Chapter XIII: The Theory of the Measurement of Mental Development (2)

The variation in age norms with different tests is shown graphically in Figures 6, 7 and 8. In order that the various tests may be plotted on the same scale, so as to compare changes in development for the different tested processes, I have used the average increase in ability from 8 to 9 years of age for each test as a common measure and arbitrarily plotted the slant of the curve between these ages at 45 degrees. The increase from 8 to 9 is represented by 10 units on the objective scale to the left of the graphs. On this basis it is possible roughly to compare changes in the absolute annual increase at different ages for the same test and for different tests. It assumes that the units in which each test is scored are equivalent for that test. An average difference between the basal ages or between any two ages cannot be assumed to be accompanied by the same distribution of increases. Moreover, the 8-year norm is at different distances from zero for the different tests so that the relative increase from 8 to 9 cannot be regarded alike for the different tests. The method, however, is sufficiently accurate for illustrating the very different forms of the developmental curves which might be expected if they were measured by absolute increases from year to year. Even the variation in the slant of the lines at the different ages gives a graphic picture which will assist in interpreting the significance of average curves of general ability. As the curves stand, they show the norms for each age for any test, as if placed on its own objective scale, and the various objective scales have been harmonized on the assumption that the norms at 8 and 9 years are accurate. We thus have a simple representation of the absolute changes in the abilities tested from age to age by the same tests relative to a single objective scale. It will not give a seriously erroneous picture for any tested ability so long as the units in which the particular test is scored may be presumed to be objectively equal.

The tests on which Figures 6, 7, and 8 were based included
practically all which were reported in the researches used. They
were as follows: Norsworthy (_159_), perception of 100-gram
weight, cancelling A's (boys), ideas remembered from four simple
sentences, memory of related and of unrelated words, part-wholes,
genus-species, opposites and reverse of opposites given the next
day, “a-t” test. J. Allen Gilbert (_108_), taps in 5 seconds,
fatigue in tapping, visual reaction time, color-discrimination
reaction time, reproduction of 2-second interval. Smedley (_51_,
No. 3), strength of right-hand grip (boys), taps in 30 seconds
(boys), ergograph; visual, auditory, audio-visual, and
audio-visual-articulatory memory for digits. W. H. Pyle, Standards
of Mental Efficiency (J. of Educ. Psychol., 1913, IV., 61-70),
uncontrolled association, opposites, part-wholes, genus-species,
digit-symbol and symbol-digit substitution, memory for concrete
and for abstract words, memory of Marble Statue selection, (only
boys' norms used for each). Pyle and Anderson combined by Whipple
(_220_) two word-building tests (boys). Anderson as given by
Whipple memory for letter squares. D. F. Carpenter, Mental Age
Tests (J. of Educ. Psychol., 1913, IV., 538-544), substitution of
colors in forms and of numbers in forms, perception time in
marking A's, concentration, _i. e._, difference in time of last
test under distraction, memory of pictures of objects, all tests
devised by Carrie R. Squire. Stenquist (_54_), construction test.
Sylvester (_191_), form-board test.

In Fig. 6 curves A and B are Smedley's tests; curve C includes in
addition Norsworthy's unrelated words, Pyle's memory for concrete
and abstract terms, Anderson's letter-squares, Carpenter's memory
for pictures, and Gilbert's for the time interval; curve E includes
Pyle's two and Carpenter's two substitution tests; curve F includes
Pyle's Marble Statue and Norsworthy's memory for related words and
for sentences; curve S is Norsworthy's; curve D is the combination
of these 17 tests.

In Fig. 7 curve H includes Gilbert's visual reaction time,
Norsworthy's A and a-t tests, Carpenter's two A tests; curve I
includes Gilbert's and Smedley's tapping tests; curve J is the
median of the central tendencies of all 40 tests; curve K includes
Norsworthy's two opposites and her part-whole and genus-species
tests, the Pyle opposites, genus-species and part-whole tests; curve
L is the same as D, curve M includes Smedley's strength of grip and
ergograph tests and Gilbert's fatigue of tapping; curve N includes
Pyle and Anderson's word building tests and Pyle's uncontrolled word
association test.

In Fig. 8 curve P is Gilbert's visual reaction time test, curve S is
Norsworthy's test for memory of unrelated words, the other curves
are the median and quartiles for the central tendencies of all 40
tests after each was expressed at each age in terms of the gain from
8 to 9 years taken as a unit.

Several points are to be noted about the nature of the curves for different tests. In Fig. 6 showing the curves for different forms of memory tests, that for the memory of digits is very different in character from that for memory of related material. The most extreme differences in the time of maturity are shown by the test for memory for digits presented orally and the substitution of color in forms, the former continues to increase so rapidly relative to the absolute increase from 8 to 9 years that it cannot be represented in the graph reaching 539 units of the scale by 14 years of age, while improvement in ability in the latter is not measured after 9 years. We cannot take time to discuss how much of the differences between the various curves may be due to the nature of the tests themselves, the form of scoring the results, or the condition under which they were given, selection of subjects, etc. The conclusion is safe, however, that when groups of three or four tests of similar type show such marked differences as those for memory of digits and memory for related material we may expect similar differences in the rates of maturity of the corresponding processes.

From Fig. 7 we may learn that tests emphasizing functions such as speed of motor or perceptual motor reaction, curves H and I, are notably different in their form from curves for tests of imaginative processes, curve N. As we group tests together covering larger ranges of activity we approach the median curve for general ability. Note the median curve for 17 memory tests (curve L) compared with the median for the 40 tests (curve J). By empirical studies we might pick out types of tests which would most closely represent the maturity of average ability. For example, the median for the substitution tests, curve E, resembles the median for the memory tests, curve D, more closely than does that of the 4 digit tests, curve B. Curve K, for 7 association tests, resembles the median for the 40 tests, curve J, much more closely than the curve for the perceptual-motor speed tests, curve H. This difference can not be explained by the use of 7 instead of 5 tests in calculating the central tendency of the group. It probably means that the sort of psycho-physical processes usually tested more closely represent on the average the abilities shown in association tests than they do the abilities shown by speed of motor reaction. The significance of this sort of analysis for those constructing a scale for measuring intellectual ability is obvious.

Fig. 8 shows the median and quartile range for the central tendencies of the 40 tests and gives examples of two extremely different tests, visual reaction time and memory for unrelated words. How closely these particular tests represent fundamental differences in the maturity of different processes, we cannot, of course, be sure without prolonged research; but nobody would question that analogous differences would be found in different processes. When we think of curves of general ability we must, therefore, keep in mind the light which might be thrown on them by an analysis of the various processes tested in the particular scale used.

Another feature of all developmental curves which is apparent as soon as the causes of development are considered, is that growth in an individual is the result of several factors. These include the native capacity, the rate at which that capacity manifests itself instinctively, and the external stimuli which encourage or retard that manifestation. To some extent these factors vary independently. Our curves of development will never completely express all the facts until they analyse out all these factors for each of the processes. In the meantime we shall be able to think of general trends of development by considering average curves. The fact that they represent combinations of unanalyzed factors must, however, make us very cautious in interpreting our norms.

(b) CHANGES IN THE RATE OF DEVELOPMENT.

There has been considerable discussion of the form of the curves of mental development. The logical aspects of the curves on the assumption of normal distribution of ability at each age and uniform age of maturity have been treated by Otis (_163_) and the bearing of these assumptions upon the Binet scale pointed out. Thorndike has plotted the developmental curves for a dozen tests on the basis of the variability at 12 years of age used as unit and gives a chapter in his Educational Psychology to the changes with maturity (_198_, Chap. XI). Bobertag suggests that the rates of development of normal and deficient children are analogous to the upward progress of two projectiles fired from such different heights that the force of gravity would retard the lower projectile more than the upper (_81_). This analogy supposes that the rate of maturity would continually decrease and that those who were feebler mentally would be arrested in their developmental earlier. Bobertag, Kuhlmann (_137_, _138_) and Otis give evidence from the results of Binet testing that the rate of development decreases with age. The percentages of older children passing certain positions on the Binet scale or certain tests taken from it were found to change less at year intervals for the older ages. This evidence is not conclusive unless we know that the positions compared are at the same point in the distributions of ability at the beginning of the periods of growth. The same percentage change at a point farther away from the central tendency would mean a larger growth than at the middle of the distribution, when judged either in reference to a physical scale or to units of deviation.

While recognizing that the complete curve of mental development is logarithmic in form Pearson contends that, when measured by Jaederholm's adaptation of the Binet scale, development is adequately represented by a straight line from 6 to 15 years of age (_164_). As this conclusion is based upon the use, as equivalent units, of years of excess and deficiency at all these ages the data lacks the cogency of a scale of equal physical units.

With the Point Scale it is not known whether the units in different parts of the scale are equivalent. Without assuming that they are equal it is impossible to discover the form of curves of development from the records of children at a series of ages. Yerkes and Wood publish a curve of the increase of intellectual ability based upon point-scale measurements, which resembles in form the hypothetical curves. They say:

“The point-scale method has the merit of indicating directly the
rate, or annual increments of intellectual growth. We do not claim
for our measurements a high degree of accuracy, especially in the
case of the early years of childhood. But even the roughly
determined curve of intellectual growth from four to eighteen years,
which we present below, has considerable interest for the genetic
psychologist and for the psychological examiner. We have ascertained
that whether measured by the ratio of the increment of increase,
year by year, to the norm for the appropriate year or by the ratio
of the extreme range of scores to appropriate year norms,
intellectual development rapidly diminishes in rate, at least from
the fifth year onward” (_169_, p. 603).

Waiving the question whether annual increases or the range of measurements relative to the age norms would be satisfactory indications of the change in the rate of growth, it seems to be fairly clear that neither of these criteria would be adequate unless we first knew that the units in which they were measured were equivalent at different portions of the scale. To show that the point scale units are even theoretically equivalent it would seem to be necessary to assume, on the basis of normal distribution of ability, that each unit of the deviation for each age distribution either equaled the same number of scale units or the same proportion of the total distance from lowest to highest ability at each age measured in the point-scale units. The originators of the scale do not seem to have planned it with this in view. Moreover, the difficulty of empirically demonstrating such equivalence of units on a point scale or any form of the Binet scale prevents its use for indicating curves of mental development, however serviceable it may be for other purposes.

The simplest demonstration of the form of the development curves is applying the same test, scored in equal physical units, to children of different ages. In Figs. 6, 7, and 8 the evidence from tests was assembled for ages 8 to 14 inclusive. It is probable, however, that the form of these development curves, when the unit of measurement was anything but time taken for the same task, has been affected by the difference in the real value of units called by the same name, _e. g._, giving the opposite of one word is not always equal to giving the opposite of another.

The best developmental curves empirically determined are probably those for the form board presented by Sylvester (_191_), Wallin (_212_) and Young (_227_) since in each of these cases the same test was presented at all ages and the scores were in equal physical units of seconds. It can hardly be supposed, however, that the form board curves alone would be typical of average mental development. To know something about the general curve of mental development we need a combination of a number of mental tests scored on scales of equal units. These may be either equal physical units or units on scales for mental development similar to those of Thorndike and others for measuring educational products, handwriting, arithmetic, spelling, _etc._

That either a straight line or a simple curve would represent the development of ability from birth to maturity is very doubtful. When we consider the entire developmental curve from birth nobody doubts that there is a change in the rate of development at the time of the arrest of instinctive changes at adolescence. There are probably fluctuations in the rate before this final arrest. Pintner and Paterson also assume a complex curve of development (_44_). Whether the fluctuations should be allowed for in the description of the borderline of deficiency is the important question in our study. With measurements of bodily growth we noted that changes in the rate of maturity are accompanied by a skewness of distribution of ability at the ages affected. The same effect may be expected with mental measurements. The percentage method of defining the borderline of deficiency has an advantage when the form of distribution at any age is uncertain (See Chap. XIV, d.). Since the changes in the rate of development are most likely to be important at the prepubertal and adolescent ages the description of the borderline in terms of deviation or quotient may be expected to be most uncertain at this period. Moreover, none of the quantitative definitions of the borderline, except the percentage method, remain equivalent if rates of development of normal and deficient children change relative to each other, a question we shall now consider.

(c) THE QUESTION OF EARLIER ARREST OF DEFICIENT CHILDREN.

It has been assumed by Bobertag (_81_), Stern (_88_), Goddard (_117_) and others that deficient children reach their maturity earlier than normal children. If this were true the curves of mental development for the average and for the deficient children should not be expected to retain their same relative positions after the idiots had begun to show arrested development. Moreover, unless this arrest were compensated by some peculiar form of accelerated growth among those above normal ability, we might expect that the distributions of ability would change in form at the various ages after arrest had begun. A relative increase in the distance of older deficients from the average as compared with younger deficients may be interpreted as meaning either the earlier cessation of growth of the deficients or a change in the relative rates of growth of individuals of different mental capacity. When fully considered the present evidence from the Binet tests fails, I believe, to demonstrate the earlier arrest of the deficients, although it is undoubtedly true that the Binet scale may not be fine enough to measure the improvement of idiots. We shall take up certain investigations that bear upon this point.

Goddard has reported tests upon the same group of 346 inmates in an institution for the feeble-minded who were tested three years in succession (_117_). The paper suggests that the idiots, as a group increased less in absolute ability than those of higher mental age. The average gain for 55 idiots who tested I or II mentally was about half a test in the two years. In order to reach our present problem, however, we must know that the idiots, for example, developed relatively less mentally than did those of the higher grades of ability in the imbecile and moron groups of _the same life-ages_. This question cannot be answered from the paper. It probably cannot be adequately answered from mental age results on account of the irregularity in the value of the year units at different points on the Binet scales.

Bobertag summarizes Chotzen's data obtained by the examination of the children in the Breslau Hilfsschulen with the Binet scale. He believes that the position on an objective scale attained by the average of these retarded children is progressively lower with advancing age relative to the average position attained by normal children, assuming that the quotient for normal children remained constant at each age. The average intelligence quotients of all the children in the special schools (exclusive of those testing III or less) was 0.79 for those 8 years of age, 0.72 for those 9 years, 0.70 at 10, and 0.67 at 11-12 (_81_, p. 534).

Stern also compiled a table from Chotzen's results which shows this decrease in intelligence quotients with life-age separately for each group of those whom Chotzen by his expert diagnosis regarded as imbeciles, morons, doubtful, and not feeble-minded although attending the special schools (_188_, p. 80). This table is reproduced here as Table XX. On the surface it suggests that the quotients of the extreme groups are nearer together at the older ages, instead of being farther apart. The objection to this evidence from the Binet scale is that the norms are not equivalent for different ages on the scale used. Since the objective norms on the Binet scale are more difficult to attain at the older ages this variation would tend to make older children show lower quotients than the same children would show at younger ages, so that such tables are quite uncertain in significance.

TABLE XX.

AVERAGE INTELLIGENCE QUOTIENTS OF CHILDREN OF DIFFERENT ABILITY. (From
Chotzen's Tables X & XI.)

─────────────┬──────────────┬──────────────┬─────────────┬─────────────
LIFE-AGE │ NOT │ DOUBTFUL │ MORONS │ IMBECILES
│FEEBLE-MINDED │ DEFECT │ │
─────────────┼──────────────┼──────────────┼─────────────┼─────────────
8 │ 0.92 │ 0.84 │ 0.76 │ 0.71
9 │ 0.85 │ 0.81 │ 0.77 │ 0.67
10 │ (0.80) │ (0.80) │ 0.74 │ 0.62
11 │ (0.73) │ (0.68) │ 0.71 │ (0.64)
12 │ (0.75) │ (0.75) │ (0.73) │ (0.61)
13 │ │ (0.73) │ │
─────────────┴──────────────┴──────────────┴─────────────┴─────────────

The Jaederholm data with his form of the Binet scale, as treated by Pearson, shows a straight regression line for the backward children which falls below the normal development line on the average four months of mental age for each additional year of life from 7-14 (_167_). Accepting Pearson's interpretation that a year of excess or deficiency and a year of growth is a constant unit, we find that the deficient group from special classes was falling continually behind the normals with increase of age a relatively greater distance from any rational reference point. Pearson accounts for this change in the distance between the two groups of normal and backward children, as I understand his paper, by supposing that with increase in age more and more normal children become deficient. It would seem that this data would be more easily explained by supposing that the distributions became skewed toward deficiency for the older ages, rather than that the distributions remained normal and became flatter.

The best evidence as to the relative positions of the curves for deficients and those for average ability would be provided by using psychological tests that could be adequately scored in terms of equal physical units for the same task. The position of various lower percentiles relative to the average or to an assumed reference point could then be compared on the same objective scale. I have reviewed studies of this type in discussing skewed distributions in Chap. XIII, A, c. I there reached the conclusion that the weight of the evidence was that the distributions were slightly skewed in the direction of deficiency, although the evidence was not conclusive. We are now raising the further question whether this skewness increases with age.

On account of the difficulty of determining the points for zero ability in terms of the physical scales used, let us see what conclusion might be reached if we calculated the relative distance of median and low ability of equivalent degree from the scores of the same higher degree of ability assumed as a reference point at the various ages. There seems to be no reason in the theory of measurement why the highest score instead of the lowest score in random samples might not be used for a reference point for comparing the distances between normal and deficient children at different ages. Instead of using the highest single score, it would be better to use the upper quartile or quintile since it would be less affected by a chance error in giving the test.

Applying this method to determining the relative position of median and retarded ability I have calculated the data for the form board test cited previously from Sylvester (_191_) and from Young (_227_). This affords the only adequate evidence of which I know, derived from tests scored in equal physical units given to sufficiently large groups to indicate whether or not the retarded group changes its relative position from the normal group at different ages. The comparison is shown in Fig. 9. With Sylvester's data the distance of the lower quartile in ability from the median is compared with the distance of the upper quartile from the median, the latter distance being taken as a unit. With Young's data for Witmer's form board the quintile is used instead of the quartile and each sex is given separately. Since Young's table shows the scores for half ages, it was necessary to take the average of the two scores, thus giving the approximate score for the middle of the complete age group. The graph discloses no pronounced tendency for the retarded group to fall relatively farther behind the median with increase in age. There are, however, notable fluctuations in the relative positions of the groups so that at 7 years with Young's data for boys and at 13 years for Sylvester's curve the retarded group is twice as far from the median relative to the distance between the median and the corresponding better group as it is at some other times. It is possible that the curves for the older groups of those of poorer ability are too high since it is likely that more of the actually deficient children tend to be dropped from the public school classes with increase in age. Nevertheless, so far as the evidence at present goes it is not sufficient to determine whether the backward and the corresponding better group show a general change in their relative distances from the median with approach to maturity.

On the other hand the curves indicate the tendency for the distributions to be skewed toward deficiency and for the relative distances to fluctuate as we should expect if the accelerations in growth occurred at different ages for those of different ability. The data of Young suggest that there may be sex differences in the age of acceleration, the backward girls showing accelerations, relative to the upper group at ages 7 and 12, a year or more before the boys. For Sylvester's data the ratio of the distance between the median and the lower quartile divided by the distance between the median and the upper quartile for each of the age groups is as follows: 5 yrs. 1.8, 6 yrs. 2.4, 7 yrs. 3.0, 8 yrs. 2.0, 9 yrs. 2.2, 10 yrs. 2.4, 11 yrs. 2.0, 12 yrs. 1.8, 13 yrs. 3.0, 14 yrs. 2.1. For Young's data the corresponding ratios are—Boys: 6 yrs. 1.5, 7 yrs. 1.9, 8 yrs. 1.5, 9 yrs. 0.8, 10 yrs. 1.6, 11 yrs. 1.2, 12 yrs. 1.4, 13 yrs. 1.0, 14 yrs. 1.3. Girls: 6 yrs. 1.7, 7 yrs. 1.0, 8 yrs. 1.5, 9 yrs. 0.9, 10 yrs. 1.0, 11 yrs. 1.3, 12 yrs. 0.9, 13 yrs. 1.5, 14 yrs. 1.4. Changes in the rate of growth causing asymmetrical distributions are to be expected throughout the periods of growth. A fundamental skewness toward deficient mental capacity, therefore, would be indicated only if it were found at maturity or at ages when the average rate is decreasing, when the more capable individuals would theoretically approach relatively nearer the deficients if the latter accelerated later.

So far as physical growth is concerned Baldwin (_74_, _75_) has shown with repeated annual measurements on the same group of children that the period of adolescent acceleration shifts from 12½ years for the tallest boy to 16 years for the shortest boy. For the tallest girl the maximum height was attained at 14½, for the shortest at 17 years, 3 months. Maturity may be reached at 11 years by a tall well nourished girl, while with a short girl light in weight it may be delayed until 16. “Children above medium height between the chronological ages of 6-18 grow in stature and in physiological maturity in advance of those below the medium height, and they may be physiologically from one to four or five years older than those below the medium height. Those above the medium height have their characteristic pubescent changes and accelerations earlier than those below; there is a relative shifting of the accelerated period according to the individuals' relative heights” (_74_).

Doll presents evidence from the physical measurements of a large feeble-minded group in institutions which he suggests shows that the shorter among them cease growing earlier. When the height of these feeble-minded is measured in relation to the Smedley percentiles of the height of normal children of their corresponding ages, he finds a correlation of -.20 between age and percentiles of height, the taller relative to normals being younger. He says: “This confirms Goddard's similar conclusion, but negatives for the feeble-minded at least, the theory affirmed by some writers, that children who grow at a retarded rate continue their growth to a later age” (_98_ p. 51). On the contrary this minus correlation is more likely to mean only that the Smedley norms on school children are too high for the older ages because of the excess of taller children who remain for the high school work. This would give the minus correlation without supposing that the taller individuals continue their growth to a later age, as he thinks.

Moreover, a total longer period of physical growth for smaller, less normal, children has been demonstrated. Boas (_80_) says: “Among the poor the period of diminishing growth which precedes adolescence is lengthened and the acceleration of adolescence sets in later; therefore, the whole period of growth is lengthened but the total amount of growth during the larger period is less than during the shorter period of the well-to-do” (_80_). A reversal in growth tendency between brain capacity and size of body, which is supposed when the mentally deficient are said to arrest earlier, would be one of the most puzzling paradoxes in the study of development. We should, therefore, be exceedingly cautious before accepting the hypothesis of the earlier maturity of deficient children.

A complicated situation is presented when we come to represent graphically the effect on the distributions of these differences in growth among those of different intellectual capacity. In the hypothetical diagrams, Fig. 5, it is shown how arrest of development might be presented graphically in relation to the distribution curves, ability being measured on the same physical scale. The earlier acceleration and earlier maturity of those of better ability are indicated. The distributions are shown as skewed at all ages after birth. Equivalent units of mental development at different ages can be found only in corresponding percentages of the groups, not in the units of the deviation or in development quotients relative to the averages at different ages. In other words the lowest 0.5% continues to be an equivalent unit while -3 S. D. measures different portions of the group and different portions of the distance from lowest to highest ability. Corresponding percentages retain one common significance, namely, that the same proportion of the group is ahead in the struggle for survival, regardless of the form of the distribution.

It is hoped that the discussion of the statistical problems connected with the quantitative study of mental development has given more meaning to the different attempts to devise scales for measuring mental ability. It should be noted that the same relative development at different ages, expressed relative to the distance from lowest to highest ability measured in equal objective units, does not correspond to the same relative development measured in percentages of the groups, as soon as the forms of the distributions change. The theoretical considerations show that we have available at once a perfectly definite and clear method of stating relative development in terms of corresponding percentages of corresponding groups. If the groups distribute normally these units are translatable into units of the standard deviation of the group. If the distributions change in symmetry the only equivalent units of deficiency available are in terms of corresponding percentages reading from either end of the group. On the other hand percentile units are not equivalent in _amount_ of change for the same distribution, so they are of most importance for comparing different age distributions of uncertain forms.

Until we have a scale of equal objective units for mental ability, it is not possible to obtain a measure of relative development which shall take into account the _amount_ of relative change. We must be content to measure the change in percentile rank (changes in serial position) of an individual relative to those of his own age.

Having clarified our conceptions of mental development and brought them into harmony with certain suppositions regarding the distribution of ability and its change from year to year, we are in a better position to evaluate in the following chapter the different objective methods of defining the borderline of feeble-mindedness.