The integration of motivational information: A conjoint measurement analysis

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  • Acta Psychologica 46 (1980) 257-269 Q North-ttolland Publishing Company

    THE INTEGRATION OF MOTIVATIONAL INFORMATION: A CONJOINT MEASUREMENT ANALYSIS *

    Geert De SOETE ** State University of Ghent, Ghent, Belgium

    Accepted June 1980

    Within Anderson's (1974a, b, c, 1978b) information integration theory, tlie integration of motivational information was investigated by means of conjoint measurement techniques. Eighteen university students were asked to judge hypothetical co-students characterized by three features (intelligence, motivation and the extent to which they study) according to their chances to pass. Both rank order data and ratings were obtained. The orderings of most subjects could be represented very well by an additive model. A polynomial regression procedure was applied to determine the shape of the response function for the ratings. As this function was quite linear for all subjects, the ratings could be said to form an interval scale.

    In all kinds of circumstances, judgments are being made about people and situations based on a wide variety of information. Both the clini- cian, the counselor and the personnel manager have to sum up the client, respectively the job applicant, by integrating several diagnostic data. But also in the daily interaction, people continuousIy attribute to each other qualities, intentions and abilities inferred from certain per- ceived features. It was originally in connection with the latter area, viz., person perception, that Anderson developed his information integra- tion theory. Soon, his theory was applied successfully to the most diverse areas (Anderson 1974a, b, c, 1978b). In this paper we shall investigate the integration of motivational information by use of con- joint measurement techniques.

    * I wish to thank Andr6 Vandierendonck for his encouragement and numerous helpful sugges- tions in preparing the manuscript. Ivan Mervielde is also gratefully acknowledged for his critical remarks on an earlier draft. ** Aspirant of the Belgian 'Nationaal Fonds voor Wetenschappelijk Onderzoek'. Presently at the L.L. Thurstone Psychometric Laboratory, University of North Carolina. Davie Hall 013 A, Chapel Hill NC 27514, U.S.A.

    257

  • 258 G. De Soete / Information integration

    Anderson's information integration theory

    With his integration theory Anderson (1974a, b,c, 1978b) wants to describe how several kinds of information are combined to make up a unitary judgment. Two operations are sequentially distinguished: valua- tion and integration. Valuation deals with the internal dimensional representation of each stimulus component (cf Matthai~s et al. 1976). Indeed, each task requires a preliminary ewduation of the meaning and relevance of each piece of stinmlus information (Anderson 1974b). Suppose the stimulus set consists of the cartesian product of three fac- tors A, B and C, which represent three different facets of the stimuli. Valuation of a stimulus (ai, bj, ck) means that scale valtles.f'~ (ai),f2(bj) and f3(ck) (with fl, f2 and f3 real-valued functions) are assigned to the stimulus components. Once the different pieces of information are evaluated, they can be integrated in an overall judgment: the scale values fl(ai), f2(bj) and .[3(ck) are combined by the integration ftmc- tion I in a latent response:

    riik = l(f,(ai), f2(bi), fa(ck)) ( 1 )

    which results in a manifest response Rij k by means of the response function M:

    Ri/k = M(ri/k) (2)

    A scheme of these processes is presented in fig. 1 which is a modified version of Anderson's (1970, 1974a, 1977) functional measurement diagram.

    STIMULUS PSYCHOLOGICAL LATENT MANIFEST COMPONENTS STIMULUS COMPONENTS RESPONSE RESPONSE

    a i . ~, f l i a i )

    b j .) f2 (b j )

    c k :> f3 (ck )

    ~- r i j k

    I N T E G R A T O R

    :)' Rij k

    VALUATION INTEGRATION RESPONSE FUNCTIONS FUN CT ION FUNCT ION

    Fig. 1. A scheme of Andersons's information integration theory (this is a modified version of Anderson's (1970, 1974a, 1977) functional measurement diagram).

  • G. De Soete / hfformation integration 259

    Integration theorists generally take the valuation for granted and concentrate primarily on the integration. Mervielde (1977) explains this by indicating that valuation itself can be seen as a result of an integra- tion operation. Anderson has repeatedly shown how simple algebraic models are apt to describe this process. Especially additive models were successful. Anderson draws a sharp psychological distinction between what he calls a simple adding and an initial impression weighted averaging model. In the common case of constant weights per factor, both models can respectively be written as:

    ri i k = w l fa (a i ) + w2f2(b/) + w3~3(ck) (3)

    and

    rijt = 00Io + ViA(ai) + v2f2(b,) + v3f3(cD (4)

    with v0 + vl + v2 + v3 = 1. In eq. (4) I0 stands for the initial impression of the subject. Contrary to the w in (3), the v in (4) are only relative weights. However, it can be proved (cf Sch6nemann et al. 1973) that whenever the data fit the following model:

    rijk = ~l(ai) + ~b2(bi) + 4~3(ck) (S)

    they also fit eqs. (3) and (4). Consequently, in this paper we do not dis- tinguish between (3) and (4) and we consider the additive model (5) only. Although the additive model has played a dominant role, several kinds of interactive models have, rather sporadically, been found.

    Integration models are usually validated by means of functional mea- surement, a methodology developed by Anderson (1970, 1974a, 1977, 1978a) parallel to his integration theory. In most information integra- tion experiments the overt responses consist of ratings, which are analyzed by means of ANOVA. Absence of significant interaction effects is interpreted as evidence for an additive model and for the validity of the assumption that the response scale is an interval scale (cf. Anderson 1977, 1978a; Klitzner and Anderson 1977). If however an interaction is detected, one is never sure whether this is due to the judg- mental process itself, or to the nonlinearity of the response function. In this case, it is safer to assume only a monotonic relationship between the latent and manifest responses and to diagnose the model by relying

  • 260 G. De Soete /hLtbrmation integration

    on the ordinal characteristics of the data. This can be done by axiomatic conjoint measurement that provides a number of qualitative tests (axioms) which are necessary and ahnost sufficient conditions for the different combination rules (Krantz et al. t971; Krantz and Tversky 1971). Once the underlying model is determined in this way, the data can be scaled according to it and the relation between the latent and manifest responses can be empirically investigated by means of poly- nomial regression.

    The integration of motivational information

    When judging the future performance of a person, people generally have not only an impression of the skills and abilities of the person, but they also know more or less his motivation. How will this motivational infor- mation influence the judgment? Intuitively, the idea of motivation as a multiplier seems appealing. This is for instance apparent from Hull's well known formula:

    Reaction Potential = Drive X Habit .

    The idea occurs also explicitly in Heider's (1958) attribution theory:

    Performance = Motivation X Ability (1958: 83). {6)

    This proposition has been empirically verified by Anderson and Butzin (1974), together with two derived formulas:

    Motivation = Performance X (Ability) -1 (7)

    Ability = Performance (Motivation)-' (8)

    ANOVAs on the ratings of two different groups of subjects indicated for (7) as well as for (8) rather an additive than a multiplicative model. Only eq. (6) has been confirmed. However, if one considers the averaged ratings with regard to (6), as plotted in the first panel of figs. 1 and 2 of Anderson and Butzin (1974), as ordinal data, one can represent them very well according to an additive model by lneans of ADDALS (de

  • G. De Socte /Information integration 261

    Leeuw et al. 1976) [11. The stress [2] value for both groups is less than 0.0001.

    Similar results have been obtained by Ullrich and Painter (1974). In their experiment students had to judge job applicants according to their ability for filling a managerial position, given their intelligence, experi- ence and achievement motivation. A conjoint measurement analysis of the results revealed an additive model for several subjects. In order to investigate further the role of motivational information we performed the following experiment. University students were asked to judge the chances to pass of a set of hypothetical co-students which were charac- terized by their intelligence (l), motivation (M) and the extent to which they studied (S). Working with three factors instead of two has sub- tantial advantages (Klitzner and Anderson 1977) of which the most important is that a better discrimination between additive and interac- tive models is possible.

    Given the findings in person perception and attribution (Anderson 1974b, c, 1978b), we can expect in accordance with the results of Ullrich and Painter (1974), an additive model I + M + S. If however Anderson and Butzin's (1974) proposal

    Future Performance = Past Performance + Motivation Ability (1974: 609)

    is valid, a dual-distributive model S + M X I is to be found because the extent to which one studies is a kind of performance, which in turn results from an interaction between ability and motivation. If on the contrary one considers S as a part of the ability to succeed, then a distributive combination rule (I + S) X M must be expected.

    The exper iment

    Method

    Su bjec ts Eighteen first year students, 7 males and 11 females, all enrolled in the Faculty of Psychological and Educational Sciences at the University of Ghent, participated

    [ 1 ] We thank Norman H. Anderson for making this data available to us. [2] The stress is a goodness of fit measure which is the square root of the normalized sum of squared deviations between tile optimally transformed data and the model. ADDALS uses Kruskal's (1965) stress formula two.

  • 262 (;. De Soete /Inbrmation in tegration

    voluntar i ly in partial fu l f i lments of certain class requirements. No one was acquainted either with conjo int measurement or with Anderson's in format ion integrat ion theory.

    Stimuli The st imuli consisted of descr ipt ions of hypothet ica l first year students, which were character ized by three features: intel l igence (3 levels: IQ = 100, 120, 140), motiva- t ion (3 levels: not, weakly, strongly mot ivated) and the extent to which they studied (2 levels: studies little, much). By combin ing the three variables factorially, 18 stimuli were obtained. Inclusion of an addit ional st imulus (IQ = 125, weakly mot ivated, studies l ittle) for convenience of design yielded a total of 19 stimuli.

    Procedure As we intended to analyze the judgments nonmetr ica l ly (axiom analyses) as well as metrical ly (po lynomia l regressions), it was desirable to gather not only ratings but also explicit ordinal data, because ties or minor order inversions in the ratings due to random judgmenta l f luctuat ions can cause many violat ions against the conjo int measurement axioms. When both rank order data and ratings must be obta ined at the same time, it is better to ask the subjects to order the stimuli first, otherwise there is a big chance that they will use their ratings while rankorder ing the stimuli. In order to be able to int roduce a number of repl ications per subject the stimuli were arranged in a balanced incomplete block design (Cochran and Cox 1957: plan 11.32), composed of 19 blocks of I0 stimuli. The subjects were requested to rankorder stimuli within blocks. In this manner each st imulus was judged ten t imes, while each pair of st imuli was presented five times.

    The subjects were run in group. Each subject received a computer generated booklet . On the first page they were instructed that the purpose of the exper iment was to investigate how well first year students in psychological and educat ional sciences were already able to predict the success of co-students at the university. This mild decept ion was inspired upon Mervielde (1977). The fol lowing pages con- ta ined the 19 blocks which had to be rankordered independent ly according to the chances the described students had to pass at the university. The order of the blocks as well as that of the s~imuli within each block was randomized per subject. Final ly the 19 stimuli were presented again and the subjects had to indicate on an eleven point scale (going from 0 to 10) how many chances to ten the st imulus students had to pass.

    The whole task took on the average 65 minutes to complete. Afterwards the sub- jects were asked to fill in a quest ionnaire about the exper iment.

    R esu Its

    Orderings and agreement among subjects One subject did not understand the task properly. His data were discarded as they were unusable. For each of the remaining 17 subjects the stochastical ly dominant ordering (Coombs and Huang 1970) over the 19 stimuli was obtained. As each pair

  • G. De Soete /Information integration 263

    of stimuli is repl icated five t imes, a st imulus x is said to dominate stochastical ly y whenever x dominates y in more than two of the five with in-b lock comparisons of (x, y). Kendal l 's (1955) coeff ic ient of concordance among the subjects is 0.753. When tested against the null hypothes is of no agreement, this value is highly signifi- cant (X 2 (18) = 230.5 2, p < 0.001). However, reject ion of this hypothesis does not imply that there are no individual differences. Indeed, the pairwise Kendall (1955) tau correlations between the several orderings range from 0.024 to 0.959. Conse- quent ly, the data had to be analyzed individually.

    Consistency and transitivity A first way to assess the consistency among the judgments of a single subject is to compute Durbin 's (1951) test statistic which under the null hypothesis that each ranking in each block is equally likely, is approximately distr ibuted as a chi-square. For each subject, the X2-value associated with the statistic is l isted in the first co lumn of table 1. As the critical value at the 0.001 level is 42.31, the null hypothe- sis could be rejected for all subjects.

    Another way of looking at the consistency is by inspecting the subsequent choices on each pair of stimuli. Coombs and Huang (1970: 328) not iced that with five repl ications of a 50/50 choice on each pair, the distr ibut ion of the dominant st imulus is a folded binomial over 3, 4 and 5 with a mean of 3.44 and a standard deviation of 0.371. A signif icant deviation f rom chance at the 0.01 level (one-tail test) for the average over the (19)= 171 pairs is 3.52 or more. The average con-

    Table 1 Consistency and transistivity.

    Subject x 2 associated with Consistency Number of circular triads Durbin's statistic

    L 162.90 4.988 0 A 161.43 4.971 0 R 162.44 4.965 0 Q 161.11 4.936 0 G 157.96 4.801 1 K 156.75 4.790 0 J 155.39 4.766 2 F 159.00 4.760 0 C 151.65 4.760 2 E 153.46 4.754 1 P 152.18 4.725 4 N 153.58 4.714 3 D 150.78 4.714 4 B 151.28 4.696 4 I 150.59 4.667 0 M 143.75 4.602 5 H 136.01 4.515 7

  • 264 G. De Soete /InJormation inte,gration

    sistency per subject is presented in the second co lumn of table 1. For all subjects the hypothes is of randomness could be rejected.

    Besides est imating the consistency of the data, it is also impor tant to evaluate the transit ivity, because, if transivit i ty does not hold, the overall ordering con- structed out of the within-block rankings is of no meaning. The transit ivity of the stochastical ly dominant choices per pair can be assessed by count ing the number of circular triads (Kendal l 1955). With 19 stimuli, the n lax imum number is 285, while the expected number assuming a 50/50 pairwise choice is 242. The number of circular triads for each subject is given in co lumn three of table 1. As is apparent from table 1, transit iv ity can be said to hold quite well for all subjects.

    Independence According to Krantz and Tversky's (1971) f lowchart for the diagnosis of three-fac- tor polynomials in the unsigned case, one has to test for independence in the first place. Factor l is said to be independent of M for a fixed level of S, if the rank order of I is the same at all levels of M. This can be assessed by comput ing Kendal l 's coeff ic ient of concordance between the rank orders of I over all levels of M (Wallsten 1976). Moreover, this can be done at each level of S.

    These coeff ic ients revealed that 14 of the 17 subjects satisfied the independence requ i rement perfectly. For one subject, M, factor M was not completely indepen- dent of I. Since the coeff ic ient of concordance, averaged over the levels of S, amounted to 0.939, only a small violat ion was involved. The two remaining sub- jects, D and H, violated the axiom more seriously. Their lowest averaged coeffi- cients of concordance were respectively 0.074 (for S independent of M) and 0.203 (for 1 independent of M). In cases like these, one has to test for sign dependence. This was not done here because the factors had no enough levels to perform the other tests required in the signed case.

    Double cancellation The double cancel lat ion axiom could only be tested for the factors I and M, as minimal ly a 3 X 3 matr ix is required. In each test of the cancel lat ion condit ion, six cells are involved. In all the di f ferent tests which are possible in a given 3 X 3 matr ix, only six di f ferent six cell combinat ions are involved. "The double cancel lat ion could be tested at each level of S. Consequent ly , 12 di f ferent six cell combinat ions could violate the condit ion. Only one subject, P, failed to satisfy the double cancel lat ion in one single six cell combinat ion . All other subjects, including subjects D and H, satisfied the axiom perfectly.

    Join t independence The jo int independence was verified next. Factors I and M are said to be jo int ly independent of S, if the rank order over all combinat ions o f /and M is the same at all levels of S. The agreement between those rank orders can again be expressed by Kendal l 's coeff ic ient of concordance. These coeff ic ients were calculated for the di f ferent jo int independence tests. A l though most subjects showed some minor violations, the results were very similar to those of the independence analyses: serious violat ions occurred only with subjects D and H.

  • G. De Soete / Information integration

    Table 2 Goodness of fit for ADDALS scalings and results of regression analyses.

    265

    Subject Kendall tau Stress r 2 F-ratio *

    L 1.000 0.000 0.956 347.21 A 1.000 0.000 0.983 967.39 R 0.987 0.000 0.782 57.48 Q 1.000 0.000 0.961 349.26 G 0.951 0.002 0.786 58.85 K 0.974 0.000 0.912 166.68 J 0.941 0.002 0.742 46.11 F 0.922 0.049 0.843 85.77 C 0.980 0.000 0.838 82.71 E 0.937 0.004 0.842 85.52 P 0.907 0.000 0.893 134.19 N 0.931 0.054 0.847 88.54 D 0.951 0.000 0.847 88.27 B 0.964 0.001 0.848 89.33 I 0.948 0.017 0.890 130.04 M 0.927 0.002 0.854 93.39 H 0.864 0.002 0.738 45.15

    * All F-ratios have one degree of freedom for the numerator and 16 for the denominator, ex- cept for subject A who forgot to rate one stimulus, so that the dffor the denominator for this subject become 15.

    All F's are highly significant (p < 0.001).

    Model characterization Fi f teen of the 17 subjects satisfied the independence, double cancellation and joint independence axioms very well. Consequently, their judgments could best be described by the additive model 1 + M + S. A discussion of the resurts for subjects D and H is postponed for a while.

    Because an additive combinat ion rule seems appropriate for most subjects, we have scaled each stochastically dominant ordering according to this model by means of the ADDALS algorithm (de Leeuw et al. 1976) [3]. Some goodness of fit mea- sures for the additive representations are reported in table 2. The first column con- rains the Kendall tau correlations between the rank order data and the scale values, while the second column lists the stress values. The squared stress (see footnote 2) is up to a normalizat ion factor the loss funct ion minimized by the algorithm. As is evident from table 2, all. representations are quite satisfactory, even for subjects D and H.

    The normalized scale values associated with the levels of the three factors are

    [3] The maximum number of iterations allowed was set to 30, while the minimum stress improvement required for continuation was set to 0.0001.

  • 266 G. De Soete /hzbrmation in tegration

    Table 3 Normalized AI)DALS scale values.

    Sub- I 1 12 I3 S1 $2 M1 M2 M3 ject

    L -0.216 0.006 0.222 -0.999 0.999 -0.672 0.006 0.677 A -0.187 0.005 0.192 --0.288 0.288 1.188 0.025 1.162 R --0.760 -0,029 0.788 -0.909 0,909 0.272 0.001 0.271 Q --0.942 0,017 0.959 -0,744 0.744 0.215 0.017 0.198 G -0.669 0.074 0.744 --0.710 0.710 0,662 ---0.074 0.736 K -0,409 0.000 0.409 -1.005 1.005 -0.510 0,101 0.611 J -0.369 0.178 0.192 0.276 0.276 -1.138 0.023 1.161 F -0.978 0.291 0.687 -0.548 0.548 0.679 0.028 0.651 C -0.369 0.038 0.331 1.009 1.009 0.529 -0.121 0.649 E -0.802 0.009 0.793 - 0.803 0.803 -0.290 0.250 0.540 P 0.707 -0.001 0.708 0.707 0.707 0.706 --0.001 0.707 N 1.002 -0.097 1.099 -0.267 0.267 0.508 0.098 0.606 D 1.167 0.045 1.122 0.000 0.000 0.409 0.045 0.455 B 0.375 -0.058 0.433 0.401 0.401 0.983 - 0.178 1.161 I -0.914 0.036 0.950 0.576 0.576 0.413 0.206 0.619 M -0.787 0.045 0.832 0.812 0.812 0.256 0.239 0.495 tt 0.000 0.001 0.001 0.004 -0.004 -1.203 0.042 1.245

    presented in table 3. In order to allow for both within-subject and between-subject comparisons, not only the sum of scale values per factor but also the total sum of squared scale values was made constant for each subject. By inspecting the range of the scale values per factor, one can see that although all subjects except D and H, took account of the three variables, the relative importance of each factor varied from subject to subject. This f inding justifies our emphasis on analyzing the data at the individual level.

    F rom the scale values of subjects H and D, it can be inferred that these individ- uals only dif ferentiated the stimuli on the basis of respectively one and two factors (motivat ion and intell igence). This has given rise to random order inversions which caused the many axiom violations of these subjects. As in these cases one can hardly speak of an imp!icit combinat ion rule, these subjects cannot be said to dis- confirm a theory which predicts an additive model.

    Validation o f the rating scale The scale values obtained by ADDALS can be regarded as estimates of the latent responses. As outl ined in the first section, the latent responses are transformed by the response funct ion into manifest responses. If this funct ion is linear, the mani- fest responses, i.e., the ratings, can be said to form an interval scale. We have applied a stepwise polynomial regression procedure to determine the shape of this function. For all subjects a linear regression accounted for a substantial part of the variance of the scale values, as is apparent from the squared correlation coeff icients

  • G. De Soete / Information integration 267

    presented in the third co lumn of table 2. The last co lumn of that table gives the F- rat ios associated with the best l inear fit. In no case addit ion of higher order poly- nomial terms, up to the third degree, improved the fit significantly. Since the l inearity of the response funct ion has been assessed by this procedure, the ratings could be considered as interval data.

    As a final check of the validity of our two main conclusions, viz., the appropri - ateness of the additive model and the l inearity of the response scale, a repeated measures ANOVA was performed on the ratings [4]. If both our assertions were true, signif icant main effects and nonsigni f icant interact ion effects of the st imulus factors were to be obtained, unless of course the assumptions under ly ing the meth- od were too seriously violated. The ANOVA results were exactly as expected. The main effects of S, 34 and I were highly signif icant (resp. F(1 ,16) = 24.44, F(2 ,16) = 16.54, F (2 ,16)= 29.91, all p < 0.001), whereas none of the F-values associated with the interact ion terms approached significance. These results provide addit ional evidence for the validity of our conclusions!

    Discussion and conclusions

    Conjoint measurement techniques have proved to be quite useful for determining the appropriate model. Although this approach lacks an adequate error theory, this did not cause any particular difficulty in interpreting the results of the axiom analyses. The judgments of all sub- jects who differentiated the stimuli on the basis of the three factors, could adequately be represented by an additive model. A question put forward by a referee in connection with this result concerns the power of the present 3 3 2 design to reject the additive model. Indeed, as in most conjoint measurement applications, the design was minimal, since only such a small design allows for testing the consistency and transitivity at the individual level in a meaningful way. This advantage of the 3 3 2 design is of course worthless when the design lacks enough power to validate the proposed models. Fortunately, previous experimental applications of conjoint measurement where the very same design was used show it to have the required power (cf. Coombs and Huang 1970; Ullrich and Painter 1974).

    Our conclusion to an additive model contrasts with the intuitive idea of motivation as a multiplier and suggests that at least Belgian students integrate motivational information in a very similar manner as other

    [4] The one missing rating (cf. table 2) was estimated by means of linear regression using the latent responses as obtained by ADDALS.

  • 268 G. De Soete / b~formation integratioJt

    kinds of information are processed (Anderson 1974b, c). The discrep- ancy between the present results and those of Anderson and Butzin (1974) may possibly be attributed to cultural differences, as a recent replication of the Anderson and Butzin study in India, conducted by Singh et al. (1979), evidences.

    Anderson and his colleagues have very often tried to prove the linearity of the response ftmction by heavily relying on ANOVA tech- niques (e.g., Anderson 1977, 1978a; Klitzner and Anderson 1977). We, on the contrary, have assessed this linearity and consequently the validity of the response scale, by relying only on the ordinal character- istics of the data for the model diagnosis and without making prelim- inary distributional assumptions. This can be very useful for fttrther research in the area.

    Although some methodologically innovative methods have been applied, we have only been able to prove the descripti l ,e validity of the additive model. Le. the model is only an as i f model. What a person really does when making a judgment that is like an addition, is not known! Much more research will be needed to answer this question!

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    Anderson, N.H., 1974b. 'Cognitive algebra: integration theory applied to social attribution'. In: L. Berkowitz (ed.), Advances in experimental social psychology. Vol. VII. New York: Academic Press. pp. 1 - 101.

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