On LLMs and LPMs: Does the LL in LLM Stand for Linear Literalism?

 I've blogged in the past about what I call linear literalism and fundamentalist econometrics. And I've blogged a bit about linear probability models (LPMs). Recently I have had some concerns about people outsourcing their thinking to LLMs and the use of these tools like Dunning-Kruger-as-a-Service (DKaaS) where the critical thinking and actual learning starts and stops with prompt engineering and a response. Out of curiosity I asked ChatGPT about the appropriateness of using linear probability models. Although the overall response was thoughtful about thinking more carefully about causality, it still gave the canned 'thou shalt not'  theoretically correct fundamentalist response. My prompt could have been more sophisticated, but I tried to prompt from a user's prospective, someone who may not be as familiar with applied statistics work, or who may have even read my blog and wanted to question something about the use of LPMs and may not be thinking about the tradeoffs or who may be unfamiliar with the social norms and practices related to their use.  As has been noted before on this blog, in applied work, there is no consensus among practitioners that nonlinear models (like logistic regression) are 'better' than LMPs when estimating treatment effects. If anything this illustrates at best, a response from an LLM about applied econometric analysis could be just as good as having another expert in the room, but an experienced practitioner understands that experts often disagree, and that disagreement comes with a lot of nuance, and is often as much the result of social norms and practices as theory. Perhaps someone could take the fundamentalist response from this prompt and do their analysis and solve their problem and there is no harm at the end of the day. But there is danger in fundamentalism, if this leads them to ignore great work and potential learnings derived from LPMs, or prevents them from getting more actionable and interpretable results vs. stumbling through the mental gymnastics sometimes required to interpret things like odds ratios vs. marginal effects! 

Note: As I refined my prompt I got slightly better answers, but not one that really captures the nuance I would like to have. Again....it's like arguing with an expert that knows the theory but may not be as familiar with all the nuances of different disciplines and the practical implications of the tradeoffs involved. I have to say - to give a response close to what I would expect from another practitioner that maybe outside the field of economics, is impressive! 

While it did a great job articulating the pros and cons of LPMs vs logistic regression or other models, I think these responses would leave the casual reader with the impression that they should be wary of relying on LPMs to estimate treatment effects in most situations. And as noted above that could be an issue depending on the situation.  Because I disagree with the tone of the response doesn't mean chatGPT is a bad tool, there are great textbooks and econometricians that I think would certainly endorse these responses! Its the missing nuance around these differences in opinion that concern me and while its not super critical in this situation, I'd be concerned about others. 

And I have to disagree that some of the 'cons' listed about LPMs are really as big of an issue that a reader might think if they take this response very seriously. Most of the listed 'cons' can often be addressed easily and in my experience despite all the alleged 'cons' LPMs generally lead to the same decision you would make when properly interpreting and understanding results from a logistic regression, while providing all the very practical benefits you get from an LPM.  The nuance is, the 'cons' in theory don't always have implications that matter in any practical sense. As Angrist and Pischke state: "there is a lot to be said for sticking to a linear regression function as compared to a fairly arbitrary choice of a non-linear one!"

Prompt: Is it OK to use a linear probability model to measure causal effects?

Answer:

Using a linear probability model to measure causal effects can be problematic and is generally not recommended. A linear probability model (LPM) is a linear regression model where the dependent variable is a binary outcome, typically coded as 0 or 1. In this model, the predicted values can be outside the [0, 1] range, which can lead to biased estimates and incorrect interpretations.

There are several issues with using a linear probability model for causal effects:

1. Heteroscedasticity: The variance of the error term in the linear probability model is not constant across all values of the independent variables, violating one of the assumptions of linear regression. This can lead to inefficient estimates and incorrect standard errors.

2. Predicted probabilities outside [0, 1]: Since linear regression does not constrain the predicted values to be between 0 and 1, it can produce probabilities that fall outside this range, which is not meaningful for binary outcomes.

3. Non-linearity: The relationship between the independent variables and the probability of the binary outcome is often non-linear. Using a linear model may not capture this relationship accurately.

To address these issues, researchers typically use alternative methods for estimating causal effects with binary outcomes. The most common approaches are logistic regression and probit regression. These models are specifically designed for binary outcomes and provide estimates that are valid probabilities between 0 and 1.

Additionally, for estimating causal effects, researchers often use methods such as randomized controlled trials (RCTs) or quasi-experimental designs (e.g., difference-in-differences, matching methods, instrumental variables) to establish causality more convincingly.

In summary, while linear probability models might be used for exploratory analysis, they are generally not suitable for measuring causal effects with binary outcomes. Logistic regression or probit regression should be preferred when dealing with binary dependent variables, and careful consideration of study design and potential confounders is essential for establishing causality.

Additional Reading:

Love It or Logit, or: Man, People *Really* Care About Binary Dependent Variables. Marc Bellemare. http://marcfbellemare.com/wordpress/9024

Whether to probit or to probe it: in defense of the Linear Probability Model. World Bank Blogs. Jed Friedman. https://blogs.worldbank.org/impactevaluations/whether-to-probit-or-to-probe-it-in-defense-of-the-linear-probability-model 

Modeling Claims with Linear vs. Non-Linear Difference-in-Difference Models

Previously I have discussed the issues with modeling claims costs. Typically medical claims exhibit non-negative highly skewed values with high zero mass and heterskedasticity. The most commonly suggested approach to addressing these distributional concerns in the literature call for the use of non-linear GLM models.  However, as previously discussed (see here and here) there are challenges with using difference-in-difference models in the context of GLM models. So once again, the gap between theory and application presents challenges, tradeoffs, and compromises that need to be made by the applied econometrician.

In the past I have written about the accepted (although controversial in some circles) practice of leveraging linear probability models to estimate marginal effects in applied work when outcomes are dichotomous. But what about doing this in the context of claims analysis? In my original post regarding the challenges of using difference-in-differences with claims I speculated:

"So as Angrist and Pischke might ask, what is an applied guy to do? One approach even in the context of skewed distributions with high mass points (as is common in the healthcare econometrics space) is to specify a linear model. For count outcomes (utilization like ER visits or hospital admissions are often dichotomized and modeled by logit or probit models) you can just use a linear probability model. For skewed distributions with heavy mass points, dichotomization with a LPM may also be an attractive alternative."

 I have found that this advice is pretty consistent with the social norms and practices in the field.

In their analysis of the ACA Cantor, et al (2012) leverage linear probability models for difference-in-differences for healthcare utilization stating:

"Linear probability models are fit to produce coefficients that are direct estimates of the relevant policy impacts and are easily interpreted as percentage point changes in coverage outcomes. This approach has been applied in earlier evaluations of insurance market reforms (Buchmueller and DiNardo 2002; Monheit and Steinberg Schone 2004;  Levine, McKnight, and Heep 2011;  Monheit et al. 2011). It also avoids complications associated with estimation and interpretation of multiple interaction terms and their standard errors in logit or probit models (Ai and Norton 2003)."

Jhamb et al (2015) use LPMs for dichotomous outcomes as well as OLS models for counts in a DID framework.

Interestingly, Deb and Norton (2018) discuss an approach to address the challenges of DID in a GLM framework head on:

"Puhani argued, using the potential outcomes framework, that the treatment effect on the treated in the difference-in-difference regression equals the expected value of the dependent variable for the treatment group in the post period with treatment compared with the hypothetical expected value of the dependent variable for the treatment group in the post period if they had not received treatment. In nonlinear models, the treatment effect on the treated equals the difference in two predicted values. It always has the same sign as the coefficient on the interaction term. Because we estimate many nonlinear models using a difference-in-differences study design, we report the treatment effect on the treated in all tables of results."

In presenting their results they compare their GLM based approach to results from linear models of healthcare expenditures. While they argue the differences are substantial in supporting their approach, I did not find the OLS estimate (-$323.4) to be practically different from the second part (conditional on positive) of the two part GLM model (-$321.4), although the combined results from the two part model had large practical differences from OLS. It does not appear they compared a two-part GLM to a two-part linear model (which could be problematic if the first part OLS model gave probabilities greater than 1 or less than zero). In their paper they cited a number of authors using linear difference-in-differences to model claims you will find below.

See the references below for a number of examples (including those cited above).

Related: Linear Literalism and Fundamentalist Econometrics

References:

Cantor JC, Monheit AC, DeLia D, Lloyd K. Early impact of the Affordable Care Act on health insurance coverage of young adults. Health Serv Res. 2012;47(5):1773-90.

Modeling Health Care Expenditures and Use
Partha Deb and Edward C. Norton
Annual Review of Public Health 2018 39:1, 489-505

Buchmueller T, DiNardo J. “Did Community Rating Induce an Adverse Selection Death Spiral? Evidence from New York, Pennsylvania and Connecticut” American Economic Review. 2002;92(1):280–94.

Monheit AC, Cantor JC, DeLia D, Belloff D. “How Have State Policies to Expand Dependent Coverage Affected the Health Insurance Status of Young Adults?” Health Services Research. 2011;46(1 Pt 2):251–67

Amuedo-Dorantes C, Yaya ME. 2016. The impact of the ACA’s extension of coverage to dependents on young adults’ access to care and prescription drugs. South. Econ. J. 83:25–44

Barbaresco S, Courtemanche CJ, Qi Y. 2015. Impacts of the Affordable Care Act dependent coverage provision on health-related outcomes of young adults. J. Health Econ. 40:54–68

Jhamb J, Dave D, Colman G. 2015. The Patient Protection and Affordable Care Act and the utilization of health care services among young adults. Int. J. Health Econ. Dev. 1:8–25

Sommers BD, Buchmueller T, Decker SL, Carey C, Kronick R. 2013. The Affordable Care Act has led
to significant gains in health insurance and access to care for young adults. Health Aff. 32:165–74

Marginal Effects vs Odds Ratios

Models of binary dependent variables often are estimated using logistic regression or probit models, but the estimated coefficients (or exponentiated coefficients expressed as odds ratios) are often difficult to interpret from a practical standpoint. Empirical economic research often reports ‘marginal effects’, which are more intuitive but often more difficult to obtain from popular statistical software. The most straightforward way to obtain marginal effects is from estimation of linear probability models. This paper uses a toy data set to demonstrate the calculation of odds ratios and marginal effects from logistic regression using SAS and R, while comparing them to the results from a standard linear probability model.

Suppose we have a data set that looks at program participation (for some program or product or service of interest) by age and we want to know the influence of age on the decision to participate. Our data may look something like the excerpt below:

participation     age
1                       25
1                       26
1                       27
1                       28
1                       29
1                       30
0                       31
1                       32
1                       33
0                       34

Theoretically,  this might call for logistic regression for modeling a dichotomous outcome like participant, so we could use SAS or R to get the following results:

                   Estimate Std. Error z value Pr(>|z|) 
(Intercept)  5.92972    2.34258   2.531   0.0114 *
age            -0.14099    0.05656  -2.493   0.0127 *

                    OR               2.5 %       97.5 %
(Intercept)   376.049897 6.2769262 7.864410e+04
age              0.868502 0.7641126 9.595017e-01

 While the estimated coefficients from logistic regression are not easily interpretable (they represent the change in the log of odds of participation for a given change in age),  odds ratios might provide a better summary of the effects of age on participation (odds ratios are derived from exponentiation of the estimated coefficients from logistic regression -see also: The Calculation and Interpretation of Odds Ratios) and may be somewhat more meaningful. We can see the odds ratio associated with age is .8685 which implies that for every year increase in age the odds of participation are about (.8685-1)*100 = -13.15% or 13.5% less.  You tell me what this means if this is the way you think about the likelihood of outcomes in everyday life!

Marginal effects are an alternative metric that can be used to describe the impact of age on participation. Marginal effects can be described as the change in outcome as a function of the change in the treatment (or independent variable of interest) holding all other variables in the model constant. In linear regression, the estimated regression coefficients are marginal effects and are more easily interpreted (more on this later). Marginal effects can be output easily from STATA, however they are not directly available in SAS or R. However there are some adhoc ways of getting them which I will demonstrate here.  (there are some packages in R available to assist with this as well). I am basing most of this directly on two very good blog posts on the topic:

https://statcompute.wordpress.com/2012/09/30/marginal-effects-on-binary-outcome/ 
https://diffuseprior.wordpress.com/2012/04/23/probitlogit-marginal-effects-in-r-2/ 

One approach is to use PROC QLIM and request output of marginal effects. This computes a marginal effect for each observation’s value of x in the data set (because marginal effects may not be constant across the range of explanatory variables). Taking the average of this result gives and estimated ‘sample average estimate of marginal effect’:  -.0258

This tells us that for every year increase in age the probability of participation decreases on average by 2.5%.  For most people, for practical purposes, this is probably a more useful interpretation of the relationship between age and participation compared to odds ratios.  We can calculate this more directly (following the code from the blog post by WenSui Liu) using output from logistic regression and the data step in SAS. Basically for each observation in the data set calculate:

MARGIN_AGE = EXP(XB) / ((1 + EXP(XB)) ** 2) * (-0.1410);

Where -.1410 is the estimated coefficient on age from the original logistic regression model. We can run the same analysis in R, either replicating the results from the data step above, or using the mfx function defined by Alan Fernihough referenced in the diffuseprior blog post mentioned above or the paper referenced below.

The paper notes that this function gives similar results to the mfx function in STATA. And we get almost the same results we got from SAS above but additionally provides bootstrapped standard errors :

marginal.effects   standard.error
      -0.0258330        0.6687069

Marginal Effects from Linear Probability Models

Earlier I mentioned that you could estimate marginal effects directly from the estimated coefficients from a linear probability model. While in some circles LPMs are not viewed favorably, they have a strong following among applied econometricians (see references for more on this). As Angrist and Piscke state in their very popular book Mostly Harmless Econometrics:

"While a nonlinear model may fit the CEF (population conditional expectation function) for LDVs (limited dependent variables) more closely than a linear model, when it comes to marginal effects, this probably matters little"

Using SAS or R we can get the following results from estimating a LPM for this data:

 Coefficients:
                   Estimate    Std. Error  t value    Pr(>|t|)  
(Intercept)  1.700260   0.378572   4.491     0.000111 ***
dat1$age    -0.028699   0.009362  -3.065   0.004775 **

 You can see that the estimate from the linear probability model above gives us a marginal effect  (-.028699) almost identical to the previous estimates derived from logistic regression, as is often the case, and as indicated by Angrist and Pischke.

In the SAS ETS example cited in the references below, a distinction is made between calculating sample average marginal effects (which were discussed above) vs. calculating marginal effects at the mean:

“To evaluate the "average" or "overall" marginal effect, two approaches are frequently used. One approach is to compute the marginal effect at the sample means of the data. The other approach is to compute marginal effect at each observation and then to calculate the sample average of individual marginal effects to obtain the overall marginal effect. For large sample sizes, both the approaches yield similar results. However for smaller samples, averaging the individual marginal effects is preferred (Greene 1997, p. 876)”

For a step by step review of the SAS and R code presented above as well as an additional example with multiple variables see:

Matt Bogard. "Comparing Odds Ratios and Marginal Effects from Logistic Regression and Linear Probability Models" Staff Paper (2016)
Available at: http://works.bepress.com/matt_bogard/30/ 

References: 

Simple logit and probit marginal effects in R.  https://ideas.repec.org/p/ucn/wpaper/201122.html


SAS/ETS Web Examples Computing Marginal Effects for Discrete Dependent Variable Models. http://support.sas.com/rnd/app/examples/ets/margeff/ 

Linear Regression and Analysis of Variance with a Binary Dependent Variable (from EconomicSense, by Matt Bogard).

Angrist, Joshua D. & Jörn-Steffen Pischke. Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press. NJ. 2008.

Probit better than LPM? http://www.mostlyharmlesseconometrics.com/2012/07/probit-better-than-lpm/ 

Love It or Logit. By Marc Bellemare. marcfbellemare.com/wordpress/9024

R Data Analysis Examples: Logit Regression. From http://www.ats.ucla.edu/stat/r/dae/logit.htm   (accessed March 4,2016).

Greene, W. H. (1997), Econometric Analysis, Third edition, Prentice Hall, 339–350.

Linear Probability Models for Skewed Distributions with High Mass Points

There are a lot of methods discussed in the literature related to modeling skewed distributions with high mass points including log transformations, two part models,  GLM etc. In some previous posts I have discussed linear probability models in the context of causal inference.  I've also discussed the use of quantile regression as a strategy to model highly skewed continuous and count data. Mullahy (2009) alludes to the use of quantile regression as well:

"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"

The focus on high end parameters  using linear probability models is mentioned in Angrist and Pischke (2009) :

"COP [conditional-on-positive] effects are sometimes motivated by a researcher's sense that when the outcome distribution has a mass point-that is, when it piles up on a particular value, such as zero-or has a heavily skewed distribution, or both, then an analysis of effects on averages misses something. Analysis of effects on averages indeed miss some things, such as changes in the probability of specific values or a shift in quantiles away from the median. But why not look at these distribution effects directly? Distribution outcomes include the likelihood that annual medical expenditures exceed zero, 100 dollars, 200 dollars, and so on. In other words, put 1[Yi > c] for different choices of c on the left hand side of the regression of interest...the idea of looking directly at distribution effects with linear probability models is illustrated by Angrist (2001),...Alternatively, if quantiles provide a focal point, we can use quantile regressions to model them."

References:

Mostly Harmless Econometrics. Angrist and Pischke. 2009

Angrist, J.D. Estimation of Limited Dependent Variable Models With Dummy Endogenous Regressors: Simple Strategies for Empirical Practice. Journal of Business & Economic Statistics January 2001, Vol. 19, No. 1.

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

The Oregon Medicaid Experiment and Linear Probability Models

I just recently discussed the methodology used in some recent papers analyzing the Oregon Medicaid expansion (see: http://econometricsense.blogspot.com/2014/01/the-oregon-medicaid-experiment-applied.html ). This was one of the papers:

"The Oregon Experiment--Effects of Medicaid on Clinical Outcomes," by Katherine Baicker, et al. New England Journal of Medicine, 2013; 368:1713-1722. http://www.nejm.org/doi/full/10.1056/NEJMsa1212321 

If you read the supplementary appendix you will find the following:

In all of our ITT estimates and in our subsequent instrumental variable estimates (see below), we fit linear models even though a number of our outcomes are binary. Because we are interested in the difference in conditional means for the treatments and controls, linear probability models would pose no concerns in the absence of covariates or in fully saturated models (Angrist 2001, Angrist and Pischke 2009). Our models are not fully saturated, however, so it is possible that results could be affected by this functional form choice, especially for outcomes with very low or very high mean probability. We therefore explore the sensitivity of our results to an alternate specification using logistic regression and calculating average marginal effects for all binary outcomes, and are reassured that the results look very similar (see Table S15a-d below). 

http://www.nejm.org/doi/suppl/10.1056/NEJMsa1212321/suppl_file/nejmsa1212321_appendix.pdf

You will find a similar methodology in the more recent article in Science previously discussed. This weaves well with some of my past posts:

Linear Regression and Analysis of Variance with Binary Dependent Variables

Regression as an Empirical Tool (matching and linear probability models)

Linear Regression and Analysis of Variance with a Binary Dependent Variable

(see also my posts related to Logistic Regression) 

If for instance Y is dichotomous or binary, Y = { 1 if ‘yes’  0 if ‘no’}, would  you consider it valid to do an analysis of variance or fit a linear regression model?

We might not think so based on traditional assumptions, because besides assuming that Y is continuous….

1)      ANOVA / linear regression both work under the assumption of a uniform (homoskedastic) error term ‘e’

2)      For a dichotomous y, the expected value E(y|X)  =  ‘probability’ and may be continuous, but the  error terms follow a binomial distribution with mean ‘p’ and a variance that is a function of the mean, which is inherently heteroskedastic , var(error) ~ n*p*(1-p)   

3)      Therefore the assumption of uniform variance is violated, so the F test and other tests based on standard errors based on this assumption are questionable

Some people will argue that violations of this assumption may only matter by degree, and under certain conditions ANOVA and linear regression using least squares is OK with a binary dependent variable (Lunney 1971, D'Agostino 1971,Astin & Dey 1993, Angrist & Pischke 2008).

I recall from mathematical statistics and econometrics, the lectures (and test questions) related to properties of estimators including efficiency, consistency, unbiasedness etc. We also spoke some about robustness, but in the theoretical work, its not so easy to 'prove' robustness as it is to show that a certain estimator is unbiased or consistent. In this way, I think a lot of times, robustness to assumptions isn't given a lot of credence by students or practitioners. ( However, Angrist and Pischke in their book 'Mostly Harmless Economics' spend a lot of time discussing ideas related to the robustness of the least squares estimator).  Robustness to assumptions related to the distribution of the error terms (under OLS / ANOVA are discussed in the following:

LITERATURE RELATED TO REGRESSION AND ANOVA  WITH A DICHOTOMOUS DEPENDENT VARIABLE

ANALYSIS OF VARIANCE CONTEXT--------------------------------------------

A Second Look at Analysis of Variance on Dichotomous Data

Author(s): Ralph B. D'Agostino

Source: Journal of Educational Measurement, Vol. 8, No. 4 (Winter, 1971), pp. 327-333

link: http://www.jstor.org/stable/1433756

“probably a safe rule of thumb for deciding when the I x J ANOVA techniques may be used on dichotomous data with equal sample sizes in each cell is; the sample proportions for the cells should lie between .25 and .75 and there should be at least 20 degrees offreedom for error. This rule combines Lunney's results along with standard rules (Snedecor & Cochran, 1967, p. 494). The reason for the .25 and .75 lies in the fact that for this range there is little change between the within cell variances, p(l - p), and so there is a sufficient homogeneity of variances. Given that this rule is satisfied a standard procedure for analysis is the ANOVA on the original data. In this range (.25 to .75) it is doubtful if any alternative valid procedure, such as an analysis of transformed data, would lead to different conclusions”

Using Analysis of Variance with a Dichotomous Dependent Variable: An Empirical Study

Author(s): Gerald H. Lunney

Source: Journal of Educational Measurement, Vol. 7, No. 4 (Winter, 1970), pp. 263-269

Link http://www.jstor.org/stable/1434469

“The findings show the analysis of variance to be an appropriate statistical technique for analyzing dichotomous data in fixed effects models where cell frequencies are equal under the following conditions: (a) the proportion of responses in the smaller response category is equal to or greater than .2 and there are at least 20 degrees of freedom for error, or (b) the proportion of responses in the smaller response category is less than .2 and there are at least 40 degrees of freedom for error.”

REGRESSION CONTEXT---------------------------------------------------

Dey ,Eric L. and Alexander W. Astin. Statistical Alternatives For Studying College Student Retention: A Comparative Analysis of Logit, Probit, and Linear Regression. Research in Higher Education, Vol. 34, No. 5. 1993. link http://www.jstor.org/stable/40196112  

"These results indicate that despite the theoretical advantages offered by logistic regression and probit analysis, there is little practical difference between either of these two techniques and more traditional linear regression. While this may not always be the case, these and other analyses show that for variables that are moderately distributed (say, within a .75/. 25 split; for example, see Cleary and Angel, 1984) there is little practical difference in obtained results upon which to make a decision about one technique or another, especially in large samples."

Angrist, Joshua D. & Jörn-Steffen Pischke. Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press. NJ. 2008.

Link http://www.mostlyharmlesseconometrics.com/

"While a nonlinear model may fit the CEF (population conditional expectation function) for LDVs (limited dependent variables) more closely than a linear model, when it comes to marginal effects, this probably matters little"

BOTH CONTEXTS---------------------------------------------

The Analysis of Relationships Involving Dichotomous Dependent Variables

Author(s): Paul D. Cleary and Ronald AngelSource: Journal of Health and Social Behavior, Vol. 25, No. 3 (Sep., 1984), pp. 334-348

link  http://www.jstor.org/stable/2136429  

"If the researcher wishes to estimate the probability of an outcome as a linear function and, A. If the sample size is moderately large and the dependent variable is not too skewed (.25 < p < .75), then OLS regression or ordinary ANOVA is adequate."