campbell, 1972), internal-external (Schachter, 1971) or genetic-environmental (Stricker, 1978). Finally, the appearance of sex differences in intake both in laboratory and nonlaboratory lunches provides additional validation that the laboratory lunch is indicative of normal eating behavior. It is perhaps too much to expect that there would have been also a correlation in intakes between laboratory and non-laboratory lunches, since in the non-laboratory situation there was a tremendous variety in nutritive content, sensory quality, and circumstances of eating. This variability is presumably the basis of the greater variability in non-laboratory intake. It is interesting to note, however, that the intakes in the laboratory and outside it were remarkably close (within 30 kcal for men and 90 kcal for women). This result suggests that in spite of the variable conditions, the amount consumed may be mainly determined by physiological factors, since, presumably, these would be operating more strongly in the laboratory situation, where much of the nutritional and environmental variability has been stripped away. In summary, this study, based on a small group of intensively studied subjects, introduces several new concepts and correlations, and it is hoped that it will provide a sound methodology for further studies of the control of food intake in man.

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Adams, N ., Ferguson, J., Stunkard, A. J., & Agras, S.The eating behav ior of obese and nonobese women . Behavioral Research and Therapy, 1978,16,225-232. Barr, A. J., Goodnight, J. 1-1 ., & Service, J. W. A User's guide to S AS-76. Raleigh, NC: SAS Institute, 1976. Booth, D . A. Conditioned satiety in the rat. Journal of Comparati ve and Physiological Psychology, 1972, 81,457--471. Bousefield, W. H . Qu antit at ive indices of the effects of fasting on eating beha vior. J ournal of Genetic Psychology, 1935,46,476--479 . Bray, G. A. Obesity in Perspective. Vol. 2. Washin gton , D .C.: National Inst itutes of Health , 1975. Davis, J. D ., & Levine, M. W . A model for the control of ingestion . Psychological Reviews, 1977, 84, 379--412. Dixon, W. J., & Brown, M. B. BMDP-79 Biomedical Comput er Programs P Series. Berkeley,CA: University of California Press, 1979. Draper, N., & Smith, H. Applied Regression Analysis, 2nd Ed . New York: Wiley, 1981. Herman, C. P., & Mack, D . Restrained and unrestrained eating . Journal ofPersonality, 1975,43, 647-660. Jordan, H. A., Wieland, W. F ., Zebley, S. P., Stellar, E., & Stunk ard , A. J. Direct measurement of food intake in man : A method for objective study of eating behavior. Psychosomat ic Medicine, 1966, 28, 836--842. Kissileff, H . R., Klingsberg , G ., & Van Hallie, T. B. A universal eating monitor for continuou s recording of solid or liquid consumption in man . American J ournal ofPhysiology, 1980,238, RI4--R22. Kissileff, H . R., & Thornton , J. Facilitation and inhibition in the cumulati ve food intake curve in man. In A. J. Morri son & P. Strick (Eds.), Chanq inq Con cept s of the N ervous System . Pp . 585-60 7. New York : Academic Press , 1982. Kissileff, H. R., Pi-Sun yer, F. X., Thornton, J., & Smith, G . P. C-terminal octapeptide of cholecystokinin decreases food int ake in man. Th e Am erican J ournal of Clinical N utrition, 1981,34,154--160. Le Magnen , J. Habits and food intake . In C. F . Code (Ed .), Ham/book of Physiology, Section 6, The Alimentary Canal, Vol. 1, Food and Water Intake . Pp. 11-30. Washington , D .C .: American Ph ysiological Society, 1967. Levitz, L. The susceptibility of human feeding beha vior to external controls. In G. Bray (Ed.), Obesity in Perspective . Pp . 53-60. Washington, DC : Nat ional Institutes of Health , 1975. McCleery, R. H. On satiation curves. Animal Beha vior, 1977, 25, 1005-1015. Meyer, J. E., & Pudel, V. Experimental studies on food intake in obese and normal weight subjects. Journal of Psychosomatic Research, 1972,16,305-308 . Moon, R. D. Monitoring of human eating patterns during the ingestion of non-liquid foods. International Journal of Obesity, 1979, 3, 281-288. Ostle, B., & Mensing, R. W. Statistics in Research 3rd ed . Ames, Iowa: Iowa State Univer sity Press, 1975. Peryam, D. R., & Pilgrim , F . J. Hedonic scale method of measuring food preferences. Food Technology, 1957,11(9), Suppl. 9-14. Pudel, V. Food-Dispenser. Eine Methode zur Untersuchung des spontanen appetitverhalten s. Z eits chrift fur Ernahrunqswissenschaft, 1971 , 10, 382-393 . Pudel, V. Experimental feeding in man . In T. Silverstone (Ed .], App etit e lind Food Inta ke. Pp. 264--275 . Berlin: Abakon , p. 76. Pudel, V. E., & Oetting, M. Eating in the laboratory: behavioral aspects of positive energ y balance. International J ournal of Obesity, 1977,1 ,369-386. Ralston, A. A Fir st Course in N umerical Analysis. New York : McGraw Hill, 1965. Rolls, B.J .,Rolls, E.T., & Rowe, E. A. Sensory specificappetite and satiety . International J ournal of Obesit y, 1979, 3, 397- 398. Rolls, B. J ., Rowe, E. A., & Rolls, E. How flavor and appearance affect human feeding. Proceedinq s of the Nutrit ion Society, 1982,4/ ,109-117. Schachter, S. Some extraordin ar y fact s about obese humans and rat s. American Psycholoqist , 1971 , 26,.129-144.

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Skinner , B. F . D rive and reflex strength. J ournal of General Psychology, 1932, 6, 22-47 . Stellar, E., & Hill, J. H. The rat s rate of drinking as a function of water deprivation . Journal of Comparative and Physiological Psychology, 1952, 45, 96-102. Stricker , E. M . H yperphagia . New England Journal of M edicine, 1978, 298, 1010-1013. Teghtsoonian, M ., Becker, E., & Edelman , B. A psychophysical analysis of perceived satiety: Its relat ion to consumm at or y behavior and degree of overweight. A ppetite, 1981, 2,217- 229. Van Hallie, T. B., & Ca mpbell, R. C. Multidisciplinary approach to the problem of obesity. Journal of the Am erican Dietetic Ass ociat ion, 1972, 61, 385-390. Walike; B. C , Jordan, H. A., & Stellar, E. Preloading and the regulation of food intake in man . J ournal of Comparative and Physiological Psychology, 1969, 68,327-333 . Warner, K . E., & Balagura, S. Intrameal eat ing of obese and non obese humans. J ournal of Comparat ive and Physiological Psychology, 1975, 89,778-783. Wing, R. R., Carroll, C , & Jeffrey, R. W. Repeated observation of obese and normal subjects eating in a natural environment. Addictive Behavior, 1978,3, 191-196. Winer, 8. J. Statistical Principles in Experimental Design, 2nd ed . New York: McGraw Hill, 1971. Witherly, S. A., Pangborn, R. M., & Stern, J. S. Gustatory respon ses and eating duration of obese and lean adults. Appet ite, 1980, 1, 53-65. Wooley, S. C Physiologic versus cognitive factors in short term food regulation in the obese and nonobese . Psycho somatic M edicine, 1972,34, 62-68 .

ApPENDIX

Preliminary Tes ts and Instru ctions to the Subj ect s

Each potential subject came to the laboratory in the morning between 8 a .m . and 10.20a .m. in the fasting sta te (no food since retiring the previou s evening). The subject was given a written description of the stud y before signing the consent form. The description was modified onl y slightl y from pre viou sly publi shed instructions (Kissilelf et al. 1980). Its major point s were: (1) The purpose of the study was to obtain people 's reactions to various foods and food combinations we were de veloping to foster improved nutrition . (2) Becau se these reports vary with the time since one last ate, subjects were asked to eat a sta ndardized meal and not to eat again until they returned 3 h later. (3) The diet was described . (4) In order to assess the subjects' reaction s at a specific time after eat ing, they will be instructed to eat until they feel satisfied and to place their straw on a special holder provided. A device will record this act. (5)Subject s are asked to keep a record of their eating habits and ph ysical activities on the day preceding and day ofeach experimental session. (6) Any que stions about the study not answered at the screening session would be answered in a follow-up interview at the end ofthe study. (7)The study would last from four to eight non-consecutive days, on which two meals would be served each day. After the subject signed the consent form he/she was given que stionnaires to fill out, including the Herman and Mack (1975) "restraint" que stionnaire. The average restraint scores are included in Table 1. Next the subject was given a taste test (Kissilelf et al. 1981) in which the following eight items were sampled at 1min intervals and the subject rated how much he/she liked or disliked each on a 9-point scale (Per yam and Pilgrim, 1957): 6'1% sucrose, distilled water, cinnam on flavored Breakfast Square s (General Mills), apple juice, liquified yogur t diet (the test meal food), Ensure (Ross Laboratories), milk, and Instant Breakfast (Carn ation). Solid s were given in 3 g and liquid s in 5 ml portions. The first two item s were given in order, but the last six were arranged in Latin square order, randomized for every six subjects . The prete st meal (see "Experimental Setti ng and Test Food s") was given next while the subject filled out a questionn aire recording the preference and frequency of eat ing some typical lunch eon

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items. Subjects were reminded not to eat again and to return again at a time calculated to be 175 min after they had started the pretest meal. The test meal was served 180min after the pretest meal had started. Just before the test meal was served, the subject listened to a set of tape recorded instructions which was essentially the same as those used in a previous study (Kissileff et al. 1980), except that "container" was inserted instead of "bowl" and the subject was told to place a straw instead of a spoon on the specialized holder indicating that he/she was finished eating. Rationale for Selection of the Test Meal

The test meal was selected because it could be served in solid chewable form as well as liquified. The present results could then be compared with results using other means of serving (e.g., spoon and bowl) and other diet consistencies (cf. Kissileff et al. 1980; Kissileff & Thornton, 1982). The diet was served in liquified form through a straw because smoother curves which would permit more accurate modelling would be obtained, than would be obtained when subjects ate from a bowl with a spoon, owing to the increased weight placed on the UEM when the spoon was placed in it. In addition, the effect of reservoir visibility could be observed by contrasting meals eaten with, or without a cover on the opaque reservoir. Data Collection and Analysis

The computer controlling the UEM was programmed to indicate meal termination to the experimenter when the weight on the table was approximately constant (i.e., within normal oscillation ofthe balance under non-eating conditions, usually ± 4 g) for 15 min. Because of occasional instability of output, owing possibly to air currents, or electrical noise, which could cause 5 g fluctuations that could be interpreted by the computer as continuation of eating, strings of uniform points at the end of a meal were trimmed to no more than four (i.e., 1 min of no eating) if the computer had not truncated these automatically. Finally large spikes resulting from lifting of the straw, pulling upward on the viscous diet remaining at the end of the meal, were discarded. These were detected as steps in excess of 15g at the last computer recorded point when the UEM indicated meal termination after 15 min of no eating. Received 28 July, 1981; revision 2 March, 1982