Some cool website on study design and biostatistics

This month (November 2024), I wanted to take a break from writing tutorial and articles. Instead, I wanted update myself on (and share) some very helpful/useful online resources.

A colleague of mine introduced me to website called Datamethods. It’s mainly a discussion forum, but it has some useful resources. This particular post contains references that are very useful for anyone who is interested in study design and biostatistics (link). It is a collection of papers and articles that addresses common myths and practices regarding the application of biostatistics in study designs.

Another great website is Scott Cunningham’s Mixed Taped Sessions. He has a book called Mixed Taped Session about causal inference, and he has regular workshops. I attended his Causal Inference Part 2 workshop, and it was amazing. We learned about the basic difference-in-differences methods (coding in R and Stata), and the innovations surrounding these methods (e.g., Callaway & Sant’Anna’s staggered difference-in-differences approach). Scott also provides the historical perspectives on these methods, which are insightful as they are entertaining. Moreover, he conducts interviews with prominant econometricians, which he posts on his YouTube channel.

Hopefully, these sites are useful for you as they have been for me.

Staggered difference-in-differences using R

I was interested in learning how to apply the Callaway & Sant'Anna staggered difference-in-differences framework to my work. After reading several papers and watching the video by Sant'Anna, I wrote a short tutorial on how to apply this framework to a simulated data. The tutorial is located on my RPubs site.

This is a unique method that used the R “did” package, which is based on the paper by Callaway & Sant’Anna.

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Mediation analysis using R

It’s not uncommon to see covariates in a regression model that should not be there. For example, measurements that occur after the treatment assignment are included into a regression model as baseline covariates. Rather, one should consider a mediation analysis.

I wrote a tutorial on how to perform mediation analysis using R on my RPubs site (link).

I know that I make this mistake at times. This tutorial helped me to carefully consider which covariates to include in a regression model and which ones to consider for mediation analysis.

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MEPS tutorial on interrupted time series analysis in R

I wrote a short tutorial on how to perform an interrupted time series analysis in R. I had a challenging time working on this because I wasn’t familiar with all the nuances of the ITSA. More importantly, I wasn’t able to leverage my Stata skills to do this in R. I’m used to the Stata margins command, which is great for creating constrasts. R has its own version of the margins command, but it lacks some of Stata’s features such as the pwcompare, which I use a lot in Stata. However, I found a workaround with linear splines, and I have uploaded this to my RPubs site (link). I hope you find this useful. I also saved my R Markdown code on my GitHub site (link).

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MEPS tutorials on linkage files and trend analysis

I create two MEPS tutorials recently. One is on the use of condition-event linkage files to capture the disease-specific costs. I used migraine as a motivating example. In this tutorial, I go through the steps to identify migraine-related costs assocaited with office-based visits and inpatient night stays. In the second tutorial, I review how to perform simple trend analysis with linear regressio models. I pooled MEPS data from 2016 to 2021 and apply the approriate primary sampling units and strata from the pooled file.

The first tutorial is located on my RPubs page (MEPS Tutorial 4 - Using condition-event link (CLNK) file: A case study with migraine). The R Markdown code to create the tutorial is located in my GitHub repository (link).

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The second tutorial is also located on my Rpubs page (MEPS Tutorial 5 - Simple Trend Analysis with Linear Models). The R Markdown code to create the tutorial is located in my GitHub repository (link).

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Interrupted time series analysis (ITSA) with Stata

Interrupted time series analysis (ITSA) is a study design used to study the effects of an intervention across time. An important feature of the ITSA is the time when the intevention occurs. The time before and after the intervention are of interest because we want to visualize if the trends are similar or different. Additionally, we want to visualize the change immediately after the intervention is implemenated. I call this period the index date.

In this article, I’ll review the single-group ITSA and multiple groups ITSA. Then I’ll review how to perform an ITSA in Stata.

You can view the complete tutorial on my RPubs site.

Tweedie GLM model in R for Cost Data

I wrote a tutorial on using a Tweedie distribution for a GLM gamma model for cost data in R. Unlike Stata, R is very particular with zeroes when constructing GLM models. Hence, I opted to use the Tweedie distribution to mix and match the link function with the Gamma distribution. I settled on the identity link because it doesn’t involve retransformation and is each to interpret.

My tutorial is available on my RPubs site and GitHub site.

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Two-part models in R - Application with cost data

I created a tutorial on how to use two-part models in R for cost data. I used the healthcare expenditures from the Medical Expenditure Panel Survey in 2017 as a motivating example. Normally, I use Stata when I construct two-part models. But I wanted to learn how I could do this in R. Fortunately, R has a package called twopartm that was developed by Duan and colleagues. You can find their document for the twopartm package here.

The tutorial I created is located on my GitHub page and RPubs page.

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Interpreting regression models

I wrote a short explanation on how to interpret regression models.

I have posted this on RPubs, and the code is saved on my GitHub site.

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Cobb-Douglas production function and costs minimization problem

Update 2: This article was updated on 12 August 2023 when Dimanjan Dahal (Twitter account) identified a better way to present the Lagrangian functions. I updated this to better reflect the minimization problem and set the partial derivative solution to 0. Thank you, Dimanhan.

Update 1: This article was updated on 11 October 2021 when an anonymous reader identified an error with the example used at the end. The error was the negative value generated for the output elasticity of capital. In the previous example, I used R to generate a set of random numbers that were used in a regression model. The beta coefficient generated a negative value which was used in the linear form of the Cobb-Douglass equation. Since the output of elasticity should be between the values of 0 and 1, this negative coefficient should not be possible. Hence, I’ve updated the data frame used in the example to avoid this issue. Appreciation goes out to the anonymous reader who identified this error.

INTRODUCTION

The Cobb-Douglas (CD) production function is an economic production function with two or more variables (inputs) that describes the output of a firm. Typical inputs include labor (L) and capital (K). It is similarly used to describe utility maximization through the following function [U(x)]. However, in this example, we will learn how to answer a minimization problem subject to (s.t.) the CD production function as a constraint.

The functional form of the CD production function:

where the output Y is a function of labor (L) and capital (K), A is the total factor productivity and is otherwise a constant, L denotes labor, K denotes capital, alpha represents the output elasticity of labor, beta represents the output elasticity of capital, and (alpha + beta = 1) represents the constant returns to scale (CRS). The partial derivative of the CD function with respect to (w.r.t) labor (L) is:

Recall that quantity produced is based on the labor and capital; therefore, we can solve for alpha:

This will yield the marginal product of labor (L). If alpha = 2, then a 10% increase in labor (L) will result in a 20% increase in output (Y).

The partial derivative of the CD function with respect to (w.r.t) labor (K) is:

This will yield the marginal product of capital (K).

The CD production function can be converted to a linear model by taking the logarithm of both sides of the equation:

This will allow for OLS regression methods, which is commonly used in economics to understand the association between inputs (L and K) on production (Y).

However, what happens when we are interested in the marginal cost with respect to (w.r.t.) production (Y)? This becomes a constraint (cost) minimization problem where the firm can control how much L and K they will use. In other words, we want to minimize the cost subject to (s.t.) the output

Cost becomes a function of wage (w), the amount of labor (L), price of capital (r), and the amount of capital (K). To determine the optimal amount of inputs (L and K), we solve this minimization constraint using the Lagrange multiplier method:

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Solve for L

Substitute L in the constraint term (CD production function) in order to solve for K

Now, we can completely solve for L (as a function of Y, A, w, and r) by substituting for K

Substitute L and K into the cost minimization problem

Simplify

Final cost function

Let’s see how we can use the results from a regression model to give us information about the total costs w.r.t. to the quantity produced.

Recall the linear form of the Cobb-Douglas production function:

I simulated some data where we have the capital, labor, and quantity produced in R.

## Use the following libraries:
library(jtools)
library(broom)
library(ggstance)
library(broom.mixed)

## Generate random data for the data frame (cddata)
set.seed(1234)

production <- sample(100:600, 30, replace=TRUE)

labor <- sample(50:350, 30, replace=TRUE)

capital <- sample(6000:7000, 30, replace=TRUE)

## Cost function parameters: wage and price constants
wage <- 35.00
price <- 30.00

## Set up the data frame (cddata):
cddata <- data.frame(production = production, labor = labor, capital = capital, wage = wage, price = price)

## Name rows using some timeline from 1988 to 2017 (30 years for 30 observations for each variable):
row.names(cddata) <- 1988:2017

Then I perform a regression model using OLS

## Setting up the model, where log(a) is eliminated due to it being the intercept.
cd.lm <- lm(formula = log(production) ~ log(labor) + log(capital), data = cddata)

summary(cd.lm)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.96586 -0.25176  0.06148  0.37513  0.67433 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)
(Intercept)   4.44637   17.41733   0.255    0.800
log(labor)    0.14373    0.23595   0.609    0.548
log(capital)  0.05581    2.00672   0.028    0.978

Residual standard error: 0.5065 on 27 degrees of freedom
Multiple R-squared:  0.01414,   Adjusted R-squared:  -0.05888 
F-statistic: 0.1937 on 2 and 27 DF,  p-value: 0.8251

After running the model, I stored the coefficients for use later in the production function.

## Store the coefficients
coeff <- coef(cd.lm)

## Assign the values to the production function parameters where Y = AL^(alpha)K^(beta)
intercept <- coeff[1]
alpha <- coeff[2]
beta <- coeff[3]

From the parameters, we can get A (intercept), alpha (log(labor)), and beta (log(capital)).

This will give us the quantity produced (Y) for given data on labor (L) and capital (K).

We can get the total costs (C) based on the quantity produced (Y) using the cost function:

I set up my R code so that I have the intercept, alpha, beta, labor, wage, and price of the capital set up. I estimated each part of the cost function separately and then multiply the parts at the end.

## Cost
PartA <- (production / intercept)^(1 / alpha + beta)
PartB <- wage^(alpha / alpha + beta)
PartC <- price^(beta / alpha + beta)
PartD <- as.complex(alpha / beta )^(beta / alpha + beta) + as.complex(beta/ alpha)^(alpha / alpha + beta)

costs <- PartA * PartB * PartC * PartD

Note: R has a problem with performing complex operations with exponents that were defined using arrays or vectors. If you try to compute something like x^{alpha}, you will get an error where the value is “NaN.” I don’t have a complete understanding of the problem, but the solution is to make sure your root or base term is preceded by “as.complex(x)” to resolve the issue.

I plot the relationship between quantity produced and cost. In other words, this tells us the lowest costs needed to produce the quantities on the plot.

plot(production, costs)

CONCLUSIONS

Using the Cobb-Douglas production function and the cost minimization approach, we were able to find the optimal conditions for the cost function and plot the outcome relative to the quantity produced. As production increases, the minimum cost needed increases in a non-linear, exponential fashion, which makes sense given that Y (quantity produced) is in the numerator on the right-hand side of the cost function and positively related to the cost.

This was a fun exercise that made me think about the usefulness of the Cobb-Douglas production function, which I learned to optimize multiple times in my Economics courses. I was excited to find a pleasant utility for it using simulated data and will probably explore more exercises like this in the future.

REFERENCEs

I used a lot of resources to write this blog, which are provided below.

A site dedicated to the discussion of economics called EconomicsDiscussion.net was a great resource.

These papers were incredibly helpful in preparing the example in R:

• Lin CP. The application of Cobb-Douglas production cost functions to construction firms in Japan and Taiwan. Review of Pacific Basin Financial Markets and Policies Vol. 5, No. 1 (2002): 111–128.

• Larriviere JB, Sandler R. A student friendly illustration and project: empirical testing of the Cobb-Douglas production function using major league baseball. Journal of Economics and Economic Education Research, Volume 13, Number 3, 2012: 81-92

• Hu, ZH. Reliable Optimal Production Control with Cobb-Douglas Model. Reliable Computing. 1998; 4(1): 63-69.

I encountered some issues regarding complex numbers in R. Fortunately, I found some great resources about it.

• I found a great discussion about R’s calculation of exponents and “NaN” results and why complex numbers can mess up your math in R.

• Another good site (R Tutorial: An Introduction to Statistics) explaining complex numbers in R.

• John Myles White wrote a nice article about complex numbers in R.

Acknowledgements: I would like to thank the user who reached out to me about the coefficient errors for the output elasticity of capital. This helps me to learn my mistakes and correct them. Without the support and guidance from the community, I would not achieve my own goals of being a lifelong learner. Thank you.