## Saturday, March 28, 2015

### Using the R MatchIt package for propensity score analysis

Descriptive analysis between treatment and control groups can reveal interesting patterns or relationships, but we cannot always take descriptive statistics at face value. Regression and matching methods allow us to make controlled comparisons to reduce selection bias in observational studies.
For a couple good references that I am basically tracking in this post see here and here. These are links to the pages of the package authors and a nice paper (A Step by Step Guide to Propensity Score Matching in R) from higher education evaluation research respectively.

In both Mostly Harmless Econometrics and Mastering Metrics Angrist and Pischke discuss the similarities between matching and regression. From MM:

"Specifically, regression estimates are weighted averages of multiple matched comparisons"

In this post I borrow from some of the previous references, and try to follow closely the dialogue in chapter 3 of MHE. So, conveniently the R matchit propensity score matching package comes with a subset of the Lalonde data set referenced in MHE. Based on descriptives, it looks like this data matches columns (1) and (4) in table 3.3.2. The Lalonde data set basically consists of a treatment variable indicator, an outcome re78 or real earnings in 1978 as well as other data that can be used for controls. (see previous links above for more details). If we use regression to look at basic uncontrolled raw differences between treatment and control groups, it appears that the treatment (a job training program) produces negative results (on the order of \$635):

R code:
summary(lm(re78~treat,data=lalonde))

Output:
Coefficients:
Estimate Std. Error t value            Pr(>|t|)
(Intercept)     6984        361   19.36 <0.0000000000000002 ***
treat           -635  <---      657   -0.97                0.33

Once we implement matching in R, the output provides comparisons between the balance in covariates for the treatment and control groups before and after matching. Matching is based on propensity scores estimated with logistic regression. (see previous post on propensity score analysis for further details). The output below indicates that the propensity score matching creates balance among covariates/controls as if we were explicitly trying to match on the controls themselves.

R Code:
m.out1 <- matchit(treat ~ age + educ + black + hispan + nodegree + married + re74 + re75, data = lalonde, method = "nearest", distance = "logit")

Output:
Summary of balance for all data:
Means Treated Means Control SD Control  Mean Diff   eQQ Med  eQQ Mean   eQQ Max
distance        0.5774        0.1822     0.2295     0.3952    0.5176    0.3955    0.5966
age            25.8162       28.0303    10.7867    -2.2141    1.0000    3.2649   10.0000
educ           10.3459       10.2354     2.8552     0.1105    1.0000    0.7027    4.0000
black           0.8432        0.2028     0.4026     0.6404    1.0000    0.6432    1.0000
hispan          0.0595        0.1422     0.3497    -0.0827    0.0000    0.0811    1.0000
nodegree        0.7081        0.5967     0.4911     0.1114    0.0000    0.1135    1.0000
married         0.1892        0.5128     0.5004    -0.3236    0.0000    0.3243    1.0000
re74         2095.5737     5619.2365  6788.7508 -3523.6628 2425.5720 3620.9240 9216.5000
re75         1532.0553     2466.4844  3291.9962  -934.4291  981.0968 1060.6582 6795.0100

Summary of balance for matched data:
Means Treated Means Control SD Control Mean Diff  eQQ Med eQQ Mean    eQQ Max
distance        0.5774        0.3629     0.2533    0.2145   0.1646   0.2146     0.4492
age            25.8162       25.3027    10.5864    0.5135   3.0000   3.3892     9.0000
educ           10.3459       10.6054     2.6582   -0.2595   0.0000   0.4541     3.0000
black           0.8432        0.4703     0.5005    0.3730   0.0000   0.3730     1.0000
hispan          0.0595        0.2162     0.4128   -0.1568   0.0000   0.1568     1.0000
nodegree        0.7081        0.6378     0.4819    0.0703   0.0000   0.0703     1.0000
married         0.1892        0.2108     0.4090   -0.0216   0.0000   0.0216     1.0000
re74         2095.5737     2342.1076  4238.9757 -246.5339 131.2709 545.1182 13121.7500
re75         1532.0553     1614.7451  2632.3533  -82.6898 152.1774 349.5371 11365.7100

Estimation of treatment effects can be obtained via paired or matched comparisons (Lanehart et al, 2012; Austin, 2010-see previous posts here and here)

Rcode:
t.test(m.data1\$re78[m.data1\$treat==1],m.data1\$re78[m.data1\$treat==0],paired=TRUE)

Output:
t = 1.2043, df = 184, p-value = 0.23
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-579.6904 2396.0948
sample estimates:
mean of the differences
908.2022 <---

This indicates an estimated treatment effect of about \$900.00, which is quite a reversal from the raw uncontrolled/unmatched comparisons. In Mostly Harmless Econometrics, as part of the dialogue relating regression to matching, Angrist and Pischke present results in table 3.3.3 for regressions utilizing data that has been 'screened' by eliminating observations where ps > .90 or < .10. Similar results were obtained in R below:

Rcode:
summary(lm(re78~treat + age + educ + black + hispan + nodegree + married + re74 + re75, data = m.data3))

Output:
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)    23.6095  3920.6599    0.01    0.995
treat        1229.8619 <---  849.1439    1.45    0.148
age            -3.4488    45.4815   -0.08    0.940
educ          506.9958   240.1989    2.11    0.036 *
black       -1030.4883  1255.1766   -0.82    0.412
hispan        926.5288  1498.4450    0.62    0.537

Again we get a reversal of the values from the raw comparisons, and a much larger estimated treatment effect than the non-paramterically matched comparisons above.

Note: Slightly different results were obtained from A&P, partly because I am not sure the exact specification of their ps model, which may have impacted the screening and ultimately differences in the data used.  Full R code is provided below with the ps model specification. Also, I have replicated similar results in SAS using the Mayo Clinic %gmatch macro as well as using approaches outlined by Lanehart et al (2012).  These results may be shared in a later post or white paper.

R Program:

```# *------------------------------------------------------------------
# | PROGRAM NAME: ex ps match mostly harmless R
# | DATE: 3/24/15
# | CREATED BY: MATT BOGARD
# | PROJECT FILE: Tools and References/Rcode
# *----------------------------------------------------------------
# | PURPOSE: Use R matchit and glm to mimic the conversation in
# | chapter 3 of Mostly Harmless Econometrics
# | NOTE: because of random sorting within matching application different results may be
# | obtained with each iteration of matchit in R
# *------------------------------------------------------------------

rm(list=ls()) # get rid of any existing data

ls() # view open data sets

options("scipen" =100, "digits" = 4) # override R's tendency to use scientific notation

# raw differences in means for treatments and controls using regression

summary(lm(re78~treat,data=lalonde))

t.test(lalonde\$re78[lalonde\$treat==1],lalonde\$re78[lalonde\$treat==0],paired=FALSE)

#-----------------------------------------------------
# nearest neighbor matching via MatchIt
#-----------------------------------------------------

# estimate propensity scores and create matched data set using 'matchit' and lalonde data

m.out1 <- matchit(treat ~ age + educ + black + hispan + nodegree + married + re74 + re75, data = lalonde, method = "nearest", distance = "logit")

summary(m.out1) # check balance

m.data1 <- match.data(m.out1,distance ="pscore") # create ps matched data set from previous output

hist(m.data1\$pscore) # distribution of propenisty scores
summary(m.data1\$pscore)

#-----------------------------------------------------
# perform paired t-tests on matched data
#-----------------------------------------------------

t.test(m.data1\$re78[m.data1\$treat==1],m.data1\$re78[m.data1\$treat==0],paired=TRUE)

#-----------------------------------------------------------------------------------------
# lets look at regression on the ps restricted data on the entire sample per MHE chapter 3
#-----------------------------------------------------------------------------------------

m.data2 <- lalonde # copy lalonde for ps estimation
dim(m.data2)

# generate propensity scores for all of the data in Lalonde
ps.model <- glm(treat ~ age + educ + black + hispan + nodegree + married + re74 +
na.action=na.pass)

summary(ps.model)

# add pscores to study data
m.data2\$pscore <- predict(ps.model, newdata = m.data2, type = "response")

hist(m.data2\$pscore) # distribution of ps
summary(m.data2\$pscore)
dim(m.data2)

# restrict data to ps range .10 <= ps <= .90

m.data3 <- m.data2[m.data2\$pscore >= .10 & m.data2\$pscore <=.90,]
summary(m.data3\$pscore)

# regression with controls on propensity score screened data set
summary(lm(re78~treat + age + educ + black + hispan + nodegree + married + re74 + re75, data = m.data3))

# unrestricted regression with controls
summary(lm(re78~treat + age + educ + black + hispan + nodegree + married + re74 + re75, data = lalonde))```
Created by Pretty R at inside-R.org

### Applied Propensity Score Analysis with SAS: SAS Global Forum Papers

Below are two interesting SAS Global Forum papers implementing propensity score methods.

Paper 314-2012
PROPENSITY SCORE ANALYSIS AND ASSESSMENT OF PROPENSITY SCORE APPROACHES USING SAS® PROCEDURES
Rheta E. Lanehart, Patricia Rodriguez de Gil, Eun Sook Kim, Aarti P. Bellara, Jeffrey D. Kromrey, and Reginald S. Lee, University of South Florida

This paper is a very good paper with a succinct review of the literature and very clear SAS code for implementation of several propensity score methods including caliper based matching, inverse probability of treatment weighted regression, and analysis of covariance.

Paper 220-2013
Propensity Score-based Analysis of Short-Term Complications in Patients with Lumbar Discectomy in the ACS-NSQIP Database
Yubo Gao The University of Iowa Hospitals and Clinics, Iowa City, Iowa

What I find interesting about the above paper is the use of the hash object in the matching implementation (vs. arrays used in the previous paper)

I am currently working on a suite of SAS programs that utilize the Mayo Clinic %gmatch macro as well as parametrization of the programs presented in Lanehart (suitable for a macro call)

## Friday, March 27, 2015

### Propensity Score Matching: Optimal Caliper Width and Paired Comparisons

Below are abstracts from some really good propensity score matching papers (among many). Sure there are a lot of good papers out there (think Rosenbaum/Rubin etc.) but these are nice applied papers that I appreciate as a practitioner.

Optimal caliper widths for propensity-score matching when estimating differences in means and differences in proportions in observational studies. Pharmaceut. Statist. 2011, 10 150–161
Peter C. Austin

In a study comparing the effects of two treatments, the propensity score is the probability of assignment to one treatment conditional on a subject’s measured baseline covariates. Propensity-score matching is increasingly being used to estimate the effects of exposures using observational data. In the most common implementation of propensity-score matching, pairs of treated and untreated subjects are formed whose propensity scores differ by at most a pre-specified amount (the caliper width). There has been a little research into the optimal caliper width. We conducted an extensive series of Monte Carlo simulations to determine the optimal caliper width for estimating differences in means (for continuous outcomes) and risk differences (for binary outcomes). When estimating differences in means or risk differences, we recommend that researchers match on the logit of the propensity score using calipers of width equal to 0.2 of the standard deviation of the logit of the propensity score. When at least some of the covariates were continuous, then either this value, or one close to it, minimized the mean square error of the resultant estimated treatment effect. It also eliminated at least 98% of the bias in the crude estimator, and it resulted in confidence intervals with approximately the correct coverage rates. Furthermore, the empirical type I error rate was approximately correct. When all of the covariates were binary, then the choice of caliper width had a much smaller impact on the performance of estimation of risk differences and differences in means.

Comparing paired vs non-paired statistical methods of analyses when making inferences about absolute risk reductions in propensity-score matched samples. Statist. Med. 2011, 30 1292--1301
Peter C. Austin

Propensity-score matching allows one to reduce the effects of treatment-selection bias or confounding when estimating the effects of treatments when using observational data. Some authors have suggested that methods of inference appropriate for independent samples can be used for assessing the statistical significance of treatment effects when using propensity-score matching. Indeed, many authors in the applied medical literature use methods for independent samples when making inferences about treatment effects using propensity-score matched samples. Dichotomous outcomes are common in healthcare research. In this study, we used Monte Carlo simulations to examine the effect on inferences about risk differences (or absolute risk reductions) when statistical methods for independent samples are used compared with when statistical methods for paired samples are used in propensity-score matched samples. We found that compared with using methods for independent samples, the use of methods for paired samples resulted in: (i) empirical type I error rates that were closer to the advertised rate; (ii) empirical coverage rates of 95 per cent confidence intervals that were closer to the advertised rate; (iii) narrower 95 per cent confidence intervals; and (iv) estimated standard errors that more closely reflected the sampling variability of the estimated risk difference. Differences between the empirical and advertised performance of methods for independent samples were greater when the treatment-selection process was stronger compared with when treatment-selection process was weaker. We recommend using statistical methods for paired samples when using propensity-score matched samples for making inferences on the effect of treatment on the reduction in the probability of an event occurring.

Elizabeth Stuart has lots of addtional resources and links on her page, for instance:

Papers

Software (R/SAS etc.)

## Thursday, March 26, 2015

### To Explain or Predict

Some people may not make the important distinctions between prediction vs inference when it comes to modeling approaches/methodologies/data handling/assumptions.  I recently ran across a blog post by Rob J. Hyndman that pointed to the following article in the journal Statistical Science:

Statist. Sci.
Volume 25, Number 3 (2010), 289-310.

"Statistical modeling is a powerful tool for developing and testing theories by way of causal explanation, prediction, and description. In many disciplines there is near-exclusive use of statistical modeling for causal explanation and the assumption that models with high explanatory power are inherently of high predictive power. Conflation between explanation and prediction is common, yet the distinction must be understood for progressing scientific knowledge. While this distinction has been recognized in the philosophy of science, the statistical literature lacks a thorough discussion of the many differences that arise in the process of modeling for an explanatory versus a predictive goal. The purpose of this article is to clarify the distinction between explanatory and predictive modeling, to discuss its sources, and to reveal the practical implications of the distinction to each step in the modeling process."

This is a nice article which I think complements Leo Brieman's paper discussed here before regarding two cultures of predictive modeling. Rob gives a nice synopsis of some of the main points from the paper:

1. The AIC is better suited to model selection for prediction as it is asymptotically equivalent to leave-​​one-​​out cross-​​validation in regression, or one-​​step-​​cross-​​validation in time series. On the other hand, it might be argued that the BIC is better suited to model selection for explanation, as it is consistent.
2. P-​​values are associated with explanation, not prediction. It makes little sense to use p-​​values to determine the variables in a model that is being used for prediction. (There are problems in using p-​​values for variable selection in any context, but that is a different issue.)
3. Multicollinearity has a very different impact if your goal is prediction from when your goal is estimation. When predicting, multicollinearity is not really a problem provided the values of your predictors lie within the hyper-​​region of the predictors used when estimating the model.
4. An ARIMA model has no explanatory use, but is great at short-​​term prediction.
5. How to handle missing values in regression is different in a predictive context compared to an explanatory context. For example, when building an explanatory model, we could just use all the data for which we have complete observations (assuming there is no systematic nature to the missingness). But when predicting, you need to be able to predict using whatever data you have. So you might have to build several models, with different numbers of predictors, to allow for different variables being missing.
6. Many statistics and econometrics textbooks fail to observe these distinctions. In fact, a lot of statisticians and econometricians are trained only in the explanation paradigm, with prediction an afterthought. That is unfortunate as most applied work these days requires predictive modelling, rather than explanatory modelling.

Rob also links Galit Shmueli's web page, (the author of the article above) who apparently has done some extensive research related to these distinctions.  Lots of additional resources (blog) here in this regard. Galit states:

"My thesis is that statistical modeling, from the early stages of study design and data collection to data usage and reporting, takes a different path and leads to different results, depending on whether the goal is predictive or explanatory."

I touched on these distinctions before, but did not realize the extent of the actual work being done in this area by Galit.

Analytics vs Causal Inference
Big Data: Don't throw the baby out with the bath water

## Sunday, March 8, 2015

### Are observational studies off the table or can you have your eggs and eat them too?

From the New York Times:

"But the primary problem is that nutrition policy has long relied on a very weak kind of science: epidemiological, or “observational,” studies in which researchers follow large groups of people over many years. But even the most rigorous epidemiological studies suffer from a fundamental limitation. At best they can show only association, not causation. Epidemiological data can be used to suggest hypotheses but not to prove them."

I remember being in a discussion once about the safety of GMO foods and someone made the remark that epidemiological studies were unreliable. I really was perplexed that someone could throw an entire field like epidemiology under the bus. They were most likely thinking about some of the claims made in an article I have referred to before here recently:

Deming, data and observational studies: A process out of control and needing fixing. 2011 Royal Statistical Society.  (link)

“Any claim coming from an observational study is most likely to be wrong.” Startling, but true. Coffee causes pancreatic cancer. Type A personality causes heart attacks. Trans-fat is a killer. Women who eat breakfast cereal give birth to more boys. All these claims come from observational studies; yet when the studies are carefully examined, the claimed links appear to be incorrect. What is going wrong? Some have suggested that the scientific method is failing, that nature itself is playing tricks on us. But it is our way of studying nature that is broken and that urgently needs mending, say S. Stanley Young and Alan Karr; and they propose a strategy to fix it."

This criticism of observational studies is interesting. I have discussed this before in relation to wellness program evaluation studies and generally concluded that those criticisms sort look like straw man arguments trying to hold observational studies against the golden standard of the randomized clinical trial. Obviously you would expect results from a RCT to be more reliable and replicable. Does this mean that we refuse to ask interesting questions or analyze important policy decisions just because a RCT is too expensive or impractical, when there are quasi-experimental approaches that could be used to advance our knowledge and understanding of important issues? Or in these cases are we just better off basing our understanding of the world on circumstantial and anecdotal evidence alone?

"What about the health impacts of meat consumption?  It is true that many observational, epidemiological studies show a correlation between red meat eating and adverse health outcomes (interestingly there is a fair amount of overlap on the authors of the dietary studies and the environmental studies on meat eating).  But, this is a pretty weak form of evidence, and much of this work reminds of the kinds of regression analyses done in the 1980s and 90s in economics before the so-called "credibility revolution."

So, perhaps some bad practices associated with p-hacking and lack of identification has given epidemiological research, and observational studies in general, a worse reputation than they deserve. And if Jayson is correct, a lot of these poor study designs lacking identification may have misled us about our diets and healthy food choices.

## Saturday, March 7, 2015

### Pinning P-values to the Wall

In recent headlines we hear about one journal that is 'banning' the use of p-values. In part this is justified on the basis that there can often be widespread misuse or mis-understanding of p-values in empirical applications:

"The p value is the probability to obtain an effect equal to or more extreme than the one observed presuming the null hypothesis of no effect is true."

"In other words, it is the probability of the data given the null hypothesis. However, it is often misunderstood to be the probability of the hypothesis given the data...."

Which leads to many people with a thought process leading them to think:

"if you reach the magical p-value then your hypothesis is true."

Similarly the journal recognizes the misunderstanding of confidence intervals (no surprise since a 95% confidence interval can be used to equivalently test any hypothesis at the 5% level of significance).

"In the NHSTP, the problem is in traversing the distance from the probability of the finding, given the null hypothesis, to the probability of the null hypothesis, given the finding. Regarding confidence intervals, the problem is that, for example, a 95% confidence interval does not indicate that the parameter of interest has a 95% probability of being within the interval. Rather, it means merely that if an infinite number of samples were taken and confidence intervals computed, 95% of the confidence intervals would capture the population parameter. Analogous to how the NHSTP fails to provide the probability of the null hypothesis, which is needed to provide a strong case for rejecting it, confidence intervals do not provide a strong case for concluding that the population parameter of interest is likely to be within the stated interval."

But despite these interpretation issues, it may be the case in many instances that the same results are reached regardless of a bayesian or frequentist approach. (barring those situations where bayesian methods may offer an advantage). The bigger issue perhaps is p-hacking.

In relation to p-hacking, I recently ran across an article (likely one very familiar to a lot of readers) that discusses issues related to p-hacking and research methodologies in general that are 'out of control' so to speak:

Deming, data and observational studies: A process out of control and needing fixing

In this article, the authors lay out several best practices to bring scholarly research back 'in control.'

0 Data are made publicly available
1 Data cleaning and analysis separate
2 Split sample: A, modelling; and B, holdout (testing)
3 Analysis plan is written, based on modelling data only
4 Written protocol, based on viewing predictor variables of A
5 Analysis of A only data set
6 Journal accepts paper based on A only

In a previous post, I discussed how the Quarterly Journal of Political Science was now requiring each article submission to have a research package, consisting of data and analysis code:

I really like some of these ideas and agree without the data and code, replication is often a joke, and it really is in so many ways trust me science. This is why it is hard to get excited about any headline relating to some finding from a  so called study without careful scrutiny.

Given the credibility revolution in applied micro-econometrics, the guidelines outlined in the Royal Statistical Society article and requiring model packages with code (like the QJPS), I would think there is ground for good work to be done, short of totally banning p-values.

### Bayesian Statistics, Confidence Intervals, and Regularization

"Bayesian statistics offers a framework to handle uncertainty that is based on a more intuitive mental model than the frequentist paradigm."

"Bayesian regression has close ties to regularization techniques while also giving us a principled approach to explicitly expressing prior beliefs. This helps us combat multicollinearity and overfitting."

This post has some very nice visuals and graphics illustrating Bayesian inference and what exactly a confidence interval does and does not tell us. And how, often, a Bayesian conclusion is what we have in mind:

"From the posterior distribution we can get the average and the credible interval (i.e. the uncertainty). On the surface, a credible interval looks just like the type of confidence interval we calculated above with an upper and lower limit. But here you can make probability statements such as P(trueheightininterval)=95%, which is often what we are seeking. We want to know the probability that the parameter lives in the interval and not the proportion of times we'd expect the true parameter to fall within the interval bands if we ran the experiment a large number of times. This is a more intuitive way to think about uncertainty."

Also, how Bayesian estimators are similar to shrinkage estimators:

"In fact, Bayesian regression has close ties to regularization techniques. Ridge regression can be thought of as a Bayesian regression where slope priors are normally distributed with means equal to 0. Lasso regression can be interpreted as a Bayesian regression with Laplace priors on the slopes."