Monday, April 28, 2014

How is it that Structural Equation Models Subsume Potential Outcomes?

I have been trying to figure out, under what conditions can we identify causal effects via SEMs, particularly, is there a framework similar to the Rubin Causal Model or potential outcomes framework that I can utilize in this attempt?  In search of an answer I ran across the following article:

Comments and Controversies
Cloak and DAG: A response to the comments on our comment
Martin A. Lindquist, Michael E. Sobel
Neuroimage. 2013 Aug 1;76:446-9

On potential outcomes notation:

"Personally, we find that using this notation helps us to formulate problems clearly and avoid making mistakes, to understand and develop identification conditions for estimating causal effects, and, very importantly, to discuss whether or not such conditions are plausible or implausible in practice (as above). Though quite intuitive, the notation requires a little getting used to, primarily because it is not typically included in early statistical training, but once that is accomplished, the notation is powerful and simple to use. Finally, as a strictly pragmatic matter, the important papers in the literature on causal inference (see especially papers by the 3R's (Robins, Rosenbaum, Rubin, and selected collaborators)) use this notation, making an understanding of it a prerequisite for any neuroimaging researcher who wants to learn more about this subject."

The notation definitely takes a little time getting use to, and it is also true for me that it was not discussed early on in any of my graduate econometrics courses. However, Angrist and Pischke's Mostly Harmless Econometrics does a good job making it more intuitive, with a little effort.  Pearl and Bollen both make arguments that SEMs 'subsume' the potential outcomes framework.While this may be true, its not straightforward to me yet. But I agree it is important to understand how SEMs relate to potential outcomes and causality, or at least understand some framework to support their use in causal inference, as stated in the article:

"Our original note had two aims. First, we wanted neuroimaging researchers to recognize that when they use SEMs to make causal inferences, the validity of the conclusions rest on assumptions above and beyond those required to use an SEM for descriptive or predictive purposes. Unfortunately, these assumptions are rarely made explicit, and in many instances, researchers are not even aware that they are needed. Since these assumptions can have a major impact on the “finndings”, it is critical that researchers be aware of them, and even though they may not be testable, that they think carefully about the science behind their problem and utilize their substantive knowledge to carefully consider, before using an SEM, whether or not these assumptions are plausible in the particular problem under consideration."

I think the assumptions they provide 1-4b seem to lay a foundation in terms that make sense to me from a potential outcomes framework, and the authors hold that these are the assumptions one should think about before using SEMs for causal inference. Apparently Pearl had some issues with this approach.

See also:

Monday, April 14, 2014

Perceptions of GMO Foods: A Hypothetical Application of SEM



Suppose we were interested in understanding consumer perceptions of GMO foods. People often make decisions and form opinions about issues based on abstract constructs like fears, goals, ambitions, values, political ideology etc. These are things that may not be so easy to quantify, despite being relevant to behavior and choice. When it comes to perceptions of GMO food, perhaps they are shaped by some degree of skepticism of ‘big ag’ or chemical companies. We could call this ‘monsantophobia.’ Perceptions of biotechnology could also be shaped by the extent of one’s knowledge of basic biology, agricultural science, and genetics.  This second construct could be referred to as a ‘science’ factor. A third ‘factor’ that may shape one’s views could be based on their ideals related to the role of government and political ideology. We’ll call this the ‘political’ factor.

So how can we best quantify these ‘latent’ constructs or ‘factors’ that may be related to perceptions of biotechnology, and how do we model these interactions?  This will require a combination of techniques involving factor analysis and regression, known as structural equation modeling. We might administer a survey, asking key questions that relate to one’s level of monsantophobia,  science knowledge, and political views.  To the extend that ‘monsantophobia’ exists and shapes views on biotechnology, it should flavor responses  to questions related to fears, skepticism, and mistrust of ‘big ag.’ Actual knowledge of science should influence responses to questions related to science etc. We also may want to quantify the actual flavor of perceptions of GMO food. This could be some index quantifying levels of tolerance or preferences related to policies concerning labeling, testing, and regulation or purchasing decisions and expenditures on related goods.  To the extent that perceptions are ‘positive’ the index would reflect that on some scale related to answers to survey questions about these issues. You could also include a set of questions related to policy preferences and try to model the interaction of the above factors and their impact on the support for some policy or the general policy environment.

Suppose we ask a range of questions related to skepticism of big ag and agrochemical companies and record the responses to each question as a value for a number of variables (Xm1…Xmn), and did the same for science knowledge (Xs1…Xsn), political ideology (Xp1…Xpn), and overall GMO perception (Yp1…Ypn) and policy environment (Ye1…Yen) . Given the values of these variables will be influenced by the actual latent constructs we are trying to measure, we refer to the X’s and Y’s above as ‘indicators’ of the given factors for monsantophobia, science, politics, gmo perception, and policy environment. They may also be referred to as the observable manifest variables.

Now, this is not a perfect system of measurement.  Given the level of subjectivity among other things, there is likely to be a non-negligible amount of measurement error involved.  How can we deal with measurement error and quantify the factors? Factor analysis attempts to separate common variance (associated with the factors) from unique variance in a data set. Theoretically, the unique variance in FA is correlated with the measurement error we are concerned about, while the factors remain ‘uncontaminated’ (Dunteman,1989).

Structural equation modeling (SEM) consists of two models, a measurement model which consists of deriving the latent constructs or factors previously discussed, and a structural model, which relates the factors to one another, and possibly some outcome. In this case, we are relating the factors related to monsantophobia, science, and political preferences to the outcome, which in this case would be the latent construct or index related to GMO perceptions and policy environment. By using the measured ‘factors’ from FA, we can quantify the latent constructs of monsantophobia, science, politics ,and GMO perceptions with less measurement error than if we simply included the numeric responses for the X’s and Y’s in a normal regression.  And then SEM lets us identify the relative influence of each of these factors on GMO perceptions and perhaps even their impact on the general policy environment for biotechnology. This is done in a way similar to regression, by estimating path coeffceints for the paths connecting the latent constructs or factors as depicted below.

Equations:




References:

Principle Components Analysis- SAGE Series on Quantitative  Applcations in the Social Sciences. Dunteman. 1989.

Awareness and Attitudes towards Biotechnology Innovations among Farmers and Rural Population in the European Union
LUIZA TOMA1, LÍVIA MARIA COSTA MADUREIRA2, CLARE HALL1, ANDREW BARNES1, ALAN RENWICK1

Paper prepared for presentation at the 131st EAAE Seminar ‘Innovation for Agricultural Competitiveness and Sustainability of Rural Areas’, Prague, Czech Republic, September 18-19, 2012

A Structural Equation Model of Farmers Operating within Nitrate Vulnerable Zones (NVZ) in Scotland
Toma, L.1, Barnes, A.1, Willock, J.2, Hall, C.1
12th Congress of the European Association of Agricultural Economists – EAAE 2008

PLoS One. 2014; 9(1): e86174.
Published online Jan 29, 2014. doi:  10.1371/journal.pone.0086174
PMCID: PMC3906022
Determinants of Public Attitudes to Genetically Modified Salmon
Latifah Amin,1,* Md. Abul Kalam Azad,1,2 Mohd Hanafy Gausmian,3 and Faizah Zulkifli1



Sunday, April 13, 2014

Intuition for Fixed Effects

I've written about fixed effects before in the context of mixed models. But how are FE useful in the context of causal inference? What can we learn from a panel data using FE that we can't get from a standard regression with cross sectional data?  Let's view this through a sort of parable, based largely on a very good set of notes produced by J. Blumenstock, used in a management statistics course (link).

Suppose we have a restaurant chain and have gathered some cross sectional data on the pricing and consumption of large pizzas for some portion of the day for some period 1 across three cities, as pictured below:

Now, if we are trying to infer the relationship between price and quantity demanded using this data, we notice something odd. The theoretically implied negative relationship does not exist. In fact, if we plot the points, this seems more in line with a supply curve rather than a demand curve:
What's going on that could explain this? One explanation could be specific individual differences across cities related to taste and quality. Perhaps in Chicago, customer's tastes and preferences are for much more expensive and higher quality pizza, and they really like pizza a lot. They may be willing to pay more for more pizzas aligned with their specific tastes and preferences. Perhaps this is also true for San Francisco, but to a lesser extent, and in Atlanta maybe not so much.

What we have is unobserved heterogeneity related to these specific individual effects. How can we account for this? Suppose we instead collected the same data for two periods, essentially creating a panel of data for pizza consumption:
Now, if we look 'within' each city, the data reveals the theoretically implied relationship between price and demand. Take San Francisco for example:
This is essentially what fixed effects estimators using panel data can do. They allow us to exploit the 'within' variation to 'identify' causal relationships. Essentially using a dummy variable in a regression for each city (or group, or type to generalize beyond this example) holds constant or 'fixes' the effects across cities that we can't directly measure or observe. Controlling for these differences removes the 'cross-sectional' variation related to unobserved heterogeneity (like tastes, preferences, other unobserved individual specific effects). The remaining variation, or 'within' variation can then be used to 'identify' the causal relationships we are interested in.

See also: Difference-in-Difference models. These are a special case of fixed effects also used in causal inference.

Reference:
Fixed Effects Models(Very Important Stuff)
www.jblumenstock.com/courses/econ174/FEModels.pdf

Friday, April 11, 2014

Structural Equation Models, Applied Economics, and Biotechnology

Toma gives some nice descriptions of SEM methodology and application:

Awareness and Attitudes towards Biotechnology Innovations among Farmers and Rural Population in the European Union
LUIZA TOMA1, LÍVIA MARIA COSTA MADUREIRA2, CLARE HALL1, ANDREW BARNES1, ALAN RENWICK1

Paper prepared for presentation at the 131st EAAE Seminar ‘Innovation for Agricultural Competitiveness and Sustainability of Rural Areas’, Prague, Czech Republic, September 18-19, 2012

SEM may consist of two components, namely the measurement model (which states the relationships between the latent variables and their constituent indicators), and the structural model (which designates the causal relationships between the latent variables). The measurement model resembles factor analysis, where latent variables represent ‘shared’ variance, or the degree to which indicators ‘move’ together. The structural model is similar to a system of simultaneous regressions, with the difference that in SEM some variables can be dependent in some equations and independent in others.


A Structural Equation Model of Farmers Operating within Nitrate Vulnerable Zones (NVZ) in Scotland
Toma, L.1, Barnes, A.1, Willock, J.2, Hall, C.1

12th Congress of the European Association of Agricultural Economists – EAAE 2008

To identify the factors determining farmers’ nitrate reducing behaviour, we follow the attitude-behaviour framework as used in most literature on agri- environmental issues. To statistically test the relationships within this framework, we use structural equation modelling (SEM) with latent (unobserved) variables. We first identify the latent variables structuring the model and their constituent indicators. Then, we validate the construction of the latent variables by means of factor analysis and finally, we build and test the structural equation model by assigning the relevant relationships between the different latent variables.


See also:

PLoS One. 2014; 9(1): e86174.
Published online Jan 29, 2014. doi:  10.1371/journal.pone.0086174
PMCID: PMC3906022
Determinants of Public Attitudes to Genetically Modified Salmon
Latifah Amin,1,* Md. Abul Kalam Azad,1,2 Mohd Hanafy Gausmian,3 and Faizah Zulkifli1

Wednesday, April 9, 2014

Quantile Regression and Healthcare Costs

I thought this was a nice statement that speaks to the utility of quantile regression (which holds to any distribution with these issues not just cost data):

The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost, and is naturally adapted to the skewness and heterogeneity of the cost data.

More:

Health care cost data are characterized by a high level of skewness and heteroscedastic variances…Most of the existing literature on health care cost data analysis have been focused on modeling the conditional mean (or average) of the health care cost given the covariates such as age, gender, race, marital status and disease status. The conditional mean framework has two important limitations. First, the application of the conditional mean regression model to health care cost data analysis is usually not straightforward. Due to the presence of skewness and nonconstant variances, transformation of the response variable is often required when constructing the mean regression model and retransformation is needed in order to obtain direct inference on the mean cost. Second, the conditional mean model focuses primarily on the marginal effects of the risk factors on the central tendency of the conditional distribution. When the marginal effects vary across the conditional distribution, focusing on the marginal effects at the central tendency may substantially distort the information of interest at the tails. For example, a weak relationship between a risk factor and the mean health care cost does not preclude a stronger relationship at the upper or lower quantiles of the conditional distribution….By considering different quantiles, we are able to obtain a more complete picture of the effects of the covariates on health care cost.

From:

Weighted Quantile Regression for Analyzing Health Care Cost Data with Missing Covariates. Ben Sherwood, Lan Wang and Xiao-Hua Zhou Statistics in Medicine. 2012

 “heavy upper tails may influence the "robustness" with which some parameters are estimated. Indeed, in worlds described by heavy-tailed Pareto or Burr- Singh-Maddala distributions (Mandelbrot, 1963; Singh and Maddala, 1976) some traditionally interesting parameters (means, variances) may not even be finite, a situation never encountered in, e.g., a normal or log-normal world. Such concerns should translate into empirical strategies that target the high-end parameters of particular interest, e.g. models for Prob(y k | x) or quantile regression models.."
 
From:
ECONOMETRIC MODELING OF HEALTH CARE COSTS AND EXPENDITURES:
A SURVEY OF ANALTICAL ISSUES AND RELATED POLICY CONSIDERATIONS
John Mullahy
Univ. of Wisconsin-Madison
January 2009
See also: Quantile Regression with Count Data

Tuesday, April 8, 2014

Ambitious vs. Ambiguous Modeling


Some people believe that a conclusion reached based on solid statistically sound principles is a gold standard. But, we seldom prove anything in applied empirical analysis. This is disappointing to those that desire definitive answers.Rather than seeking proof, or absolute truth, the best we can do is inform: 

"Social scientists and policymakers alike seem driven to draw sharp conclusions, even when these can be generated only by imposing much stronger assumptions than can be defended. We need to develop a greater tolerance for ambiguity. We must face up to the fact that we cannot answer all of the questions that we ask." (Manski, 1995) 


Manski, C.F. 1995. Identification Problems in the Social Sciences. Cambridge: Harvard University Press. 

Another quote:


“…all models are approximations. Essentially, all models are wrong, but some are useful. However, the approximate nature of the model must always be borne in mind…”

— George E.P. Box

In George E. P. Box and Norman R. Draper, Empirical Model-Building and Response Surfaces (2007), 414