In our last four posts in this series, we showed you how to calculate power for a t test using Monte Carlo simulations, how to integrate your simulations into Stata’s power command, and how to do this for linear and logistic regression models and multilevel models. In today’s post, I’m going to show you how to estimate power for structural equation models (SEM) using simulations.

Our goal is to write a program that will calculate power for a given SEM at different sample sizes. We’ll follow the same general procedure as the previous two posts, but the way we’ll go about simulating data is a bit different. Rather than individually simulating each variable for our specified model, we’ll be simulating all our variables simultaneously from a given covariance matrix. Means for each of the variables can also be used to simulate the data if your SEM has a mean structure, such as in group analysis or growth curve analysis. Read more…

In my last three posts, I showed you how to calculate power for a t test using Monte Carlo simulations, how to integrate your simulations into Stata’s power command, and how to do this for linear and logistic regression models. In today’s post, I’m going to show you how to estimate power for multilevel/longitudinal models using simulations. You may want to review my earlier post titled “How to simulate multilevel/longitudinal data” before you read this post. Read more…

In my last two posts, I showed you how to calculate power for a t test using Monte Carlo simulations and how to integrate your simulations into Stata’s power command. In today’s post, I’m going to show you how to do these tasks for linear and logistic regression models. The strategy and overall structure of the programs for linear and logistic regression are similar to the t test examples. The parts that will change are the simulation of the data and the models used to test the null hypothesis. Read more…

In my last post, I showed you how to calculate power for a t test using Monte Carlo simulations. In this post, I will show you how to integrate your simulations into Stata’s power command so that you can easily create custom tables and graphs for a range of parameter values. Read more…

Power and sample-size calculations are an important part of planning a scientific study. You can use Stata’s power commands to calculate power and sample-size requirements for dozens of commonly used statistical tests. But there are no simple formulas for more complex models such as multilevel/longitudinal models and structural equation models (SEMs). Monte Carlo simulations are one way to calculate power and sample-size requirements for complex models, and Stata provides all the tools you need to do this. You can even integrate your simulations into Stata’s power commands so that you can easily create custom tables and graphs for a range of parameter values. Read more…

In this blog post, I’d like to give you a relatively nontechnical introduction to Markov chain Monte Carlo, often shortened to “MCMC”. MCMC is frequently used for fitting Bayesian statistical models. There are different variations of MCMC, and I’m going to focus on the Metropolis–Hastings (M–H) algorithm. In the interest of brevity, I’m going to omit some details, and I strongly encourage you to read the [BAYES] manual before using MCMC in practice.

Let’s continue with the coin toss example from my previous post Introduction to Bayesian statistics, part 1: The basic concepts. We are interested in the posterior distribution of the parameter \(\theta\), which is the probability that a coin toss results in “heads”. Our prior distribution is a flat, uninformative beta distribution with parameters 1 and 1. And we will use a binomial likelihood function to quantify the data from our experiment, which resulted in 4 heads out of 10 tosses. Read more…

Overview

In the first part of this post, I discussed the multinomial probit model from a random utility model perspective. In this part, we will have a closer look at how to interpret our estimation results.

How do we interpret our estimation results?

We created a fictitious dataset of individuals who were presented a set of three health insurance plans (Sickmaster, Allgood, and Cowboy Health). We pretended to have a random sample of 20- to 60-year-old persons who were asked Read more…

\(\newcommand{\xb}{{\bf x}}
\newcommand{\betab}{\boldsymbol{\beta}}
\newcommand{\zb}{{\bf z}}
\newcommand{\gammab}{\boldsymbol{\gamma}}\)We have no choice but to choose

We make choices every day, and often these choices are made among a finite number of potential alternatives. For example, do we take the car or ride a bike to get to work? Will we have dinner at home or eat out, and if we eat out, where do we go? Scientists, marketing analysts, or political consultants, to name a few, wish to find out why people choose what they choose.

In this post, Read more…

Overview

In the frequentist approach to statistics, estimators are random variables because they are functions of random data. The finite-sample distributions of most of the estimators used in applied work are not known, because the estimators are complicated nonlinear functions of random data. These estimators have large-sample convergence properties that we use to approximate their behavior in finite samples.

Two key convergence properties are consistency and asymptotic normality. A consistent estimator gets arbitrarily close in probability to the true value. The distribution of an asymptotically normal estimator gets arbitrarily close to a normal distribution as the sample size increases. We use a recentered and rescaled version of this normal distribution to approximate the finite-sample distribution of our estimators.

I illustrate the meaning of consistency and asymptotic normality by Monte Carlo simulation (MCS). I use some of the Stata mechanics I discussed in Monte Carlo simulations using Stata.

Consistent estimator

A consistent estimator gets arbitrarily close in Read more…

\(\newcommand{\epsb}{{\boldsymbol{\epsilon}}}
\newcommand{\mub}{{\boldsymbol{\mu}}}
\newcommand{\thetab}{{\boldsymbol{\theta}}}
\newcommand{\Thetab}{{\boldsymbol{\Theta}}}
\newcommand{\etab}{{\boldsymbol{\eta}}}
\newcommand{\Sigmab}{{\boldsymbol{\Sigma}}}
\newcommand{\Phib}{{\boldsymbol{\Phi}}}
\newcommand{\Phat}{\hat{{\bf P}}}\)Vector autoregression (VAR) is a useful tool for analyzing the dynamics of multiple time series. VAR expresses a vector of observed variables as a function of its own lags.

Simulation

Let’s begin by simulating a bivariate VAR(2) process using the following specification,

\[
\begin{bmatrix} y_{1,t}\\ y_{2,t}
\end{bmatrix}
= \mub + {\bf A}_1 \begin{bmatrix} y_{1,t-1}\\ y_{2,t-1}
\end{bmatrix} + {\bf A}_2 \begin{bmatrix} y_{1,t-2}\\ y_{2,t-2}
\end{bmatrix} + \epsb_t
\]

where \(y_{1,t}\) and \(y_{2,t}\) are the observed series at time \(t\), \(\mub\) is a \(2 \times 1\) vector of intercepts, \({\bf A}_1\) and \({\bf A}_2\) are \(2\times 2\) parameter matrices, and \(\epsb_t\) is a \(2\times 1\) vector of innovations that is uncorrelated over time. I assume a \(N({\bf 0},\Sigmab)\) distribution for the innovations \(\epsb_t\), where \(\Sigmab\) is a \(2\times 2\) covariance matrix.

I set my sample size to 1,100 and Read more…