Welcome to this edition of Techal! In this article, we will demystify saturated models and deviance statistics. Whether you stumbled upon the term “saturated model” in the context of McFadden’s pseudo R-squared or residual deviance, or you’re just here for some insightful knowledge, we’ve got you covered. So, let’s dive in!
Contents
Understanding Saturated Models
Imagine we’re weighing mice for a study. Let’s assume we already know the standard deviation, so all we need to estimate is the mean. We fit a normal curve to the data, and voila! We have a model. This simplest model, with just one parameter, is called the null model. We can then compare it to more complex models, such as the “fancier” model with two parameters or even the “super fancy” model with one parameter per data point, known as the saturated model.
The saturated model is called so because it maximizes the number of parameters we can estimate. It represents the best-case scenario, with a likelihood equal to 1291.5. By comparing the likelihood of different models, including the null and saturated models, we can determine how well our proposed model fits the data.
Deviance Statistics: Understanding Fit
Deviance statistics, specifically residual deviance and null deviance, play a key role in evaluating model fit. Residual deviance measures the difference between the log-likelihood of the saturated model and the proposed model. Similarly, null deviance measures the difference between the log-likelihood of the saturated model and the null model.
By calculating the chi-square value of the difference in deviances, we can determine the p-value, which indicates the significance of the difference. This allows us to assess whether the proposed model significantly differs from the saturated or null models.
Ignoring Saturated Models in Logistic Regression
In logistic regression, we’re fitting a curve to our data and calculating the log-likelihood of the data given that curve. However, with the saturated model, the curve fits the data perfectly, making the log-likelihood zero. Therefore, when performing logistic regression, we can safely ignore the saturated model.
Conclusion
In this article, we explored the concept of saturated models and deviance statistics. Saturated models provide an upper bound on log-likelihood and help calculate likelihood-based r-squared. Residual and null deviance allow us to determine the significance of the proposed model. In logistic regression, we can ignore the saturated model altogether. Armed with this knowledge, you’re now equipped to analyze and evaluate models with confidence.
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FAQs
Q: What is a saturated model?
A: A saturated model is a statistical model that maximizes the number of parameters that can be estimated. It represents the best-case scenario and is used to evaluate the fit of other models.
Q: How do deviance statistics help in model evaluation?
A: Deviance statistics, such as residual deviance and null deviance, measure the difference between the log-likelihoods of different models. By calculating the p-value of this difference, we can determine the significance of the proposed model’s fit.
Q: Why can we ignore the saturated model in logistic regression?
A: In logistic regression, the saturated model fits the data perfectly, resulting in a log-likelihood of zero. Therefore, it provides no additional information for evaluating the fit of the proposed model.
Q: How do likelihood-based r-squared and deviance statistics relate?
A: Likelihood-based r-squared measures how well a proposed model fits the data, while deviance statistics quantify the difference in fit between models. By comparing the deviances, we can calculate the p-value for the likelihood-based r-squared.
Q: Where can I find more technology-related articles and guides?
A: For more insightful content on technology and engineering, visit Techal.
Note: This article is based on the original content by Josh Starmer from StatQuest.
