The complexity effect phenomenon is an epiphenomenon of age-related fluid intelligence decline

Person. individ.Difl Vol. 16, No. 2, pp. 265-288. 1994 Elsevier Science Ltd. Printed in Great Britain 0191-8869/94 $6.00 + 0.00

Pergamon

THE COMPLEXITY EFFECT PHENOMENON IS AN EPIPHENOMENON OF AGE-RELATED FLUID INTELLIGENCE DECLINE LAZAR STANKOV Department of Psychology, The University of Sydney, Sydney, NSW 2006, Australia (Received 22 February

1993)

Summary-This paper employs the methods of covariance structure analysis to explore the relationship between complexity, age, and fluid intelligence. Fluid intelligence is defined by seven psychometric tests. Complexity is systematically varied by making two well-known measures of fluid intelligence progressively more simple. The model assumes the existence of three psychometric factors-fluid intelligence with loadings from all psychological tests including those whose complexity was manipulated, and two narrow factors (the Swaps and Triplets factors) representing only complexity levels of a given test. The results show that age is related to the general fluid intelligence factor-it does not seem to have direct links with different levels of complexity manipulations. It is argued that the often observed tendency for the magnitude of age differences in performance to increase as the task becomes more complex-known in the literature as the Complexity Effect Phenomenon-is a consequence of the fact that fluid intelligence correlates with age.

THE

COMPLEXITY AN

EFFECT PHENOMENON EPIPHENONMENON

IN AGING

IS

In an influential book on aging, Salthouse (1985) has popularized the term “Complexity Effect Phenomenon”. This refers to the tendency for the magnitude of age differences in performance to increase as the task becomes more complex. He points out that the writings of Birren (1965), Botwinick (1984), Welford (1977) and many others suggest this hypothesis. Numerous examples from the areas of perception, memory (including studies of the depth-of processing), card-sorting and reaction time, divided attention, and problem-solving can be cited to support this phenomenon. In highlighting its status, Salthouse (1988a) has stated that “. . . the Age by Complexity to represent one of the few laws of human aging phenomenon . . . might even be considered currently available . . .” (p. 426). The relationship between age and complexity appears similar to that hitherto observed between intelligence and complexity. This paper explores the nature of the relationship between these three variables and argues that the Complexity Effect Phenomenon is an epiphenomenon that arises from the age-related changes in fluid intelligence. To see this it is first necessary to describe some of the contemporary work in human intelligence. Fluid intelligence and complexity Recent studies of intelligence show a renewed interest in the construct of complexity. This interest is due to the improvements in methodology introduced through cross-fertilization between cognitive psychology and the psychometric studies of human abilities over the past 15 years. However, neither psychometric nor experimental cognitive studies have looked at the incremental nature of complexity in a systematic way. The psychometric tradition has frequently acknowledged the link between complexity and intelligence. This link was based on the observation of low loadings of sensory measures and higher loadings of the tasks that measure problem solving and higher mental processes on a general factor (see Snow, 1989; Stankov, 1978). Invariably, the tasks at the lower complexity levels were different from those of high complexity. For example, it is often assumed that success in resisting the Muller-Lyer illusion requires a simpler cognitive process than, say, solving a Matrices problem. This is the most common explanation given to the lower loadings of perceptual abilities on the general factor. In most of this work, however, it mattered little that perceptual processes (like those

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used in the Muller-Lyer illusion) may actually play an insignificant part in the solution of complex cognitive patterns (like those needed in solving the Matrices items). Nevertheless, Horn (1988), Humphreys (1979) Thomson (1939) and Thorndike (1926) and many others have acknowledged that higher-order processes contain a number of lower-order processes. Systematic decomposition of the complex task into its ingredients has not been the aim of psychometric tradition. On the other hand, experimental cognitive psychology has addressed this issue, if only in an incomplete way. Sternberg’s (1977) componential analysis, for instance, was an attempt to break down complex tasks of intelligence into their constituent processes. This was an important step in the analysis of complexity. However, the thrust of that work was to correlate individual components with an overall measure of intelligence and establish their relative importance. I have been unable to find evidence that a composite task involving two or more distinct component processes has higher correlation with intelligence than either process in isolation. Stankov and Crawford (1993) used two measures of fluid intelligence-the Triplet Numbers test and the Swaps test. These tests are also employed in the present study. Both tests were decomposed into four increasingly simpler subtasks with approximately equal reliabilities. The results with young adults showed that, in addition to having higher overall performance levels, the increasingly simpler tasks had lower correlations with measures of fluid intelligence. This is equivalent to saying that the difference between people who score high and those who score low on fluid intelligence tests increases as the complexity of the test increases. The justification of this line of research derives from a belief that manipulations involved in making the task more complex provide insights into the nature of fluid intelligence. They tell us about the ingredients of intelligence, i.e. processes within the organism that allow us to deal with tasks of varying complexity. The most salient psychological processes implied by these manipulations are easy to describe. They are the number of elements and their relations which the individual can hold in mind as he or she works on the solution to a problem. This is clearly in agreement with the definition of complexity as a more or less complicated collection or system of related things or parts. My approach to the study of intelligence and complexity is congruent with the approaches of many other psychologists including Horn (1988) Snow (1989), and Salthouse (1985). Complexity is an aspect of intelligence-perhaps the most important aspect of fluid intelligence. Intelligence represents a broad organization of many diverse abilities. By virtue of this broadness it provides an empirical basis for defining complexity of a task from a psychological (as opposed to logical or computational) point of view. Complexity of a task is revealed through its correlation with measures of intelligence. Although there is a parallel between the way we talk about the old-young and high-low scoring individuals on intelligence tests, it is necessary to note an important difference between age and intelligence variables in their relationship to complexity. An easy way to highlight the nature of this difference is as follows. Imagine, as was the case with the work of Stankov and Crawford (1993) that we have people of the same age and we give them tasks that vary in complexity. The finding that the difference between the more intelligent and the less intelligent grows larger as the task becomes more complex is not surprising. On the other hand, imagine that we have people of the same intelligence, but who differ in age. In this case, most of us would not expect the young vs old difference to increase as the complexity increases. Clearly, complexity is a psychological construct, related to cognitive psychological variables. Chronological age, however, has no intrinsic aspects that are related to test complexity. From this perspective, it is impossible to talk about the Complexity Effect Phenomenon without mentioning broad psychometrically-defined cognitive abilities. The relationship

of a third variable to both age and intelligence

During adult life, performance on measures of fluid intelligence (Gf) declines with age. The relationship between Gf and age is not linear, due to a relatively slow decline during early adulthood and an accelerated drop in peformance past the age of 65570 years. Nevertheless, linear function is often used to describe the main findings, and Pearson’s product-moment correlations between Gf and age range between -0.30 and -0.70 (see Horn, 1982, 1988; Salthouse, 1988b; Stankov, 1986, 1988a).

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Even in samples that show high correlations between intelligence and age (i.e. r = -0.60 and above) one cannot be sure that a variable that correlates highly with one has an equally high correlation with the other. For example, although it is well known that there is a pronounced decline of the sensory processes such as hearing and vision during adult aging, it has been acknowledged that this decline has little if any effect on the decline of Gf (see Horn, 1982). On the other hand, we know that the competing presentation of two complex cognitive tasks is an effective way to increase the Gf loading of a given pair of tasks, yet the current evidence does not support the hypothesis that the ability to carry out competing tasks is impaired with age. In fact, some recent evidence indicates that older people are as good as younger at competing tasks (see Stankov, 1986, 1988a). These findings can be compared with the results involving variables that do show high correlation with both age and intelligence. For example, many measures of attention belong to this group (see Horn, 1988; Stankov, 1988a). Indeed, Salthouse (1985, 1988b) has argued that the theory of attentional resources has a particularly strong explanatory power in accounting for the major findings in cognitive aging. An analogous situation exists regarding the relationship between a task’s complexity and its correlations with intelligence and age. The so-called Complexity Effect Phenomenon implies that age difference increases with increasing task complexity, while the relationship between intelligence (especially Gf) and task complexity has been highlighted by numerous authors (e.g. Horn, 1988; Snow, 1989; Stankov & Crawford, 1993). Thus, the question arises as to whether the Complexity Effect Phenomenon can be explained simply as a reflection of the well-documented decline of Gf with age. If so, this phenomenon could best be regarded as an epiphenomenon of a Complexity Effect for Gf. Alternatively, if the observed age-complexity interaction could not be explained by the relationship between age, task complexity, and Gf, then the Complexity Effect Phenomenon could be regarded as a phenomenon in its own right. When viewed from the multivariate perspective, there is a sense in which one can claim that this phenomenon exists in its own right, even though a part of it is an epiphenomenon. To understand this, it is necessary to explicate the structural model of this study. The model The variables. In this study, age and intelligence are the typical continuous organismic variables familiar to all psychologists. A general framework for this study is provided by the theory of fluid (Gf) and crystallized (Gc) intelligence. Since the relevant aspect of complexity relates to the construct of Gf, our operational psychometric definition of intelligence is in terms of fluid abilities. Complexity may be thought of as an independent variable, i.e. as different levels of a variable which are selected by the experimenter for study. The dependent variables are derived from conveniently chosen psychological tasks that generate continuous scores-in this case the Swaps and Triplet Numbers test. A fundamental requirement in choosing these tasks is that the level of the independent variable can be adjusted, i.e. to systematically increase the complexity. If this could be done we should obtain a systematic change in both the arithmetic means of the dependent variables across all levels (indicating an increase in difficulty), as well as changes in correlations (indicating an increase in the complexity levels of the tasks). More complex tasks should have higher correlations with both age and intelligence. This approach combines the typical psychometric and experimental cognitive approaches, retaining the strengths of both, and therefore having advantages over each of them. Thus, this approach allows for experimental control and manipulation which is typically absent from psychometric studies. At the same time it examines not only changes in the overall level but also looks at the traditional “bread-and-butter” data for psychometricians+hanges in measures of covariation. It also provides for a psychological and an operational definition of complexity as opposed to task difficulty. The expected structure. The aim of this study is to determine whether complexity manipulations that are known to affect correlations with intelligence, also affect correlations with age. If so, are they related to the decline in Gf with age? Alternatively, are these manipulations related to age in a way that is independent of Gf? These issues will be addressed through the application of structural equation modeling. The essential aspect of this procedure is the formulation of a model of the structural relationships among psychological variables, and examination of the fit of that model to the observed data.

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Given the existence of several psychometric tests and two experimental tasks with several levels each, the model of interest here has three latent variables or factors. The first factor is intelligence with loadings on all variables, both psychometric and experimental. The other two factors represent the experimental tasks. As such, these factors have their loadings only on the given levels of complexity of each task. Age defines a separate factor (AGE). In order to explain the logic of the approach adopted in this paper, consider a graphical version of this model involving a minimal number of manifest variables (four), and three latent variables or factors in Fig. 1. This figure is drawn in accordance with the usual conventions for path diagrams to represent structural equation models (Joreskog & Sorbom, 1988; McArdle & Bolger, 1990). The four and the three factors or latent variables are “circles”. Also, manifest variables are “squares” single-headed arrows represent parameters (in this case the arrows arising from the circles are factor loadings derived from covariances or correlations), and two-headed arrows (“slings”) between different latent variables represent correlations or covariances. Each manifest variable has a unique variance. These unique variances are indicated by “slings” that originate and terminate at the same “square”. They are not drawn in Fig. 1 in order to avoid clutter. Latent variables or factors in Fig. 1 are AGE, TASK, and Gf. Manifest variables are Age a measure of the Cognition of Figural Relations primary factor (CFR, say, Matrices test), and two task-related measures (Tl and T2) that differ with respect to complexity, i.e. T2 is more complex than Tl. In accordance with the operational definition of complexity, T2 should have higher correlation with a latent variable than Tl. This is open to empirical verification. In the following discussion we shall assume that loadings of these two manifest variables on each latent variable differ so that, in terms of Fig. 1, a < b, c < d, and e < f. In other words, T2 is more complex than Tl. At this stage it is useful to introduce the concept of a “bridge” (see McArdle & Bolger, 1990). For our purposes here, “bridge” is a connection between any two manifest variables, and consists of a sequence of arrows and slings. In the path diagram of Fig. 1 there are several bridges between any two manifest variables. Consider, for example, a couple of bridges that can be traced between

W

0

b

c

d

e

f

Fig. 1. Skeletal representation of the path diagram of the expected relationships between manifest and latent variables of this study together with a list of bridges between Age and Tl, T2 which, if present, would support the Complexity Effect Phenomenon (CEP). See text for further details. Bridges from Age to Tl and T2; Direct bridge: Age-AGE-Tl, T2 (width w*a,b) supports the CEP; Indirect bridges: arrows from AGE to Tl and T2 are absent (a = 0; b = 0). (a) All slings in the above diagram are present. CEP is supported because of the bridge Age-AGE-TASK-TI,T2 (b) Sling between AGE and TASK absent (y = 0). CEP is not supported. (c) Sling between AGE and Gf absent (x = 0). CEP is supported because of the bridge Age-AGE-TASK-TI,T2. (d) Sling between TASK and Gf absent (z = 0). CEP is supported because of the bridge Age-AGE-TASK-T1 ,T2. (e) Conjoint b. and c. conditions (y = 0; x = 0). CEP not supported. (f) Conjoint b. and d. conditions (y = 0; z = 0). CEP not supported (g) Conjoint c. and d. conditions (x = 0; z = 0). CEP is supported because of the bridge Age-AGE-TASK-TI,T2.

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Age and Tl. Thus, one bridge starts from Age, passes through AGE and reaches Tl. Another, longer bridge starts from Age, passes through AGE, then through Gf, and again reaches Tl . There are three additional bridges between Age and Tl in Fig. 1. The “width” of a bridge is defined as the product of the segments that enter into a bridge. Thus, the width of the first bridge is w*a and the width of the second bridge is w*x*e. In the traditional factor analytic sense, the first bridge represents that part of Age, Tl relationship that is due to the AGE factor. The second bridge reflects that part of the Age, Tl relationship that is due to both AGE and Gf factors, this bridge takes into account correlation (i.e. coefficient x) between factors. Now, if we consider bridges that span any manifest variable of Fig. 1 and Tl and T2, and the fact that parameters related to T2 are higher than those related to Tl, it follows that the widths of bridges related to T2 will be larger. This is another way to talk about complexity of T2 vis-a-ok Tl. In general, all slings and arrows depicted in Fig. 1 are plausible. However, depending on the actual relationship among the variables in the system, some arrows and slings and therefore some bridges may not exist. If there were no slings between the three latent variables (i.e. the orthogonal structure with lines x, y, and z absent), only one bridge would exist between Age and Tl (and T2). A solution of this type would provide clear support for the Complexity Effect Phenomenon. At the other extreme, i.e. when all slings of the system exist, the relationships between Age and Tl and T2 can be conveniently divided into two types. Direct relationship is indicated by the bridge that goes through the AGE factor only. Indirect relationships involve bridges that pass through AGE and at least one additional latent variable. The existence of the direct relationship even in the presence of the indirect relationships provides support for the Complexity Effect Phenomenon. However, if Age is linked to the experimental task variables Tl and T2 via indirect bridges only, i.e. if lines marked a and b do not exist, the meaning of the Complexity Effect Phenomenon may have to change. In the absence of a direct bridge, Age could have links with the experimental tasks through the narrow TASK factor only, e.g. either x or z or both x and z lines absent, and line marked y present. This is acceptable as an example of the Complexity Effect Phenomenon. Alternatively, Age could have an indirect link through the broad Gf factor only. In this case it would be necessary to conclude that this is a phenomenon that occurs with, and seems to result from, another, clearly an epiphenomenon. This is because the effect of AGE and therefore the relationship between Age and any other variable of the system is critically dependent on the variables that define Gf factor. Finally, Age can have a link through both TASK and Gf factors, again an epiphenomenon. Some relevant options that could emerge in the empirical data are summarized in the caption of Fig. 1. The model-testing procedure. The design of this study calls for a judicious repeated application of the structural equation modeling procedure, this is necessary in order to arrive at a model of the relationship among the variables that is both theoretically interesting and methodologically acceptable. Strategically, the model calls for a test of the following three basic assumptions: (a) there are three latent psychometric variables in the system (Gf, SWAPS, TRIPLETS) and AGE; (b) there is an increase in factor loadings of the levels of the experimental tasks on broad (Gf) and narrow task-specific (SWAPS and TRIPLETS) factors; (c) latent variables (factors) are correlated (or they have non-zero covariance). The dependent variables. Based on our previous experiences with these tasks, both accuracy and speed measures are used as dependent variables in this study. Our previous work with similar cognitive variables to those employed here leads me to expect that relevant answers to the above questions will be obtained with accuracy scores. A salient issue, due to the inclusion of several levels of the independent variables, is the possibility of ceiling or floor effects occurring at the extremes of the range of values. In empirical studies of this nature one cannot be sure that the choice of levels spans the whole range appropriate for the sample of subjects. It follows, as a rule, that some reduction of variance would occur at the extremes. If that happens, it is necessary to establish the extent to which substantive conclusions can be sustained. Work by Stankov and Crawford (1993) showed a reduction in the variance of accuracy scores, due to a ceiling effect. Hence, I shall examine the evidence in these data that may indicate the appropriateness of substantive conclusions.

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Although speed of providing answers tends to decline with task difficulty, individuals scoring low on intelligence tests do not necessarily work markedly slower or faster than high-scoring individuals (Stankov & Crawford, 1993). It will be important, therefore, to establish if there is a systematic change in correlations between age and speed of test taking as a consequence of an increase in task complexity. A positive finding, i.e. a significant difference between young and old in speed of test taking, would be relevant in the consideration of the Complexity Effect Phenomenon since this effect has been observed in many chronometric studies of cognitive processing (see Cerella, 1985). METHOD

Subjects

Participants in this study were 164 people whose age varied between 19 and 89 years (arithmetic mean = 48.41, SD = 20.05). Due to a computer malfunction, a group of 11 Ss did not have their responses recorded for two versions of the Swaps test. Some of the analyses of this paper involving the Swaps test were carried out with 153 rather than 164 Ss. The sample can be conveniently divided into three groups: (a) young adults 19 to 30 years old (N = 48); (b) middle aged 30 to 60 years old (N = 53); and (c) over 60 years old (N = 63). All Ss were recruited from within the Sydney Metropolitan area. Most participants below the age of 60 were acquiantances and friends of part-time Sydney University students or were members of a tennis club. As a consequence, from among the 48 young adults 23 people were not enroled in any post-secondary school courses. Most of the elderly Ss were recruited from an organization called the University of the Third Age. This is an educational and social establishment, where retired professionals provide teaching in areas of interest to the members, no degrees are awarded. On the average, participants in this study had 13.037 (SD = 3.896) years of formal education. They were asked to rate the general state of their health on a 4-point scale (1 = excellent) with the average being 1.865 (SD = 0.662). Correlation between age and health rating was 0.284. Information regarding educational, vocational and professional experiences, income, health conditions, exposure to stressful events in life, experience with computers, etc. was also collected from these Ss. This information was screened in order to establish if the sample was unusual in comparison to the population at large. There was no indication of particular biases in this set of data. From among the health and physical/functional variables collected for this study, the highest correlation with age was obtained for the impaired vision (r = 0.625) and hearing (r = 0.350). The experience of stressful events in life, on the other hand, had a correlation of r = -0.091 with age in this sample. Cognitive

tasks

Psychological tests of this study can be conveniently cognitive tasks characterized by systematic manipulations tests of intelligence. Experimental

divided into two groups: experimental of levels of complexity and psychometric

variables

The Triplet Numbers test. The stimulus material for all versions of the Triplet Numbers test employed in this study consists of a randomly chosen set of three different digits presented simultaneously on the computer screen. These digits changed after each response. The four versions of this test differ with respect to the instructions given to the Ss. These instructions make the ‘Two-rules’ version similar to the versions used in previous psychometric work, while all other versions were used for the first time in the Stankov and Crawford (1993) study. Instructions for the increasingly complex versions of this test are as follows:

1. The ‘Search’ Triplets (TR 1). Press the ‘Yes’ key if a particular number, e.g. 3 is present within the triplet. Otherwise, the ‘No’ key is to be presed. Time: 2 min. 2. The ‘Half-rule’ Triplets (TR2). The task here is to press the ‘Yes’ key if the second digit is the largest within the triplet. Otherwise, the ‘No’ key is to be pressed. Time: 3 min. 3. The ‘One-rule’ Triplets (TR 3). In this version, only one rule from the most complex version is used, i.e. press the ‘Yes’ key if the second digit is the largest and the third digit is the smallest. Otherwise, the ‘No’ key is to be pressed. Time: 5 min.

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4. The ‘Two-rules’ Triplets (TR4). The S has to press the ‘Yes’ key if the first digit is the largest and the second digit is the smallest or the third digit is the largest and the first digit is the smallest. Otherwise, the ‘No’ key is to be pressed. Time: 6 min. Acronyms within the parentheses are those used in Fig. 5 to label the variables in the path diagram that describes the final model. In the preliminary work with this test, all four versions were administered with the same time limit, i.e. 6 min. Stankov and Crawford (1993) found that reliable data can be obtained with the shortened versions of the easier tasks and, in order to save time, employed different time limits for the four complexity levels. The Ss did not know the length of the tasks in advance and it was assumed that differential effects of practice and fatigue will not affect the overall performance on these tasks. No evidence to date has contradicted this assumption. This means that the number of items attempted varies across versions of the task. For this reason percentages of correctly answered items are employed as the accuracy score for all versions of the Triplet Numbers test. In addition to the accuracy scores, each Ss’ average speed at providing the answers was also calculated. The Swaps test. The stimulus material for all versions of the Swaps test consists of a set of three letters, J, K, and L, presented simultaneously on the computer screen (the order of letters was varied from item to item) together with a number of instructions to interchange, or “swap”, the positions of pairs of letters. The four versions of the task differ in the number of such instructions. There were four blocks of 12 items with an equal number of swaps in each. These items were randomly intermixed to form a 48-items test. The Ss did not know how many swaps would be required on any given trial. This aspect of administration is different from our approach with the Triplet Numbers test, where blocking of items according to their difficulty was employed. Again, as with the Triplet Numbers test in these experiments, all three letters were kept on the screen until the answer was typed in. To preclude the possibility of memory influencing performance on this task, the required swap instructions were also kept visible throughout the Ss’ work. The answer consisted in completing all the swaps, then typing in the final resulting order. An example of the four increasing levels of complexity is as follows: Stimuli: J K L 1. One Swap (SWl). ‘Swap 2 and 3’. Ans: J L K. 2. ‘Two Swaps’ (SW2). ‘Swap 2 and 3’, ‘Swap 1 and 3’. Ans: K L J. 3. ‘Three Swaps’ (SW3). ‘Swap 2 and 3’, ‘Swap 1 and 3’, ‘Swap 1 and 2’. Ans: L K J. 4. ‘Four Swaps’ (SW4). ‘Swap 2 and 3’, ‘Swap 1 and 3’, ‘Swap 1 and 2’, ‘Swap 1 and 3’. Ans: J K L. The numbers following the word ‘Swap’ refer to the position of letters within the set. Each Swap instruction was presented on a separate line in the middle of the computer screen. Two performance measures were collected from this test, the number of correct answers (out of 12 items) and the average speed of answering an individual item. Psychometric

tests

In order to investigate complexity manipulations tests were used:

the relationship of the Triplet

between measures of broad intellective abilities and the Numbers and Swaps tasks, the following psychometric

1. Number Spun Forward (NSF). This is a computer-administered analog of the WAIS-R subscale. 2. Number Span Backward (NSB). This too is a computer-administered analog of the WAIS-R subscale. 3. General Knowledge test (GK). This is a 20-item test requiring answers to questions about commonly accessible factual information available to most members of contemporary Western cultures. 4. Matrices test (MAT). This is a 15item test of the Figural Relations primary factor. It represents an attempt to emulate the well-known marker tests of fluid intelligence while allowing for computer administration. The Ss’ task is to fill-in, using the computer mouse, PAID--1612-F

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the empty right-hand corner cell of a 3 by 3 grid with simple symbols (“ + ” and “0”) in accordance with the rules inherent in the remaining eight cells of the grid. This test has been used for several years in research projects at the University of Sydney and factor analyses have repeatedly shown that it is a good measure of Gf. 5. Letter Counting test (LC). This is a 12-item test of the Temporal Tracking primary factor (see Stankov & Horn, 1980). In this task, a person has to count the number of times each of the three letters R, S, and T appear successively on the computer screen. The string length was always either 10 or 12 letters with Ss required to enter their numerical response into the keyboard. 6. Letter Series test (LS). This is a 38-item version of the typical Thurstonian series completion test, which is a well-known marker of the Induction primary factor. The test was also administered with a 20 min time limit. 7. Hidden Words test (HW). This is a 20-item long computer-administered test of the Flexibility of Closure primary factor. The Ss’ task is to try to recognize a word written with incomplete letters (i.e. sections of the printed letters were missing) entering these responses on the computer keyboard. All tests were administered under the S-paced conditions. For each test separate speed (average time to answer an item) and accuracy (number-correct) scores were obtained. It is necessary to explain the reasons for choosing the above seven tests in this study. Since the most complex versions of our experimental tasks measure fluid intelligence at the second order, the majority of the psychometric tests, i.e. Matrices, Letter Counting, and Letter Series, are also marker tests for the primary factors of Gf. Another two tests, the Immediate Memory tests, are known markers of the short-term acquisition and retrieval function (SAR) which is closely linked to Gf, in original formulations of the Gf/Gc theory. The Hidden Word test defines the broad visualization factor (Gv) at the second order. However, if no other Gv markers appear in the best battery, this test too would tend to load on the Gf factor. Finally, the General Knowledge test is the measure most removed from any Gf processes in this battery, it is a clear marker for Gc. Given the nature of the psychometric test battery for this study, a single factor indicating Gf should emerge. This factor should obtain low loadings from the General Knowledge test and, perhaps also, the Hidden Words test. Procedure For these experiments, we employed Macintosh SE computers. Ss were tested in parties of 2 to 4 at the same time. All instructions and scoring were carried out by the computer. Ss were not allowed to use paper-and-pencil or notes during the testing. An experimenter was present in the room throughout the testing in order to provide further instructions if needed, and in order to run the computer system smoothly. The testing lasted between 90 and 200 min-older Ss requiring longer time to complete the tasks. Whenever requested, breaks in testing were allowed. Typically, a 10 min coffee break was given after 40-50 min of work and in about 10 cases Ss were tested in two sessions. Statistical analyses Two types of statistical procedures are employed in this paper. One set utilized the SYSTAT package: it was used to run ANOVA analyses, correlations and covariances, exploratory factor analysis, and to generate scatterplots for Figs 1 and 2. COSAN (McDonald, 1978) and LISREL 7.13 (Joreskog & Sorbom, 1988) packages were used for structural equations modeling. Even though LISREL is perhaps better known and it has been commercially available for a longer period of time, COSAN provides certain conveniences for the purposes of this paper. The main advantage is the easy and flexible way to constrain model parameters for inequalities. Thus, it is possible to stipulate that one model parameter must be greater than some other model parameter. In this paper, all reported analyses based on correlations were carried out with COSAN. A LISREL-based analysis of the variance-covariance matrix was carried-out on the selected final model. In accordance with the suggestions of Joreskog and Sorbom (1988) for both COSAN and LISREL, maximum likelihood solutions are fitted even though small to moderate departures from univariate normality are present in the experimental tasks of this study.

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Complexity, age and fluid intelligence Table 2. Descriptive statistics for the experimental tasks’

Table 1. Descriptive statistics and loading on the first factor from the psychometric tests (accuracy scores) Psychometric variable ~ Number Span Forward Number Span Backward General Knowledge Matrices Letter Counting Letter Series Hidden Words

Mean

SD

Factor loading’

6.4 5.9 13.1 6.0 5.7 7.3 9.7

1.5 1.9 3.2 2.8 3.8 3.1 4.5

0.622 0.726 0.330 0.666 0.687 0.759 0.623

‘Correlations among these measures are presented in Table 3.

MM” Swaps test One Swap Two Swaps Three Swaps Four Swaps Triplet Numbers Triplet Numbers test ‘Search’ Triplets ‘Half-rule’ Triplets ‘One-rule’ Triplets ‘Two-rules’ Triplets

10.7 9.9 9.7 9.1

0.994 0.977 0.953 0.871

SD

Reliability

2.1 2.6 2.8 3.0

0.68 0.75 0.66 0.69

0.013 0.041 0.069 0.159

0.87 0.86 0.85 0.83

‘Reliability estimates are test-retest correlations. These correlations are based on young adults (N = 47 for the Triplet Numbers test and on N = 43 for the Swaps test).

RESULTS The main part of this paper examines findings with accuracy scores. These scores are the most commonly used measures of intelligence. The speed of test-taking measures will be considered briefly at the end of this section. Findings with accuracy measures

It is useful to analyze present data with both exploratory and confirmatory analytic procedures. Descriptive statistics and the results of exploratory analysis will be considered first. Psychometric tests. Table 1 presents arithmetic means and standard deviations for seven pscyhometric tests of this study, i.e. Digit Span Forward, Digit Span Backward, General Knowledge, Matrices, Letter Counting, Letter Series, and Hidden Words. Correlations and covariances between all tests are presented in Table 3. The third column of Table 1 contains loadings on the first principal factor among the psychometric tests. In these data only the first latent root was greater than one. Inspection of the size of component loadings reveals that the highest values come from Gf and memory span measures. As expected, General Knowledge test has the lowest loading. The communality of this test is only 0.10. Thus the effect on this factor’s correlation with age can only be minimal. This test obviously contributes very little to the factor measured by the present psychometric battery, about as much as would be captured by the typical correlation between Gf and Gc. The appropriate interpretation of this factor is Gf. A principal component score for each S was calculated and correlated with scores on Swaps and Triplet Numbers tests at different levels of complexity. These correlations are presented in Figs 2 and 3. Experimental tasks: Triplet Numbers and Swaps test. Table 2 presents means and standard deviations for each of the different levels of complexity for each experimental task. The table also contains correlations between two presentations of the tasks. These test-retest reliability estimates are based on a sample of young adults. Inspection of Table 2 shows that arithmetic means decrease as the number of rules in the Triplet Numbers test increase, and as the number of swaps in the Swaps test increase. Trends in both tests are statistically significant. Thus, for the Swaps test MANOVA F(3,150) = 25.858; P < 0.05 and for the Triplet Numbers test, MANOVA F(3,150) = 32.102; P < 0.05. In both cases, the linear trend on arithmetic means is significant, indicating that the complexity manipulation leads to increasingly difficult versions of the two tasks. An important feature of the data in Table 2 is the ceiling-level performance in the easier versions of the Triplet Numbers test and, to a lesser extent, in the Swaps test, i.e. a large number of people get maximum scores on the easier versions of these tests. An accompanying feature of the easy tasks is the presence of rather small standard deviations. Again, this is particularly pronounced with the Triplet Numbers test. For the Swaps test the differences between standard deviations are not significant. This finding is not important for the use of MANOVA procedure in analyzing the repeated measures design but it may act as a qualifier of the procedures based on correlations to be used later in this paper. The presence of such discrepancies calls for a distinction between the correlational effects attributable to the size of the variances and genuine lack of covariation between

274

LAZAR STANKOV

the attributes. In other words, if lack of correlation between low complexity tasks and age can be attributed to the low variances that arise from the ceiling-level performance, substantive interpretation of the effects as due to an increase in complexity may be open to question. This may in fact indicate random responding at the easy level. Evidence in favor of systematic effects rather than random responding is needed. Exploratory examination of the relationship between age, Gf and experimental tasks: correlations and interactions. In these data, correlation between age and Gf (factor scores) is r = -0.64. Given a relatively large sample of people older than 60, this correlation is in agreement with the typical findings. In order to examine the possibility that the departure from linearity in the age-Gf relationship may be of importance, a quadratic function of age was correlated with Gf. This correlation (-0.65) was only slightly higher than -0.64 and, since “raw” age correlated 0.99 with its quadratic transformation, it was decided to use “raw” age variable in subsequent analyses.

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3

Complexity,

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age and fluid intelligence

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Correlations between age and Gf on one hand and performance at different levels of complexity for both experimental tasks on the other hand are set out in Figs 2a, 2b, 3a and 3b underneath the title for a given scatterplot. The expectation is that both age and Gf will show an increasing correlation with experimental tasks in accordance with increases in the tasks of complexity. Apart from the fact that the ‘Half-rule’ Triplets test shows a slight decline in correlation relative to the ‘Search’ Triplets test, the pattern of correlations with the Gf factor is in general agreement with this expectation. For the Triplet Numbers test, age shows the same pattern of correlation as does the Gf. However, as shown in Figs 2a and 2b, Gf does, but age does not correlate with the Swaps test, in the expected way. All correlations between Swaps test and age are about equal in size. In addition to examining the pattern of correlations it is instructive to look at the scatterplots that give rise to these correlations. Figures 2a and 2b display scatterplots for the Swaps test in its relationship to age (left-hand side) and to Gf (right-hand side). Figures 3a and 3b display scatterplots for the Triplet Numbers test in its relationship to age (left-hand side) and Gf (right-hand side). The slope of the linear best-fitting line within each scatterplot is suggestive of the strength of the relationship-the greater the departure from the horizontal line, the stronger

LAZAR STANKOV

276

the relationship between the two variables. However, these slopes are affected by the size of standard deviations and levels of the Swaps test have more nearly equal standard deviations. Thus, it is mainly for Swaps test (i.e. for Fig. 2) that we may take note of the slopes as indicative of the strength of the relationship. These slopes increase systematically as we move from the least demanding to the most demanding conditions of the Swaps test. The most important observation in the scatterplots of Figs 2 and 3 is that the older and the less able (i.e. those Ss having low scores on Gf) start making errors first, and are most affected by task requirements. A similar finding regarding the performance of the less able Ss was reported by Hunt and Lansman (1982) with their primary-secondary task paradigm. It is particularly important for our purposes, that both the easiest and the most difficult tasks show the same tendency. This observation is relevant to the claim that low correlation at the easiest level is due to statistical artefacts related to extreme differences in the size of standard deviations. It is usually assumed that errors at the easiest level are due to random processes. Similarly, at the most difficult level, the correct answers are often assumed to arise from a random process. Thus, small standard (a)

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Complexity,

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deviations are interpreted to imply an equal distribution of errors across the whole range of Gf or age, i.e. if the task is easy, both high Gf (or young) and low Gf (or old) are equally likely to give an incorrect answer. In fact, inspection of Figs 2 and 3 reveals a clear and systematic trend in these data. At all levels of complexity manipulations it is the low Gf and older people who tend to commit errors and obtain low scores. This points to the presence of systematic influences. The effects of such influences is in accordance with the theory of the role of complexity in aging and intelligence. A careful observer will notice that different symbols are used for Ss depicted in Figs 2 and 3. In the left-hand scatterplots in these figures, little triangles indicate people who have low Gf scores (lower than minus one standard deviation below the mean), circles indicate people between minus one and plus one standard deviation around the mean and black squares indicate high scoring individuals (higher than plus one standard deviation above the mean). In the right-hand scatterplots, triangles stand for people who are over 60 years of age, black squares stand for people younger than 30, and circles represent 30 to 60 year olds. Inspection of the diagram with this information in mind will lead to the same interpretation, i.e. low Gf and older people tend to obtain low scores on all versions of the experimental tasks.

LAZAR STANKOV

218

The relationship between age (and Gf) on one hand, and levels of the experimental tasks on the other, can also be displayed by plotting the results of old and young (and high Gf and low Gf) Ss for each level of these tasks. Figure 4a presents the plot of the Swaps scores and Fig. 4b presents the findings with Triplet Numbers scores. In both figures solid lines stand for high and low Gf scores, and broken lines represent old and young Ss. It is apparent from Figs 4a and 4b that differences between groups (both old-young and high-low Gf) increase as the task becomes more demanding. It is also apparent that these differences are somewhat smaller for age than they are for Gf. Finally, the interaction effects with the Swaps task are not as pronounced as interaction effects with the Triplets task. Basically, the Swaps task in this study shows a greater difference between the two Gf (high-low) groups (and between age, i.e. young-old groups) at the easier Swaps levels than what was reported by Stankov and Crawford (1993). All interaction effects in Figs 4a and 4b are statistically significant. The only exception is the age by complexity interaction for the Swaps test in the left-hand panel of Fig. 4 [F(3,454) = 1.504 P = 0.191. Overall, preliminary analyses show that the evidence for the complexity effect for the Swaps test in these data is not as strongly supported by the correlations as it is supported by the outcomes of ANOVA analyses displayed in Fig. 4 and by the ordering of slopes in Figs 2 and 3. These analyses, however, represent only a rough description of the relationships that exist in the data. The problem resides in the treatment of experimental tasks. In the preceding analyses these tasks were treated as extraneous to the remainder of the psychometric test battery. In fact, they are proper measures of Gf. The real question is whether the part of experimental tasks’ variance that is shared with the other measures of intelligence increases. Correlations between raw scores on experimental tasks and intelligence/age do not provide an adequate answer to this question. These correlations do not distinguish between common and unique parts of test scores. Testing the feasibility

of the model: structural

equations

modeling

The lower section of Table 3 contains correlations between the seven psychometric variables, age, and the two experimental tasks, the Swaps and Triplet Numbers tests, each of which contain four levels of complexity. Initially, these correlations are used to test if the model relating psychometric variables and experimental tasks is consistent with the observed data. There are two reasons for using correlations in searching for the structure among cognitive tasks. Firstly, traditional psychometric studies of intelligence are based on correlational data and interpretation of the factory analytic results is well-established. Due to the differences in scale between variables under study, interpretation of the covariance-based solutions would require more effort from the typical user. Secondly, preliminary analyses have indicated that complexity effects are not as strong in correlations as they are in ANOVA and in slopes of the regression lines. Thus, reliable findings

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test

The left-hand test.

Complexity, age and fluid intelligence

279

with correlations should be relatively hard to obtain. In order to establish invariance of the chosen solution, crucial analyses were repeated with covariances. Analyses based on correlations. Prior to arriving at the solution presented in Fig. 5, a series of analyses were carried out on correlations among the 15 psychological tests only, i.e. all the variables from Table 3 excluding age. The aim of these analyses was to establish, within the plausible alternative models, the most viable structural account of the relationship between the tests. After arriving at such a model, the age variable was brought into the system. In all the attempted solutions with the 15 psychological variables there were three factors, a factor of fluid intelligence (Gf) having loadings on all tests, and two factors with loadings on each of the two experimental tasks of Swaps and Triplet Numbers. The solutions differed among themselves with respect to the existence of correlations between all or some of the factors, i.e. whether the solution should be orthogonal or oblique, and whether it was stipulated that loadings on factors have to increase systematically. An increase in loadings was always imposed in the model for only one of the two factors on which the experimental tasks have their loadings. Summary statistics for these different solutions are presented in Table 4. A rough rule-of-thumb in structural equation modeling states that the solution may be considered acceptable if the chi-square statistic is at most twice the size of the number of degrees of freedom (Marsh, Balla & McDonald, 1989). By that criterion, the solutions of Table 4 that have a chi-square statistic greater than 200 are clearly unacceptable. In applying this rule-of-thumb to evaluate the solutions in Table 4, the reader should pay particular attention to the probability (or P-value) column. Probability values are one set of indices of goodness-of-fit that are commonly used today. Solutions with the largest probability value provide “the best acceptable model”. Finally, some of the models are nested. This means that a difference between the two chi-squared values for these models is approximated by the chi-squared distribution as well. Its significance can be tested using the difference between the degrees of freedom for the two nested solutions. In this way one can test if the parameter(s) by which the two models differ is of importance in the population. The first two analyses of Table 4 (rows 1 and 2) provide orthogonal solutions, i.e. the three factors were assumed to be uncorrelated. The difference between these two solutions derives from the fact that, in the second solution, loadings of the four Swaps tasks and of the four Triplet Number tasks on the Gf factor were required to increase. In the first solution this requirement was not imposed, i.e. loadings of these tasks were left free. The fact that indices of goodness of fit basically did not change, suggests that the increase in correlations exists in the data in a raw form, i.e. the increase is not imposed by the constraints, the correlations tend to rise. This is in accordance with the findings of the exploratory analyses, as outlined above. The solution in row 2 of Table 4 can be seen as a “baseline” model. The next three solutions are nested. The solution in row 3 assumes that there is a correlation between the Swaps and Triplets factors only. The difference between the chi-squared values in rows 2 and 3 is 13.8711 and the difference between the corresponding degrees of freedom is 1. Since, for one degree of freedom the criterion chi-square is 3.84, it may be concluded that the model of row 3 is a significant improvement on the orthogonal model. Indeed, in the end this proved to be the model of choice for the data at hand. However, before accepting this conclusion, it is necessary to demonstrate that better fits cannot be obtained with some other plausible models. Rows 4 and 5 present summary statistics for the solutions in which two factor intercorrelations (row 4) and all three factor intercorrelations (row 5) were left free. Obviously these two nested solutions do not provide a significant improvement over and above that achieved in row 3. Furthermore, comparison of the P-values for the three correlated factors solutions indicates that the most acceptable solution is the one in row 3. Rows 6 and 7 present summary statistics for two solutions that are variations of the model in row 3. In particular, the model in row 6 requires that factor loadings on the narrow factors of Swaps and Triplets increase while all the loadings on the Gf factor are left free. Again, while this solution does not represent a significant improvement over that of row 3, the P-value in row 6 suggests a poorer goodness-of-fit than that of row 3. Row 7 represents the most unsatisfactory solution of Table 4. In that model all loadings of the Swaps test on the Gf factor and all loadings of the Triplets test on the Gf factor were constrained to be equal. Obviously, the increase in the size of loadings

Number Span Forward Number Span Backward General Knowledge Matrices Letter Counting Letter Series Hidden Words swaps: One Swap Swaps: Two Swaps Swaps: Three Swaps Swaps: Four Swaps ‘Search’ Triplets ‘Half-rule’ Triplets ‘One-rule’ Triplets ‘Two-rules’ Triplets Age 10.02 0.225 0.174 0.325 0.216 0.134 0.125 0.103 0.113 0.123 0.064 0.239 0.142 0.062

1.346

1.174

1.859 3.716 0.219 0.437 0.479 0.553 0.356 0.321 0.318 0.367 0.373 0.210 0.225 0.327 0.370 0.480

2.313 0.629 0.244 0.393 0.372 0.452 0.320 0.310 0.321 0.372 0.336 0.154 0.161 0.345 0.339 0.437 2.453 2.057 8.335 0.538 0.486 0.426 0.320 0.402 0.408 0.434 0.240 0.192 0.284 0.334 0.443

1.727

4

(main diagonal) 3

variances

2

I

(lower triangle),

2.152 3.542 2.095 5.913 14.500 0.569 0.328 0.304 0.385 0.370 0.370 0.148 0.178 0.268 0.320 0.513

5 2.197 3.120 3.089 5.544 5.631 5.737 20.350 0.371 0.254 0.294 0.256 0.095 0.054 0.263 0.381 0.541

I

(upper triangle) 6

2.137 3.339 3.200 4.357 6.740 9.661 0.409 0.328 0.389 0.363 0.390 0.200 0.189 0.418 0.426 0.524

and covariances”

0.982 1.300 0.881 1.926 2.415 2.125 3.484 4.439 0.641 0.667 0.640 0.148 0.064 0.41 I 0.420 0.423

8 I.284 1.621 1.037 3.045 3.847 3.179 3.010 3.507 6.899 0.781 0.756 0.298 0.221 0.374 0.413 0.394

9

between psychometric

I.558 I.965 0.896 3.242 3.884 3.110 3.650 3.828 5.652 7.586 0.824 0.186 0.217 0.479 0.483 0.424

10

tests, experimental

I.517 2.153 1.006 3.724 4.192 3.598 3.436 3.960 5.900 6.742 8.826 0.136 0.272 0.412 0.454 0.391

0.031 0.054 0.052 0.092 0.075 0.083 0.057 0.041 0.104 0.068 0.054 0.018 0.041 0.144 0.217 0.207

12

0.099 0.178 0.082 0.224 0.275 0.238 0.098 0.054 0.235 0.242 0.327 0.002 0.164 0.472 0.451 0.198

13

0.392 0.437 0.519 0.565 0.704 0.894 0.816 0.588 0.676 0.908 0.843 0.013 0.131 0.474 0.668 0.354

14

tasks of Swaps and Triplet Numbers 11

0.817 I.139 0.715 I.531 I .933 2.101 2.727 I.388 I.721 2.111 2.140 0.046 0.290 0.129 2.517 0.486

15

tests, and ane 16 1.299 1.826 0.387 2.505 3.824 3.186 4.777 1.722 2.026 2.284 2.273 0.540 0.157 0.477 1.509 3.826

“Age variable is multiplied by - IO, all four Triplets tests (Search-, Half-, One-, and Two-rules Triplets) are multiplied by + IO. Apart from the fact that age changes the sign of its covariances and correlations, this scaling does not affect the size of correlations. However, since the efficiency and accuracy with which numeric routines underlying LISREL minimize the fitting function can be affected, it is a good practice to scale these variables to similar variances prior to structural equation modeling analysis involving covariances. Multiplication by IO achieves this latter goal for all our variables.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1I 12. 13. 14. 15. 16.

Table 3. Correlations

&? $! *I y $ 6 <

Complexity, age and fluid intelligence Table 4. Summary

Solution-type

and constraints

statistics

for different COSAN

on loadings

1S Psychological tests 1. Orthogonal solution; no constraints 2. Orthogonal solution; increasing loadings on Gf factor 3. Correlations between Swaps and Triplets factors only; increasing loadings on the Gf factor only. This solution is presented in the left-hand section of Table 5. 4. Correlations between Swaps and Triplets factors, and Gf and Swaps factors; increasing loadings on the Gf factor. 5. Correlations between all three factors; increasing loadings on the Gf factor only. 6. Correlations between Swaps and Triplets factors only; increasing loadings on the Swaps and Triplets factors. 7. Correlations between Triplet and Swaps factors only; loadings of Swaps and Triplets subtasks on the Gf factor constrained to be equal. Age and IS psychological tests 8. Age defines a separate factor that correlates with Triplet and Swaps factors only. 9. Age defines a separate factor that correlates with Triplets Swaps, and Gf factors. 10. Age defines a separate factor that correlates with Gf. This solution is presented in the right-hand section of Table 5 Il. Direct link: age and all levels of Swaps and Triplets define a factor. 12. Same as solution in row 10 except for the fact that it is based on 14 rather than 16 variables. Excluded: General Knowledge test and Hidden Words test.

solutions

281

based on correlations

from Table 3 Average residual

Root mean square

Chi-square

d/

P-value

Largest residual

132.791 132.826

82 82

0.00033 0.00033

0.1902 0.190s

0.04004 0.04034

0.05762 0.05750

118.9549

81

0.003867

0.1659

0.03087

0.04412

118.8842

80

0.083137

0.1588

0.03040

0.04348

118.8814

79

0.002499

0.1583

0.0311

119.5619

81

0.003462

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0.031

249.5966

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0.4899

0.17700

0.2245

221.5700

93

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0.08591

0.1343

138.1765

93

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0.1667

0.03220

0.04624

139.5979

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0.001982

0.1714

0.03258

0.04673

139.4895

87

0.000305

0.1715

0.03211

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93.2164

67

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0.04175

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0.04364

II

0.4364

in parallel with the increase in the complexity of the tasks is a genuine feature of the data. Also, the data indicate that the increase in factor loadings is present on the Gf factor. The most satisfactory solution with the present battery of psychological tests, generating a chi-square value of 118.9549 with df = 81, P-value = 0.003867, allows for a correlation between SWAPS and TRIPLETS factors and for an increase in correlation on the Gf factor. This solution is presented in the left-hand section of Table 5. There are two aspects to this solution which need to be underscored. Firstly, loadings of both Swaps and Triplets tests increase on the Gf factor even though both cases display the same values for successive levels of complexity. This occurs because the increases are small, and are not apparent with numbers rounded to the second decimal place. Second, narrow SWAPS and TRIPLETS factors also show increases in the size of loadings for the first three levels of complexity, but Table 5. COSAN

Psychological 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

solution

tests

Gf

Number Span Forward Number Span Backward General Knowledge Matrices Letter Counting Letter Series Hidden Words Swaps: One Swap Swaps: Two Swaps Swaps: Three Swaps Swaps: Four Swaps ‘Search’ Triplets ‘Half-rule’ Triplets ‘One-rule’ Triplets ‘Two-rules’ Triplets Age

0.65 0.73 0.34 0.67 0.68 0.76 0.54 0.48 0.51 0.53 0.53 0.26 0.26 0.48 0.54

with the 15 psychological

variables

Common

Unique factor loadings

factors

SWAPS 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.56 0.68 0.76 0.71 0.00 0.00 0.00 0.00

TRIPLETS 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05 0.51 0.67 0.62

0.57 0.47 0.88 0.55 0.54 0.42 0.71 0.47 0.28 0.14 0.21 0.92 0.68 0.32 0.33

(left panel) and with age included Common Gf 0.64 0.70 0.29 0.68 0.67 0.75 0.57 0.50 0.52 0.54 0.54 0.26 0.26 0.49 0.56 0.00

Factor intercorrelations

Summary

0.00 0.00 statistics

for these solutions

1.00 0.37 are provided

0.00 0.00 0.00 0.00

0.00

TRIPLETS 0.00 0.00 0.00 0.00 0.00

AGE 0.00 0.00 0.00 0.00 0.00 0.00

0.00

0.00

0.00

0.00

0.00

0.54 0.67 0.76 0.70 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.07 0.45 0.69 0.62 0.00

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.74

Factor intercorrelations

I .oo

Gf Swaps Triplets Aae

SWAPS

(right panel)

factors

1.00

1.00 0.00 0.00 1.00

in Table 4 (rows 3 and IO).

I .oo 0.33 0.00

1.00 0.00

1.oo

Unique factor loadings 0.59 0.49 0.91 0.56 0.53 0.43 0.67 0.46 0.28 0.15 0.2 I 0.92 0.68 0.31 0.33 0.48

282

LAZARSTANKOV

decrease at the fourth. We may note, parenthetically, that this outcome could be troublesome for the Complexity Effect Phenomenon if age were to operate indirectly through the narrow factors. In other words, when the effects due to the intelligence factor are taken into account, the most complex Triplets and Swaps tasks may have lower correlations with age than the less complex tasks. Age and structure among the psychological variables. After obtaining the above solution with the battery of psychological variables, age was brought into the system. Three COSAN analyses based on 16 variables were then carried out. (In fact, the sign of the age variable was changed which means that age should be interpreted as “the year of birth”. I will continue to use “age” in the remainder of this paper in order to keep the terminology consistent.) In terms of the factor structure of Table 5, the addition of age meant that there was a fourth factor added with a loading on age only. The correlation between age and the Gf factor was fixed to equal 1.00. Following the suggestion of McArdle (see Horn & McArdle, 1980) this constraint was used in order to overcome a tendency towards slight overestimation of the size of this correlation and other parameters of the model. This occasionally happens when one employs maximum likelihood procedures with correlational matrices. If left free, this correlation equals 1.19. That constraint was not employed with LISREL analysis of covariances. The following solutions can be seen as attempts to determine if the age factor has zero correlation with any of the three psychological factors. The first COSAN solution (row 8 in Table 4) with 16 variables postulates the existence of correlations between age and both TRIPLETS and SWAPS factors, and no correlation between age and the Gf factor. Obviously, this solution produces an unsatisfactory fit to the data. The second solution with 16 variables (row 9 in Table 4) provides more satisfactory statistics. This solution allows for correlations between age and all three psychological factors. However, inspection of the correlations with age, reveals that the values are rather low for SWAPS (r = 0.08) and TRIPLETS (r = 0.12) factors. This suggests that an acceptable solution may be one which allows for a correlation between the Gf and age factors only. Summary statistics for this solution are in row 10 of Table 4 and the relationships among observed and latent variables provided by this solution are presented in the right-hand section of Table 5. Here, age is represented as a factor, located at the same level of analysis as the Gf, SWAPS, and TRIPLETS factors. This Age factor is correlated with the Gf factor but not with the Swaps and Triplets factors which, in accordance with previous analyses, are correlated among themselves. A run with 16 variables involving only three factors (Gf, Swaps, and Triplets) with age having loadings on the Gf factor, produces exactly the same goodness-of-fit indices as those listed in row 9 of Table 4. From the analytical point of view this solution is identical to the one reported in Fig. 5. Conceptually, however, the solution of Fig. 5 is preferable since it treats age as a factor, not a variable of the same type as psychological tests. A solution that defined the 4th factor in terms of age and the eight experimental variables was obtained in order to check for the existence of the direct links between age and different levels of Swaps and Triplet Numbers tests. Everything else in the solution remained the same as that in Table 5. In other words, the solution of row 11 in Table 4 has free parameters from all eight experimental variables on the age factor. This solution is nested in that of row 10. Obviously, since the chi-squared value in row 11 is almost the same as that of row 10, while degrees of freedom in row 11 are reduced by 8, the solution in row 10 is better supported by the data. Finally, our battery of seven psychometric tests contains two tests, General Knowledge and Hidden Words, that belong to Gc and broad visualization abilities, respectively. Thus, the last COSAN analysis of row 12 presents summary statistics for a solution based on 14 variables. Exploratory principal factors solution presented in Table 1 and the solutions of Table 5, indicate that these two variables (GK and HW) have relatively little in common with the majority of Gf measures in this study. The exclusion of these two tests produced a satisfactory P-value of 0.019. Substantive interpretation of the model based on 14 variables is identical to that based on 16 variables. This solution is clearly more acceptable from the statistical point of view. However, it is my opinion that solutions of Table 5 are more acceptable on a psychological level. Low loadings of the tests that are not markers for Gf, and high loadings of the tests that are known markers of Gf, indicate some support for both discriminant and convergent validity of the accepted solution. As can be seen in Table 3 there is no systematic trend in correlations between the General Knowledge test and levels of the experimental task of this study. Further, Gf factor scores that

Complexity,

00

1.37

1.64

9.10

4.66

Z22

283

age and fluid intelligence

4.16

13.57

2.01

1.93

1.10

1.64

0.02

0.11

0.15

0.64

Fig. 5. Path diagram of the selected solution from the LISREL analysis of the covariances between age, psychometric tests, and experimental tasks (row 12 from Table 4). The main characteristics of this path diagram are: (a) the presence of four latent dimensions (Age, Gf, Swaps, and Triplets); (b) increasing size of the covariances between intelligence and tasks of higher complexity; (c) the effect of age on subtasks of Swaps and Triplets factors is through the Gf factor only.

are correlated with the experimental tasks contain a very small weight from the GK test. This was expected on the basis of discriminant validity. Analyses based on covariances. In order to check if the results based on correlations can be obtained with covariances, additional LISREL analyses involving all 16 variables were carried out. It is useful to note that inequality constraints cannot be imposed with LISREL, the presence of increasing parameter estimates reflects a ‘natural’ relationship in the data. Summary statistics for this solution are presented in row 1 of Table 6 and a path diagram is available in Fig. 5. Cudeck (1989) and Joreskog and Sorbom (1988) strongly recommend the use of covariance matrices in structural equation modeling. Essentially the same result as that presented in Table 5 is obtained in this covariance analysis. The only apparent difference between correlation- and covariance-based solutions is the presence of increasing loadings over all four experimental conditions on the SWAPS and TRIPLETS factors in Fig. 5. In the analyses of correlations this trend was present with only three conditions; loadings tended to decline at the most complex level on both TRIPLETS and SWAPS factors. Two additional solutions that are nested within the solution of Fig. 5 were obtained. Since the absence of relationship between age and the two task-related factors was originally detected with correlations, not covariances, it was deemed necessary to have another LISREL run in which slings between Age and SWAPS and TRIPLETS factors in Fig. 5 were left free. Summary statistics for this solution are presented in row 2 of Table 6. It is obvious that this solution does not represent an improvement over and above the one presented in row 1 of Table 6 and in Fig. 5. Similarly, in order to explore if the presence of direct links between age and experimental task conditions would improve the model fit, a solution in which age factor had loadings on all eight experimental variables was obtained. Summary statistics for this solution are presented in row 3 of Table 6. Again, as with correlations, the improvement in model fit is not statistically significant and we should retain the parsimonious model of Fig. 5.

Table 6. Summary Solution-type

statistics

and constraints

for different

LISREL

solutions based on covariance

on loadings

1. LISREL analysis based on covariances of Table 3. Age defines a separate factor that correlates with Gf. ‘Search’ Triplets loading on TRIPLETS factor fixed at zero. This solution is presented in Fig. 5. It corresponds to the solution presented in row IO of Table 4. 2. Age defines a separate factor that correlates with Triplets Swaps, and Gf factors. Corresponds to row 9 of Table 4. 3. Direct link: age and all levels of Swaps and Triplets define a factor. Corresponds to row 11 of Table 4.

Chi-sauare

matrix from Table 3 df

P-value

140.78

96

0.002

139.47

94

0.002

132.49

88

0.002

284

LAZAR STANKOV Table7.Speed of test-taking

measures

for the Swaps and Triplets

Descriptive Complexity

conditions

Swaps test I. One Swap 2. Two Swaps 3. Three Swaps 4. Four Swaps Triplets test I. ‘Search’ Triplets 2. ‘Half-rule’ Triplets 3. ‘One-rule’ TriDlets 4. ‘Two-rules’ Tkplets

statistics

of the solutions

testsa

Mean

SD

Age

Gf

135.6 222.2 316.8 403.6

87.2 114.0 146.0 185.3

0.576 0.549 0.544 0.508

-0.583 -0.551 -0.609 -0.548

1.9 2.1 2.6 4.7

1.1 0.9 1.4 2.9

0.48 I 0.447 0.530 0.565

-0.543 -0.500 -0.592 -0.535

“For the Swaps test the total time to complete the tests is recorded; Numbers test it is the average time to answer an item

The interpretation

Numbers

Correlations

presented

in Table

for the Triplet

5 and Fig. 5 is as follows:

Age is related to experimental tasks through its links with the Gf factor. Age does not appear to have any other significant direct or indirect links with the experimental tasks of this study. In terms of the options listed in Fig. 1, the solutions presented in Table 5 and Fig. 5 correspond to the indirect path f, i.e. there is no support for the Complexity Effect Phenomenon. Manipulations of complexity in the Swaps and Triplet Numbers tests lead to increases in the size of factor loadings on the general factor of this battery; the Gf factor. Similar increases in the narrow task-related factors were observed in the covariance-based solutions, but not in the correlation-based solutions. Stankov and Raykov (in press) carried out further and more detailed analyses with a set of 12 variables from the present study. The emphasis in that work was methodological, the aim was to study the usefulness of the SEM models in distinguishing between the constructs of difficulty and complexity. The starting matrix was the covariances/means matrix rather than correlation or covariance matrices used in this paper. The study began with a model that is similar to the one presented in Fig. 5. It proceeded with a series of nested analyses in order to establish the equality constraints on parameters of the experimental tasks affect the goodness-of-fit indices. In other words, we constrained estimates of test means to be equal, all loadings of the Swaps test on the Gf factor to be equal, then all loadings of the same test on the SWAPS factor to be equal, etc. The results do not add any new information about the Complexity Effect Phenomenon in aging. However, they do provide strong support for point (b) above: whenever we try to constrain a set of four task-related parameters to be equal, the goodness-of-fit index becomes significantly worse. This holds for both the Triplet Numbers and the Swaps task. Findings with the speed of test-taking

measures

Stankov and Crawford (1993) report that while increases in complexity make people work slower, there is no systematic change in correlations between Gf and measures of speed of test-taking. This is also apparent from the data presented in Table 7, speed measures do not behave in accordance with the Complexity Effect Phenomenon. It has been our experience that the speed of test-taking measures from a diverse battery of cognitive tests tend to have higher correlations among themselves than do the accuracy measures, and they often define a single broad factor. In order to check if the same outcome obtains with the present data, principal factor analysis was carried out with the speed measures for the seven psychometric tests of this battery. Correlations among the speed of test-taking measures and the results of factor analysis are presented in Table 8. I mention the result here in order to point out that, in this data, two tests, computerized Matrices and Hidden Words, were different from the rest of the psychometric battery. They are more difficult than the familiar paper and pencil tests. Speed scores for these two tests have low negative correlations with the other speed scores and, as a consequence, their loadings and communalities are lower than those of the other tests. It appears that the speed of test-taking with difficult tasks, requiring a considerable amount of mental

Complexity, age and fluid intelligence

285

manipulation and checking and re-checking of possible answers, do not have much in common with the other cognitive tasks of this test battery. All other tests elicited relatively quick and determined responses either because they contained multiple choice or the answer could be easily retrieved from memory. DISCUSSION

Performance on complex tasks has traditionally been linked to intelligence. Recent work on the boundary between experimental cognitive psychology and individual differences represents an attempt to identify the ingredients of complexity. Ingredients are the elementary cognitive operations that gradually increase the complexity of tasks, i.e. they lead to an increased correlation with intelligence tests. Stankov and Crawford (1993) claim that the ingredients of complexity important for the two tasks studied here, are: the number of elements and their interrelations that can be held in the mind during problem solving. In comparison to the traditional psychometric approach, the work reported here provides a more precise way to examine intelligence. Within the psychology of aging, it has been frequently observed that the difference in performance between the young and the elderly depends on the complexity of the task (see evidence reviewed by Cerella, 1985; Salthouse, 1985). Since Gf and age have a significant correlation, it is reasonable to ask if they relate to complexity in different ways. This study applied structural equation modeling procedures to examine the relationship between age, a set of psychometric measures of Gf, and two Gf tasks. The increasingly more complex tasks lead to an impoverished overall performance (i.e. they become more difficult), and to a gradually increased correlation with the Gf factor of the present battery (i.e. they become more complex). These subtasks also define two narrower, task-related (SWAPS and TRIPLETS) group factors. The size of factor loadings on these narrow factors did not increase consistently in all analyses of this paper. Of the two experimental tasks of this study the Swaps test showed a somewhat lower sensitivity to complexity manipulations. In the present study, age is related to the Gf factor only. Therefore, the Complexity Effect Phenomenon observed in many aging studies is yet another way of saying that the elderly tend to have lower scores on Gf tasks. Within this framework, the Complexity Effect Phenomenon is correctly described as an epiphenomenon of the age-related decline in Gf. Experimental complexity manipulations can be seen as theoretically more precise than a concept such as Gf. Thus, the question can be asked whether age differences derivable from complexity manipulations could be used to explain the correlations between age and Gf. The answer to this question is straightforward if we think in terms of partial correlations. Since correlation between age and Gf is r = -0.64 in this study, and increasing levels of task complexity have higher correlations with both, the answer has to be an unequivocal “Yes”. It may be argued, however, that it is useful to follow the approach of this paper and distinguish between Gf and narrow task-related factors. Basically, Gf is better defined by the more complex versions of experimental tasks. This means that the removal of these tasks from the battery will affect the AGEGf correlation more than the removal of the simple tasks. This paper also examined the sensitivity of the speed of test-taking measures to complexity manipulations. These measures are important if we adopt a limited version of the Complexity Effect Phenomenon that restricts its relevance to the speed of mental operations and if we assume that speed of test-taking taps that speed. The results indicate that the arithmetic means of the speed Table 8. Correlations

between the speed of test-taking 1

1. 2. 3. 4. 5. 6. 7.

Number Span Forward-S Number Span Forward-S General Knowledge-S Matrices-S Letter Counting-S Letter Series-S Hidden Words-S

1.oo 0.361 0.363 -0.168 0.392 0.366 -0.055

2 1.00 0.322 0.016 0.351 0.253 -0.018

measures 3

1.00 -0.107 0.580 0.378 -0.053

and loadings 4

1.00 -0.104 -0.079 -0.059

on the first principal 5

1.00 0.359 -0.087

6

1.00 -0.048

factor of this battery 1

1.oo

Factor

loadinns

0.595 0.489 0.705 -0.142 0.731 0.536 -0.084

LAZAR STANKOV

286

of test-taking measures increase as the task becomes more difficult, but their correlation with intelligence or with age does not increase systematically. It follows that the speed of test-taking measure in this study does not behave in a manner consistent with the Complexity Effect Phenomenon. It is necessary here to note a methodological issue addressed by the present study. In general, it is not easy to form a 60-item test with groupings of items into 5 lots of 12-item groups, of approximately equal difficulty. If successful, the easiest group of items often appears too easy. This is the case, for example, with the Raven’s Progressive Matrices test. In a sense, easy Raven’s items are like the easiest versions of the Swaps and Triplet Numbers tests. This means that in empirical studies of this nature the simplest items may be causing a ceiling effect and, as a consequence, the variance of tests based on these easy items may be reduced. This, of course, provides a challenge to the substantive interpretation of correlations outlined above, since low correlations at the simple levels may be the consequence of random non-systematic responding. This possibility was examined in the present paper through the inspection of scatterplots representing correlations between experimental tasks at different levels of complexity, and age and Gf. Since even in the simplest tasks it is the oldest and the least able who show less than perfect performance, the claim that low correlations are statistical artefacts is unfounded. Nevertheless, future studies of complexity should aim to eliminate, or at least reduce, the presence of ceiling or floor effects. As mentioned in the Introduction, the present view of the relationship between age, intelligence and task complexity is not radically different from views expressed by others. A multivariate way of thinking about these variables suggests that an epiphenomenon rather than a proper phenomenon may be under consideration. Structural equation modeling procedures provide a means of examining this suggestion. Although the present data support this view, it is possible that a different theoretical framework (e.g. age defined in terms of functional variables rather than chronologically), or different data within the same framework may reverse the conclusions reached here. Capacity

theories,

intelligence

and age

While it is possible to talk about the ability to deal with complexity by invoking the constructs of attentional resources or working memory-two versions of the capacity theory which are currently popular accounts of the basic processes of intelligence-there are good reasons for assuming that this is unnecessary. Since explanations of the Complexity Effect Phenomenon in aging often resort to the same two constructs, it is useful to mention briefly some of the reasons why their appeal has waned in studies of intelligence. These are: (a) There should be a decrement in performance under dual or completing tasks conditions if limited capacity systems are involved. Empirical studies have shown that there are competing tasks, notably those involving distinct perceptual processes of vision and audition, that do not lead to any decrement. Nevertheless, these competing tasks do show higher correlations with measures of intelligence than single tests (see Stankov, 1988b). Multiple resources theory may be capable of explaining the absence of decrement but it cannot provide a parsimonious account of the increase in correlations. (b) The logic of the primary-secondary task paradigm (see Hunt & Lansman, 1982) which was developed in order to test the applicability of the attentional resources theory to intelligence, does not seem to work when applied to the competing tasks. Basically, this paradigm involves too many assumptions which are difficult to satisfy in empirical studies and, therefore, is virtually untestable (see Stankov, 1987). (c) Some tasks that are supposed to tap working memory processes did not show an increase in correlation with intelligence (see Spilsbury, Stankov & Roberts, 1990). Also, several studies in which a working memory load was manipulated showed a systematic increase in correlations with intelligence up to a point, and then showed a decrease in correlation at the most difficult level (see Myors, Stankov & Oliphant, 1989; Roberts, Beh & Stankov, 1988). (d) There are several studies in which putative measures of attentional resources, shadowing tasks, competing tasks, did not show the expected high correlation with age (see Sullivan & Stankov, 1990; Stankov, 1986,1988a).

Complexity,

age and fluid intelligence

287

Some of the above reasons derive from the fact that theories in experimental psychology rely on measures of the overall levels of performance-they are not designed to deal with correlations. Further, these theories have a heavy theoretical baggage (e.g. resource- vs data-limited processes, structural interference, automatic-controlled processing, performance operating curves, etc.) which does not appear necessary if the aim is to account for individual and age differences in intelligence. The ability to deal with complexity, i.e. with the increasing number of elements and relations, has the same explanatory power as do the constructs of attentional resources and working memory. Thus, while it can be claimed that the increase in task complexity in the present study is due to an increased demand on processing resources this should be understood as a figure of speech. Our task should be to search for cognitive operations that can reliably increase correlations with intelligence and build out theories about intelligence, and cognitive aging, on that basis. Acknowledgements-1 am grateful to Dr David Grayson for his suggestions on the strategic issues in the use of Structural Equation Modeling procedures of this paper. I am also grateful to Dr Tenko Raykov for his advice regarding the use of LISREL and for his comments on an earlier draft of this paper.

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