Abstract
Dependency networks—models of networks of potentially causal relationships—capture the spirit of what Cox and Wermuth refer to as level-two causality, or causality as an explanation of a process (1996, 220; 2001, 70–74; 2004, 288, 299–300). Social scientists can develop such models on the basis of their knowledge of the subject matter and by synthesizing relationships that are invariant when tested through the procedures of robust dependence (i.e., stable association) or controlled intervention (i.e., potential outcomes). This chapter discusses four types of dependency networks: graphical models, association graphs for loglinear models, generative mechanisms, and structural economic models. Graphical models emphasize linear relationships; loglinear models detect interaction effects; path-analytic generative mechanisms allow the quantification of total, direct, and indirect effects; and structural economic models underscore the importance of theory. This chapter’s detailed explications supplement contemporary discussions of notions of causality that do not provide extensive empirical examples. This chapter leaves open the question “Are these multivariate dependencies truly causal?”
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Notes
- 1.
*This SAS code will replicate the proportional odds model for the three voting choices, given the relevant data set ThreeClusters123
It applies proc logistic illustrating the use of three weighted files
to produce true estimates of the three class latent structure;
*In addition to Springer’s website, these data are also available at the internet journal Case Studies in Business, Industry, and Government Statistics (CSBIGS) at Bentley University. See Volume 2, number 2, the article by Smith 2009;
*Table 4.1. Panel 1;
proc logistic;
class trueclas partyid libcon minority
/param = reference;
model choice3 = trueclas libcon partyid minority/
scale = none aggregate lackfit ctable rsquare;
weight newvar;
run;
*This code replicates the continuation logits model for Clinton vs. All others;
*Table 4.1. Panel 1.1;
proc logistic descending;
class trueclas partyid libcon minority
/param = reference;
model clin1vot = trueclas libcon partyid minority/
scale = none aggregate lackfit ctable rsquare;
weight newvar;
run;
*This code replicates the continuation logits model for Perot vs. Bush;
*Table 4.1. Panel 1.2;
proc logistic descending;
class trueclas partyid libcon
/param = reference;
model pero1vb0 = trueclas libcon partyid/
scale = none aggregate lackfit ctable rsquare;
weight newvar;
run;
- 2.
This SPSS syntax will find this best-fitting loglinear model given the relevant data set threeclusters123.sav. Just open the data file, paste the syntax into the syntax window, and run it. In addition to Springer’s website, these data are available at the internet journal Case Studies in Business, Industry, and Government Statistics (CSBIGS) at Bentley University. See vol. 2, no. 2, the article by Smith 2009:
USE ALL.
COMPUTE filter_$ = (fileid = 1).
VARIABLE LABEL filter_$ ‘fileid = 1 (FILTER)’.
VALUE LABELS filter_$ 0 ‘Not Selected’ 1 ‘Selected’.
FORMAT filter_$ (f1.0).
FILTER BY filter_$.
EXECUTE.
HILOGLINEAR
econreg1(2 4) hreformX(2 4) envirvip(1 2) charnvip(1 2) choice3(1 3)/METHOD = BACKWARD
/CRITERIA MAXSTEPS(10) P(.05) ITERATION(20) DELTA(.5)
/PRINT = FREQ RESID
/DESIGN.
- 3.
Goodman ([1972] 1978, 91) notes that loglinear models do not establish causation; to give a causal interpretation to the system of variables, certain assumptions must be made: “For example, we must assume that the system under study is closed in the sense that, if variables outside the system…have any effect on the variables in the system, it is to produce stochastic disturbances corresponding to the usual random variation of the observed frequency f ijkl from its expected value F ijkl (when sampling is random). In addition, we must make other assumptions (e.g., assumptions as to which variables are prior to which other variables) in order to select one causal system from among the various causal systems that are consistent with a given model.”
- 4.
*The following SAS syntax creates the dichotomous Clinton versus Bush response variable, calculates the main effects, and the predicted values. Then it calculates the interaction effect and the predicted values. It uses the first file of threeclusters123;
data trythree; set trytwo;
*this creates a dichotomous response variable Clinton vs. Bush;
if choice3 =2 then delete;
if choice3 =3 then choice3 =2;
*this calculates the main effects dichotomous response;
proc logistic;
class character envirvip econreg1 hreformx/param = ref;
model choice3 = character envirvip econreg1 hreformx
/scale = none aggregate lackfit ctable rsquare;
output out = predict pred = prob;
run;
*this calculates the predicted values for the main effects model;
data bob; set predict;
if character =. then delete;
if envirvip =. then delete;
proc sort; by character envirvip;
proc means; var prob;
by character envirvip;
run;
*this calculates the interaction effect and the main effects;
proc logistic;
class character envirvip econreg1 hreformx/param = ref;
model choice3 = character| envirvip econreg1 hreformx
/scale = none aggregate lackfit ctable rsquare;
output out = predict1 pred = prob1;
run;
*this calculates the predicted values for the interaction effects model;
data bobs; set predict1;
if character =. then delete;
if envirvip =. then delete;
proc sort; by character envirvip;
proc means; var prob1;
by character envirvip;
run;
- 5.
The predicted values provide the following interpretations for this interaction effect. For the main effects model when both the character issue and the environmental issue favor the Republican positions (C- N-) then the predicted proportion choosing Clinton (+) is 0.27. When the environmental issue changes to N+, then the proportion for Clinton is increased to 0.58, an increase of 0.31. For the model with an interaction effect the analogous proportions are C- N- = 0.23 and C- N+ = 0.63, for a larger increase of 0.40. However, when the character issue initially favors the Democratic candidate (+) and the environmental issue does not (−) the main effects model exhibits the largest gain when there is consonance: For the main effects model C + N- = 0.64, C + N+ = 0.85 for a gain of 0.21 for Clinton. For the interaction effects model C + N- = 0.68, C + N+ = 0.80 for a gain of 0.12 for Clinton. The difference between the differences = |0.09 |. Such interaction effects should be interpreted with extreme caution (Littell et al. 2006, 551).
- 6.
Lazarsfeld (1955b, xi) suggests that in addition to the time ordering of variables, the general to the specific ordering principle can be applied, e.g., patriotism is more general than saluting the flag, so the latter is a manifestation of the former. He also suggests that structural levels can be distinguished: e.g., membership in an occupational group, employment in a specific factory, and membership in a special clique within the shop.
- 7.
Hyman (1955, 326) suggests that the investigator’s subject matter knowledge governs when to stop inserting intervening variables into a chain of relationships, he states: “We stop inserting new links into this chain only when we feel psychologically satisfied that the underlying process has been clarified.”
- 8.
Classic examples of generative mechanisms are Hyman’s (1955, 324–326) analysis of the variables that intervene between prejudice (x) and misunderstanding of a cartoon theme (y); Glock and Stark’s (1966, 134–135) explication of how hostility (t 1) interprets the relationship between religious dogmatism (x) and anti-Semitism (y); and (Glock’s 1967) codification of this method. Currently, Morgan and Winship (2007, 219–242) review the roots of mechanistic explanations, analyze hypothetical mechanisms using graphical methods, and discuss the movement toward such explanations in sociology.
- 9.
In early 2004, Lucas formulated these and other survey questions and included these items in several surveys of engineering and science students at MIT and in the United Kingdom.
- 10.
The AIC and the BIC are essentially values of the log likelihood that have been penalized for the number of parameters estimated; the BIC imposes a heavier penalty than the AIC.
- 11.
An internally valid index is composed of items that are more strongly correlated with each other than with other items not included in the index. Each item should exhibit strong, expected correlations with other survey questions that logically follow from the meaning of the index. Cronbach’s alpha (α) provides an indication of the index’s internal validity. An externally valid index has strong correlations with actual behaviors that logically follow from the meaning of the index. For example, students who self-report high levels of basic research skills should actually perform better in research than those who self-report low levels of basic research skills. Students who participate in entrepreneurial competitions should exhibit higher scores on venturing self-efficacy than those who do not participate.
- 12.
A simple answer may be that male scientists treat their younger female colleagues less well than they treat their younger male colleagues. If this were true, then women who are treated nicely should blossom, which seems to be the case. Zeldin and Pajares (2000, 237–243) find that the self-efficacy beliefs of successful women scientists concerning their mathematics ability were nurtured by favorable familial, academic, peer, and work-related influences. The encouragement they received and the vicarious experience of watching and learning from others, which their participation in research provided, enhanced their self-efficacy beliefs.
- 13.
The more detailed fit statistics confirm that the alternative model fits very slightly better than the reported model. Note that the BIC statistic picks the more parsimonious of these two models whereas the AIC does not (the AIC difference of 1.6 is < 2):
Box 4.1 Measures of goodness of fit
Reported
Alternative
Model
Model
Model chi square
12.85
13.25
Degrees of freedom
10
11
Probability
0.23
0.28
Noncentrality parameter
2.85
2.25
Comparative fit index
0.988 > 0.90
0.991 > 0.90
Root mean square residual
0.005
0.005
RMSEA
0.048
0.041
LO 90
0
0
HI 90
0.115
0.108
PClose
0.46
0.52
AIC
48.85
47.25
BIC
99.61
95.19
- 14.
The noncentrality parameter, designated as δ, expresses the degree of misspecification of the model. It has this form (the M refers to the model being tested):
$$ {{\hat{\delta }_M} = { \max }\left( {\chi_M^2 - d{f_M},0} \right)} $$The estimate of \( {\hat{\delta }_M} \) is which ever is larger: zero or the difference between the chi square for the model minus the model’s degrees of freedom (Kline 2005, 137–138).
- 15.
The RMSEA is the root mean square error of approximation. It estimates the amount of error of approximation per model degree of freedom and takes sample size into account. The formula takes the square root of the noncentrality parameter for the model and divides it by the product of the model’s degree of freedom times the sample size minus 1: RMSEA = (δM/ df M (N − 1))1/2. Values close to zero (RMSEA < or = 0.05) and a confidence interval between 0 and 0.10 indicate a close fit of the model to the data (Kline 2005, 137–140).
- 16.
The students’ home university and gender exhibit different means on confidence in basic research skills: on a 0to1 scale MIT students have a mean of 0.58 compared with a mean of 0.51 for Cambridge students; but this difference is not statistically significant (b = 0.06, t = 1.7, p = 0.09). Young men report more confidence in their basic research skills than young women; 0.59 − 0.50, for a significant difference (b = 0.09, t = 2.5, p = 0.01). When these regressors are used to simultaneously predict basic research skills, both become statistically significant: MIT students have more confidence in their basic research skills (b = 0.095, t = 2.61, p = 0.01) and young men have more confidence than young women (b = 0.12, t = 3.2, p = 0.002). However, the variable “self-reported research experience” interprets the effect of the university: with this control the difference between MIT and Cambridge disappears (b = −0.03, t = −0.60, p = 0.54), but the gender difference in favor of men remains (b = 0.12, t = 3.5, p = 0.0007). However, the direct effect of prior research experience on basic research skills is considerably larger than these effects and very significant (b = 0.21, t = 3.5, p = 0.0007).
- 17.
Research on change over time for the 2004–2005 cohort of exchange students supports the view that consideration of use is malleable. MIT students after an academic year at Cambridge significantly reduced their level of reported research experience (p = 0.0008) and, concomitantly, their consideration of use (p = 0.0006). After an academic year at MIT, the Cambridge students reported a significant change in their research experience (p = 0.0004) but no significant drop in their level of consideration of use.
- 18.
Smith and Lucas’s memo of November 7, 2005 tested this hypothesis. Using instrumental variables, a model with reciprocal causation between consideration of use and confidence in basic research skills was estimated. The effect of basic research skills on consideration of uses was large (b = 0.48) and statistically significant (p = 0.04) whereas the effect of consideration of use on basic research skills was negative (b = −0.11) and not significant (p = 0.85). Additionally, the recursive model that assumes that basic research skills drive consideration of use produced a range of fit statistics that were superior to the alternative recursive model that assumes that consideration of use drives basic research skills. Apparently, basic research skills are prior to consideration of use, as Figures 4.4 through 4.7 depict.
- 19.
Anecdotal evidence supports this view (Thomson 2005). When commenting on his selection for a Nobel Prize in chemistry, Richard Schrock of MIT credited basic research as catalyst to his success. He defines basic research as the exploration of an interesting new area that may have potential. As Schrock pointed out, his work had applications that he did not anticipate that time.
- 20.
Robins and Wasserman (1999, 306–307) stress that in an observational study, given a large enough sample, almost any x → y relationship is vulnerable to the effects of some unmeasured confounder U k. Some rather obvious potential confounding background variables that may be correlated with consideration of use are the students’ departmental major (no effect), age (r = 0.19, p = 0.13), female gender (r = −0.07, p = 0.55), and MIT home university (r = 0.13, p = 0.15). In contrast to these small, insignificant correlations, note that confidence in one’s basic research skills is strongly and significantly correlated with consideration of use (r = 0.39, p = 0.001). These data are from the 2004–2005 cohort of exchange students at baseline as reported by Smith and Lucas (July 15, 2005, 3).
- 21.
Pearl ([2000] 2009, second edition, 164) defines direct and indirect as follows:
Definition 5.4.2. (Total Effect)
The total effect of X on Y is given by P(y | do(x)), namely, the distribution of Y while X is held constant at x and all other variables are permitted to run their natural course.
Definition 5.4.3. (Direct Effect)
The direct effect of X on Y is given by P(Y | do(x), do(s XY )), where S XY is the set of all observed variables in the system except X and Y.
He then states:
In linear analysis, Definitions 5.4.2 and 5.4.3 yield, after differentiation with respect to x, the familiar path coefficients in terms of which direct and indirect effects are usually defined.
- 22.
- 23.
Figure 4.6 indicates slight spurious components due to MIT females and Cambridge females. The compound path from MIT females to basic research skills (−0.28) and then to self-rated innovative ability (0.56) and from MIT females to research experience (0.70) and then directly to consideration of use (0.18) changes the spuriousness by −0.02 = −0.28 × 0.56 × 0.70 × 0.18. The analogous effect for Cambridge females has even less effect:.001 = −.25 ×.56 × −.04 ×.18. Because of the binary coding, these indirect effects and those involving MIT males may be uncertain.
- 24.
This exposition of path-analytic models estimated by AMOS of determinants of consideration of use built on earlier analyzes followed a recursive modeling strategy. Recent work by Pearl ([2000] 2009, second edition, 133–171), Morgan and Winship (61–74, 182–183), and others have developed “back door” and “front door” strategies for determining what variables in a system should be conditioned on for identifying causal effects of variables in directed acyclic graphs (DAGs) such as those in this chapter. The reader is invited to apply these approaches to the data and models of this chapter and to compare the differences.
- 25.
Robins and Wasserman (1999, 318–319) emphasize the necessity of prior subject matter knowledge as a guide to the inference of causal effects.
- 26.
However, generative mechanisms must be carefully specified. If no variables were hypothesized to intervene between research experience and consideration of use, then the direct effect of research experience on consideration of use would be larger. Intervening variables weaken the direct effects of antecedent variables. If a student’s research experience and innovative ability were dropped from the model, then there would be a direct effect of confidence in one’s basic research skills on consideration of use. Also, prior variables may create spurious components of effects of intervening variables.
- 27.
Heckman conducts use-inspired basic research as defined by Stokes (1997, 73). See his Nobel Prize acceptance letter present on his web site ca. May 12, 2009.
- 28.
Heckman’s three tasks of definition, identification, and estimation also apply to the development of computer simulation models. For example, the author defined his model of attitude change in Nazi Germany on the basis of a formalization of Goldhagen’s detailed study (1996) and data about the time period being simulated (Smith 1998). Parameter studies of the model identified key parameters and hypothetical relationships (Smith 2010b). Ongoing research will attempt to estimate the parameters of the model empirically and to generate and test its new implications.
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Acknowledgements
The author derived the exposition of the generative mechanism, which links the students’ home university to their consideration of use, from unpublished research reports by Smith and Lucas (July 15, 2005; August 12, 2005; November 7, 2005; June 19, 2006) and Lucas et al. (March 2006). This research was conducted under the auspices of the Cambridge-MIT Institute (CMI) and is reused here with the consent of former CMI Directors. The author based the exposition of graphical modeling of voting on portions of his article, Issues matter: A case study of electoral voting, in Case Studies in Business, Industry, and Government Statistics: An Internet Journal (vol. 2, no. 2, 127–146; © 2009 Robert B. Smith).
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Smith, R.B. (2011). Dependency Networks. In: Multilevel Modeling of Social Problems. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9855-9_4
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