MAGNITUDE ESTIMATION: NOTES ON WHAT, HOW, WHEN, AND WHY TO USE IT

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1 MAGNITUDE ESTIMATION: NOTES ON WHAT, HOW, WHEN, AND WHY TO USE IT HOWARD R. MOSKOWITZ MPI Sensory Testing 770 Lexington Ave. New York, New York Received for Publication Dec. 28, 1977 INTRODUCTION Magnitude estimation refers to a class of psychophysical scaling procedures which were developed by S. S. Stevens at Harvard University in the 1950 s. In its simplest form, the method allows panelists to assign numbers to sensory/attitudinal stimuli, without restriction, so that the ratios of the numerical assignments reflect ratios of sensory perceptions or of attitudinal levels. Using the magnitude estimation method, an experimenter who finds that two products are assigned ratings of 30 and 10 for sweetness (by magnitude estimation) can conclude and state that they are perceived to lie in a 3/1 sensory ratio. In the very earliest studies, Stevens (1953) instructed naive (unpracticed) panelists to assign numbers to the brightness of lights of different luminances and to the loudness of sounds of different energy levels. A range of luminance levels and a range of energy levels for sound yielded a corresponding set of numerical estimates. The data showed that: (1) A repeatable function emerged from this exercise. (2) The function could be approximated quite well by a power function of the form: Magnitude Estimates = k( Luminance or Sound Level, respectively)n. Another way of writing this is the logarithmic form: Log Magnitude Estimate = log k 4- N (log Luminance or log Sound energy level, respectively). (3) The exponent, N, was not affected by the size of the magnitude estimates, but rather by the relative ratios of the magnitude estimates, compared to ratios of luminances or sound pressure levels. (4) The data appeared to possess ratio-scale properties. In the subsequent quarter century of research, numerous experimenters have argued about the reproducibility of the power function, whether or not the data obey ratio-scale properties, etc. Despite the arguments, however, continuing research by investigators points to the magnitude estimation method as yielding (a) reproducible scale values (b) which appear to have ratio scale properties. This paper concerns some of the more practical nuts and bolts in- Journal of Food Quality 3 (1977) All Rights 1978 by Food & Nutrition Press, Inc., Westport, Connecticut 195

2 196 HOWARD R. MOSKOWITZ formation that an experimenter needs to conduct a magnitude estimation study. The approach represents a compromise between the requirements of strict laboratory control, wherein the magnitude estimation method has found its.most important proponents, and the exigencies and requirements of commercial testing, wherein lies its greatest future promise. The paper is divided into the following sections: (1) Panel selection for the study (2) Panel instruction for the study (3) Appropriate analysis methods (4) Conclusions which can be made Biases which are to be expected, and how they can be avoided (6) A typical worked example, from a study (Appendix) PANEL SELECTION Historically, magnitude estimation was founded upon the data generated by inexperienced, or naive panelists. Stevens study, and most psychophysical studies since then have avoided using practiced panelists, and have almost exclusively used naive individuals. Where practice has played a role, it has been in the introduction of the method. A typical consumer in some tests may participate in a magnitude estimation study lasting several days, but rarely is any training involved other than the initial instruction. Any improvement, therefore, in panelist performance is due to experience with the method, but not due to training on what numbers on the scale to assign. Number of Panelists In the scientific literature typically small panels are used. Typical numbers range from approximately 8 to 30 or 40. These numbers suffice to yield reproducible psychophysical functions relating sensory intensity to physical intensity; cf. Stevens (1975) for a review of experiments. For consumer evaluation tests, as few as 15 respondents may suffice for an accurate assessment of sensory magnitude. More respondents must be used if the aim of the experiment is to obtain affective, or hedonic information, in which there are substantial interindividual differences. Panelist Orientation Most individuals will not have had any previous exposure to mag-

3 MAGNITUDE ESTIMATION 197 nitude estimation. Over the years experimenters have devised training or orientation procedures to illustrate to panelists (a) what is the concept of magnitude (b) how to assign numbers to match sensory magnitude. Panelist orientation has been divided into the following major approaches. (a) No instructions at all, save a statement to the panelist to make the numbers he/she assigns proportional to, or match, the sensory intensity of the stimulus. This method allows the panelist the maximum amount of flexibility in magnitude estimation and makes no presupposition about whether ratios of magnitude estimates reflect ratios of sensory intensities. Stevens (1975) reported that this simple method produced reliable results which were quite similar to those magnitude estimates obtained under more rigid instructions. (b) Panelists can be instructed to match numbers to stimuli so that the ratios of their numbers reflect ratios of sensory intensity. Here the emphasis is upon the ratio-scale properties of magnitude estimates (see Appendix I). In addition to orientation instructions, panelists may be either led through an exercise, or simply started on the magnitude estimation task without any additional orientation. From various studies it appears best to spend a little time orienting the respondents in a rather rigorous, stepwise method described below. This orientation (a) ensures more reliable data (b) screens out respondents who do not understand the concept of matching numbers to stimuli to indicate magnitude. (These respondents can be further instructed afterwards if the experimenter wishes to keep their data and maintain their participation in the panel.) A Worked Orientation Method For inexperienced panelists, the approach below has been successful in orientating them rather quickly (i.e., in about min). Appendix I1 provides the actual questionnaire that the panelist fills out and a copy of the forms. Briefly, the orientation proceeds as follows: (a) A test interviewer reads out what the consumer is to do in terms of assigning numbers to match sensations. (b) The interviewer provides the panelist with a set of cards, on which are forms of different types and of different measured areas. The forms comprise squares, circles and triangles. Eighteen forms, covering six levels each of every shape, usually are quite sufficient. (c) The panelist assigns numbers to reflect perceived area, one number for each shape. The shapes are presented in a different randomized order to every panelist.

4 198 HOWARD R. MOSKOWITZ (d) After the 18 figures have been rated (requiring around 4-5 minutes) the panelists are ready to learn how to use magnitude estimation to evaluate hedonic tone (liking/disliking). (e) The experimenter presents to the respondent a word, and asks for an initial estimate of whether or not the respondent likes the word or concept (in which case the respondent fills in the letter L) or dislikes the word or concept (in which case the respondent fills in the letter D). This initial rating is a classification. Then, the experimenter asks the respondent to estimate the degree of liking or disliking, just as if the scale were unipolar (Appendix 111). Sample Presentation THE ACTUAL PROCEDURE The most typical method for presenting samples is to provide a totally randomized design (whether a complete block design or a balanced incomplete block design). That is, each panelist begins with his own unique first sample. (a) The first sample is called the standard. (b) The first number assigned is called the modulus. The following reasons have been suggested for randomizing the order of presentation of samples, as well as randomizing the first sample. (a) Psychophysical functions for sensory perception show particular biases if the first sample is either a very high level or a very low level. (b) There is a round-number tendency. Panelists tend to prefer using round numbers (1, 2, 4, 5, 10, 20, 50, 100). If the first sample is the same for all panelists, then there will be an increased chance for the round-number tendency to occur. As a result, many of the magnitude estimates for samples will artificially fall on the same round numbers (i.e., if most panelists begin with the estimate 10, and they do this for the same sample, then there will be a bias in the ratings as a result). Attribute Evaluation There are at least two different types of attributes which magnitude estimation can quantify. These are unipolar attributes and bipolar attributes. They are quite different. Unipolar attributes (e.g., sweetness, loudness) possess a fixed 0 reflecting none or no sensation at all. Numbers which increase imply increasing levels of the attribute away from 0. A ratio of magnitude estimates, e.g., 40 and 5 is meaningful. Both numbers lie on the same scale, and can be compared to each other directly. In contrast, bipolar scales (e.g., liking/disliking) possess an

5 MAGNITUDE ESTIMATION 199 intermediate 0 reflecting neither one aspect nor another. Numbers on opposite sides away from the center (e.g., 0) show increasing levels of the attribute. A direct ratio of magnitude estimates is meaningful if the numbers lie on the same side of the scale (i.e., in a liking/disiiking bipolar scale, a and a +20 mean that one is liked five times more than the other). Numbers on opposite sides of the bipolar scale cannot be as easily compared. Is a +500 the same as a -500 on the scale? Does +500 reflect a stimulus which is twice as liked as a stimulus rated -250 is disliked? For practical purposes, one can treat the bipolar scales either as: (a) Equal and opposite, in which case a +500, and a -500 are equal, and algebraically opposite. Adding them together would produce a 0. (b) Belonging to two different unipolar scales (i.e., a separate scale for liking and for disliking, respectively) and thus treating them as entirely different attributes. One can also change a bipolar scale into a unipolar scale. As a consequence, disliking would be 0, or the complete absence of liking. Increasing levels would reflect increasing levels of the attribute (i.e., increasing levels of liking). The scale would thus be unipolar, with one of the poles being 0. One cannot, however, add a constant to the numbers in a bipolar scale to make all of them positive. Ratios are destroyed in that case, although differences are not. The First Stimulus (Standard Stimulus) Since the magnitude estimation scale is unfettered in terms of scale points and aids for the panelist, the selection of the first stimulus is important. Several different methods for selecting the first sample have been suggested. (a) First sample randomly selected. The most accepted method for selecting the first sample is to randomize it, across the panel of respondents, so that each panelist is presented with a different first sample. This way, each panelist has a different starting stimulus. That should not affect the ratios of magnitude estimates for numbers assigned by one individual, but it will affect the actual numbers panelists use. If the panelist begins with the weakest stimulus and assigns that a 10, then all of the other stimuli will be assigned numbers 10 or above. On the other hand, if the panelist is presented with the strongest stimulus and calls that 10, then the remaining magnitude estimates will be 10 or less. (b) The first stimulus lies in the middle of the range and is fixed. This alternative is preferred if the panelists are naive, and if the range of stimulus variation is so great that there would be severe problems when

6 200 HOWARD R. MOSKOWITZ a panelist begins with an extremely high or an extremely low standard stimulus. The First Number (Modulus) The numbers that panelists choose can be fixed on the first presentation when the standard stimulus is presented (fixed modulus) or the number can be free to vary, according to the respondent s desire (free modulus). With a fixed modulus, the panelist may be instructed to call the first stimulus 10 and then to relate all other stimuli to that first number. With a free modulus, the panelist is allowed to assign to the first stimulus any number within certain allowable ranges. (For instance, if the panelist is judging area and all stimuli possess some finite area, then the first number can be any non-zero, positive number.) The fixed modulus method, like the fixed standard method, often leads to a characteristic clustering and over-representation of certain numbers (the round-number tendency). In addition, by forcing the panelist to use a specified first number, the experimenter often makes the panelist compromise between the numbers he/she would like to use, and the scale unit which the experimenter designates. Over the course of such magnitude estimates, the resulting distribution of numbers may exhibit a compromise distribution between what would be gotten were the respondents to use their own scales, and the numbers that would be expected were the panelists to adhere strictly to the scale unit designated by the experimenter. Randomizations Versus Fixed Order of Stimuli Researchers (e.g., Cross 1973) have found that there are sequential dependencies in magnitude estimation, just as there are in any other type of scaling. If there is only one fixed sequence of stimuli, then some part of the ratings will be ascribable to the sensory variations and respondent perceptions, while the other part will be biased, owing to contrast and convergence effects. In order to reduce the sequential dependencies, and to prevent undue influence by one stimulus on the ratings for another, the usual remedies are to: (a) Totally randomize the order of the stimuli; (b) Collect more than 1 replicate rating (i.e., present the same stimulus to the panelist on several occasions, or several times in the same session). How to Instruct The Panelist There are several variations in how to instruct the panelist who is to

7 MAGNITUDE ESTIMATION 201 test stimuli. The instructions must address the following: (a) What is the range of numbers to use. Can 0 and negative numbers even be used and if so, when? (b) What is the best strategy to use. Should the panelist try to remember the first stimulus and compare all succeeding stimuli to that first stimulus? Should the panelist compare each stimulus with the preceding stimulus? Or should the panelist begin to generate his/her own scale, and then simply match numbers to sensory intensity, without worrying about comparing each estimate to the previous one, or to the first one? There are no fixed answers to these, except to note the following: (1) For unidimensional scaling (e.g., amount of something, such as sweetness) one should specifically instruct the panelist ahead of time that positive numbers, only, are to be used, and that 0 is to reflect the absence of the particular attribute (e.g., in the evaluation of sweetness, only positive numbers can be used, and 0 reflects no perceptible sweetness at all). (2) For bipolar scaling, + numbers reflect one aspect of the polar scale (e.g., liking), whereas - numbers reflect the opposite aspect, and 0 reflects intermediacy (neither + nor -). (3) Occasionally panelists doing magnitude estimation scaling for the first time may be nervous, and may ask the experimenter for a recommended scaling strategy. It is best at that time to tell them to freely match numbers, and to indicate that as the panelist progresses along during the study, assigning numbers to stimuli, the scale will develop. Alternatively, by telling the panelist specifically to concentrate on either the first stimulus or the previous stimulus, the experimenter can allay the fear. Note, however, that during the actual experimental run the panelist will soon cease referring to the previous stimulus, or to the first stimulus, despite the experimenter s specific instructions to the contrary. Other Scaling Methods Besides Numbers Magnitude estimation typifies one specific aspect of a more general approach to ratio scaling known as cross-modality matching (see Appendix IV). The underlying principle behind all cross-modality matches (magnitude estimation included), is that the panelist adjusts the physical intensities of a specific continuum so that the resultant sensory intensities match the sensory (or attitudinal) intensities of a criterion modality. Panelists can adjust the physical level of noise so that loudness matches the desired criterion (e.g., match noise to equal taste or smell intensity), or match handgrip to smell intensity, etc. Appendix IV provides a set of references for such cross-modality evaluations, which show the range of different continua which the subject

8 202 HOWARD R. MOSKOWITZ has under his/her control, and whose intensities can be varied. The advantages of cross-modality matching over magnitude esthnation (one variant of cross-modal matching) are: (1) The experimenter does not have to worry about the round number tendency, wherein panelists tend to cluster their ratings around a few favorite numbers, such as 5, 10,25, 100, etc. Since the panelist adjusts the non-numerical stimuli, there is unlikely to be any easily remembered anchor points, which distort the data. (2) The experimenter does not have to worry about the panelist s concept of numbers independent of magnitude. Rather, the only concept that must be brought out clearly is that the respondent is to match one stimulus to another. As a result, the experimenter can run children and illiterates, who have no conception of numbers, or of ratio values. Moskowitz et al. (1975) used line matching to assess taste intensity and taste hedonics, and Bond and Stevens (1969) instructed children to match lights and sounds. In both instances the results could be predicted by knowing the relation between lines to numbers and lights to numbers, but in each instance the experimenter avoided the biasing use of numbers. In all instances, however, the experimenter must cautiously avoid using a modified category scale, which is continuous, but which has end-points. If instead of numbers the panelist is told to draw a line to reflect sensory intensity, then the line should not be bounded. The panelist should feel free to draw lines as long as he/she wishes, without having to worry about fitting the lines into a small bounded range. DATA ANALYSIS OF MAGNITUDE ESTIMATES Magnitude estimation allows each panelist to assign numbers based upon his or her own frame of reference. As a result, individuals will differ on the sizes and ranges of the numbers which they assign. This section lays forth what is known, what is mathematically proper, and what is often implemented in the analysis of magnitude estimates. Normalization Normalization refers to methods which are used to bring into common the scales that different individuals use. A variety of normalization methods exists. The following list and precis of each method will illustrate the most typical methods. No normalization. Here, the experimenter simply uses the ratings as

9 MAGNITUDE ESTIMATION 203 they come in, without attempting to bring into common the various scales. No normalization is commonly used when the experimenter presents a fixed standard (first stimulus), to which the panelists assign a fixed number (modulus). By fixing the standard and modulus, the experimenter has already normalized the individual s size of numbers. However, the experimenter has not fixed the range of panelist numbers. Some individuals may use a large range of numbers, whereas others may use a smaller range. Modulus equalization (Lane et al. 1961). The premise of modulus equalization is that (a) all panelists must evaluate the same set of stimuli, (b) the ratings for each panelist are each multiplied by a fixed, constant multiplier so that the geometric mean of those ratings is equal to the geometric mean of the group data. Modulus equalization was the original method used to normalize data, and was incorporated into one of the first computer programs [PSYCHOFIT: Panek and Stevens (1965)l designed specifically for the statistical analysis of magnitude estimation data. Modulus equalization, the method that Lane, Catania & Stevens proposed, requires the following prerequisites: (a) All panelists have evaluated all of the stimuli an equal number of times; (b) There have been no 0 s in the ratings (else there would be a geometric mean of 0). Modulus Normalization (Moskowitz 1970). The premise of modulus normalization is that in a series of studies encompassing different stimuli, the experimenter may wish to evaluate a whole host of different stimuli, far more than could be accomodated in one session, or with a small, fixed group of panelists. In order to guarantee (as much as possible) that the numbers across studies are commensurate, the experimenter selects a common stimulus, or set of stimuli, which is to be embedded in each experiment. These common stimuli are not identified to the panelist, who evaluates the full set. For analysis, the core set of common stimuli is extracted, and its geometric mean for a given panelist is made to equal a fixed constant (e.g., 10). To equalize the means across all panelists, each panelist must be assigned a unique multiplier, which multiplies all of his ratings. Then, the geometric mean of the fixed, small set of internal standards, will equal the fixed level. By doing so, the experimenter can maintain the size of numbers across all panelists approximately equal (and definitely equal for the set of normalizing stimuli). Examples of modulus normalization include the following: (a) Cain (1969) assigned a fixed number to one stimulus odorant in each odor scaling experiment. The odorant was acetone, and it served as a reference (unbeknownst to the panelist). (b) Moskowitz (1970, 1971a,b) suggested that the normalizing stimulus be

10 204 HOWARD R. MOSKOWITZ a set of 5 stimuli. In taste tests with sugar solutions, the normalizing set was M, 0.25 M, 0.5 M, 1.0 M and 2.0 M glucose, whose geometric mean sweetness rating was forced to equal 10. For sourness, it was selected concentrations of citric acid. Neither modulus normalization nor modulus equalization can guard against the possibility that contextual factors will influence the magnitude of the ratings. If, for instance, a set of 5 glucose (sweet tasting) solutions is used in a set of artificial sweeteners, all of which produce a lower sweetness perception than the 5 glucose blind references, then there may be a contrast effect, so that the glucose set is relatively uprated. In contrast, if the same 5 glucose solutions are used in the context of a generally sweet set of carbohydrate sweeteners (e.g., high concentrations of sucrose), then these 5 solutions may tend to be downrated. Despite these possibilities of contrast effects, the use of a set of varying concentrations, which are the basis for normalization, is definitely to be preferred to the use of one single concentration. With several concentrations, the experimenter can at least vary the range, and normalize the ratings on the basis of a wide variation in the core set of stimuli. One should guard against the temptation to standardize the magnitude estimates, for the following reasons: (1) In linear values, standardization will, by its nature, destroy the ratios among the magnitude estimates. Standardization comprises first the subtraction of the mean estimate from each magnitude estimate, and then the division of that difference by the standard deviation. All magnitude estimates are then expressed as deviates from the mean, e.g., new standardized rating = (OLD RATING - MEAN RATING) (STANDARD DEVIATION OF RATINGS) (2) In logarithmic coordinates, the standardization will dramatically change the range of magnitude estimates, and it wil change the ratios by a power transform. For instance, the standardization of the estimates implies the following in logarithmic values: [(log A - Mean log)] + (standard deviation) A = magnitude estimate. This is equivalent to: log A - log Mean - log (A/Mean) - 1 STD DEVIATION- STD DEVIATION STD DEVIATION X

11 MAGNITUDE ESTIMATION 205 In linear terms: [A/MEAN] ustd REPLACES A. External Calibration: Moskowitz and colleagues have recently suggested an alternative method by which one can coalesce the various scales used by different individuals (Moskowitz and Chandler 1977). The calibration comprises the following steps: (1) Present to the panelists the test stimuli, and have them go through the typical magnitude estimation sequence, assigning numbers to stimuli to reflect sensory intensities (1 or more stimuli). (2) Instruct the same panelists to estimate, using the numerical scale that they have just been using, what numbers correspond to extremely strong, very strong, moderately strong, strong, weak, very weak and not at all noticeable. (The last term is usually, by convention, assigned a magnitude estimate of 0.) It is critical for the panelists to assign to these concepts numbers as if the concepts themselves reflect sensory stimuli of different intensities. (Occasionally, panelists feel uneasy about the initial assignment, just as they do about the initial assignment of magnitude estimates, but after 1 or possibly 2 assignments, that uneasiness departs.) (3) The experimenter may feel free to select any number of different words in the calibrating scale. For instance, extremely strong, strong, weak and very weak would just do as well. What is necessary is one or more external anchoring attributes. Moskowitz et al. (1976) were able to calibrate hedonic responses using a line length scale, with a single external line drawn to reflect moderate. (4) The experimenter can obtain, for each calibrating scale, a pivot number in the following manner: (a) Add up the non-zero numbers corresponding to the words. (b) Average them, obtaining a separate average for each panelist, for each replicate, for each calibrating scale. (c) Divide that pivot number into each magnitude estimate, where appropriate (i.e., the sweetness pivot value is divided into the sweetness magnitude estimates). Each panelist has a unique pivot for each attribute. (d) Divide the pivot value back into the verbal scales as well (i.e., if the pivot = 20, then divide 20 into the other verbal descriptors, such as extremely sweet, etc.). (e) Multiply the results by a constant (e.g., 100) just to make the numbers larger, with fewer decimals (optional). The result of this normalization is a set of ratings which have the following properties: (a) The ratios of each panelist s ratings are maintained since the ratings were all previously multiplied by the same constant. (b) If the panelist used bipolar scales, then the signs on the scales are maintained. (c) The experimenter has associated with the ratings verbal descriptors which illustrate the description of each magnitude estimation level.

12 206 HOWARD R. MOSKOWITZ Appendix V illustrates a set of data before and after with this calibration. Averaging Data How to average magnitude estimates poses a difficult problem. One of the most important things about magnitude estimates is that they have presumed ratio-scale properties. One must find an averaging procedure which will maintain this ratio scale property. Another thing to keep in mind is that 0 magnitude estimates must be accounted for, and when bipolar scales are used, then negative magnitude estimates must be accounted for. Two decades ago, J.C. Stevens (Stevens 1957) suggested that the magnitude estimation data are distributed log-normally. A log-normal distribution appears skewed. However, the distribution is normal, if the logarithms of the magnitude estimates are the measures, instead of the magnitude estimates themselves. Traditionally, the log-normal distribution required the geometric mean as the most efficient average, with the median magnitude estimate as an appropriate, but less efficient measure of central tendency. In many studies, the geometric mean and the median are almost equivalent to each other. Traditional psychophysical studies have used the geometrical mean, or the median. In those experiments (cf. Stevens 1975; Marks 1974) the experimenter was sure to present to the panelist stimuli that were clearly perceptible, such as lines of different lengths, shades of different darknesses, shapes of different areas, etc. No 0 or negative estimates was permitted. Today, with the evaluation of complex systems, where 0 and negative judgments are permitted, the geometric mean is an infeasible measure of central tendency. The median, while being correct as a measure of central tendency (average) is inefficient, and calls for difficult-to-apply, and less efficient statistics. As a result, a return to the traditional arithmetic mean may be necessary, even if the arithmetic average is not the most appropriate mean for the distribution of magnitude estimates. Based upon a number of unpublished studies, using the arithmetic mean instead of the geometric mean, it appears that both means describe the data adequately. The geometric mean always provides a lower average estimate than the arithmetic mean. But more importantly, the arithmetic mean can account for 0 s and negative numbers, whereas the geometric mean cannot. In addition, the arithmetic mean provides numbers which are amenable to the wide range of statistics including analysis of variance and regression analysis.

13 MAGNITUDE ESTIMATION 207 TYPES OF RESULTS WHICH ARE OBTAINED The magnitude estimation method provides very versatile data, which can be used for a number of different types of conclusions. Among them are the following: Relative Intensities Magnitude estimates possess ratio-scale properties. As a result, if product X is assigned an estimate of 20, for example, and product Y is assigned an estimate of 50, then one may conclude that Y is 2.5 x as strong as X, on the specific attribute being scaled. Furthermore, if the scale is bipolar (i.e., as it can be for liking/disliking), then other conclusions are possible. For instance: If X is rated +30 and Y is rated +15, then Y is 50% as liked as X (or conversely, panelists like X twice as much as they like Y). If X is rated -30, and Y is rated +30, then panelists like Y and dislike X, and furthermore the intensity or degree of affect (liking/ disliking), independent of the type of affect, is equal. If X is rated as -30, and Y is rated as +15, then we can state that X is disliked, Y is liked, and the intensity of feeling for X is twice as strong as the intensity of feeling for Y (even though one is liked as the other is disliked). Among the published scale information for degree of sensory intensity, using single index numbers, are the following compilations: relative sourness of different organic acids, Moskowitz (1971b); relative sweetness of sugars, Moskowitz (1970,1971a). The reader should note that when using the magnitude estimation procedure to assess foods or fragrances, the experimenter does not have to know the constituents of the product being evaluated. As long as the panelist uses the magnitude estimation method correctly, the experimenter can still compute the ratio of magnitude estimates, and thus be able to conclude that stimulus X is so many times greater or less than stimulus Y. Dose-Response Relations Traditionally, magnitude estimation has been used to develop doseresponse relations between measured physical intensity and subjective sensory responses. A host of such functions has appeared during the past quarter century, and are summarized by Gescheider (1976), Marks (1974) and Stevens (1975). The major results thus far are as follows: For sensory intensity of simple continua, the magnitude estimates are related to physical intensities by curvilinear functions, which

14 208 HOWARD R. MOSKOWITZ become straight lines in log-log coordinates. As a result, it appears that over the middle range of physical intensities, the sensory intensity function is a power function, which can be written as: ME = k(pi)n or log ME = n (log PI) + log K (ME = magnitude estimate, PI = measured physical intensity, K = constant multiplier, n = exponent, or slope of the line in log-log coordinates.) If n = 1, then the magnitude estimates grow commensurately with physical intensity. A 1O:l increase in physical intensity is rated as a 1O:l increase in magnitude estimates. Line length, for example, is governed by an exponent of 1.0. If n is less than 1, then the magnitude estimates grow less rapidly than physical intensity. A 1 O:l increase in physical intensity is rated as less than a 1O:l increase in sensory intensity. Odor intensity, sourness, loudness and brightness, for example, are governed by exponents less than 1.0. If n exceeds 1.0, then the magnitude estimates grow more rapidly than physical intensity. A 1O:l increase in physical intensity is rated as more than a 1O:l increase in sensory intensity. The perceived pain of electric shock and the apparent heaviness of lifted weights are governed by exponents exceeding 1.0. For sensory intensity evaluated in regions around threshold, where sensory perception is an on-off affair, the function that magnitude estimation generates is no longer linear in log-log coordinates, but becomes steeper (Marks and Stevens 1968). In general, however, the region around threshold is usually better evaluated by standard threshold measures than by magnitude estimation. Appendix VI presents a list of power function exponents for various sensory continua which experimenters have scaled. Note that in each study, the experimenter used a series of stimuli graded in physical intensity. Other than that, and other than the fact that the averages derive from a series of different subjects, the methods used to obtain the functions differed from one experimenter to another. Re-Calibration of Other Scales In the foregoing section, one of the methods mentioned for normalization consisted of the use of a series of external anchor words to reflect verbal levels of sensory intensity (e.g., extremely sweet, very sweet etc.). These verbal descriptors provide an insight into the relation between the numerical scale and the verbal equivalents of magnitude.

15 MAGNITUDE ESTIMATION Appendix VII presents the results of a series of experiments in which the panelists assigned magnitude estimates to stimuli to reflect either sensory intensity or hedonics (liking/disliking). At the end of the session, the same panelists assigned numbers to reflect verbal levels of magnitude. Among the major results in Appendix VII are: (a) Verbal scales often are not equal interval scales. The magnitude estimation values corresponding to the different verbal categories do not correspond to equal differences or equal ratios. (b) The hedonic scale is not equal interval (for adjacent categories), nor are the verbal descriptors of pleasantness and unpleasantness equal and opposite. That is, likely extremely is not equal and opposite in sign to dislike extremely, etc, BIASES AND REMEDIES Despite the best attempts to produce an adequate sensory scale, and despite all precautions which an experimenter can take to insure that panelists are exposed to stimuli in the proper order, a number of recurring biases and problems emerge. The purpose of this final section is to consider some of the biases and to recommend some remedies. Bias 1: The Round-Number Tendency If panelists are instructed to match numbers to stimuli, then they tend not to use the entire continuum of available numbers. There is, instead, a tendency to use a few favorite numbers. As a result, a frequency distribution of magnitude estimates will exhibit several peaks in terms of numbers. The round numbers, 1, 5, 10, 20, 50 and 100 come up more often than the other numbers. For instance, Appendix VIII shows a distribution of numerical assignments from two panels of 60 individuals each (children and adults) who judged 30 products for sweetness on three occasions. The round-number tendency was described by Stevens (1975) who recommended that the first stimulus (standard) and the first numerical assignment (modulus) be randomized across individuals. In this way even though there is a preponderance of l s, 5 s, 10 s etc., these favorite numbers will tend to distribute along the entire set of stimuli. In contrast, were the first stimulus and number always to be fixed, then one would tend to see a pile-up of these round numbers on the same stimuli, creating an artificial narrowing of the variability of the data and a non-representative mean value for the ratings.

16 210 HOWARD R. MOSKOWITZ Unfortunately, as long as individuals are trained in the decimal system and as long as they have a predilection for using only a limited portion of the number scale, there will continue to be the roundnumber tendency. Only when the adjusted continuum is non-verbal (e.g., line length, tane sound pressure), and where there are not salient levels, can the experimenter hope to eliminate this round-number bias. Bias 2: The End-Effect Panelists, when presented with an unlimited range of magnitude estimation numbers, tend to constrict themselves to a restricted number range, inside which range they feel free to operate. The range of numbers differs among individuals. Some people choose numbers between 1 and 10 and operate as if they were working on a category scale with equal intervals rather than equal ratios. Others choose a range between 0 and 100 and operate in the same way. Finally, others choose numbers between 0 and 1,000 (or some other range). The end-effect can be seen when these individuals use their number range. Simply stated, the respondents tend not to use their highest nor lowest magnitude estimates. Unfortunately, the exact value of their highest estimate and lowest must be determined and is not always clear from their judgments. Does an individual who uses numbers between 1 and 1,000 have as the highest value 1,000, or 1,500, or 2,000? The true boundaries are open to experimental investigation. Fortunately, since magnitude estimates are unbounded, they do not suffer the additional experimenter-induced bias, which limits the range arbitrarily to 9 points. Bias 3: The Regression Effect Stevens and Greenbaum (1966) reported that when individuals assigned numbers to match the sensory intensities of a criterion stimulus (e.g., tones of varying sound pressure), the psychophysical function was flatter (i.e., exhibited a lower exponent) as compared to the inverse situation, when the same individuals adjusted the criterion stimulus (tone loudness) to match numbers. The existence of a flatter magnitude estimation curve (numbers varied) and a steeper magnitude production curve (criterion stimulus varied to match numbers) is called the regression effect. It is similar to the statistical regression towards the mean found in regression analysis by statisticians. Stevens and Greenbaum discussed the cause of the effect. Some individuals are conservative in assigning numbers to match stimuli. As a direct consequence, these individuals will yield a shorter than expected

17 MAGNITUDE ESTNATION 211 range of numbers to match the fixed range of physical stimuli. Other individuals are not as constricted, nor as conservative, and assign a higher range of numbers. That higher range produces a consequently higher slope for the same continuum. The two individuals may operate with identical sensory functioning, but their number-matching behavior is quite different. There are two remedies to the regression effect. (1) The experimenter can tum around the experiment, when ~ossible, Not only does the panelist match the numbers to the criterion stimulus (i.e., adjust the numbers), but also matches the criterion stimulus to the numbers (i.e., adjust the criterion stimulus). One operation will produce steeper functions than the other. The average sensory function from the joint operation will produce a truer estimate of the sensory function, by cancelling out some (albeit not all) of the regression tendency. (For experimental data on the regression effect, consultant: Stevens and Greenbaum 1966; Meiselman et al ) (2) The experimenter may have the panelist match a third criterion stimulus (e.g., handgrip) to both numbers (criterion modality I) and to tone (criterion modality 2). The two sets of matches will produce two distinct functions: a number-handgrip function, and a handgrip-tone function. By equating the two functions together, one can eliminate that one modality, handgrip, which the individual varied and obtain a purer estimate of the number-tone function. This method was suggested by Moskowitz (1971) to secure a better estimate of the sensory sweetness function. Dawson and his colleagues (cf. Dawson and Brinker 1971) at Notre Dame have used this approach in what they call multi-modality matching to obtain valid scales of opinion and to counteract some of the biases in simple magnitude estimation. Bais 4: Unusual Numerical Matching Panelists without experience usually have no marked problem in using magnitude estimation to scale. On some occasions, however, their numerical assignments exhibit ratios that are entirely out of bounds with their previous numbers. For instance, if a panelist is presented with a series of tones of varying loudness and uses numbers in the range between 1 and 160, then occasionally a very soft tone far out of the range may provoke a fraction such as 0.01 or (1/100 or 1/1000). An extremely strong tone, much louder than previous ones, may provoke an estimate of 1,000 or 2,000. This bias towards using extremely unusually large or small ratios to express outlaying stimuli that are very strong or very weak reflects the

18 212 HOWARD R. MOSKOWITZ fact that the panelist has no conception of numerical ratios at that high or low end. Rather, the numbers there are markers to indicate extremes, but the numerical values must be interpreted with considerable caution. Otherwise, the mean rating would be biased, either upwards or downwards, respectively. The extreme number bias can be counteracted in several ways: (a) The experimenter can ask the panelist whether the panelist meant to assign the number with that ratio. By pointing out what the ratios really mean, the experimenter has an opportunity to probe into the numbers that the panelist uses, interrogate the panelist as to the meaning of the numbers, and be satisfied that the panelist understands that the magnitude estimates are to reflect sensory ratios. (b) The experimenter can use number techniques other than number matching to validate the numbers, viz., the method of cross-modality matching (Stevens 1975). By comparing on a 1:l basis the mean numbers the panelists (as a group) give with the mean adjusted stimulus (i.e., line matching), then the experimenter can see what number assignment should have been put in place of the extreme number. This remedy requires that the experimenter use number matching and another matching method in the same study. The matched methods allow for the development of magnitude estimation and cross-modal matching functions. Where the magnitude estimates go off scale, the cross-modal matching function can be used to indicate what should have been the correct level of magnitude estimation. his 5: The Categorizing Scaler Quite often the experimenter comes across an individual who uses the option of magnitude estimation to (a) limit the range of numbers, often quite dramatically, and (b) use the numbers to describe specific levels. For instance, an individual may use numbers from 0 to 100, in increments of 25 (0, 25, 50, 75 and 100). The individual, when questioned, usually states that he or she can only perceive four gradations, corresponding to 4 categories (none, low, moderate, high). In this bias, there is usually little that can be done. The panelist has locked himself or herself into a scheme of numbers, in order to facilitate the magnitude estimation task, and has chosen a convenient set of a few numbers to use as category markers. HOW TO WORK WITH CHILDREN Adapting magnitude estimation to children poses a unique, chal-

19 MAGNITUDE ESTIMATION 213 lenging and ultimately rewarding problem. Children s use of numbers varies widely, and is correlated with their school experience. The following is a guideline, which we at MPI Sensory Testing have found useful in our product evaluation with children. It can be modified to suit the needs of particular research projects. First, one should note that children beyond the age of 9 or so exhibit similar magnitude estimation behavior to adults. They may be somewhat more erratic than adults in their assignment of numbers, and may pay less attention than adults to instructions, but once they are involved in the estimation task, their data conforms to that which one obtains for adults. See Appendix VIII. For children as young as 6, we have found the following method to yield quite satisfactory results. For judgments of magnitude, present to the children squares of various sizes, whose areas have already been predetermined. Then, instruct the child to point to the square that best reflects how strong the magnitude is. (This is equivalent to having the child use a category scale, thus far.) At the end of the session, the child is instructed to draw lines reflecting (a) the size of the squares (match line lengths to squares), and (b) in the same language and scale, match line lengths to concepts or verbal descriptors, of extremely large, very large, moderately large etc. The result of this exercise is: (a) The squares corresponding to the evaluations of the products; (b) Line lengths (which take the place of numbers) corresponding to the squares; (c) A series of verbal calibrating lengths, corresponding to concepts. Through this set of evaluations, the experimenter can... (a) Normalize the squares in terms of the verbal descriptors. (b) Once having pivoted and normalized the squares, then proceed to replace, in each child s data form, the equivalent, normalized values (or equivalent magnitude estimates) for the square. The data will then appear identical to that in Appendix V. AN OVERVIEW TO MAGNITUDE ESTIMATION The foregoing paper deals with magnitude estimation from the practitioner s point of view. It stresses methods and approaches to data acquisition and data analysis, without inquiring too deeply into what data is being acquired. With practice, however, the practitioner of any scaling method, magnitude estimation notwithstanding, soon builds up a compendium of lore about what panelists can and cannot do, what kinds of biases the method induces, and what limitations the panelists

20 214 HOWARD R. MOSKOWITZ themselves bring to bear in the study. Magnitude estimation as a scaling method is twenty-five years old. Scientifically, the data acquisition has increased almost exponentially, with each group of scientists bringing its own approaches. Variants of the methods discussed here almost certainly will appear in various psychological journals, as basic researchers continue to use the method to learn about the nature of the human senses, and the varieties of sensory and attitudinal scales. Magnitude estimation as an adjunct to industrial evaluation of products is probably no more than 3 years old. The exigencies of real-world requirements, the need for efficiency, and the need to use magnitude estimation in contexts where there is little or no control over the experiment will probably generate more methods and produce an even greater variety of procedures than those suggested here. The future of magnitude estimation is a bright one, for it has already proven its use to a generation of research scientists. Let us hope that with the coming generations of researchers, the method will continue to prove its mettle to tackle applied problems; and that with increasing varieties of problems, that the magnitude estimation method wil become a living, changing and constantly improving scaling procedure. We should bear in mind that the gift of the quarter century of research started at Harvard may be research s first valid, usable ratio scale. Only the future studies can indicate the fulfillment of that promise.

21 MAGNITUDE ESTIMATION 21 5 APPENDIX I Typical Instructions for Magnitude Estimation No assigned modulus. In front of you is a series of paper cups filled with stimulus solutions. Your task is to tell how sweet they seem by assigning numbers proportional to sweetness. If the second stimulus is nineteen times as sweet as the first, assign it a number nineteen times as large. If it seems one-eleventh as sweet, assign it a number one-eleventh as large, and so forth. Use numbers, fractions, and decimals, but make each assignment proportional to the sweetness as you perceive it. APPENDIX I (PART B) Suggested Instruction - Without Emphasis on Ratios (From Stevens 1975, p. 30) You will be presented with a series of stimuli in irregular order. Your task is to tell how intense they seem by assigning numbers to them. Call the first stimulus any number that seems appropriate to you. Then assign successive numbers in such a way that they reflect your subjective impression. There is no limit to the range of numbers that you may use. You may use whole numbers, decimals, or fractions. Try to make each number match the intensity as you perceive it. Assigned Modulus APPENDIX I (PART C) In front of you is a series of... solutions. Your task is to tell how sweet they seem. Call the first stimulus 10 (reader - you may assign any number here you wish for the modulus, and simply modify instructions accordingly). If the second stimulus is nineteen times as sweet as the first, assign it the number 19 times as large. That number is 10 X 19 = 190. If it seems one-eleventh as sweet as the first, then assign it a number 1/11 as large. That number is lo/ll or around 0.9. Use numbers, fractions, and decimals, but make each assignment proportional to the sweetness as you perceive it. Remember, 10 is your reference number, and reflects the sweetness of the first sample.

22 # Name: (1-3) (4) Replicate 0 (5) TRAINING SECTION Please pick up the card deck which has been given to you. DO NOT LOOK through the deck, but pay attention only to the top card, which has a shape labeled. Now, you are going to assign numbers to show how large the shapes you will see in this card deck seem to you. Give the first area any number you wish. Write this number on the line labeled SIZE under the 1st shape at the bottom of the page. REMEMBER: You will be using this first number to compare the size of the first shape to the size of other shapes, which could be larger, smaller or the same size as the 1st shape. Therefore, there are no upper limits on the size of the number you use, but the number should not be so small that you cannot easily divide it into smaller portions (not smaller than 10 for instance). Now, flip to the 2nd shape on the second card. Give it a number which represents the area or size of the 2nd shape as compared to the 1st shape. For instance: If you give a number of 28 to the 1st shape and the 2nd shape seemed to have approximately the same size or area, you would also give the 2nd shape a rating of 28. If the 2nd shape seemed only one-half as large as the 1st shape, then you would give it a rating of 14. If the 2nd shape appeared to you to be four times as large as the 1st shape, then you would give it a rating of 112. Now, flip to the 3rd shape on the third card and evaluate the size of this shape in the same manner as above. Then do the same for all the other cards in the deck. Please DO NOT look ahead, but evaluate each shape as you come to it. RATING OF SHAPES SHAPE A SHAPE D SHAPE G SHAPE J SHAPE M Size : Sue : Size: Size: Size: (6-9) (18-21) (30-33) (4245) (54-57) SHAPE B SHAPE E SHAPE H SHAPE K SHAPE N Size: Sue : Size: Size: Size: (10-13) (22-25) (34-37) (4649) (58-61) SHAPE C SHAPE F SHAPE I SHAPE L SHAPE 0 Sue: Size: Size: Size: Size. (14-17) (26-29) (38-41) (50-53) (6 2-66) E Q,

23 # Name: I.D.# Day (2-4) Perfume ):t TRAINING SECTION Replicate: As another exercise, we would like you to express your liking or disliking of different words. A list of words is shown below. Using another scale of your own design, show how you feet about each word. You will use numbers to show how you feel about each word. If you like a word, write an (L) next to it or a (D) next to it if you dislike it. Then indicate just how much you either like or dislike the word by also writing in a number. A large (L) number means you like it a lot, while a large (D) number means you dislike it a lot. A small (L) number means you like it a little, while a small (D) number means you dislike it a little. If you feel indifferent or neutral about a word, give it a zero (0). Let s begin. Give the first word an L or a D to show if you like it (L) or dislike it (D). If you like it a lot, give it a large L number. If you dislike it a lot, give it a large D number. EXAMPLE: Say you give the first word an LlO to show how you feet about it, but you like the second word twice as much. You would then give the second word an L20. If you dislike the third word, you should give it a D and a number to show how much you dislike it. If you dislike it just a little, you might give it a D30 but if you dislike it a lot, you might give it a D150. Match numbers to show how you feel about the words. Please remember that the scale that you use is entirely your own. There are no limits on the size of the scale that you use, and no one s scale is more right than anyone else s. You may use any number you wish to show how much you like or dislike a particular word. Simply assign each word the number that you feel represents how much you like (L) or dislike (0) the word. Please keep in mind that large L numbers mean you like a word a lot and large D numbers mean you dislike a word a lot. WORD YOUR RATING WORD YOUR RATING WORD YOUR RATING L/D IHOW MUCH? L/D I HOW MUCH? L/D I HOW MUCH? Flowers Mud F (56) (37-40) zzihetti Rattlesnake E:m Kiss (61) (16) (17-20) (41) (42-45) (66) Murder Love PUPPY (21) (22-25) (47-50) Sex Sun Pollution (26) (27-30) (71) (51) (52-55) (76) (5) (57-60) (62-65) (67-70) (72-75) (77-80)

24 H TRAINING SECTION You have now used two different types of scales to rate the shapes and words. 1. A scale starting at zero and using bigger and bigger numbers to represent 1 something that is larger or greater than zero. This may be used for size or strength or intensity (see scale below). If you give one shape a 10 and the next a 20, it would mean the second is twice as large as the first. +loo I +50 3: 5 s g $ t J a j l? a z - $ s s Y E A scale showing two (opposite) kinds of feelings, liking and disliking, for instance: Zero is the middle of this scale and numbers get larger in both directions. For instance, if you like something a little, you give it a small positive number (Example L12). If you like something else twice as much, you would give it a rating twice as high (Example: L24). If YOU dislike it slightly, you might give it a (D5) for example, while you might give it a (D500) if you dislike it extremely. If you give a zero, it would mean you are neutral about something. This scale looks like this, N f--- D40 D20 D10 0 L10 L20 L40 I I I I I I I I I I I I I

25 MAGNITUDE ESTIMATION 219 References on Cross-Modality Matching APPENDIX IV MOSKOWITZ, H.R Intensity scales for pure tastes and for taste mixtures. Perception & Psychophysics 9, STEVENS, J. C., and MARKS, L. E Cross-modality matching of brightness and loudness. Proceedings of the National Academy of Sciences 54, STEVENS, S. S Cross-modality validation of subjective scales for loudness, vibration, and electric shock. Journal of Experimental Psychology 57, STEVENS, S. S Matching functions between loudness and ten other continua. Perception & Psychophysics 1, STEVENS, S. S On predicting exponents for cross-modality matches. Perception & Psychophysics 6, STEVENS, S. S., MACK, J. D. and STEVENS, J. C Growth of sensation on seven continua as measured by force of handgrip. Journal of Experimental Psychology 59, Some Taste Examples The table below shows results in which experimenters matched both numbers and white noises ( Hz) to the taste of pure aqueous solutions and mixtures (sweet, salty, sour and bitter). Both matches produced power functions of the form: Number (Magnitude Estimate = k (Concentration)* (1) Noise (Dynes/Cm2) = k (Concentration)B (2) In addition, from other experiments, it has been found that... Numbers (Magnitude Estimates) = k(noise)m (3) The table shows the prediction of the number-concentration exponent (a) from the factor (0.64 X B).

26 220 HOWARD R. MOSKOWITZ APPENDIX IV Parameters of the Power Law for Saltiness: S = kca Exponents 0.64X Magnitude Noise Noise Experiment Estimate Match Match Mean Pure Salt Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiment 5 Experiment 6 Mean With Quinine Sulfate Series % QS % QS % QS04 Mean Series % QS % QS % QS04 Mean Grand Mean-QS With Sucrose Series 1 1.0% Sucrose 2.0% Sucrose 4.0% Sucrose Mean Series 2 4.0% Sucrose 8.0% Sucrose 15.0% Sucrose Mean Grand Mean With Acid Series % Acid 0.04% Acid 0.08% Acid Mean Series % Acid 0.15% Acid 0.30% Acid Mean Grand Mean

27 Worked Example. Sample Magnitude Estimation Data Form. (A) Linear, Raw Data Sweetness Calibration (Sweetness) Panelist Extremely Very Moderately Slightly Not At Sample Sweet sweet sweet Sweet All Sweet A B C Mean O Std Dev M/Std Dev (B) Pivoted Data [Pivot Value = (Extremely + Very + Moderately + Slightly)/4] (B) Normalized ( Calibrated ) Data Panelist Sample f Number A B C Pivot Extremely Very Moderately Slightly Not At Sweet Sweet Sweet Sweet All Sweet F 3 - G g C 0 3 e : +I j l g c 5 8 z Mean Std Dev MeanISD N

28 ~ ~ ~ 222 HOWARD R. MOSKOWITZ APPENDIX VIa Representative Exponents (n) of the Power Functions Relating Psychological Magnitude to Stimulus Magnitude. Continuum Exponent Stimulus Condition Loudness Loudness Brightness Brightness Lightness Smell Smell Taste Taste Taste Temperature Temperature Vibration Vibration Duration Repetition rate Finger span Pressure on palm Heaviness Force of handgrip Vocal effort Electric shock Tactual roughness Tactual hardness Visual velocity Visual length Visual area -~ binaural monaural 5O target - dark - adapted eye point source - dark - adapted eye reflectance of gray papers coffee odor heptane saccharin sucrose salt cold - on arm warmth - on arm 60 cps -on finger 250 cps - on finger white-noise stimulus light, souod, touch and shocks thickness of wood blocks static force on skin lifted weights precision hand dynamometer sound pressure of vocalization 60 cps -through fingers felt diameter of emery grits rubber squeezed between fingers moving spot of light projected line of light projected square of light

29 MAGNITUDE ESTIMATION 223 APPENDIX VIb Power Function Exponents for Odor Intensity Odorant/Flavor Exponent Diluent Eugenol Ethyl Salicylate Butyl Acetate Trans Cinnamaldehyde N Caproic Acid Clove Oil Spearmint Oil Lemon Oil California Orange Lemon Oil, California Grapefruit Nonanoic Acid Anise Strawberry Chocolate Wintergreen Odorant Amy1 acetate Anethole 1-Butanol 1-Butanol 1-Butanol Butyl acetate Butyric acid Coumarin Citral Ethyl acetate Eugenol Eugenol Geraniol Guaiacol 1-Heptanol 1-Hexanol D Menthol Methyl salicylate 1-Octanol 1 -Pentanol Phenylethanol Phenyl acetic acid 1-Propanol Iso-valeric acid Exponent Liquid Liquid Liquid Air Air Air Liquid Air Liquid Liquid Liquid Air Air Liquid Liquid Air Liquid Liquid Liquid Liquid Liquid Liquid Air Liquid Diluent

30 224 HOWARD R. MOSKOWITZ APPENDIX VIc Some Relevant Exponents for Power Functions - Texture Hardness Attribute 1. Vs Modulus of Elasticity (Space Cubes) 2. Vs Ultimate Strength (Space Cubes) 3. Vs Force/Indentation Ratio (Rubber) Force 1. Vs Dynes/Cm2 (Handgrip Dynamometer) 2. Vs Dynes/Cmz (Large muscle group against the floor) Viscosity 1. Vs Centipoises (Gum Solutions, Suspensions) 2. Vs Centipoises (Silicone Oils) Crunchiness 1. Vs Modulus of Elasticity (Space Cubes) 2. Vs Ultimate Strength (Space Cubes) Chunkiness 1. Vs Size of Grind (Hamburger) Roughness 1. Vs Grit Size (Sandpaper) Exponent

31 MAGNITUDE ESTIMATION 225 APPENDIX VII A Sample of Calibration Scales for Magnitude Estimates. 1. Sweetness of sauce with 3 replicate sessions (pivot = Average Magnitude Estimate) Replicate 1 Replicate 2 Replicate 3 Extremely Sweet Very Sweet Moderately Sweet Slightly Sweet Barely Sweet Flavor Intensity (pivot = Extremely Strong) (Product = Sauce) Extremely Strong Very Strong Moderately Strong 53.4 Slightly Strong 34.1 Not at all Perceivable 0 3. Degree of Consistency (pivot = Average Magnitude Estimate to all Categories) (Product = Sauce) Extremely Very Moderately Slightly Degree of Color Deepness (pivot = Extremely Strong ) (Product = Pudding) Extremely Very 81.8 Moderately 61.7 Slightly 20.2 None at all 0 5. Liking/Disliking (pivot = Average of Magnitude Estimates) Unipolar Bipolar Like Extremely Like Extremely Like Very Much 99.7 Like Very Much Like Moderately 85.3 Like Moderately Like Slightly 73.9 Like Slightly 59.3 Like Barely 49.3 Neither Like Nor Dislike 0 Do Not Like At All 0 Dislike Slightly Dislike Moderately Dislike Very Much Dislike Extremely

32 226 HOWARD R. MOSKOWITZ APPENDIX VIII Percentage and Cumulative Percentage of Magnitude Estimates of Sweetness - Adults Vs Children (n = 90 each) Adults Children Magnitude Cumulative Cumulative Estimate Percent Percent Percent Percent

33 MAGNITUDE ESTIMATION 227 REFERENCES BOND, B. and STEVENS, S. S Cross-modality matching of brightness to loudness by 5 year olds. Perception & Psychophysics 6, CAIN, W. W Odor intensity: Differences in the exponent of the psychophysical function. Perception and Psychophysics 6, CROSS, D. V., Sequential dependencies and psychophysical judgments. Perception & Psychophysics, 14, DAWSON, W. E. and BRINKER, R. P Validation of ratio scales of opinion by multimodality matching. Perception and Psychophysics 9, GESCHEIDER, G Psychophysics: Method and Theory. Erlbaum Associates, Hillsdale, N. J. LANE, H. L., CATANIA, A. C. and STEVENS, S. S Voice level: Autophonic scale, perceived loudness, and effect of side tone. J. Acoust. SOC. America 33, MARKS, L. E., Sensory Processes: The New Psychophysics, Academic Press, New York. MARKS, L. E. and STEVENS, J. C The form of the psychophysical function near threshold. Perception & Psychophysics 4, MEISELMAN, H. L., BOSE, H. E., and NYKVIST, W. F Magnitude production and magnitude estimation of taste intensity. Perception & Psychophysics 12, MOSKOWITZ, H. R., Ratio scales of sugar sweetness. Perception & Psychophysics 7, MOSKOWITZ, H. R (a). The sweetness and pleasantness of sugars. Am. J. Psycho. 84, MOSKOWITZ, H. R (b). Ratio scales of acid sourness. Perception & Psychophysics 9, MOSKOWITZ, H. R., SHARMA, K. N., KUMRAIAH, V., JACOBS, H. L. and SHARMA, S. D Cross cultural differences in simple taste preferences. Science 187, MOSKOWITZ, H. R., DRAVNIEKS, S., and KLARMAN, L Odor intensity and pleasantness for a diverse set of odorants. Perception & Psychophysics 19, PANEK, D. W. and STEVENS, J. C PSYCHOFIT: A problem for the analysis of psychophysical scaling data. Unpublished computer program, Laboratory of Psychophysics, Harvard University, Cambridge, Mass. STEVENS, J. C A comparison of ratio scales for the loudness of white noise and the brightness of white light. Doctoral Dissertation, Harvard University, Cambridge, Mass. STEVENS, S. S On the brightness of lights and the loudness of sounds. Science 118, 576. STEVENS, S. S Psychophysics: An Introduction to Its Perceptual, Neuml and Social Prospects. John Wiley & Sons, New York. STEVENS, S. S. and GREENBAUM, J Regression effect in psychophysical judgment. Perception & Psychophysics 1,