direct estimation: effect of methodological factors on scale tupe

Scand. J . Psychol., 1975,16, 19-29

Direct estimation: effect of methodological factors on scale type

HENRY MONTGQMERY

Abstract.-An experiment was performed to investigate the importance of methodological differences between magnitude estimation and category rating. It was found that the form of a scale varied with (0) the range of responses used by the subject and (b) with his freedom of choosing a highest number. Other factors investigated played a minor role. A narrow range and fixed upper scale value yielded the typical category scale, a wide range with no restriction on the highest value the typical magnitude scale. The same factors that determined the form of the scale also affected the Weber functions.

The methods used for direct estimation of subjective magnitude are often partitioned into two general classes-the ratio estimation methods and the category rating methods (cf. Stevens & Galanter, 1957; Torgerson, 1960). The fundamental difference between these two classes of methods is usually assumed to lie in the instructions for assignment of responses t o subjective magnitudes. In ratio estimation methods the subject is instructed t o assign numbers to stimuli so that they reflect the ratios between subjective magnitudes and in category rating t o assign numbers or labels so that they reflect the differences (or distances) between subjective magnitudes (cf. e.g. Torgerson, 1960; Eisler, 1%2 a). Ratio estimation methods, typically re- presented by the method of magnitude estimation, and category rating have been compared on several continua and, unfortunately from the point of view of scaling theory, the two kinds of methods give scales which are clearly curvi-linearly related for most continua studied. The function relating the category scale t o the magnitude scale (the C-M-relation) is often approximately logarithmic.

The attempts which have been made to explain the non-linear C-M-relation usually appear to be based on the assignment rule given in the instructions. It is supposed that the subject interprets subjective differences in category rating in a way which is incon-

university of Goteborg, Sweden

sistent with ratio judgments. According t o Tor- gerson (1%1) the subject is unable t o distinguish equal subjective differences from equal subjective ratios. Ekman, Goude & Waern (1961) assume that equal differences in category rating reflect equal degrees of similarity between stimuli. Stevens (1957) is of the opinion that the subject confuses discriminability with psychological distance. For similar points of view, see Attneave (1%2), Galanter & Messick (1961), Eisler (1962a) and Treisman (1964).

Some authors, however, have questioned the validity of the idea that judgments of differences are inconsistent with ratio judgments. Warren & Poulton (1962) reported a linear relation between a ratio scale and a scale obtained by bisections, the latter requir- ing judgments of differences between subjective magnitudes. The non-linear C-M-relation was assumed to be due to failure t o consider artifacts introduced by incidental differences in experimental procedure. Also Strangert (1965) has presented experimental data which corroborate Warren & Poulton’s findings and Poulton (1%8) has listed methodological differences usually found between category rating and magnitude estimation in order to explain the non-linear C-M-relation.

Consequently, there are two kinds of explanations of the non-linear C-M-relation: (a) The source of the non-linear C-M-relation is in the assignment rule given in the instructions, i.e. instruction t o let the numbers reflect subjective ratios or subjective differences. If assumption (a) is valid, direct estimation methods should preferably be partitioned into ratio estimation methods and difference estimation methods. (b) The non-linear C-M-relation is due t o other methodological differences. If assumption (b) is the correct one the terms ratio estimation and difference estimation are somewhat misleading and arbitrary, since the difference between the scales obtained were due to other factors than the instruc-

Scand. J . Psychol. 16

20 H . Montgomery

tion t o estimate ratios or differences. However, the evidence presented by Warren & Poulton and Stran- gert is not conclusive as t o assumption (a ) or (b) . In order to demonstrate the correctness of assumption (6) it must a t least be shown that a difference estimation method which in all other possible respects is similar t o typical magnitude estimation gives a scale which is linearly related t o the typical magnitude scale, and that a ratio estimation method which in all other possible respects is similar to typical category rating gives a scale which is linearly related t o the typical category scale. The experiment to be reported below was a n attempt to fulfill these require- ments. The general purpose of the experiment was t o determine the effects of all factors in which the methods of category rating and magnitude estimation usually differ. The two kinds of methods were considered to differ in the following four respects.

I . Rule of assignment of responses to subjective magnitudes. In magnitude estimation the subject is typically presented with a standard stimulus and instructed t o assign numbers so that they reflect the subjective ratio between the stimuli given and the standard stimulus. In category rating a lower and a n upper limit of a response scale are defined and the subjects a re instructed t o assign scale values so that equal differences between scale values reflect equal differences (intervals, distances) between subjective magnitudes.

2. Openness of set of responses. By openness is meant the degree of freedom given t o the subject to choose a lowest and a highest number. If a set of responses has n o openness it will be called closed. In category rating the set of responses is usually closed, since the lowest and highest number to be used by the subject is anchored a t the lowest and highest of the stimuli t o be judged. In magnitude estimation the standard stimulus is usually located in the middle of the series of stimuli given. Consequently, the choice of the lowest and highest number to be used is left t o the subject. It may, however, be said that the choice of the lowest number is somewhat restricted, since the subject is not allowed t o use numbers lower than zero (cf. Poulton, 1968).

3. Range of numbers. This factor denotes the range of numbers between the lowest and the highest number used by the subject. In category rating the range of numbers is usually defined by the numbers given t o the two standard stimuli, e.g. 1 and 7. In magnitude estimation the range of numbers is usually chosen by the subject since the set of responses


is open. The range of numbers used in magnitude estimation is, however, affected by the range of stimuli, position of standard stimulus, and number assigned t o the standard stimulus (cf. e.g. Engen & Levy, 1955; Hellman & Zwislocki, 1%1; Strangert, I%]) . Usually, the range of numbers used by the subject in magnitude estimation is much wider than the range of numbers used in typical category rating.

4. Discrete vs. continuous set of numbers. In category rating the subject is usually allowed t o use only integers while in magnitude estimation all posi- tive rational numbers are permitted.

It should be stressed that in some cases rather extraordinary experimental conditions-particularly in the choice of stimuli-were employed. It goes without saying that scales obtained under such extreme conditions could not be considered to give valid measures of subjective magnitudes. However, it was necessary t o make use of these experimental conditions in order t o achieve the aim of the present study.

METHOD

Design Thirteen experimental conditions were employed. In all conditions the subject was instructed to judge the loudness of intensities of white noise. The design of the experiments is described in Table 1.

The experimental factors referred to in Table 1 consisted of the following treatments.

1. Ratio estimation (R). The subject was instructed to assign numbers which reflected the ratio between the loudness of the noise presented and the loudness of a standard noise.

2. Difference estimation (D). The subject was instructed to use a scale ranging from 1 to some greater number, e.g. 100, for his judgments. Intensities representing the end points of the scale were presented to the subject. It was stressed that equal differences between numbers given by the subject should reflect equal differences between loudnesses.

3. Wide and open set of responses (WO). In the ratio estimation experiments, the standard stimulus was located in the middle of the series of stimuli given. The stimulus value of the standard stimulus was 80 dB in Cond. RWOl a, 70 dB in Cond. RW02 and 60 dB in Cond. RW0*2. The standard stimulus was called 10 in Conds. RWOla and RW02 and 2 0 in Cond. RW0*2. For the difference estimations a relatively open set of responses was created by allocating the standards below and above the series of stimuli to be judged.

4. Wide and closed set of responses (WC). A relatively closed set of responses for ratio estimations was provided by using the strongest stimulus as standard stimulus. The standard was called 100. In the difference estimation condi-

Eflect of methodologicalfacrors on scale type 21

Table I , Experimental design. R: ratio estimations, D: difference estimations, W: wide range of responses, N: narrow range of responses, 0: open set of responses, C: closed set of responses. The apostrophe ( e . g . DNC’2) denotes estimation in integers, the star * ( R W 0 * 2 ) indicates that the standard stimulus in magnitude estimation M’US called 20

Rule for assignment of responses to stimuli WO wc NC

Range and openness of set of responses

R D R

D

RWOl a DWOl a DWCl b DNC’I b RWO*2 RWC2 RWO2 Spacing 2

Spacings l a and Ib

DWC2 DNC2 DNC’2

R RWC3 RNC3

D DNC’3 Spacing 3

tions the weakest and strongest of stimuli to be judged were used as standard stimuli. The standards were called 1 and 100.

5. Narrow and closed set of responses (NC). In the ratio estimation condition in this treatment (Cond. RNC3). the strongest stimulus was used as standard and was called 10. The weakest stimulus was chosen by means of a preliminary experiment to yield a subjective value of about unity, so that approximately the same range should be obtained as in the corresponding category rating condition (Cond. DNC’3), in which the standards were called 1 and 1 1. Three additional difference estimation conditions were employed in treatment NC, viz. Conds. DNC’I b, DNC2 and DNC’2. Cond. DNC’2 was assumed to be equivalent to a category rating experiment carried out by Eisler (1%26) (cf. below). The standards were called 1 and 7. The same numbers were assigned to the standards in Cond. DNC2. In Cond. DNC’I b the standards were called 1 and 9. In all difference estimation experiments in treatment NC, the weakest and the strongest of the stimuli to be judged were standards, as in Condition WC.

Treatment NO (narrow and open set of responses) could not be realized. In three conditions (Conds. DNC’Ib, DNC’2 and

DNC’3), the instructions indicated that the subject should use a discrete set of numbers (integers). In all other conditions as well as one DNC-condition (DNCZ), the subject was instructed to use a continuous set of numbers.

As may be seen from Table 1, four different spacings of stimuli were used (spacings l a , 1 b, 2, and 3). The spacings are given in Table 2. Different spacings were used in order to investigate the generality for some results over spacings and because all conditions could not be realized with the same spacing of stimuli. The six middle stimuli in spacing 1 b were the same as the simuli in spacing 1 a. This was to make it possible to compare scales in the WO- and WC-treatments (cf. Table 1). Spacing 2 was a spacing used by Eisler (1%2b) in a series of category rating and magnitude estimation experiments. This spacing was chosen in order to compare the results from Conds. RW02 and DNC’2 with the results

from two similar experiments in Eider’s investigation. Spacing 3 was designed for ratio estimations in the NC- treatment.

Procedure Before the start of the experiment the instructions were given orally and the standard stimulus or standard stimuli was (were) presented to the subject for as long as he wished. When there were two standard stimuli the weaker one was presented first. The experiment started with six trials which were not used for computation of scale values. The subject was not informed of this. The stimuli corresponding to these preliminary trials were presented in random order and were selected from the same set of stimuli that was used in the rest of the experiment. In the experiment proper the stimuli were presented in five SUC-

cessive blocks where each block consisted of all stimuli presented in random order. Consequently, each stimulus was judged five times. Different orders of presentation were used for each subject.

Apparatus The same apparatus was used in all conditions. It consisted of a white noise generator, adjusted to a SPL of 110 dB (re O.ooOo2 microbar), that fed a pair of earphones (Beyer, DT 48,5) with a bandpass filter (74-2400 cps), an attneuator (Marconi, smallest step0.1 dB), a transformer and a switch.

Subjects 10 subjects participated in each condition. Different subjects were used in all conditions. The subjects were under- graduate students of psychology.

Instructions Instruction RWO (Ratio estimation, wide and open set of responses): “Yourtask is tojudge how loud youexperience noises which will be presented to you.

You are to judge the loudness of the noise in relation to a standard noise which will be presented to you. This noise is called 10. For each noise which is presented to you, you are

S c a d . J . Psychol. 16

22 H . Montgomery

w

M

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to say a number which reflects the ratio between the loudness of the noise presented and the loudness of the noise called 10. If, forexample, you experience that anoise is twice as loud as the standard noise, you are to say 20 and if, for example, you experience that the loudness of a noise is 26% of the loudness of the standard noise, you are to say 2.6, etc.

You are permitted to use any numbers which you feel appropriate, thus also numbers ending with decimals."

Instruction RWO was used in Conds. RWOI a, RWO*2 and RW02.

The following instructions are given without the intro- ductory passage.

Instruction RC (Ratio estimations, closed set of responses): ". . . You are to judge the loudness of the noises in relation to a loud standard noise which will be presented to you. This noiseis called 100( 10). For each noise which is presented to you, you are to say a number which reflects the ratio between the loudness of the noise presented and the loudness of the noise called 100 (10). If, for example, you experience that a noise is half as loud as the standard noise, you are to say 50 (5 ) and if, for example, the loudness of a noise is 13% of the loudness of the standard noise, you are to say 13 (1.3). etc.

You are permitted to use any numbers which you feel appropriate, thus also numbers ending with decimals."

Instruction RC was used in Conds. RWC2, RWC3 and RNC3. In Cond. RNC3 the number assigned to the standard and the numbers used in the numerical examples were replaced by the numbers within parentheses.

Instruction DW (Difference estimations, wide range of numbers): " . . . You are to use a scale ranging from 1 to 100 for your judgments, where 1 represents a weak noise which will be presented to you and 100 a loud noise which also will be presented to you. If, for example, you experience that the loudness of a noise is half-way between the noise called 1 and 100, you are to say 50 or to be exact 50.5. It is very important that you use the scale so that the difference between e.g. 12 and 22 is the same as the difference between e.g. 75 and 85, that is that equal differences between numbers you say correspond to equal differences between loudnesses.

You are permitted to use all numbers between 1 and 100, thus also numbers ending with decimals."

Instruction DW was used in Conds. DWOI a, DWCl b and DWC2.

Instruction DN (Difference estimations, narrow set of responses): I s . . . You are to use a scale going from 1 to 7 (9, 11) for yourjudgments where 1 represents a weak noise which will be presented to you and 7 (9, 11) a loud noise which also will be presented to you. If, for example, you experience that anoise is half-way between the noise called 1and7(9, Il)youaretosay4(5,6). Itisveryirnportantthat you use the scale so that the difference between e.g. 2 and 3 is the same as the difference between e.g. 5 and 6 (7 and 8,9 and lo), that is that equal differences between the numbers you say correspond to equal differences between loudnesses."

Instruction DN wasusedin Conds. DNC'I b, DNC2and DNC'3. In Cond. DNC2 the numerical examples were 1.2 and 2.2 and 5.5 and 6.5 and the instruction was concluded with the following sentence: "You are permitted to use all

Scand. J. Psychol. 16

Eflect of methodological factors on scale type 23

Table 3. Polynomial regression analysis on eflects of methodological factors

Percentage of total sum of squares

Experimental Independent Dependent Linear Quadratic Cubic factor variable variable term term term

Rule

Range

Openness

RWOl a DWOl a 99.85 0.04 .01 RWCZ DWC2 99.84 0.01 .oo RNc3 DNC‘3 99.70 0.23 .oo DWCl b DNC’ I b 98.19 1.49 .04 DWCZ DNC2 98.35 1.34 .33 DWCZ DNC‘2 98.22 0.77 .80 RWC3 RNC3 99. I4 0.22 .44 DWOI a DWCl b 98.16 I .42 .4 1 RWO*2 RWC2 98.15 1.55 .23 RW02 RWCZ 98.68 1.13 .09

Discrete vs. continuous DNC2 DNC‘2 99.73 0.04 .08

numbers between 1 and 7, thus also all numbers ending with decimals”.

RESULTS

Subjective scales and descriptive statistical analysis Table 2 gives subjective scales computed from the judgments made by the subjects in each condition. In the following the scales will be denoted as the conditions from which they derive. The scales for those conditions which could be considered as typical magnitude estimation experiments (the RWO- -conditions) were computed by taking the arithmetic meanofthe fivejudgments made by each subject, for each stimulus, and the geometric mean of the arithmetic mean of all subjects. This procedure corresponds to a method for computing intraindividual standard deviations (cf. Eisler, 19626) which was used on the results to be reported below (cf. section “Intraindividual variability”). The customary procedure t o compute the geometric mean over ail values for each stimulus would not give very different results since the variability within observers is comparatively small. The judgments obtained in the other conditions were averaged by taking the arithmetic mean for each stimulus over all trials and subjects. The arithmetic means were used since they did not seem to deviate systematically from the medians.

In order to study the relation between the scales obtained, first, second, and third degree polynomial regressions were computed for pairs of scales obtained from the same set of stimuli. The contribu-

tion of the linear term to the total sum of squares was used as a measure of the degree of linearity, whereas the contributions of the quadratic and cubic terms were used for evaluating the effect of methodological variables. The polynomial equations were fitted by computer according to the BMD program No. O5R. The contributions of the terms in the fitted polynomial equations are given in Table 3.

An alternative way of analyzing the data is t o study the functions that relate judgments t o stimulus values. However, several experiments have shown that the relationship between judgments and stimulus values can be affected by the spacing of the stimuli (Garner, 1954; Pradhan & Hoffman, 1963; J. C. Stevens, 1958; Stevens & Galanter, 1957), whereas the relationship between different types of subjective scales seems t o be largely independent of stimulus spacing (Eisler, 1%2b).

The author wishes to point out that the statistical analysis chosen was used only as a purely descriptive method for evaluating the results. Due t o lack of homogeneity of variances, no statistical tests could be carried out for relationships between the scales obtained.

Effect of methodological factors Rule (Difference versus ratio estimations). Fig. 1 A shows the relation between scales obtained from difference and ratio estimation conditions which were similar in range of numbers and openness (DWOl a vs. RWOl a , DWCZ vs. RWC2, DNC’3 vs. RNC3). (No effects were assumed for discrete vs. continuous set of numbers, cf. below). In all three


24 H . Montgomery

9 -

7 .

5 -

3 -

R W O l a

0 0

0 . 61

0 0

0 0

oo 0 0

10

80 I 1 . 1 7 9 ,

c1 C

0 60 .

0 LO -

0

0 2o - 0

1 ' " " ' ~ -

O W C l b

7 -

5 -

3 .

1 -

R W C 3

3 0 0

0 c?

2 0

0

oo 0

N V z 0

D N C 2

R W C 2 ,Io2 3 O ? " 1 0

0

20 LO 60 80 1 0 0

D W C Z

N

u Z 0

R N C 3

7 v

3 :i 1 0 20 LO 60 80 100

DWC 2

0 = R WO*2 A = R W O 2

0 20 40 60 80 100 f 0 1 0 20 3 0 40 50 R W 0 2

D W O l a

Fig. 1. The relation between scales for loudness under varying methodological conditions (A: difference vs. ratio estimation, B: narrow vs. wide set of responses, C: closed vs. open set of responses, D: discrete vs. continuous set of responses).

plots the trend is essentially linear. As for scale DWCZ vs. RWC2 the scales practically coincide. The contribution of the linear term in the polynomial regression equations for the relations shown in Fig.

1 A range from 99.70% to 99.85 %. Range (Narrow vs. wide set of responses). Fig.

1 B exhibits the relation between narrow and wide scales, obtained from conditions which were identi-


Effect of methodological factors on scale type 25

95 dB was five times greater than the estimate of 70 dB in Cond. RWC3. but only three times greater in Cond. RWC2. These results can be seen as a de- monstration of the well known effect of the range of stimuli on magnitude estimates (cf. e.g. Poulton, 1968; Strangert, 1%1) which implies that the sub-

jects tend to use a constant range of numbers for different ranges of stimuli.

cal in rule and openness (DNC’I b vs DWC1 b, DNC2 vs. DWC2, DNC’2 vs. DWCZ and RNC3 vs. RWC3). (No effect was assumed for discrete vs. continuous set of numbers, cf. below). In all plots the trend is slightly concave downward with a reversal at the upper extreme for scales DNC2 vs. DWC2, DNC’2 vs. DWC2 and RNC3 vs. RWC3. The contribution of the linear term in the polynomial regression analysis vanes from 98.19% to 99.14 %.

The contributions of the second and third degree terms indicate a greater effect of range than of rule. The second degree term contributes more to the total variance than the third degree term with exceptions for scales DNC’2 vs. DWCZ and RNC3 vs. RWC3, where the contributions of the second and third degree terms are approximately equal.

Openness (Closed vs. open set of responses). The effect of openness could best be evaluated by comparing scale RWC2 and scale RW0*2, which cover approximately the same range of numbers. Fig. 2 C2 shows that the relation between the two scales is slightly concave downward. The curvature is most accentuated at the upper extreme. Fig. 2C2 also shows that scale RWC2 has about the same relation against scale RW02. A relation which is concave downward also obtains between scales DWCl b and DWOl a as shown in Fig. 2Cl . This relation could, however, be due to the fact that the total set of stimuli to be judged was different in Conds. DWCl b and DWOl a. The contribution of the linear term in the polynomial regression analysis is 98.15 % for scales RWO*2 vs. RWC2,98.68% for scales RW02 vs. RWC2 and 98.16% for scales DWOI a vs. DWCl b. The second degree term mainly accounts for the effect of openness. The effect of openness is about as great as the effect of range.

Discrete vs. continuous set of responses. Fig. 2 D shows that scale DNC2 is essentially linearly related to scale DNC’2. The contribution of the linear term in the polynomial regression equation which relates the two scales is 99.73%.

Spacing of stimuli. It can be seen from Table 2 that the ratio between the same pair of stimuli can be judged very differently in conditions which correspond to different spacings but are similar in other respects. Consider e.g. the stimulus pair 95 and 70 dB. In Cond. RWOla the estimate of 95 dB was twelve times greater than the estimate of 70 dB whereas inCond. RW02 the 95 dB stimulus was only judged as four times louder than 70 dB. In the two RWC conditions (RWC2 and RWC3), the estimate of

The relation between category and magnitude scales With spacing 2 one typical category rating condition, DNC’2, and two typical magnitude estimation conditions, RW02 and RWO*2, were employed. Conds. DNC‘2 and RW02 were supposed to be equivalent to a magnitude estimation and category rating experiment reported by Eisler (cf. above). Fig. 2 shows the relation between scale DNC2 and scale RW02 as well as the relation between Eisler’s category and magnitude scale. Both relations are concave downward. However, the relation between Eisler’s category and magnitude scale is more curvi-linear. Plots of scale RW02 against Eisler’s magnitude scale and scale DNC‘2 against Eisler’s category scale (not shown here) indicated that the two category scales were approximately linearly related whereas a curvi-linear relation obtained between the two magnitude scales. The author re- - alu - - c

: m

7

5

Q 3

1

0 V

0

~- 0 10 20 30 40 5 0

Magnrtude scale

Fig. 2. Scales for loudness as a function of the magnitude scale for loudness. (Circles: scale DNC‘2 vs. RW02, squares: scale DWC2 vs. RW02, triangles (A): scale RWC2 vs. RW02, triangles (V): a category scale from an experiment reported by Eisler (1%2b) vs. a magnitude scale from the same experiment). Scales RW02, DWCZ and Eisler’s category scale have been normalized by a linear transformation to make them correspond to scale DNC’2.


26

C 0

4 > c 0

.- c

.-

? 0 C 4 c m - rg 3 0 .- 0 c .- I c c -

H . Montgomcvry

04

R W O l a D W O l a DWC 1 b D N C ' 1 b RWO.2

04 04

0 2 02

R W 0 2 RWC 2 DWC 2 D N C 2 D N C ' 2

RWC 3 R N C 3 DNC'3

Fi.p. 3. Intraindividual SDs as a function of scale values of loudness under varying methodological conditions (see

frains from speculating about the cause of this dis- crepancy.

Fig. 2 also shows scales RWC2 and DWC2 as a function of scale RW02. The former scales seem to be a compromise between scales DNC'2 and RW02, the typical category and magnitude scale, since their relation to scale RW02 is less concave downward than the relation between scale DNC'2 and RW02.

Intraindividual variability Previous experiments have shown that the Weber function of the category scale (intraindividual SDs as a function of scale values) is different of that of the magnitude scale (cf. Eisler, 1963, 1965 and Eisler & Montgomery, 1974). It will be shown below that this difference seems to be due to pre- cisely those methodological factors which also affect the type of scale.

In Fig. 3 intraindividual SDs are plotted against the scale values from each experiment. For the RWO-conditions (typical magnitude estimation), the SDs were calculated according to a method described by Eisler (1%2b). This method was used since the distributions were skewed (cf. Eisler,

text). The curves are parabolas which were fitted by a variation of the method of least squares (see text).

1%2b). The SDs for the other conditions were calculated by averaging the individual variances. The curves in Fig. 3 are parabolas (cr=k(x -a) (b -x) where o=intra-individual standard deviation, x=subjective scale values and k, a and b constants). The parabolas were fitted by minimizing the sum of the squares of the relative deviations Z((y - y ' ) / ~ ) ~ , 0, =empirical and y'=computed value) (cf. Eisler, 1%2b). The rationale for using this criterion was that a, was assumed to be approximately propor- tional toy (cf. McNemar, 1958, p. 81). Fig. 3 demon- strates that a parabola gives a good description for all experiments.

Table 4 gives the parameters of the fitted parabolas. It can be seen that k , the constant of proportionality, exhibits great variation. However, k(b-a);where a and b denote the left and right intersection of the parabola with the X-axis, is relatively constant as is shown in Table 4. (The coeffi- cient of variability for k(b -a) is .1625). This is to say that k is approximately porportional to l/(b -a). No systematic effects of methodological factors on k(b -a) can be seen.

In order to facilitate the interpretation of the


Effect of methodological factors on scale type 27

Table 4. Parameters describing intraindividual S.D.s as a function of subjective scale values (see text)

Experiment k U b k(b-a) A, A*

RWOI a DWOl a DWCl b DNC‘I b RWO*2 RW02 RWC2 DWC2 DNC2 DNC’2 RWC3 RNC3

.01049

.00340

.00302

.03610

.MI76 ,00694 .00341 ,00538 ,05923 ,05025

0.73 -0.62 -5.56 -0.00

0.32 0.07

-0.08 -0.44

0.72 0.57

42.20 124.47 107.32

9.55 161.09 58.66

111.65 103.23

7.48 7.72

,4350 ,4253 ,3409 ,3448 ,284 1 ,4066 ,3810 ,5577 ,4004 .3593

.0656

.I059

. I 1 0 0 ,1528 ,0249 ,0156 .0391 .0243 ,0532 .0776

,3440 .6559 ,1034 .0842 3557 .43 1 I ,0980 ,0557 ,1063 .I552

.00384 2.79 113.01 ,4232 .0745 ,1428 ,02966 -0.05 11.27 .3358 .I384 .I568

DNC’3 .03004 -0.72 12.20 ,3881 .2264 . I538

parameters a and b , two measures Al=(xmin-a)/ (xmax-xmi,J and A,=(b -xmax)/(xmax-xmi,,) were con- structed. xmin and xmax denote the subjective scale values of the weakest and strongest stimulus and A thus shows the difference between a ( b ) and lowest (highest) scale value in normalized form. The two measures are found in Table 4. As to the effect of methodological factors on A, and Az, the following observations could be made (see Table 4).

Rule. No systematic trend for rule can be seen (cf. Conds. RWOI a vs. DWOl a, RWC2 vs. DWC2 and RNC3 vs. DNC3).

Range of numbers. A, is somewhat greater for Conds. DNC2 and RNC3 than for the corresponding conditions in the W-treatment (Conds. DWC2 and RWC2). The same results also obtains for Conds. DNC‘1 b vs. DWCl b and DNC’2 vs. DWC2. How- ever, the trend for A1 vs. range of numbers seems too weak to permit any definite conclusions. No systematic trend can be seen in Az.

Openness. From the values of Az for Conds. RW0*2 and RWC2, which were similar in all factors except openness, it can be seen that (b-x,A is approximately as great as the entire subjective range for scale RWC*2 whereas ( b -x,J for scale RWC2 is only approximately one-tenth of the subjective range. It can also be seen that Az is greater for all open scales than for the remaining scales. Thus openness seems to have affected the value of Az. No systematic trend can be seen for A1 which is not surprising since openness was mainly manipulated in the upper end of the scale.

Discrete vs. continuous set of numbers. The variation in the values of Al, on the one hand, and A$, on the other hand, is rather small for Conds. DNC’2 and DNC2.

Spacing of stimuli. Comparing the value of A, for similar conditions from different spacings, it can be seen that A1 is always smallest for spacing 2. This result may be related to the fact that the physical magnitude of the weakest stimulus was closer to zero in spacing 2 than in the other spacings.

DISCUSSION

The results indicate that the rule given in the instructions for assignment of responses to subjective magnitudes contributes very little or not at all to the non-linear relation between the category and magnitude scale. This is shown by the essentially linear relations obtained between ratio and difference estimation scales. It is worth noting that alinear relation between the two kinds of scales was obtained both when the independent variable was a typical magnitude scale (scale DWOla vs. scale RWOla) and when the dependent variable was a typical category scale (scale DNC’3 vs. RNC3).

The results obtained for range and openness of set of numbers indicate that those factors contribute to the non-linear relation between the category and magnitude scale. When range or openness was made similar to the range or openness of typical magnitude estimation, the resulting scales were related to narrow or closed scales by functions which were essentially concave downward, which is the kind of relation also found between the typical category and magnitude scale. Wide and closed scales, which from the methodological point of view may be regarded as a compromise between typical category rating and magnitude estimation, were found to be also a compromise between the scales obtained from typical category rating and magnitude estimation.


28 H . Montgomery

This finding seems tohold also for the single subject, since plots of individual scales from Conds. RWC2 and DWC2 (not shown here) did npt deviate systematically from the corresponding group scales.

The effect of openness deserves some further attention. Openness actually corresponds t o two factors, viz. the freedom given t o the subject to choose ( a ) a lowest number and (b) a highest number. In all difference estimation experiments, the subject had practically n o freedom t o choose a lowest number withtheexceptionofCond. DWO1 a. In the ratio estimation experiments, the choice of a lowest number was given t o the subject with the restriction that he could not use numbers below zero. The linear relation which was obtained for scales DWC2 vs. RWC2 and DNC3 vs. RNC3, indicates that the freedom t o choose lowest number does not contribute t o the non-linear relation between the category and magnitude scale. The freedom to choose highest number, on the other hand, probably accounted for the curvi-linear relation between the scales from Conds. RWC2 and RW02, since these conditions were clearly different as t o the freedom t o choose highest number, and practically identical in all other respects.

The present study bears some resemblance with a recent study by Gibson & Tomko (1972). The subjects in Gibson & Tomko’s study scaled the apparent intensity of electrotactile stimuli by using one of three scaling procedures: (1) magnitude estimation, (2) category rating using the whole numbers 2-8, or, (3) category rating using the whole numbers 2-50. As in the present study, the wide category scale was related t o the narrow category scale by a function which was concave downward. In contrast to the present study, the wide category scale was linearly related t o the magnitude scale. It should be noted, however, that the subjects did not rate the standard stimuli in Gibson & Tomko’s category rating conditions. I f the numbers assigned t o the standard stimuli are included in the plot of the wide category scale against the magnitude scale (cf. Fig. 4 in Gibson & Tomko’s paper), it can be seen that the number assigned to the upper standard stimulus is much greater than would have been expected from the linear trend in the original plot. This gives rise t o some suspicion as t o whether or not the subjects actually related their judgments t o the number assigned t o the upper standard stimulus. Perhaps the subjects regarded 50 as a n unnatural upper end-point and instead carried out magnitude estimations with


the number assigned t o the lower standard stimulus as their modulus. Further research is evidently needed t o clarify this issue.

The Weber functions in the present experiment seem to have been affected by the same factors that determined the form of the scale. This result sug- gests that the form of the Weber functions may shed some light on the psychological mechanism behind the effect of methodological factors on scale type. However, Weber functions may be subjected t o distortions which are independent of the form of the scale (cf. Eisler & Montgomery, 1974). This seems t o be particularly true for category scales (Mont- gomery & Eisler, 1974). A possible point of de- parture for the study of distortions in Weber functions is the General Psychophysical Differential Equation (GPDE) (Eisler, 1%3, 1%5; Eisler, Holm & Montgomery, 1973) which predicts the relation between subjective scales from their Weber functions provided that the Weber functions are free from distortions (Eisler & Montgomery, 1974). However, the tests of the GPDE have been insensitive to the exact form of the Weber functions (Eisler & Mont- gomery, 1974). Attempts t o test the GPDE with more sensitive methods are now being made in this laboratory and, hopefully, these attempts will pro- vide a common framework for the interpretation of the effect of methodological factors on direct scales and their Weber functions.

This work was supported by the Swedish Council for Social Science Research. Free computer time was made available by the Swedish Office for Rationalization and Economy. I am indebted to Dr Hannes Eisler for valuable suggestions.

REFERENCES Attneave, F. (1962). Perception and related areas. In S.

Koch (Ed.), Psychology: A study of science, vol. 4. New York: McGraw-Hill.

Eisler, H. (1962~). On the problem of category scales in psychophysics. Scand. J. Psychol. 3 , 81-87.

Eisler, H. (19626). Empirical test of a model relating magnitude and category scales. Scand. J. Psychol. 3, 88-96.

Eisler, H. (1963). A general differential equation in psychophysics: Derivation and empirical test. Scand. J. Psychol. 4. 265-272.

Eisler, H. (1965). On psychophysics in general and the general psychophysical differential equation in particu- lar. Scand. J . Psychof. 6 , 85-102.

Eisler, H., Holm, S. & Montgomery, H. (1973). Is the general psychophysical differential equation an approx- imation? [Rep. Psychol. Lab., Univ. Stockholm, No. 386.1

Effect of rnethodologicul fac tors on scale type 29

Eisler, H. & Montgomery, H. (1974). On theoretical and realizable ideal conditions in psychophysics: Magnitude and category scales and their relation. Perr. Psychoph. 16, 157-168.

Ekman, G., Goude, G. & Waern, Y. ( l % l ) . Subjective similarity in two perceptual continua. J. exp. Psycho/.

Engen, T. & Levy, N. (1955). The influence of standards on psychophysical judgments. Percept. Mot. Skills 5 ,

Garner, W. R. (1954). Context effects and the validity of loudness scales. J. exp. Psychol. 48, 218-224.

Gibson, R. H. & Tomko, D. L . (1972). The relation between category and magnitude estimates of tactile intensity. Perc. Psychoph. 12, 135-138.

Hellman, R. P. & Zwislocki, J. (l%l). Somefactorsaffect- ing the estimation of loudness. 1. Acousr. Soc. Amer. 33, 687-694.

Marks, L. E. (1%8). Stimulus-range, number of categories, and form of the category scale. Amer. J. Psycho/. 81. 467-479.

McNemar, Q. (1955). Psychological statistics. New York: Wiley.

Montgomery, H. & Eisler, H. (1974). Is an equal interval scale an equal discriminability scale? Perc. Psychoph. 15, 441-448.

Poulton, E. C. (1968). The new psychophysics: Six models for magnitude estimation. Psychol. BUN. 69, 1-19.

Pradhan, P. L. & Hoffman, P. J . (1963). Effect of spacing and range of stimuli on magnitude estimation judgments. J. exp. Psycho/. 66, 533-541.

Stevens, J. C. (1958). Stimulus spacing and the judgment of loudness. J. exp. Psychol. 56, 246-250.

Stevens, S. S. (1957). On the psychophysical law. Psychol. Rev. 64, 153-181.

Stevens, S. S. & Galanter, E. H. (1957). Ratio scales and category scales for a dozen perceptual continua. J . exp. Psychol. 54, 37741 I .

Strangert, B. (I%]). A validation study of the method of ratio estimation. [Rep. Psychol. Lab., Univ. Stock- holm, No. 95.1

Strangert, B. (1965). Invariansproblematik och direkta psykofysiska skattningsmetoder. Unpublished thesis for thefil. lic. degree, Department ofpsychology, Univ. Stockholm.

Torgerson, W. S. (1960). Quantitative judgment scales. In H. Gulliksen & S. Messick (Eds.), Psychological scaling: Theory and applications. New York: Wiley.

Torgerson, W. S. ( l%l) . Distances and ratios in psychophysical scaling. Acra Psycho/. 19, 201-205.

Treisman, M. (1%4). Sensory scaling and the psychophysical law. Quarterly J. exp. Psychol. 16, 11-22.

Warren, R. M. & Poulton, E. C. (1%2). Ratio and partition judgments. Amer. J . Psycho/. 75, 109-1 14.

61, 222-227.

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Postal address:

H. Montgomery Dept. of Psychology University of Goteborg Fack S-40020 Goteborg 14, Sweden

Scond. J . Psychol. 16

direct estimation: effect of methodological factors on scale tupe

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