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PSYCHOMETRIKA--VOL. 30, NO. 3 SEPTEMBER, 1965

A MODEL FOR SERIAL VERBAL LEARNING*

CLARA KUNO

KEIO UNIVERSITY, JAPAN

A model for analyzing the learning process with a special emphasis on serial-position effect is proposed. This model consists of two analyses, one being an analysis of the learning process of each item in a list by a stochastic method, and the other being an analysis of serial-position effect in terms of pro- and retroactive inhibitions, and of forgetting. The model is experimentally verified, and moreover, it is found that the model permits prediction of the results of many experiments with lists of various lengths and varying difficulty.

1. Introduction

1.1 Historical Review

In serial verbal learning of nonsense material which is sequentially presented, the items near the beginning and the end of the series are, in general, easier to learn than those in the middle. This phenomenon has been well known since the 19th century and is called the serial-position effect, or primacy effect at the beginning and recency effect at the end, respectively.

More than thirty papers on experimental studies have appeared on this subject. The problems discussed in the earlier papers mainly concern the predominancy of the primacy effect or recency effect. Among these and among other papers for other experimental purposes, some present definite results on the predominancy of either effect in the serial-position effect. After categorizing these papers, it is found that the predominancy of the primacy effect is supported mainly by the experiments on the serial-anticipation method, i.e., those by Ebbinghaus [6], Robinson and Brown [19], Warden [25], Lepley [13], Ward [24], Hovland [9], Malmo and Amsel [14], and McCrary and Hunter [15]; the predominancy of the recency effect is supported by the experiments on the free-recall method or the paired-association method, i.e., those by Calkins [5] and Raffel [20]; and the equal effectiveness of the two effects is supported mainly by the experiments on the free-recall method, i.e., those by Bigham and Munsterberg [2], Smith [231, Finkenbinder [7], Foucault [8], and Shipley [22].

On the other hand, very few theoretical analyses of the serial-position

*The author wishes to acknowledge help received during discussion with Prof. T. Indow.

323

324 PSYCHOMETRIKA

effect have been carried out. Foucault [8] and Ono [18] explained this effect by pro- and retroactive inhibition; Leplcy [12, 13], Hull et aL [10], and Bugelski [3] did so by forward and backward associations; and Atkinson [I] by the probabilistic model.

Foucault's model is rather crude and other models are constructed upon too complicated assumptions or by using too many unmeasurable intervening variables. Furthermore, all of these models deal with only one kind of experiment, e.g., Foucault's model deals with only the experiment on reproduction procedure (free-recall procedure) and all other models deal with only the experiment on the serial-anticipation method. However, both are serial verbal learning experiments with human subjects and the data on them contain both the primacy and the recency effects, more or less. Therefore, the process underlying both experiments seems to be the same and it would be undesirable that each model should deal with only one kind of experiment.

Atkinson's model seems to be clear but it has no conceptual relation to the usual psychological terms. The purpose of the present paper is to construct a generalized model analyzing serial-position effect in any kind of serial learning experiment in terms of well-known concepts.

Recently, stochastic [4] and information-theoretical [21] learning models have been proposed. These models deal only with the average properties of many words in a list which are assumed to be homogeneous, namely, all the words are supposed to be on the same level of difficulty of learning. Hence, these models are applicable only to a special experimental situation where the positions of the words in a list are randomized at each trial.

1.2 The Problem

Since the items in a list are presented in a fixed order in an ordinal learning experiment and consequently the difficulty of learning each item in a list is not on the same level, it is desirable to construct a model for ordinal serial learning processes which permits analysis of both the process of learning each item in a list and the serial-position effect of the items contained in that list, in terms of well-known concepts, irrespective of the method of experiment.

When such a model has been constructed, its validity should be verified by examining the following three points.

(i) How well do the theoretical curves derived from the model fit the experimental data?

(ii) To what extent can the model predict other learning processes? (iii) Do the curves derived from the model fit the data better than

the curves derived from other models? The plan set forth in this paper is to begin with a general description

of the construction of the model. Following this, estimation procedures of

UI,ARA KUNO 325

the parameters contained in the model will be explained, and thcn the model will be applied to experiments on free-recall procedure and the serial-anticipation method. We can then use the model to predict the results of learning experiments yet unperformed. Finally, goodness of fit of the model will be statistically tested against experimental data.

2. The Model

This model consists of two analyses, one being an analysis of the learning process of each item in a list (Part I), and the other being an analysis of the serial-position effect in the list (Part II).

2.1 The Model ]or the Process o] Learning Each Item (Part I)

This section will give in detail a statistical model which is related to the Miller-McGill learning model [16]. All notation in this section is analogous to that of the Miller-McGill two-parameter case [16].

In Part I of this model, we assume that rk, the probability of recalling an item after k previous recalls, is given by

r~+l -- r~ = a(l -- ~k),

with the initial condition ~o = Po, where a, the parameter of fixation, and Po, the parameter of memorization, are constants and 1 => a, Po ~ O. This equation means that the increase of r~ for each recall is proportional to the possibility of ascension of rk , namely (1 -- r~). The above equation is re- written in the form

rk+, -- (1 -- a)r~ = a,

which is a nonhomogeneous difference equation with the initial condition ro = Po The solution of this equation is given by

(1) r, = 1 -- (1 --p0)(1 -- a) *.

This expression is the same as that of Bush-Mosteller's fixed point form [4] and Miller-McGill's two-parameter case [16]. The two parameters contained in this expression, a and Po , will be estimated in the next section by the method of maximum likelihood.

Now, in our experiments, all the figures plotting po against the position of items in a list showed a U-shaped tendency as illustrated by o in Fig. 1, and no apparent tendency was observed in all the figures plotting a as illustrated by in Fig. 1. From these tendencies, which will be statistically verified in Section 4.1.2, we have the conclusion that the serial-position effect appears primarily in the parameter Po and not in a.

2.2 The Model for Serial-Position Effect (Part I I)

The purpose of Part I I is, therefore, to construct a theory to analyze the serial-position effect appearing in Po in terms of pro- and retroactive

326 PSYCHOMETRIKA

1.00

0.80 a

0.60

0.40

0.80

0.60

P. 0.40

0.20

0.000

0

Q O

Q

O

O

0

0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0

0

2 4 6 8 10 12 14 16 18 20 serial position

FIGURE 1

Empirical Values of at and p~0 Obtained from Free-Recall Experiment of Typical Subject (The ordinate of a~ (e) is shifted vertically from that of p~0 (o).)

inhibition, and of forgetting. We shall introduce the following six parameters. ~, is the theoretical amount of recall immediately after presentation

of an item defined apart from any of the inhibitions mentioned below. In the usual learning experiments, the empirical amount of learning is limited to 1.00, so that the value -~ cannot be measured directly in an experiment unless the value is equal to 1.00 (see Experiment I I I in section 5.4.1).

is the amount of forgetting during the period of time from the presentation of one item to the presentation of the next. Therefore, if we denote by F the amount of forgetting of the ith item immediately after presentation of the last item on the list with I items, we have

(2) F = ~( l - -0 .

a is the amount of inhibition arising from an item affecting the memorization of the succeeding item, namely, the parameter of the proactive inhibition.

is the amount of inhibition arising from an item affecting the retention of the preceding item, namely, the parameter of the retroactive inhibition.

), is the decreasing rate of the proactive inhibition a. Consequently, ak is the amount of inhibition that affects the memorization of the item following the succeeding one; aM -2 is the amount of inhibition affecting the memorization of the jth item counted from the one in question. There-

VLAaA XVNO 327

fore, the amount of the proactive inhibition which the kth item of the list affects the ith item of the list, Fk~, is

(3) Fk, ---- aX '-k-l,

where k < i. On the other hand, the ith item of the list is affected by all the items previously presented. Since the total amount of the proactive inhibition which affects the ith item, I, , is the sum of I~, over k (< i), we have

i - - I i - -1 i - -2

(4) I , = Z I~, = Z -x ' -~- ' = - Z x' = . (1 - ~'- ' ) .

is the decreasing rate of the retroactive inhibition ft. As in the case of )~, the total amount of the retroactive inhibition which affects the ith item of the list, I~, is given by

' - ' - ' ~(! - ~,'-') (5) I : = ~ ~ =

h-o 1 - - /~

If l denotes the number of items comprising the list, then for the free- recall procedure experiment, P,o (the Po of the ith item of the list) is the amount of memorization of the ith item immediately after the presentation of the last item on the list, and is written

(6) P,o = '~- - E - - I , - I:.

Substituting (2), (4), and (5) into (6), we obtain

(7) P ,o = "1 - 5 (1 - i ) - a(1 - k ' - ' ) _ fl(1 - ut - ' ) . 1 - -k 1 - -~

We shall estimate the values of parameters contained in (I) and (7) in the next section.

3. Estimation of Parameters

3.1 The Maximum-Likelihood Estimation o] the a and po in Equation (1)

The maximum-likelihood method for the estimation of the parameters a, and P~o is used in order to avoid the introduction of a tendency to artifi- ciality.

The estimates are obtainable from the frequency distribution of n,k , where n~ is the number of unrecalled trials of item i between its kth reca]~ and its (k + 1)st recall. Let the state that item i has not been recalled on n,k successive trials after its kth recall be A,~ ; then P~k , the probability of occurrence of state A ~k, is given by

p,~ = r,~(l - T,~)"'L

If the p,0's for all k are assumed to be independent, then the maximum likelihood for item i is given by

328 PSYCHOMETRIKA

i}

L, = I~ r ,k (1 - r,O"', kffiO

where ~ is a number which is arbitrarily fixed as common to all items. Substituting r~k of (1) into the above equation, the likelihood L~ is

expressed as a function of a~ and p~o In order to obtain the values of a, and P~o which maximize L~ , the partial derivatives of log L~ with respect to a~ and P~o are made equal to zero, and the following equations are obtained.

(1 p~o)(1 - a , ) ' ' Z1 - - ~11---~o>~11 - - a,> '= Zn,k , kffi0 kffi0

The right-hand members of these equations are determined from experimental data and the left-hand members are functions of a~ and P~o only.

\

Y\

TABLE i

Table for Calculating Values of a and Po from Two Values of X and Y

Value of a

i 2 3 4 5 6 7 8 9

1.00 i.00 1.00 i.00 i.00 1.00 1.00 i.00 i.00 .37 .53 .58 .59 .6i .6i .62 .63 .63 .03 .33 .4i .45 .46 .47 .48 .49 .50

. i6 .30 .35 .38 .39 .40 .41 .42

.03 .20 .27 .30 .33 .34 .35 .35 .iO .20 .25 .27 .29 .30 .3i .Oi . i3 . i9 .23 .25 .26 .27

.07 . i4 . i8 .22 .23 .24

.02 .I0 .14 .18 .20 .21 .06 .11 . i4 , i7 .t9

Value of Po

i 2 3 4 5 6 7 8 9

.50 .33 .25 .20 .17 .14 .13 . l l .iO

.68 .42 .30 .23 .19 .16 .14 .12 . l i

.82 .51 .35 .26 .21 . i8 .15 .13 . l i .62 .40 .30 .23 .19 .16 . i4 .12 .69 .47 .34 .26 .21 .17 .i5 .13

.55 .39 .29 .23 .19 .16 .14

.6i .44 .33 .25 .20 .17 .15 .49 .37 .29 .22 .18 .i6 .53 .40 .32 .24 .20 .17

.44 .34 .26 .22 .i8

ULARA KUNO 329

Let them be X and Y, respectively. In case ~ = 4, when the values of the right-hand members are given, we can find the values of a~ and P~o from Table 1.

3.2 A Tentative Method ]or Estimating the Six Parameters in Equation (7)

The procedure for estimating parameters contained in (7) is rather cumbersome. I t appears impossible to estimate the values either by the least-square or the maximum-likelihood method. I f one wishes to estimate these parameters by trial and error, the calculation would become too tedious and there might be some risk of subjective preiudices arising. In this paper, a tentative method by which the estimation may be considered to be objective within a certain range of error is adopted.

We shall explain this method in the case of 20 items (l = 20). First, we plot the values of P~o against i, and assume the most probable values of p,o at i = 1, 5, 10, 15, and 20. Let these values be pl.o, Ps.o, Pio.o, p~5.o, and p20.o , respectively. Substituting these P,o values into those of (7), we have the following five equations.

P~.o = ~' -- 198 ~(1 -- ~9) 1 - - .u '

1 - -X - - - - - i ' - - # '

~(1-x g) ~(1 - ~o) (8) p,o.o = ~ - 10~ - - l - - -X - - - -1 - - u '

~,o = "y - 5~ a(1 - X" ) te l - u ~) , 1 - -X 1 - -u

,~(1 - ~) . ~2o,o = ~/ - - 1 - - k

Since 1 ~ X, u > O, neglecting those terms higher than the eighth power of X and u, the above equations can be reduced to

p l .O ~-- 5" - - 195 B , l - - ta

~(1 - - X ~) P~.o --- ~/ -- 155 ~ X

(9)

1 - - p. ~

Plo,o = 'Y -- 105 8

1 - -~ 1 - -~z '

P15.o = ")' -- 55 ol B(1 -- ,u 5) 1 - -X 1 -- ,~ '

Ol

P2o.o = ~- - 1 -- X"

330 PSYCHOMETRIKA

In fact, the values of X and t~ which were used in our experiment are less than 0.75, and hence their eighth-power terms are less than 0.07.

Now we have five equations with six unknown parameters. The process of estimating these parameters is as follows.

(i) Since the values of ~ show little variation between subjects, we estimate it by trial and error.

(ii) The value of ~ is decided from the following relation, which is easily obtained from (9).

Plo.o = Pl.o + P~0.0 + 10~ -- %

(iii) The value of a/(1 -- X) is calculated by substituting ~, into the equation for P2o.o

(iv) The value of f~/(1 -- ~) is obtained from the equation for p~ .o (v) By using these values of % 8, a/(1 -- X), and fl/(1 -- p), the value

of X is calculated from the equation for ps.o (vi) The value of ~ is calculated from the equation for P~.0. (vii) Since the values of a/(1 - X) and fl/(1 - ~) are already known,

we have the values of a and B. In this way, we can estimate the values of all parameters % ~, a, f~, X, and ~.

4. Experimental Verification o] the Model

4.1 The Free-Recall Procedure Experiment (Exp. I)

4.1.1 Brie] Description

The material for this experiment is a group of almost random numbers of from two to six digits. Each list to be learned contains 5, 10, or 20 items of this type. Nine kinds of lists of medimn difficulty of learning were chosen from these 5 X 3 = 15 combinations. Four lists of each kind were prepared so that we had a total of 9 X 4 = 36 lists. The subjects of these experiments were students taking a course in psychology, two males and two females. The individual cards (items) forming a list were presented successively, and after presentation of the last the subject wrote down all the items he could recall. Each trial was repeated ten times without pause.

By the term "correct response," we mean a response in which all the digits in one item were correctly recalled in correct order. All other types of responses were classified as false, even if a part of an item was correctly recalled. The values of ~k n~ and ~k kn,~ were obtained from the results of each experiment for each subject, and the values of the a~'s and p~o's were obtained by using Table 1.

Let the average values of a, and p~o over the four experiments of the same kind be d~ and/5~o, respectively. In Fig. 1, d~ and P~o are plotted against

~ARA XUNO 331

i (position in list), in a case where I (length of list) = 20, m (number of digits) = 3, for Subject Y. We obtained 9 X 4 = 36 similar figures for the four subjects corresponding to the nine kinds of experiments. Now, it is recognized by inspecting the figures that the serial-position effect appears primarily in parameter P,o, as illustrated by (o ) in Fig. 1, and not in a~, as illustrated by (e) in Fig. 1.

4.1.2 Statistical Test of the Significance o] the Serial-Position Effect

In order to make definite the point mentioned above, the tendency contained in the data was tested by the following three steps.

(i) Estimating the Probability o/the Occurrence o] Each Run

Experiments with 20 items in a list were statistically tested, since the number of degrees of freedom in these cases is large. The 20 plotted points in a figure were divided into two groups by a straight horizontal line so that each group contained just 10 points, and then the number of runs in each group was counted. An exact treatment was employed for the case using the following equation [17].

(n, - 1)! 1).' / n.T P

332 PSYCHOMETRIKA

the a plots show no definite tendencies and Po plots show certain tendencies. Moreover, the fact that the number of runs in the lower group of po plots is less than that of the upper group implies a U-shaped tendency in Po plots.

4.1.3 Analysis o] the Serial-Position Effect The U-shaped tendency of parameter Po was analyzed by our model

(Part I I) and the values of the six parameters in (7) were estimated by the method described in Section 3.2.

From this analysis, we can recognize the following properties. (i) a, f~, h, and tL take almost constant values throughout the exper-

iments for each subject. (ii) ~ does not depend on the difficulty of the items (number of digits),

but it decreases as the list increases in length. (iii) ~ does not depend on the length of the list, but it decreases as

the items increase in difficulty (number of digits). In view of these properties, the following re-estimations of the values

of the six parameters seem to be possible. (i) We adopt the average values of parameters a, B, ~, and tL for all

the lists as their estimated values for each subject. (ii) We adopt the average value of ~ for all lists of the same length

as its estimated value for each length and for each subject. (iii) We adopt the average value of ~/for all lists with the same degree

of difficulty of items as its estimated value for each degree of difficulty of items and for each subject.

All of these values are shown in Table 2. Although it contains individual

TABLE 2 Estimated Values of Parameters in Experiments on Free-RecaU Procedure

Subject

I K N Y

.35 .25 .14 .18 ), .20 .50 .73 .70 B .52 .40 .48 .31 tL .20 .13 .20 .24 7 (2-digit numbers) 1.60 1.65 1.73 1.61 ~. (3-digit numbers) 1.44 1.45 1.53 1.46 ~, (4-digit numbers) 1.34 1.35 1.43 1.31 ~, (5-digit numbers) 1.34 1.25 1.28 1.06 ~, (6-digit numbers) 1.20 1.20 1.18 1.06

(5-item lists) .08 .10 .10 .10 (10-item lists) .03 .04 .05 .04 (20-item lists) .01 .02 .02 .02

ULARA XUNO 333

Po

1.00

0.80

0.60

0.40

0.20

0.000

o

O/ ! I ! ] ! I l 1 I

2 4 6 8 10 12 14 16 18 serial position

FIGURE 2 Theoretical Curve of (7) with Parameters in Table 2 and p~0 Plots of

Free-Recall Experiment of Typical Subject

t 2O

differences in the values of parameters, when these values are substituted in (7), the curves fit the data fairly well, as illustrated in Fig. 2; and the model has the added advantage of predicting the results of unperformed experiments by the same subject (see section 5).

The curve in Fig. 2 was obtained from (7) by using the averaged estimated values of parameters. As seen from this figure~ all the theoretical curves showed that they fit the experimental data.

The relations between the values of parameters in Table 2 and the shapes of curves are as follows.

(i) The larger the value of parameter a, the steeper the declivity of the curve at the beginning (primacy effect).

(it) The larger the value of parameter ~, the steeper the acclivity of the curve at the ending (recency effect).

(iii) The larger the value of parameter ~, the larger the upper shift of the curve.

(iv) The larger the value of parameter ~, the steeper the upward in- clination to the right of the curve.

On the other hand, it was found from the x 2 test (see 4.1.2) and p test (between subjects) that parameter a has no apparent tendency in all experiments irrespective of its position in the list, the difficulty of each item, and the length of the list.

4.2 Serial-Anticipation Method Experiment (Exp. II) Ordinary methods of verbal learning experiments, aside from the free-

recall procedure, are the serial-anticipation method and the paired-association

334 PSYCHOMETRIKA

method. These two methods are the same in experimental procedure except that the response to an item is or is not the same as the next stimulus. There- fore, although we performed only the experiment of the serial-anticipation method, the same model may be used to analyze the paired-association experiment.

In this experiment, a part of the material in Exp. I was used, and the subjects for this experiment were three of the four subjects in Exp. I. In this experiment, we used 6-item and l 1-item lists, but the first item in a list served as a cue and the last item in the list served to check the final response, and hence the numbers of responses were 5 and 10 for lists with 6 and 11 items, respectively.

By the term "correct response," we mean a response in which all the digits of the item following the one presented are correctly anticipated in correct order. Parameters a~'s and p~o's were calculated by the same method as in Exp. I. Although the number of points is too small for a statistical test of the tendency in this case, all the a~ plots seem to show no definite tendency and all the p~o plots seem to show the U-shaped tendency.

In Fig. 3, the values of a,- and P;o are shown with the theoretical curve in the case where 1 = 11, m = 4, for Subject N.

All the procedures of estimating parameters, except ~, are the same as that for Exp. I. The values of parameter ~ vanish in this case, since ir~ this experiment all the time intervals between the presentations of stimuli and the respective responses are the same. All of the theoretical curves fit the experimental data well, as illustrated in Fig. 3.

5. Predictability o] This Model

Let us consider the predictability of this model for the free-recall experiment, since in this case we have many kinds of data..

5.1 The Prediction o] P,o with Known Parameters

First, the method for the prediction of P;o values in cases where all the parameters are already known is discussed. In the previous section, it was concluded that the parameters a, f~, h, and ~ take constant values throughout all the experiments for each subject, ~ depending only on the length of the lists and ~/depending only on the number of digits in the items. Consequently, in a case where all parameters are already known, we can predict the values of P~o for an unperformed experiment by the simple substitution of these values in (7). For example, the P~o for Subject I, with a list of 20 items of 6-digit numbers, is given by

pio = 1.21 -- .01(20 -- i) -- .35(1 - - .20 ~-1) _ .52(1 - - .20 ~- ' ) .80 - - .80

VLARA xuNo 335

(/

Po

1.00

0.80

0.60

0.60

0.40

0.20

0.00

o o

...... I t I i I I t I ~ I 1 2 3 4 5 6 7 8 9 10

serial position

FIGURE 3 Theoretical Curve of (7) and Empirical Values of a~ and p~0 Obtained from

Serial-Anticipation Experiment of Typical Subject (The ordinate of ai(*) is shifted vertically from that of pio(o).)

5.2 Prediction o] r~

I t was also found in our experiment that the parameter a has no apparent tendency for all experiments irrespective of its position in the list, changes in difficulty of items, and the length of the list. Hence, we may regard the average of all available values of parameter a~ as the estimated value of parameter a. Thus when the values of parameter P,o are obtained, then the values of r~, can be calculated by substituting these values of parameters a and P~o in (1).

By using these parameters, the theoretical curves derived from (1) fit all the experimental data as illustrated in Fig. 4.

5.3 Prediction o] P~o with Some o] the Parameters Unknown

Next we shall discuss cases where some of the parameters are unknown. If we plot ~, in logarithmic scale against the number of digits m as shown in Fig. 5, we see that all the data lie in a straight line. From this, we have

(10) ~ = ~oe -~,

where ~'o and i" are constants. Similarly, if we plot ~ in logarithmic scale against the logarithm of the length of lists l, we see that all the data lie in a straight line (see Fig. 6). From this, we have

336 PSYCHOMETRIKA

(I1) ~ = ~o/-',

where ~o and ~ are constants. As was concluded from our experiment, the other four parameters

a, f~, ~, and ~ take constant values throughout the experiments for each subject. Subst i tut ing (10) and (11) in (7), we have

(12) P ,o = 7oe - r ' " - - ~o l - ' ( l - - i ) - - a( I -- X':I) 8(I -- u~-'). 1 - - ) , 1 - -~

We can predict many cases by this equation so long as (10) and (11) are valid. Equat ion (12) for the four subjects in Exp. I may be writ ten as follows.

S : I p~o = 1.848e - ' ' '9~ - .360/ - '~7( / - Q

.35(1 - .20' - ' ) .52(1 - .20' - ' ) i - - - .-20 - 1 - .20 '

S :K P~o = 1.880e -'7s9'' -- .695l-H9(l -- i)

.25(1 -- .50'- ' ) .40(1 -- .13 ~-') 1 -- .50 1 -- .13

1.00

0.80

0.60

'rk

0.40

0.20

,, I I I .......... I 0"000 1 2 3 4

k

FIGURE 4 Theoretical Curve of (I) and r~k Plots of Typical Subject

ULnA KVNO 337

1.8

1.6

1.4

0 ~ 0

1.2

1.06 1 I ........... I 2 4 6

Fmum~ 5 7 Plots of Typical Subject and Fitted Straight Line

S :N

S :Y

P~o = 2.130e -'9s3" -- .597l-1"11(/ -- i)

_ .14(1 - .Ta'- ') _ .4s(1 - .2o ~-') 1 - .73 1 - .20 '

P~o = 1.980e -'l~ - .597l-1"11(/- i)

_ .18(1 - - .70 ' - ' ) _ .31(1 - - .24 ' - ' )

1 - - .70 -1"--:---- .24

5.4 Range o/Validity of Parameter 7 Let us examine the range of validity of (10). I f the value of m correspond-

ing to ~ = 1.0 is known, it must coincide with the immediate memory span, since it is the critical value at which all the digits contained in an item are recalled in correct order after only one isolated presentation of the item. An experiment was performed, as set forth in the next section, to verify this coincidence.

338 PSYCHOMETRIKA

5.4.1 Experiment o] Immediate Memory (Exp. I I I)

The material for this experiment was a group of almost random numbers of 6, 8, 10, and 12 digits. Ten cards of each kind were prepared so that we had a total of 10 X 4 = 40 cards. All four subjects tested in Exp. I partic- ipated in this experiment. The cards (items) were shuffled and each item was presented. Immediately after presentation, the subject wrote down as many digits as he could possibly remember from the item presented.

By the term "correct response," we mean a response in which all the digits in one item were correctly recalled in correct order. All other cases were classified as false even if a part of an item was recalled correctly. The immediate memory span was calculated by the method of complete series for each subject. The resulting values were 7.6 for Subject I, 8.4 for K, 7.8 for N, and 9.0 for Y.

On the other hand, by extrapolating the values of ~ in Table 2, using (10), the expected values of m corresponding to ~/ = 1.0 are known. They are 8.5 for Subject I, 8.1 for K, 7.7 for N, and 6.5 for Y. In three (I, K, and N) of these four cases, these two values coincided closely. From these three of the four cases, therefore, it was verified that the extrapolation of the values of x by (10) is valid, at least up to ~, = 1.0.

5.5 The Range O] Validity of Parameter As to 5, it seems impossible to determine the range of validity by a

simple experiment. However, we can use a straight line to show the relation between ~ and l, at least from 1 = 5 to l = 20. Finally, we came to the conclusion that we can predict the value of P,o for every case in the range m = 2 to 6 and I = 5 to 20 by (12).

5.6 Minimum Number o] Experiments to Determine Parameters

The minimum number of experiments to determine all of these parameters is two, because, from any one experiment, the parameters a, B, A, and u can be estimated; the two parameters "to and ~ of (10) are estimated from two experiments with differing numbers of digits in each item; and the two parameters $o and n of (11) are estimated from two experiments with lists of differing lengths. Therefore, after two experiments with varying numbers of digits in the items and lists of different lengths are carried out, all the cases mentioned above, m = 2 to 6, t = 5 to 20, can be predicted by this model.

6. Comparison Between Curves Derived ]rom This Model and ]rom Polynomials

We shall compare the goodness of fit for curves derived from this model and curves derived from other expressions of nearly the same degree of complexity.

ULARA ~UNO 339

A polynomial was used for the alternative expression, since it is the most common form and has no special character. In this case the degree of complexity means the degree of the polynomial; in our model a polynomial of the fifth degree was chosen corresponding to six parameters.

In general, a polynomial is fitted to the data by the method of orthogonal polynomials. Since the calculating procedure is rather tedious, however, the fifth-degree polynomials are fitted to only those two cases in which the sums of the squares of the differences between empirical data and the curves derived from our model are the largest and the smallest, with 20-item lists.

The sum of the squares of the differences between empirical data and the curves derived from our model seems to be distributed in x 2 form. The same situation will hold for the sum of the squares of differences between empirical data and curves derived from polynomials. Therefore, if we assume the standard deviations to be the same for both, two ratios of the respective sums of the same data are tested by the F test of d.f. 14 vs. 14. By this test, we found no significant difference in the case where the sum of the squares is the largest. On the other hand, in the case where the sum of the squares is the smallest, the difference between the polynomial curve and the data was significantly larger at the 1 per cent level than the difference between the curve of our model and the data. Therefore, it was concluded that the curves derived from our model fit the data better than the best-fitted polynomials of the same degree of complexity.

7. Summary

A model analyzing the learning process having a serial-position effect was constructed. This model consists of two analyses, one being an analysis of the learning process of each word in a list by a stochastic model of the form in (1), and the other being an analysis of the serial-position effect appearing in (7) in terms of pro- and retroactive inhibitions, and of forgetting.

Two experiments were carried out in order to verify this model. The first experiment was performed by the free-recall procedure, while the second experiment was performed by the serial-anticipation method. The model from both of these experiments fitted the data, and moreover, the model permitted prediction of the results of many experiments with lists of varying lengths and varying difficulty. Another experiment was performed by the immediate-memory method in order to estimate the range of validity of the parameter in this model.

Equation (7) could be developed into (12) by the substitution of parameters obtained from the data of the first experiment, and it was concluded that, by (12), we can predict all cases such as m (number of digits in an item) = 2 to 6 and 1 (items in a list) = 5 to 20, from the results of only

340 PSYCHOMETRIKA

0.20

0.10

0.08

0.06

0.04

0.02

0.011 ' t ,J I 5 10 20

l

FIGURE 6 Plots of Typical Subject and Fitted Straight Line

two experiments. The curves derived from this model showed that they fit the data better than those of the best-fitted polynomials.

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Manuscript received $/6/64 Revised manuscript received 7 /P8/65