The Importance of Model Specification for Causal Inference in Social Network Analysis�Kenneth Frank (Michigan State)�Ran Xu (University of Connecticut)

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Causal inference in network analysis can be complicated by interdependencies in the data. But we argue that fundamental principles of causal inference apply nonetheless. In particular, careful attention to model specification, especially using longitudinal data, can mitigate bias more so than can the estimation technique. We attend particularly to estimation of the influence model, which has received relatively less attention in the statistical literature than the selection model. We present a simulation and algebraic proof that ordinary least squares regression can yield unbiased estimates of influence models that carefully leverage longitudinal data. Recognizing that specification cannot reduce all bias, we encourage the use of sensitivity analyses to quantify how much of an estimate must be due to bias

to invalidate an inference.

*Frank, K.A. and *Xu, Ran. (2020). Causal Inference for Social Network Analysis. Pages 288-310 in R. Light and J. Moody (Eds.). The Oxford Handbook of Social Networks. New York, Oxford. * Equal first authors.

Concerns about Causality: Influence

Teacher’s outcome behavior

Teacher’s own prior behavior

Prior behavior of network members

Selection

Influence

Control for prior behavior: longitudinal data

Model outcome as function of prior behavior of network members

Behavior might be curricular implementation

Concerns about Causality

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Outcome behavior

Prior behavior

Exposure: Prior behavior of network members

selection

Influence (ρ)

W_ii’=F|y_it-1-y_i’t-1|

y_it-1

y_it

γ

Answer: Control for Prior Behavior!

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There  are heightened concerns about potential dependencies in estimating any social network model (e.g., Robins et al., 2007; Steglich, Snijders and Pearson, 2010 ). Regarding model (1) estimated influence is biased if the errors are not independent of the network exposure term (see Ord, 1975, equations 1.2-1.4); the estimate of influence will be positively biased if there is some unexplained aspect of enforcement behavior that is related to the network exposure. The most compelling source of such dependencies would be if people choose to interact with others whose behaviors are similar to their own, known as selection in the network literature. Those who tended to engage in enforcement at time 1 might have chosen to interact with similar others between time 1 and time 2, and also would have been inclined to engage in enforcement behaviors at time 2. Because the network exposure term is likely confounded with prior enforcement behavior, model 1 includes a control for prior enforcement behavior.

A second concern in the influence model would arise if the model of a fisherman’s behaviors was a function of the contemporaneous behaviors of his/her network members. This would essentially put the outcome on both sides of the model in which case the errors would be directly related to the exposure term. It is for this reason that we model enforcement behavior as a function of the previous behaviors of others in one’s network. This avoids creating dependenices beteween the errors and predictors by putting the same variables on both sides of the model. Even given our approach there may still be concerns about omitted variables that create dependencies between the errors and the exposure term. Therefore we quantify the robustness of our inferences to potential omitted variables (Frank, 2000 ).

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 Steglich, Christian E.G. Tom A.B. Snijders, and Michael Pearson (2010). Dynamic Networks and Behavior: Separating Selection from Influence. Sociological Methodology, 40, 329-392.

Ord, Keith. "Estimation methods for models of spatial interaction."Journal of the American Statistical Association 70.349 (1975): 120-126.

Robins, Garry L., Tom A.B. Snijders, Peng Wang, Mark Handcock, and Philippa Pattison. Recent developments in exponential random graph (p*) models for social networks. Social Networks 29 (2007), 192-215.

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Frank, K.A. and Xu, Ran. 2020. Causal Inference for Social Network Analysis. James Moody and Ryan Light edits. Oxford Handbook of Social Network Analysis. Oxford, UK.

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See also:

The value of controlling for pretests

Verify with Simulation (student Ran Xu)

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Frank, K.A. and Xu, Ran. 2018. Causal Inference for Social Network Analysis. James Moody and Ryan Light editors. Oxford Handbook of Social Network Analysis. Oxford, UK.

Simulation examples where a common trait that codetermines selection and influence is observed

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Both the prior term and contagion effects in the influence model are identified and unbiased using OLS. The Blue line represents the bias of mean estimates where each point is a result of 500 simulations. Dashed lines represent the 95% confidence intervals.

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Must Control for Prior

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Must Control for Prior

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Influence and selection is confounded through the unobserved variable “risk-taking tendency”.

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Situations under which influence effect can be identified using OLS

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Identification for Cross-sectional Data

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From: Causal Inference for Social Network Analysis

Kenneth A. Frank* and Ran Xu*. Forthcoming in Handbook for Social Network Analysis. James Moody and Ryan Light. Oxford press.

*Equal co-authors.

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Errors correlated with Predictors for Cross-sectional Data

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Behavior of person 1 at time 1=f(behavior of person 2 at time 1)

+ error for person 1 at time 1

Behavior of person 2 at time 1=g(behavior of person 1 at time 1)

+ error for person 2 at time 1

Behavior of person 1 at time 1=f(behavior of person 2 at time 1)

+ error for person 1 at time 1

Behavior of person 1 at time 1=f(g[behavior of person 1 at time 1

+error for person 2 at time 1]

+error for person 1 at time 1

Predictor correlated with error term

Identification for Longitudinal Data

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From: Causal Inference for Social Network Analysis

Kenneth A. Frank* and Ran Xu*. Forthcoming in Handbook for Social Network Analysis. James Moody and Ryan Light. Oxford press.

*Equal co-authors.

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Must Control for the Prior

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What to do about Omitted Variables?

• Not unique to influence model
• Quantify robustness to omitted variables
• Latent space models may help

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It’s all in how you talk about it!

• Do best method you can
• Include relevant controls!
• But science is as much in the nature of the discourse as the method
• Virtue epistemology
• Greco, 2009; Kvanig, 2003; Sosa, 2007
• What would it take to invalidate the inference?
• How much bias must be present to invalidate an inference?
• https://www.msu.edu/~kenfrank/research.htm#causal
• Frank, K.A., Maroulis, S., Duong, M., and Kelcey, B. 2013.  What would it take to Change an Inference?: Using Rubin’s Causal Model to Interpret the Robustness of Causal Inferences.   Education, Evaluation and Policy Analysis.  Vol 35: 437-460.

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In R Shiny app KonFound-it! (konfound-it.com/)

Estimated effect

Standard error

Number of observations

Number of covariates

Replacement Cases Framework

Conclusion

Correlational Framework

overview

Thresholds for inference and % bias to invalidate

The counterfactual paradigm

Internal validity example: kindergarten retention

External validity exampleOpen Court curriculum

Extensions of the framework

Correlational Framework

Impact of a Confounding variable

Internal validity

Example: Effect of kindergarten retention

External validity

example: effect of Open Court curriculum

Frank, K.A., Maroulis, S., Duong, M., and Kelcey, B. 2013.  What would it take to Change an Inference?: Using Rubin’s Causal Model to Interpret the Robustness of Causal Inferences.   Education, Evaluation and Policy Analysis.  Vol 35: 437-460.��

Abstract

We contribute to debate about causal inferences in educational research in two ways. First, we quantify how much bias there must be in an estimate to invalidate an inference. Second, we utilize Rubin’s causal model (RCM) to interpret the bias necessary to invalidate an inference in terms of sample replacement. We apply our analysis to an inference of a positive effect of Open Court Curriculum on reading achievement from a randomized experiment, and an inference of a negative effect of kindergarten retention on reading achievement from an observational study. We consider details of our framework, and then discuss how our approach informs judgment of inference relative to study design. We conclude with implications for scientific discourse.

Keywords: causal inference; Rubin’s causal model; sensitivity analysis; observational studies

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Errors not correlated with predictors for longitudinal data

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Behavior of person 1 at time 2=g(behavior of person 2 at time 1,

behavior of person 1 at time 1)

+ error for person 1 at time 2

Behavior of person 2 at time 2=g(behavior of person 1 at time 1,

behavior of person 2 at time 1)

+ error for person 2 at time 2

Behavior of person 1 at time 2=g(behavior of person 2 at time 1)

+behavior of person 1 at time 1

+ error for person 1 at time 2

Behavior of person 1 at time 2=g[f(baseline behavior of person 2 at time 1)

+ error for person 2 at time 1])

+behavior of person 1 at time 1

+ error for person 1 at time 2

Behavior of person 1 at time 1=f(baseline behavior person 1) + error person 1 at time 1

Behavior of person 2 at time 1=f(baseline behavior person 2) + error person 2 at time 1

Terms not inherently correlated

Terms not inherently correlated

Christakis & Fowler: Contagion of Obesity

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C&F:�Methods

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Lagged controls?

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Christakis & Fowler Model

• Should we use simultaneous or staggered behavior?
• They use both:
• y_it = ρ₁∑_i’w_ii’ty_i’t/∑_i’w_ii’t + ρ₂∑_i’w_ii’ty_i’t-1/∑_i’w_ii’t + γ y_it-1 +e_it
• Obesity₂₀₀₀=ρ₁obesity of friends₂₀₀₀ +ρ₂obesity of friends₁₉₉₇ + γ own obesity _it-1 +e_t :
• Lyons: ρ ₁ and ρ₂ have opposite signs. Hmmmm.
• Collinearity problems?

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Christakis and Fowler Debate: Lyons

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Lyons, Russell

The spread of evidence-poor medicine via flawed social-network analysis, Stat., Politics, Policy 2, 1 (2011), Article 2. DOI: 10.2202/2151-7509.1024

See Andrew Gelman: http://themonkeycage.org/blog/2011/06/10/1-lyonss-statistical-critiques-seem-reasonable-to-me-there-could-well-be-something-important-that-im-missing-but-until-i-hear-otherwise-for-example-in-a-convincing-reply-by-christakis-and-f/

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Critique of

Christakis and Fowler

“influence” model pages 5-6

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Christakis & Fowler Model

• Should we use simultaneous or staggered behavior?
• They use both:
• y_it = ρ₁∑_i’w_ii’ty_i’t/∑_i’w_ii’t + ρ₂∑_i’w_ii’ty_i’t-1/∑_i’w_ii’t + γ y_it-1 +e_it
• Obesity₂₀₀₀=ρ₁obesity of friends₂₀₀₀ +ρ₂obesity of friends₁₉₉₇ + γ own obesity _it-1 +e_t
• Same data on right and left hand sides of model
• Lyons: ρ₁ and ρ₂ have opposite signs: hmmm
• Collinearity problems?
• Why not just control for prior behaviors?

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C&F:�Methods

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directionality

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Christakis & Fowler: Directionality Results

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Are they statistically

different from

one another?

No.

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Articles on Causality

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Articles on Causality

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