The step method: A new adaptive psychophysical procedure

Transcription

1 Perception & Psychophysics 1989, 45 (6), The step method: A new dptive psychophysicl procedure WILLIAM A. SIMPSON York University, North York, Ontrio, Cnd A new dptive psychophysicl method, the step method, is introduced. Simultions show the method to be less bised nd more efficient thn constnt stimuli or Pentlnd's dptive method for fewer thn 40 trils. An experiment using discrimintion of dot number, however, filed to find ny differences mong the three methods in either bis or efficiency. In the lst 20 yers there hs been considerble interest in improving the efficiency of psychophysicl mesurement (Emerson, 1986; Levitt, 1970; Liebermn & Pentlnd, 1982; Tylor & Creelmn, 1967; Wtson & Pelli, 1983). The min ide behind the new methods is tht trils should be plced t the experimenter's current estimte of the threshold. Such methods re termed dptive methods becuse the rnge of stimuli presented dpts to the subject's responses. In contrst, the clssicl method ofconstnt stimuli presents the sme set ofstimuli no mtter how the subject responds. This pper hs two ims. First, I will present new dptive method which hs dvntges over some others currently in use. Second, I will give both simultion nd experimentl dt on the performnce of this method s compred with constnt stimuli ( nondptive method) nd Pentlnd's mximum-likelihood dptive method (Liebermn & Pentlnd, 1982; Pentlnd, 1980). THE STEP METHOD The step method is similr to Pentlnd's mximumlikelihood estimtion (MLE) technique. However, there re twomin differences. First, the ssumed psychometric function is step function insted of logistic (Eqution 1). Second, the step method uses lest squres insted of mximum likelihood s the criterion for fitting function to the response dt. The rtionle for using the step function is tht Pentlnd's method fits only one ofthe two prmeters ofthe logistic function; his method ssumes fixed vlue for the slope. Ifwe re fitting only one prmeter, we might s well use psychometric function with just one prmeter- step hs vlue of0 below threshold nd 1 bove, with the threshold being the only prmeter. The dvntge here is tht there cn be no errors in the ssumed vlue of the slope. Such errors hve been shown (Emerson, 1984; Simpson, 1988) to in- Correspondence my be ddressed to Willim A. Simpson, Deprtment ofpsychology, York University, 4700 Keele Street, North York, Ontrio M3I 1P3, Cnd. crese both the bis nd the vribility of the resulting threshold estimtes. Becuse the step is discontinuous function, mximum likelihood cnnot be used; thus lest squres, which is symptoticlly equivlent to mximum likelihood, is used insted. After ech tril, which will result in 0 or 1 y vlue (in forced-ehoice methods, incorrect or correct responses, respectively), step is fit by lest squres tothe dt points gthered thus fr. The loction ofthe step is the threshold; this x vlue is presented on the next tril. A BASIC progrm tht implements the step method for two-lterntive forced-ehoice (2AFC, the cse considered in this pper) is given in Figure 1. The number of trils nd the upper nd lower limits ofthe estimted threshold region re input. In the progrm, 21 stimulus levels re used. First the x vlues for the 21 levels re clculted nd stored in the X rry; simultneously the sum of the squred error (SSE) rry is set to O. The current threshold estimte is X(M%). Initilly, M% is set to the midpoint of the rnge. The progrm presents X(M%), collects the response, then clcultes M% for the next tril. The min work is in the clcultion of M%. The progrm uses n updte method similr to tht of Liebermn nd Pentlnd (1982). Ech stimulus level is successively tken s possible threshold loction. If the just-tested x level is greter thn this threshold loction nd the response is incorrect, then the SSE rry t tht level is incremented by WI. Conversely, ifthe just-tested x level is less thn this threshold loction nd the response is correct, then the SSE rry t tht level is incremented by W2. Becuse the responses re 0 or 1, the squred devition will be 1 in either cse. The squred devitions re multiplied by weights. The weights will be WI = trget percentge nd W2 = 1 - trget percentge. As long s the weights re in the proper proportion, ny vlues cn be used (lthough integers will be fster). Here the trget percentge is 75 %; therefore WI =.75 nd W2 =.25 (or 3 nd 1, respectively). The level with the lowest SSE is indicted by Pl. If the SSE function hs flt minimum, theotherendpointofthe vlley is indictedby P2. The vlue of M% is clculted s the point 90% of the wy between PI nd P2. Simultions hve shown tht Copyright 1989 Psychonomic Society, Inc. 572

2 , step.method, psychophysicl procedure bsed on lest squres, fit of step function to 1/0 dt, currently set up for 2AFC--chnge weights w1 nd w2 for, other trget \s: 2 weights dd up to 1.0 (or use integer rtio), e.g..5,.5 = 1,1.75,.25 = 3,1.8,.2 = 4,1 DIM x(20), sse(20) set.windows: WINDOW 1, "step.method", (0,0)-(617,185),31 I nl\ = 20 : w1=3 : w2=1 CLS INPUT "~ of trils";n\ INPUT "lower limit";ll INPUT "upper limit";ul j=(ul-lll/nl\ FOR! = 0 TO nl\ xli) = 11 : 11 = ll+j sse ( i) = 0 NEXT i m\ nl\l2 END FOR tr PRINT INPUT GOSUB NEXT tr 1 TO n\ x(m\) "response (1/0)";ys\ clc clc: mn = FOR i = 0 TO nl\ IF x(m\) )= xli) AND ys\ = 0 THEN sse(!) IF x(m\) <= xli) AND ys\ = 1 THEN sse(i) IF sse(!) < mn THEN mn sse(!) pl=! IF sse(!) = mn THEN p2 = i NEXT! m = pl + (p2-pl)*.9 m\ = INT(m+.5) RETURN sse(!) + wl sse (!) + w2 Figure 1. An Amig BASIC progrm tht implements the step-dptive method for twolterntive fon:ed-choice (2AFC). THE STEP METHOD 573 for 2AFC this vlue results in miniml bis. For trget percentge of 50%, use the vlue.5 (the vlue will lwys be between 0 nd I). The lgorithm is very fst (BASIC is more thn dequte) nd is not prey to rounding or out-of-rnge errors. Pentlnd's method cn blow up due to either the exponentil or the log functions. Emerson's (1986) Byesin method is too slow to be implemented in BASIC. It is hrd to evlute Emerson's method, however, since he hs not published the lgorithm. SIMULATIONS The conditions ofthe simultions were s follows. The simultion dt for constnt stimuli nd Pentlnd's MLE method were determined s previously described (Simpson, 1988). It ws ssumed tht the psychometric function ws the 2AFC scled logistic: y =.5 +.5/{1 +exp[*(b-x)]}, (1) where (the slope) ws Ilogit nd b (the threshold) ws ologits (p =.75). The subject's responses were modeled s conforming to this function nd hving Bernoulli vrince. For constnt stimuli nd Pentlnd's method, slope estimte is required to clculte likelihoods; the dt here were obtined using the true slope ( vlue of 1). In using ny of the methods, n experimenter hs to specify n upper nd lower bound for the threshold region. This ws simulted by rndomly choosing n upper bound between 0 nd R logits nd lower bound between 0 nd -R logits (R ws 2.5, 5, or 10). The stimulus levels hd n verge rnge of R, with minimum pproching 0 nd mximum rnge of 2R (the vlues of the rnge given in the figures). On verge, the trils were plced symmetriclly bout the threshold; t their most symmetricl they could ll be on one side of threshold. Results The simultion results for the bis of the threshold estimtes re given in Figure 2. For constnt stimuli nd Pentlnd's method, bis declines with the number oftrils nd increses with the rnge of stimuli. The step method is unbised, showing only smll, rndom fluctutions

3 574 SIMPSON '0, en 0 constnt stimuli c en step -0.9 Pentlnd Rnge: 5 logits Trils ~ Q - '0, -0.3 en c en -0.9 Rnge: 10 logits Trils - '0, -0.3 en c en -0.9 Rnge: 20 logits Trils Figure 2. Bis of the threshold estimte (in logits) s function ofthe number oftrils nd the stimulus rnge for constnt stimuli, the step method, nd Pentlnd's mximum-likelihood estimtion (MLE) method. Ech point is bsed on 500 simultion runs. round the true threshold vlue; constnt stimuli nd Pentlnd's method show progressively more bis s the rnge increses. Pentlnd's method seems especilly prone to negtive bis with lrge stimulus rnges. As cn be seen in Figure 3, the stndrd devition of the threshold estimte declines s the rnge decreses nd s the number of trils increses. The vribility of ll three methods is quite similr. The step method hs n dvntge for runs ofless thn 40 trils. For runs ofmore thn 40 trils, Pentlnd's method is most efficient, lthough it is not much better thn constnt stimuli. Discussion Overll, the methods re quite similr in performnce. In previous study (Simpson, 1988), I found tht the constnt stimuli method ws bout s efficient s Pentlnd's method. This finding is confirmed here using different dptive method. The step method hs n edge in efficiency for smll numbers of trils, nd it is unbised. In the ltter respect it is similr to Emerson's (1986) Byesin method. However, both Pentlnd's nd Emerson's methods require n estimte of the psychometric function's slope. In the cse of Pentlnd's method, it is known tht errors in slope estimtes will dversely ffect the bis nd vribility of the threshold estimtes (Emerson, 1984; Mdign & Willims, 1987; Simpson, 1988). The step method voids these problems. The step method's superior efficiency for smll numbers oftrils is probbly due to the fct tht the step function will fit the dt perfectly for smll numbers oftrils, but the logistic will not. As the number oftrils increses, however, the fit of the step becomes incresingly worse thn the logistic. It is not cler whether this would be problem in experimentl situtions, since typicl use of dptive methods is to tke the men nd stndrd devition of severl blocks of 40 or so trils. It is not good ide to use lrger blocks; if the stimulus rnge is inpproprite, thethresholdwill simply hit ceilingor floor. 0 constnt stimuli c step 1.6 Pentlnd.~ Rnge: 5 logits c 0.4 cc U5 Trils c 1.6.~ ~ Rnge: 10 logits c 0.4 c c U5 Trils Rnge: 20 logits g c 1.6 ~ 0.8 "E c 0.4 cc U5 Trils Figure 3. Stndrd devition ofthe threshold estimte (in logits) s function ofthe number oftrils nd the stimulus rnge for constnt stimuli, the step method, nd Pentlnd's MLE method. Ech point is bsed on 500 simultion runs. =i

4 THE STEP METHOD 575 This sme observtion holds for constnt stimuli; severl blocks of bout 40 trils should be used. In order to test the perfonnnce ofthe different psychophysicl methods in rel sitution, the following experiment ws performed. EXPERIMENT The step method, Pentlnd's MLE dptive method, nd constnt stimuli were compred in n experiment on discrimintion of the number of rndomly plced dots. In other studies tht compred different psychophysicl methods (Hesse, 1986; Mdign & Willims, 1987; Shelton, Picrdi, & Green, 1982), the threshold for only one experimentl condition ws obtined. Typiclly, however, we re interested in determining threshold chrcteristic-the threshold s function of some prmeter. It is importnt to test psychophysicl methods in their determintion of threshold chrcteristic becuse bis in method cn distort the shpe ofthe chrcteristic (Lming, 1986). In this experiment, therefore, I determined the difference threshold for dot number t five points for ech of the three methods. In the present cse ofdot-number discrimintion, it hs been estblished by Burgess nd Brlow (cited in Lming, 1986) tht the threshold chrcteristic (here, the discrimintion function) is power function: difference threshold = k * dot number e. (2) The effects of bis will be reflected in the obtined vlues of k or e, or in deprtures of the dt from stright line on log-log coordintes. The efficiencies of the methods will be reflected in the stndrd devitions of their threshold estimtes for given number of trils. Method Subjects. The uthor nd nive observer, both with norml or corrected-to-norml vision, served s subjects. Stimuli. On ny given tril, the disply consisted of two fields (left nd right) of rndomly plced dots on CRT screen. Ech field's re ws 131 x 180 pixels. The subject viewed the disply from chinrest 57 cm from the screen; t this distnce, the dots were.05 squre nd the fields were 7 (horizontl) x 9 (verticl). The subject fixted.3 -squre mrk in the center of the screen. The inner edge of ech dot field ws I wy from the fixtion mrk. Five different numbers of dots (10, 20, 40, 80, nd 160) were used s the stndrd. The comprison field in ech tril contined more dots. The loctions of the comprison nd the stndrd were rndomly vried between trils. The subject's tsk ws to press button corresponding to the field tht contined the greter number of dots (the comprison). Procedure. Ech block consisted of40 trils. On ech tril, two dot fields were presented for 540 msec. The subject pressed button corresponding to the field tht contined the greter number of dots. There ws I-sec puse between trils. The three methods were given in rndom order. All vlues of the stndrd were given using one method before using the next. The level of the stndrd ws determined rndomly. In ll, four mesures were mde for ech level of the stndrd using ech method. Ech dtum point is thus bsed on 160 trils. Three psychophysicl methods were used to determine the difference thresholds: constnt stimuli, the step method, nd Pentlnd's MLE method. Since Pentlnd's method requires n estimte of the slope of the psychometric function, two different slope estimtes were used (I nd.2). Ech method required n upper nd lower limit for the threshold estimte. The dptive methods were s described bove. The constnt-stimuli method plced ech tril rndomly within the specified limits. The threshold estimte for the two dptive methods ws the vlue fter the lst tril. The threshold estimte for constnt-stimuli method ws clculted using lestsqures fit of 2AFC logistic function to the dt. The stndrd devitions of the thresholds were bsed on the vribility of the vlues given by the four 4O-tril blocks. Results The dot-number discrimintion functions re plotted in Figure 4. For both subjects, the dt re well fit by stright lines Oest-squres fits) on the log-log plots. Hence, the dt re described by the power function s given in Eqution 2. The vlues ofthe prmeters s determined by lest squres were.2576 (A.H.) nd.8444 (W.S.) for k, nd.8878 (A.H.) nd.5815 (W.S.) for e. 100 "'0 o (f) -~ 10 c -- i:5 "'0 o (f) o constnt stimuli o step Pentlnd slope=1 Pentlnd slope= Number of dots Number of dots AH 160 ws 160 Figure 4. DHferem:e tbresboids for dot number obtined using c0nstnt stimuli, the step method, nd Pentlnd's method. Ech point is the men of four 48-tril blocks. The stright lines re the lestsqures fits of power function to the dt.

5 576 SIMPSON Subject A.H. W.S. Tble 1 Stndrd Devition of Threshold Estimte s Function of Psychophysicl Method Constnt Stimuli Psychophysicl Method Pentlnd Step Men Note tht the prmeters re not identicl for the 2 subjects (unlike Burgess & Brlow's [cited in Lming, 1986] result). As Burgess nd Brlow found, the exponent of the discrimintion function is just under 1 (n exponent of 1 would be predicted by Weber's lw). Only single curve is plotted for ech subject's dt in Figure 4 becuse n ANOVA indicted no difference between the discrimintion functions s mesured by the different psychophysicl methods. Tht is, method ws neither significnt min effect [A.H., F(3,36) = 1.482, n.s.; W.S., F(3,36) = 2.466, n.s.] nor did it interct with number of dots [A.H., F(l2,36) =.344, n.s.; W.S., F(12,36) =.557, n.s.]. Thus, there ws no significnt bis ofthe psychophysicl methods reltive to one nother. An ANOVA compring the stndrd devitions of the threshold estimtes for the methods lso found no significnt differences [A.H., F(3,12) = 1.585, n.s.; W.S., F(3,12) = 2.115, n.s.]. As cn be seen in Tble 1, the stndrd devitions for the different methods re very similr. Discussion The three psychophysicl methods ll seem to give threshold estimtes tht re bout the sme in both bis nd vribility. No evidence ws seen for ny method being more bised or less efficient thn ny other. If this experiment cn be regrded s typicl, the differences in bis nd vribility between the methods s found in the simultions re hrd to detect when using rel subjects. Other experiments tht hve compred psychophysicl methods lso found the methods to be more-or-iess equivlent (Hesse, 1986; Mdign & Willims, 1987; Shelton et l., 1982). In this experiment, the rnge of presented stimuli ws probbly smll (s it likely is in ll experiments). The simultion results for bis (Figure 2) nd for stndrd devition (Figure 3) with rnge of5 logits show only smll differences between the methods t 40 trils. The messge from the experiment, then, s it ws from the simultions, is tht there is little, if ny, difference in efficiency between constnt stimuli nd dptive methods. Moreover, the differences in bis re probbly too smll to detect under norml (smll-rnge) circumstnces. in efficiency between the clssicl method of constnt stimuli nd the newer, dptive methods. According to the simultions, the step method is slightly more efficient thn constnt stimuli for smll nwnbers oftrils, nd Pentlnd's method is more efficient thn constnt stimuli for lrge nwnbers of trils (the crossover point is bout 40 trils). The experimentl dt, however, show no difference in efficiency for the methods. Although the step method ws shown in the simultions to be less bised thn constnt stimuli or Pentlnd's method, the experiment reveled no differences in reltive bising. Both the simultions nd the experiment show the step method to work quite well, despite its use ofn unrelistic psychometric function. It is hertening tht one need not be overly concerned bout which psychophysicl method one chooses. The threshold estimte will tke on the sme vlue nd hve the sme vribility no mtter which method is used. However, we should ber in mind tht in n experiment we do not know the vlues ofimportnt vribles, wheres these vribles cn be mnipulted in simultions. In the simultions, the step method is shown to be less bised nd to be slightly more efficient thn constnt stimuli or Pentlnd's method. Even in the simultions, though, these differences re smll. REFERENCES EMERSON, P. L. (1984). Observtions on mximum likelihood method ofsequentil threshold estimtion nd simplified pproximtion. Perception &: Psychophysics, 36, EMERSON, P. L. (1986). Observtions on mximum-likelihood nd Byesin methods of forced-ehoice sequentil threshold estimtion. Perception &: Psychophysics, 39, HESSE, A. (1986). Comprison of severl psychophysicl procedures with respect to threshold estimtes, reproducibility, nd efficiency. Acustic, 59, LAMING, D. (1986). Sensory nlysis. Toronto: Acdemic Press. LEVITr, H. L. (1970). Trnsformed up-down methods inpsychophysics. Journl ofthe Acousticl Society ofameric, 49, LIEBERMAN, H. R., '" PENTLAND, A. P. (1982). Microcomputer-bsed estimtion of psychophysicl thresholds: The best PEST. Behvior Reserch Methods &: Instrumenttion, 14, MADIGAN, R., '" WILUAMS, D. (1987). Mximum-likelihood procedures in two-lterntive forced-ehoice: Evlution nd recommendtions. Perception &: Psychophysics, 42, PENTLAND, A. (1980). Mximum likelihood estimtion: The best PEST. Perception & Psychophysics, 28, SHELTON, B. R., PiCARDI, M. C., '" GREEN, D. M. (1982). Comprison ofthree dptive psychophysicl methods. Journl ofthe Acousticl Society ofameric, 71, SIMPSON, W. A. (1988). The method of constnt stimuli is efficient. Perception &: Psychophysics, 44, TAYWR, M. M., '" CREELMAN, C. D. (1967). PEST: Efficient estimtes on probbility functions. Journl ofthe Acousticl Society of Americ, 41, WATSON,A. B., ",PELU, D. G. (1983). QUEST: A Byesin dptive psychometric method. Perception &: Psychophysics, 33, GENERAL DISCUSSION The conclusion to be drwn both from the simultions nd from the experiment is tht there is little difference (Mnuscript received July 11, 1988; revision ccepted for publiction November 28, 1988.)