GlmmTMB Vs Emmeans: Why P-values Differ?

Have you ever found yourself scratching your head over seemingly conflicting results from statistical analyses? It's a common experience, especially when dealing with complex models and multiple comparisons. Today, we're diving into a fascinating case: the difference in p-values reported by glmmTMB and emmeans. Let's break it down, guys, and figure out what's going on!

The Case of the Conflicting P-values

Imagine you're a researcher studying environmental DNA (eDNA) detections of rainbow trout. You've collected data using three different sampling methods: PES-SP, PES-EtOH, and CN-EtOH. Your main goal is to figure out if there are any significant differences between these methods in terms of how well they detect trout DNA. This is where statistical modeling comes in, and you've chosen to use a generalized linear mixed model (GLMM) because it's perfect for handling this kind of data – accounting for both fixed effects (like the sampling method) and random effects (like variation between sampling locations).

You run your GLMM using the glmmTMB package in R, a powerful tool for fitting complex models. Everything seems to go smoothly, and you get your model output. But wait! When you use the emmeans package to perform pairwise comparisons between the sampling methods, the p-values you get are different from what you expected, and different from what you might get directly from the glmmTMB output. This discrepancy can be super confusing and might even make you question your results. But don't worry, it's not necessarily an error; it's often a matter of understanding how these tools calculate p-values and what assumptions they make.

So, what's the deal? Why do glmmTMB and emmeans sometimes give us different p-values? The answer lies in the nuances of statistical inference and how different methods handle uncertainty. glmmTMB, at its core, is focused on fitting the model and estimating the parameters. It provides p-values based on the model's overall structure and the estimated standard errors of the coefficients. On the other hand, emmeans (Estimated Marginal Means) is designed specifically for making comparisons between groups or treatments. It takes the model results and calculates estimated means for each group, along with confidence intervals and p-values for pairwise comparisons. The key difference is often in how the degrees of freedom are calculated and how multiple comparisons are adjusted for. This is super important because when you're comparing multiple groups, you need to adjust your p-values to avoid falsely claiming significant differences (a concept known as the multiple comparisons problem). Different methods, like Tukey's HSD or Bonferroni correction, handle this adjustment in different ways, leading to variations in p-values. This is why understanding the specific methods used by each package is crucial for interpreting your results correctly. We'll dive deeper into the reasons for these differences and how to navigate them.

Understanding the Tools: glmmTMB and emmeans

To really get to the bottom of these p-value differences, we need to understand what each of these tools, glmmTMB and emmeans, is doing under the hood. Let's start with glmmTMB. This R package is a powerhouse for fitting generalized linear mixed models. GLMMs are incredibly versatile because they can handle different types of response variables (like counts, proportions, or continuous data) and account for both fixed and random effects. Fixed effects are the things you're specifically interested in comparing, like our sampling methods in the trout eDNA example. Random effects, on the other hand, represent sources of variability that you're not directly interested in but need to account for, such as differences between sampling locations or individual observers. glmmTMB is known for its speed and flexibility, making it a go-to choice for many researchers working with complex data.

When you fit a GLMM using glmmTMB, the package estimates the model parameters, including the coefficients for the fixed effects. It also calculates standard errors for these coefficients, which are a measure of the uncertainty in the estimates. From these standard errors, glmmTMB can calculate p-values for each fixed effect. These p-values tell you the probability of observing the data you did if there were truly no effect of that predictor. A small p-value (typically less than 0.05) suggests that the effect is statistically significant. However, it's important to remember that these p-values are based on the overall model and the specific contrasts being tested. They might not always be the most appropriate for making pairwise comparisons between groups, especially when you have multiple comparisons to consider. Now, let's shift our focus to emmeans. This package is specifically designed for estimating marginal means and making comparisons between them. Marginal means, also known as least-squares means or adjusted means, represent the predicted mean response for each group or treatment, taking into account the other predictors in the model. emmeans is incredibly useful because it allows you to compare groups while controlling for the effects of other variables. For example, in our trout eDNA study, emmeans can estimate the mean eDNA detection rate for each sampling method, while accounting for any differences in detection rates that might be due to other factors, such as water temperature or habitat type. The real strength of emmeans lies in its ability to perform pairwise comparisons and adjust p-values for multiple comparisons. When you compare multiple groups, the chance of finding a statistically significant difference just by chance increases. Multiple comparisons adjustments, like Tukey's HSD or Bonferroni, help to control this risk by making it harder to declare a difference significant. This is a crucial step in any analysis involving multiple groups, and it's where emmeans really shines. By understanding the core functions of glmmTMB (model fitting and parameter estimation) and emmeans (marginal means and multiple comparisons), we can start to see why their p-values might differ. The next section will explore the specific reasons behind these discrepancies and how to interpret them.

Key Reasons for P-value Differences

Alright, guys, let's get down to the nitty-gritty and explore the key reasons why glmmTMB and emmeans might report different p-values. This is where things get interesting, and understanding these nuances is crucial for accurate interpretation of your results. The first major reason lies in degrees of freedom. Degrees of freedom (df) are a fundamental concept in statistics, representing the amount of independent information available to estimate a parameter. In simpler terms, it's related to the sample size and the number of parameters you're estimating in your model. Different methods for calculating degrees of freedom exist, and glmmTMB and emmeans might use different approaches, especially when dealing with mixed models. This can have a direct impact on the p-values. For instance, glmmTMB often uses Satterthwaite or Kenward-Roger approximations for degrees of freedom, which are designed to provide more accurate estimates in complex models. On the other hand, emmeans might use a different method or rely on the degrees of freedom reported by the underlying model object. These differences in df calculation can lead to variations in the t- or F-statistics used to calculate p-values, ultimately resulting in different significance levels. Think of it like this: imagine you're trying to measure the weight of an object using a scale. If you have more independent measurements, you'll have a more precise estimate of the weight. Similarly, higher degrees of freedom generally lead to more precise estimates and potentially smaller p-values. The second crucial factor is multiple comparisons adjustment. As we touched on earlier, when you're comparing multiple groups or treatments, the risk of making a Type I error (falsely declaring a significant difference) increases. To control this, we need to adjust the p-values. emmeans is specifically designed to handle multiple comparisons and offers a variety of adjustment methods, such as Tukey's HSD, Bonferroni, Sidak, and more. Each method has its own way of controlling the error rate, and they can lead to different adjusted p-values. glmmTMB, on the other hand, typically provides p-values based on single comparisons, without any adjustment for multiple testing. This means that the p-values you get directly from the glmmTMB output might be inflated if you're making multiple comparisons. To illustrate, let's say you're comparing five different treatments. If you perform 10 pairwise comparisons without any adjustment, you have a higher chance of finding a significant difference just by chance compared to if you used a method like Tukey's HSD, which controls the family-wise error rate. This is why it's so important to use emmeans or another multiple comparisons method when you're making multiple comparisons. Finally, the types of contrasts being tested can also contribute to p-value differences. glmmTMB typically provides p-values for the overall effect of a predictor or for specific contrasts defined in the model formula. emmeans, on the other hand, allows you to define specific comparisons of interest, such as pairwise comparisons, comparisons to a control group, or more complex contrasts. The way these contrasts are defined and tested can influence the resulting p-values. For example, if you're interested in comparing each treatment to a control group, the p-values you get from emmeans might be different from the p-values you get from glmmTMB if you're simply looking at the overall effect of the treatment variable. To make sure you're getting the most accurate and meaningful results, it's important to carefully consider the contrasts you're testing and how they align with your research questions.

Navigating the Discrepancies: Best Practices

Okay, so we've uncovered the main reasons behind the p-value discrepancies between glmmTMB and emmeans. Now, let's talk about how to navigate these differences and ensure you're drawing the right conclusions from your analysis. Here are some best practices to keep in mind, guys. First and foremost, clearly define your research question. This is the foundation of any good analysis. What exactly are you trying to find out? Are you interested in the overall effect of a predictor, or do you need to make specific comparisons between groups? Knowing your research question will guide your choice of statistical methods and the interpretation of your results. If your primary goal is to compare specific groups or treatments, emmeans is generally the preferred tool. It's designed specifically for this purpose and provides the necessary adjustments for multiple comparisons. Remember, ignoring multiple comparisons can lead to inflated Type I error rates, meaning you might falsely conclude there's a significant difference when there isn't one. On the other hand, if you're interested in the overall effect of a predictor and not necessarily focused on pairwise comparisons, the p-values from glmmTMB might be sufficient. However, even in this case, it's always a good idea to consider the potential for multiple comparisons, especially if you're exploring your data and looking for interesting patterns. Always consider multiple comparisons adjustment when performing post-hoc tests. This is a golden rule in statistical analysis. Whether you're using emmeans or another method, make sure you're controlling the error rate appropriately. Choose an adjustment method that aligns with your research question and the number of comparisons you're making. Common methods include Tukey's HSD, Bonferroni, Sidak, and Benjamini-Hochberg (FDR). Each method has its pros and cons, so it's worth understanding their properties. Report both unadjusted and adjusted p-values. This provides transparency and allows readers to assess the impact of the multiple comparisons adjustment. In your results section, you can report the unadjusted p-values from glmmTMB alongside the adjusted p-values from emmeans. This gives a more complete picture of your findings. Understand the degrees of freedom. As we discussed earlier, differences in degrees of freedom can contribute to p-value discrepancies. Be aware of how glmmTMB and emmeans calculate degrees of freedom and how this might affect your results. If you're unsure, consult the documentation for each package or seek advice from a statistician. Consider effect sizes and confidence intervals. P-values are just one piece of the puzzle. It's equally important to consider effect sizes, which quantify the magnitude of the effect, and confidence intervals, which provide a range of plausible values for the effect. A small p-value doesn't necessarily mean the effect is practically significant, and a large p-value doesn't necessarily mean there's no effect at all. Effect sizes and confidence intervals provide a more nuanced understanding of your results. Consult with a statistician. If you're feeling overwhelmed or unsure about your analysis, don't hesitate to seek help from a statistician. They can provide valuable guidance on model selection, interpretation of results, and best practices for statistical analysis. Statistics can be tricky, and it's always better to get expert advice than to make mistakes that could compromise your findings. By following these best practices, you can navigate the p-value discrepancies between glmmTMB and emmeans and draw accurate, meaningful conclusions from your data. Remember, statistical analysis is a tool to help you answer your research questions, and understanding the nuances of these tools is key to using them effectively.

A Practical Example: Trout eDNA Revisited

Let's bring it all together with a practical example, revisiting our trout eDNA study. Imagine you've fitted a GLMM using glmmTMB to model the eDNA detection rate as a function of the sampling method (PES-SP, PES-EtOH, and CN-EtOH), and you've included a random effect for sampling location. Now, you want to compare the effectiveness of these three methods. Here's how you might approach this using both glmmTMB and emmeans, and how to interpret the results. First, you'd run your GLMM using glmmTMB, something like this (in R code):

model <- glmmTMB(eDNA_detection ~ sampling_method + (1|sampling_location), data = trout_data, family = binomial)
summary(model)

This will give you the overall model summary, including p-values for the effect of sampling_method. However, these p-values tell you whether there's a significant difference between the methods in general, but not which specific methods differ from each other. This is where emmeans comes in. You can use emmeans to perform pairwise comparisons between the sampling methods, adjusting for multiple comparisons. Here's how:

library(emmeans)
emmeans_model <- emmeans(model, ~ sampling_method)
pairwise_comparisons <- pairs(emmeans_model, adjust = "tukey")
summary(pairwise_comparisons)

In this code, we first create an emmeans object, specifying that we want to estimate marginal means for each sampling_method. Then, we use the pairs() function to perform pairwise comparisons, and we specify `adjust =