A mathematical model of performance on a simple reaction time test

Ncurotoxrcology

and Teratology. Vol. IX. No. 5, pp. 5X7-.593. 1996 Copvright a’ 1YYhElsevier Science Inc. Printki in the USA. Ail rights reserved 0892.0.162196 $ IS.00 + .O(I

PIISO892-0362(96)00077-3

ELSEVIER

A Mathematical Model of Performance Simple Reaction Time Test EDWARD Division

of Biomedical

F. KRIEG.

JR.,

and Behavioral

Received

DAVID

W. CHRISLIP

AND

JOHN

M. RUSSO

Science, Nationrrl Institute for Occupational Cincinnati. OH 45226

7 February

199.5:Accepted

27 January

on a

Safety cmd Health,

1996

KRIEG. JR., E. F.. D. W. CHRISLIP AND J. M. RUSSO. A rrrr/rl~,~ricnl n~tiel of/~fi~rr~~c~c o/r t, vi,rt/r/c rcrcc.rior~ ti,rre rest. NEUROTOXICOL TERATOL 1X(S) 5X7-93, 1906.-A nonlinear function with components for learning and fatigue was used to model individual performance on a simple reaction time test. The relationships between the parameters 01 the model and the mean and variance of the reaction times are discussed. The function is used to analyze data from a field study of agricultural workers exposed to organophosphatc pcsticidcs. Exposure had a significant effect on the relationships between education level and initial performance, age and fatigue rate. and age and performance variability. Parameter cstimates from the model were able to distingtish hctwcen effscts that the mean and standard deviation of the reaction times were unable to distinguish.

Simple reaction

time

Nonlinear

model

Organophosphate

A recent review of behavioral testing in the workplace reported that tests of reaction time have been employed in more than 80 workplace studies (4). These studies have revealed that reaction time performance is affected by many substances encountered in the workplace, including lead (19.22) mercury (5) manganese (34) pesticides (43). carbon disulfide (25,41). styrene (15.29) anesthetic gases (16). and solvent mixtures (9,26). The usefulness of the reaction time test as a tool for workplace testing has led to its being incorporated in a number of test batteries, including the Finnish Institute of Occupational Health (FIOH) Battery (17). the World Health Organization’s Neurobehavioral Core Test Battery (WHO NCTB) (21) the London School of Hygiene Battery (10). the Williamson Battery (42) the Armed Forces Cooperative Performance assessment Battery (UTC-PAB) (13). and the Ncurobehavioral Evaluation System (NES) (24). A number of individual characteristics are also known to affect reaction time. These include age (1,38), sex (8,28), degree of physical fitness (37) mental retardation (6) fatigue (18) and sleepiness (12). Reaction times can also be affected by physical activity (38), alcohol intake (12,30). and cigarette smoking (30). Reaction time data are typically analyzed by dropping out the reaction times of the initial trials and calculating the mean and

_

Kequests tor reprtnts should be addressed cupational Safety and Health, 4676 Columbia

pesticides

Intrnclass

regression

model

SD of the remaining ones. Information is lost about the rate of acquisition of the reaction time response as well as any other trend over trials. The pattern of response over trials can vary substantially from subject to subject. Some subjects acquire the response skill more rapidly, some subjects show a gradual increase in reaction times as the number of trials increases, and the performance of some subjects is more variable than others. The trends in reaction time over trials can be modeled mathematically. Parameters that reflect these trends can be estimated and used as variables in analyses done to determine the effects of toxic exposure, age. or other variables. A nonlinear model is described below: it is compared to the mean and variance of the reaction times, and it is applied to data from a field study of agricultural workers exposed to organophosphate pesticides. DESCRlPTlON

OF THE

MODFI.

The model has two components, one to account for learning (the initial decrease in reaction time) and one to account for fatigue (the gradual increase in reaction time over trials). The reaction time at trial t can be written as a product of these two components: R(t) = L(t) X F(t).

to Edward F. Krieg. Jr.. Division of Biomedical Parkway. MS C-22. Cincinnati. OH 45226.

and Behavioral

Science.

National

Institute

for Oc-

58X

KRIEG.

The learning

function

is

L(I) = L~~(L~-I,,)(l

-/)‘--’

for I 2 I and 0 5 I 5 I. 1 is the Icarning rate. L, i5 the initial performance level, and I.,. is the performance limit. If I = 0 thcrc is no learning, and if I = I there is one-trial learning: maximum performance is reached by the second trial. L(r) is undefined when I= 1 and t = I. The fatigue function is i-(r) = ( I + ,l)‘_ ’ for t 2 I and .f” 0. f‘is the fatigue rate. If f’= 0. there is no fatigue. If /‘> 0. there is no fatigue on the first trial, but as the trial5 progress fatigue increases without bound. Combining the two functions one gets

R(t) is undefined

when I= I and I = I.

RI:LA’I IONSIIIPS

A statistical

version

K)

THE’ MEAN

AND

VARIANCE

oc the model can be written

as

R, = R(t) + c, . where f, is a variable representing a random disturbance at time t. If E[e,] = 0, then E[R,] = R(f). Var(R(r)] = 0. so Var[R, ]= Var[F,]. The sample mean c&the reaction times over a set of T trials has expected value E[R] = (l/r) C R(t) and variance

By examining the equation for R(r) one can see that x increases as I__, increases. as L, increases relative to I,.,,. and as f incrcascs. R decrcascs as 1 increases. The sample variance of the reaction times. using a divisor of 7‘ - I. has expected value

CHRISLIP

AND

RUSSO

with organophosphate poisoning were accompanied by documented choline,terase inhibition. An additional 46 cases of probable organophosphate poisoning lacked evidence of cholincstcr-ase inhibition, but exhibited compatible symptoms accompanied by spccificd medical signs or by a history of direct overexposure to an ol-ganophosphatc pesticide. Another 45 subjects wcrc idcntificd as having had one or more prior cpisodes of cholincsterase inhibition without symptoms of posoning. Ninet) J adult male friends and neighhot-s of the cxposed casts who were not currently exposed to pesticides constituted the noncxposcd control group. The sub.jects in the study wcrc given an asses5mcnt battery that included a clinical neurological examination. evaluation of ncrvc conduction velocities. measurement of’ vibrotactile thresholds. and computerized tests of postural balance. mood. finger tapping speed, sustained visual attention. hand-eye coordination. symbol~dipit matching. short-term pattern mernory. short-term serial digit learning. and simple reaction time. The simple reaction time test was from the NES. In this test. the subject is asked to pi-cs a button coch time a large square appears on the computer screen. The number of trials can be set by the test giver and is typically set between 50 and 100 trials (in this study subjects wcrc given 90 trials). ‘l‘hc intcrtrial interval varies randomly between 2.5 and 7.5 s to prcvent the subject from anticipating the next stimulus. Stccnland et al. (39) analyzed the definitely and probably poisoned groups, but not the group with inhibIted acetvlcholincstcrasc and no symptoms. This group was included -in the present analysis. Twelve subjects from the dcfinitc group, thl-cc l’i-om the probable group, two from the inhibited go-oup, and four controls did not have reaction time data. Atlditionally. data from ~OLITdefinite. four probable. and eight control subjects were not included in the analysis because their data was collected during pilot testing.

The reaction times (u,) of individual subjects were fit using the Gauss-Newton algorithm [all calculations were done with the SAS System (SAS Institute. Inc., Gary, NC)]. The first rcaction time (r,) was used as a starting value for f>,. the arithmetic mean of reaction times I I through 60 was used as an initial estimate of L _,,and d was calculated as d = L,, - L,. The initial value of the learning rate was calculated as I= (rI ~ I,,)/ (I,,, --- L,), and the initial value for the fatigue rate was calculatcd as

If I = 0 and .f’= 0 then R(f) is constant and will not contributc to .S$ otherwise the first term to the right of the cqual sign will bc greater than 0 and S$ will contain variability due to the trend. AN EXAMPLE The data used here are from a study of agricultural workers exposed to organophosphate pesticides (39). Workers with medically documented episodes of acute poisoning by one or more organophosphate pesticides were examined 2-10 years after apparent recovery for signs of residual neurobehavioral deficits. The study subjects were selected from medical surveillance records maintained by the State of California Department of Food and Agriculture, and were restricted to casts in which males, age I6 years or older. had sought medical attention after a definite or probable episode of organophosphate pesticide ovcrcxposure. A total of 83 definite poisoning cases were identified in which symptoms compatible

Only values greater than 100 ms and less than 700 ms were included in the calculation of the initial values of L,, and f Conditional statements were used to adjust extreme initial values: if I was less than 0.2 or greater than I, then it was set to 0.6: if d was greater than -SO, then d and I were set to 0: and if .f’was less than or equal to 0 or greater than or equal to 0.01, then it was set to 0.001. Constraints were also placed on the parameter estimates: I+ 2 100, d 5 0.0 5 15 I, and ,f 2 0. A Ljung-Box (27) test was performed on the residuals of each of the fits. This test indicated significant autocorrelations for 21% of the subjects. The correlations for lags greater than zero ranged from -0.33 to 0.57. Their average was -0.01. Examining the autocorrelation functions and fitting autoregrcssive-moving average models to the residuals indicated that

REACTION

TIME

589

MODEL

some

of the residuals varied cyclically. some appeared to be an autoregressive process (negative and positive). and some were a combination of both; about one-third of the 21 ‘Y/owere in each category. We did not develop a model for the residuals because of the size and inconsistency of the correlations. and because correlated residuals have been shown not to bias parameter estimates (11). The standard error of the estimate (SEE) was calculated for each regression and was used as a measure of performance variability. For purposes of comparison. the mean (R) and SD (S,() of the last 80 reaction times of each subject were also caculated. Examples of fitted regression lines are shown in Fig. 1.

A linear model was used to test for the effect of exposure, the linear effects of age and education level (grade). as well as age X exposure and grade X exposure interactions on the parameter estimates. The model is called an intraclass regression model because a different regression is assumed for each class or Froup (36). The equation for the model can be written as I = 23

f = .aB7

SEE = 59.357

y/, = P + a, + PJ,,,

+ D’,Zl,,+El,

I’,, is the value of the dependent variable for the jth subject from the ith group. Z,,, and Z,,, are the values of the covariates age and grade. and E,, represents a random disturbance. p. + 01, is the intercept for the ith group. and p,, and pz, are slope coefficients. The design matrix for this model included a column of ones for the intercept and four columns to code for the groups. A sub,@_? was coded as I if’ he was in the group or 0 otherwise. Age and grade were included as continuous variablcs and columns for the interactions were calculated by multiplying the age and grade columns by the group columns. If a main effect or an interaclion was sigiificant. contrasts were done to compare the groups. If an interaction was significant, the slope coefficient of each group was tested to see if it was significantI\ different from xro.

Summary

shown

statistics

in l‘ablc

for the variables usccl in the analysis are I. a summary table of the linear models is pre-

I = .67

I = .w16 SEE = 9.483

I = 1.03

f = NH4

1303

I = .87 I = .Moo SEE = 105.470

SEE = 41.886

‘9

3

IO

a,

30

40

TM

50mmm9lolo20mommmmm

W

FIG. I. Examples of data from individual suhjzcts and their fitted regression linrs (solid). 95% prediction intervals (dash) and confidence intervals (dot) al-e alw included. I is the estimated learning rate. f’is the estimnred faiiguc rate. and SEE is the standard error of the estimate.

KRIEG. TABLE SIJMMARY

7x 43 61 3’) 7X 43 67 .TJ 7x 4.3 67 34 7x 4.3 67 ?J 7X 4.3 07 3’) 7x 43 67 3’) 7X 4.; 67

STATISTICS

AND

RUSSO

I

BY EXPOSIJRE

GROIJP

713 792 758 700 263 264 277 231 0.640.5 U.h4X3 0.6593 0.6740 0.00 144x 0.0012Y2 0.0015Y1 O.OOI7X2 72.40 70.37 76.70 67.X9 2X.? 2X.5 301 26-l 70 hS 76

17x WJ 23’)

I Iii6

312 I75 lOI) IX6

I I50

I IlJh I 1x8

IO0 0.0000 (1.0000

1l.0000 0.0000 0 0 0 0 20.Y4 24.6s 23.71 IS.16 202 20X 210 I Y’J IY

i .i1000 I 01)00 I .01)00 1H105020 Il.01 I I IO 0.00762 I 1~.013015 I7X.Y7 I ‘J7.hI 222.1 I .3O.‘JU 35s JJ.5 71lh JWJ 176 I’JI)

74

2.M

I ;i

65

IS

7s

?‘).‘J

-13

ix.1

67 IV 77 43 65

32.X ‘5.2 10.7 10.7

.YJ

12.7

I7 I ‘J 20 ‘I .J 1 0 (3

‘J. I

-16X 422 5X33 377 I SJIJOO

12

30

sented in Table 2, and regression coefficients are shown in Table 3. For L, thcrc was a significant grade X exposure interaction. The slope of the control group was significantly greater than the inhibited @ = 0.0372) and definite (p = 0.0046) groups. The slope of the definite group was significantly less than zero (~1= 0.0019) (see Fig. 2). Thcrc were no statistically significant effects for L, and 1. The age X exposure interaction was significant for ,f The slope of the definite group was significantly greater than the control (J = 0.01 I I) group, and it was also significantly greater than zero 0) = 0.0107) (set Fig. 2). For SEE there were significant age X exposure and grade X exposure interactions. For age, the slope of the definite group was significantly greater than the control (~9 = 0.000X). inhibited 0, = 0.0161). and probable (~1 = 0.0007) groups. The slope of the definite group was significantly greater than zero (I) = U.0066) (see Fig. 2). For grade. the slope of the probable group was significantly less than the control 0~ = 0.0044). inhibited (p = 0.0080). and definite @ = 0.0103) groups. and it was significantly less than zero (JJ = (X)$13). The effect of grade was significant for R, as the grade increased the mean reaction time decreased. The age X expo-

CHRISLIP

67 67 (10 hi IS I (1 Ii I7

sure interaction was also significant. The slope of the definite group was significantly greater than the control @ = O.OOSO), inhibited 0, = O.OOSS),and possible (11= 0.0220) groups, and it was also significantly greater than Lero 0, = 0.0009). For SK the age x exposure interaction was significant. The slope of the definite group was significantly greater than the control o-) = (1.0004), inhibited (II = 0.005~1). and probable 0, = 0.0016) groups, and it was significantly greater than zero (p = 0.0012). The grade X exposure interaction was also significant The slope of the probable group was significantly less than the control 0, = 0.0076). inhibited (r, = 0.0077), and definite (r> = O.Ol.lS) groups. and it was significantly less than zero @ = 0.0023). IIIS(‘l!SSIC)N

The components family of exponential

of the performance model equations of the form c;(t)

come from a

= A + B( 1 + ,)‘l(‘)

which can be used to model other types of performance. continuous the natural exponential form can be used: G(t)

= A + BP,“(‘)

If f is

REACTION

TIME

591

MODEL TABLE LINEAR

MODEL

2

SCJMMARY

TABLE

Age Grade Azc X c~posure C;riKte x exposure Exptrsurc Age Grade Age x evposure

II? 212 212 212 ?I? 212 ?I2 212 ?I? 212 213 212 212 212 ?I? 212 212 ?I2 212 112 212 712 212 212 212 212 112 212 112 712 ?I2 21-2 ?I2 ?I?

Grade

?I2

Expoaurc A% Grade Age x exposure Grade x exposure Exposure A_ee Grade Age x exposure Grade x exposure

Expo\ure A@t! tirade Age X exposure Grade x exposure EXpOSW Age Grade Age x exposure Grade x exposure Exposure Age Grade Age x exposure Grade x exposure Expowrc

x exposure

Aldridge (3) has used a natural exponential equation as a model of mastery learning. In either case, r represents a rate and A and B are scaling parameters. A, B. r. and h(t) can be positive, negative, or zero. Hyperbolic (40) and power (31) functions have also been used to fit learning curves. By equating these functions to L(t), one can show that the hyperbolic function is undefined when there is no learning, and that the power function is undefined when there is one-trial learning. L(t) is defined in both of these circumstances. Bittner (7) used a quadratic polynomial to fit learning curves to estimate long-term individual and system performance. This polynomial provides an estimate of asymptotic performance unconfounded by learning. as does L(r), but it does not provide a learning rate. If the mean of a measurement is taken over several trials, a change in the mean could represent an overall shift that occurs at each trial or a change in the trend over trials. A change in the SD could represent a change in the variability around a trend or a change in the trend itself. Even if the initial trials are removed from a test, learning can still have an effect on the mean and SD if the learning rate is low or if the difference between initial performance and asymptotic performance is large. In the example, it was possible to separate the information contained in the mean and SD. There was an effect of education level on mean reaction time that reflected an effect on

I .m 0.00 6.73 (I.67 3.25 0.59 0.18 3.00 I.52 32 0.49 l.h2 0.01 0.x4 0.31 2.71 3.3s 2.71 2.66 1.25 5.06 0.0x S.63 5.11 2.93 I .Ji 2.1 I 7.64 -3.32 I .9? 4.69 0.00 7.21 5.23 7.73

0.2154 0.9543 0.010 0.5727 0.0227 l).Qli il.6701 I).Ofi4h l).2W!, W760 ll.hX67 0.2048 t1.91411 0.4708 O.Xl6’1 tl.OJh3 0.06X5 l1.100')

Il.0494 0.2922 0.0021 0.77 I7 0.0037 0.0020 O.O.:JJ 0.235 I O.I4XII 0.0062 0.02Oh 0.1273

IU~O.i-1 0.9568 0.0078 0.0017 0.04.5 I

the initial performance level. and there was a significant age x exposure interaction for mean reaction time that reflected an increase in the fatigue rate. The age X exposure interaction was also significant for the SD of the reaction times, reflecting an effect on the SEE. and the grade X exposure interaction was significant for the SD, reflecting an effect on the initial performance level and the SEE. One can be more certain that there was an actual effect on performance variability if the trends are taken into account. It should be noted that the focus of the analysis in the example was on the effect of exposure on the linear relationships between age and education level and the dependent variables. If analysis of covariance had been used. the focus would have been on the group means adjusted for the covariates. Such an analysis would have been inappropriate for several of the dependent variables because the assumption of homogeneous slopes was violated. In the example analysis the definite poisonings showed an increase in the fatigue rate and in performance variability as their age increased, and their initial reaction times were higher if their education level was lower. This group also showed an increase in the mean and SD of the reaction times as age increased. A previous study did not find an effect of organophosphate pesticides on reaction time (35). In their analysis of the data used in the example. Steenland et al. (39) also found no effect. The negative results may be explained by

KRIEG, TABLE hSTlMATES

OF THE

KEGKESSION

AND

RUSSO

3

(‘OEFFICIENTS

FIG. 2. The effect of education level (grade) on the initial performance level (L,). and the effect of age on the fatigue rate (f) and performance variability [as measured by the standard error of the estimate (SEE)] in the definite exposure condition.

CHRISLIP

FOK

I’HE I,C)UR EXPOSIJRF

(;KOUPS

smaller sample sizes or by different methods of analyzing the data. These investigators looked for differences in group means, or group means adjusted for covariates. At the present time the biological interpretation of the parameters in the model is not clear. The reaction time task used hcrc involves the eyes. the peripheral and central nervous systems. and striate muscle. The effect of a variable on a model parameter could reflect an action on one or more of these areas. The two trends in the reaction time curves have been provisionally called learning and fatigue because these seem to be the likely processes underlying the trends. It is possible that other processes or a combination of processes may cause the trends, especially the one called fatigue. Following the interpretation of temporary work decrement by Kohl et al. (23), fatigue in the present reaction time model may, in part. rellect the influence of reactive inhibition. a response-decrementing process viewed by some to reside strictly in the ncuromotor apparatus, and by others to be a state of ncgativc motivation capable of supporting inhibitory associative relationships (14.20). Because the usual method of detecting this type of decrement-producing variable is to measure its dissipation over time. it should be possible to design reaction time intervals to control the influence of reactive inhibition. The fatigue component of the present reaction time model would then be expected to differentiate between long and short intertrial intervals. A second process capable of influencing fatigue in the reaction time response has been termed “inhibition of return” (33), an increased difficulty in directing attention to a prcviously attended location. Because there is considerable evidence indicating slower reaction times to previously attended targets (32). this potentially combined perceptual-motor proccss (2) may have influenced the present results independently of other types of inhibition. Regardless of the details of interpreting the reaction time decrement, future studies may find it useful to apply the present mathematical model to uncover local dccrementing trends in the reaction time response. Apart from addressing questions about the mechanisms controlling this response, the technique would permit more sensitive detection of untoward chemical effects on reaction time than has been possible prcviously.

REACTION

TIME

593

MODEL REFERENCES

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Kohl.

R. M.: Roenker.

D. L.; Turner.

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