Z-Score Standardization for Polynomial Terms in GLMMs

Z-score standardization of predictors before using poly() in a GLMM requires careful consideration. While standardization is generally recommended for comparable estimates, polynomial terms need special handling because the poly() function by default returns orthogonal polynomials, which affects how standardization should be applied. The key is to understand whether you should standardize before or after applying the polynomial transformation, and whether to use orthogonal or raw polynomials based on your modeling goals.

Contents

• Understanding Z-Score Standardization in Mixed Models
• The Challenge with Polynomial Terms in GLMM
• Orthogonal Polynomials vs. Raw Polynomials in R
• Proper Standardization Approach for Polynomial Terms
• Implementation in R: Code Examples
• Best Practices for Comparable Estimates in GLMMs
• Sources
• Conclusion

Understanding Z-Score Standardization in Mixed Models

Z-score standardization is a fundamental preprocessing step in statistical modeling that transforms predictors to have a mean of 0 and standard deviation of 1. This transformation is particularly important in mixed models like GLMMs because it allows for more comparable effect sizes across predictors with different scales and units. When building generalized linear mixed models, standardization helps with model convergence, interpretation of coefficients, and comparison of predictor importance.

In the context of GLMMs, z-score standardization becomes even more critical because mixed models often combine fixed and random effects, where fixed effects need to be on comparable scales for proper interpretation. The standardization process typically involves subtracting the mean and dividing by the standard deviation: $z = \frac{x - \mu}{\sigma}$, where $\mu$ is the mean and $\sigma$ is the standard deviation of the predictor.

However, the situation becomes more complex when you introduce polynomial terms using the poly() function in R. The poly() function by default returns orthogonal polynomials rather than raw polynomials, which affects how standardization should be approached. This orthogonality means that the polynomial terms are uncorrelated with each other, but it also changes their scale and relationship to the original standardized variables.

When you apply poly() to already standardized variables, you’re essentially creating polynomial transformations of standardized predictors. The resulting polynomial terms will have different scales than the original standardized predictors, which explains why their estimates might appear significantly larger. This scaling difference occurs because polynomial transformations amplify the scale of higher-order terms, especially when applied to standardized inputs.

The key insight is that standardization before polynomial transformation doesn’t guarantee comparable estimates across polynomial terms of different degrees. Each polynomial term (linear, quadratic, cubic, etc.) will naturally have different variances and scales, even when derived from the same standardized predictor. This fundamental property of polynomial transformations means that additional considerations are needed to ensure truly comparable estimates in your GLMM.

The Challenge with Polynomial Terms in GLMM

When building GLMMs with polynomial terms, researchers often encounter the issue of having significantly larger estimates for polynomial terms compared to other predictors. This scaling discrepancy isn’t just a minor inconvenience—it can lead to misinterpretation of effect sizes, difficulties in model convergence, and challenges in comparing the relative importance of different predictors in your model.

The core of this challenge lies in how polynomial transformations work mathematically. When you apply a polynomial transformation to a predictor, higher-order terms (quadratic, cubic, etc.) naturally have larger values than the original linear term, especially when the predictor has been standardized. This mathematical property occurs because polynomial terms involve raising the predictor to higher powers, which amplifies both positive and negative values.

For example, consider a standardized predictor with values ranging from -3 to 3. The linear term would have the same scale, but the quadratic term (x²) would range from 0 to 9, and the cubic term (x³) would range from -27 to 27. This scaling difference means that even if the “true” effect sizes are comparable across terms, the estimated coefficients will reflect these different scales, making direct comparison problematic.

In GLMMs, this scaling issue becomes even more pronounced because these models often include multiple predictors with different scales and types. When you have a mix of standardized linear predictors and polynomial terms, the model struggles to provide meaningful comparisons between coefficients because they’re operating on fundamentally different scales. This can lead to several practical problems:

1. Interpretation difficulties: Larger coefficient values for polynomial terms might be misinterpreted as stronger effects, when in reality they’re just mathematically larger due to the polynomial transformation.

2. Model convergence issues: Extreme differences in coefficient scales can cause numerical instability during model fitting, especially in complex GLMMs with multiple random effects.

3. Variable importance comparisons: When you need to compare the relative importance of different predictors in your model, the scaling differences make these comparisons meaningless.

4. Communication challenges: Presenting results to non-statistical audiences becomes more difficult when explaining why some coefficients appear much larger than others, even when the underlying effects are comparable.

The poly() function in R compounds these challenges by default returning orthogonal polynomials rather than raw polynomials. Orthogonal polynomials are designed to be uncorrelated with each other, which is useful for statistical inference, but they don’t preserve the original interpretation of the polynomial terms. This orthogonality changes the relationship between polynomial terms and the original predictor, further complicating the standardization process.

Understanding these mathematical and statistical nuances is crucial for properly addressing the scaling challenges when working with polynomial terms in GLMMs. The solution isn’t as simple as applying standardization before polynomial transformation—it requires a more sophisticated approach to ensure truly comparable estimates across all predictors in your model.

Orthogonal Polynomials vs. Raw Polynomials in R

The poly() function in R is a powerful tool for creating polynomial terms, but it’s essential to understand the distinction between orthogonal and raw polynomials to properly standardize predictors in GLMMs. This understanding forms the foundation of how you should approach standardization when working with polynomial terms.

By default, poly() returns orthogonal polynomials rather than raw polynomials. Orthogonal polynomials are designed to be uncorrelated with each other, which means that each polynomial term is independent of the others in the set. This property is valuable for statistical modeling because it eliminates multicollinearity between polynomial terms, making coefficient estimates more stable and interpretable.

The orthogonality is achieved through a Gram-Schmidt orthogonalization process, which transforms the raw polynomial terms to make them mutually orthogonal. While this statistical advantage is significant for model fitting and inference, it comes with important implications for standardization and interpretation:

• Loss of direct interpretation: Orthogonal polynomials don’t correspond directly to the familiar linear, quadratic, cubic, etc. terms that you might expect. Instead, they’re mathematical transformations designed for statistical properties.

• Different scaling: Orthogonal polynomials have different scales than raw polynomials, which affects how they interact with standardized predictors.

• Dependence on data: The specific form of orthogonal polynomials depends on the data used to generate them. This means that orthogonal polynomials created from one dataset won’t necessarily work well when applied to new data.

When you set raw=TRUE in poly(), the function returns raw polynomials instead of orthogonal ones. Raw polynomials are the standard polynomial terms you might expect: linear (x), quadratic (x²), cubic (x³), etc. These raw polynomials preserve the direct interpretation of polynomial effects but suffer from multicollinearity between terms.

The choice between orthogonal and raw polynomials has significant implications for standardization in GLMMs:

Orthogonal polynomials (default):

• Pros: Eliminate multicollinearity, improve numerical stability, provide independent polynomial terms
• Cons: Harder to interpret directly, scales are different from raw polynomials, dependent on the specific dataset

Raw polynomials (raw=TRUE):

• Pros: Direct interpretation of polynomial effects, consistent across datasets, easier to communicate
• Cons: Multicollinearity between terms, potential numerical instability in complex models

In the context of your GLMM question about standardization, the poly() function’s default behavior of returning orthogonal polynomials means that standardization before applying poly() doesn’t fully solve the scaling problem. Even with standardized predictors, the orthogonal polynomial transformation will create terms with different scales, leading to the larger estimates you’re observing.

Understanding this distinction is crucial because it informs your approach to standardization. If you use orthogonal polynomials (the default), you need to consider additional steps to ensure comparable estimates. If you prefer raw polynomials for easier interpretation, you’ll need to address multicollinearity through other means while potentially simplifying the standardization process.

The poly() function also includes important parameters that affect how polynomials are generated:

• degree: Specifies the degree of the polynomial (e.g., 2 for quadratic, 3 for cubic)
• coefs: Allows you to specify coefficients for the polynomial, rather than generating them from data
• simple: If TRUE, uses simple polynomials rather than orthogonal ones (similar to raw=TRUE)

When working with GLMMs, these parameters become important considerations in your standardization strategy. For example, specifying coefficients allows you to create consistent polynomial transformations across different datasets, which can be valuable for standardization purposes.

The documentation for poly() emphasizes that when raw=FALSE (the default), the function doesn’t attempt to orthogonalize to machine accuracy, and the degree must be less than the number of unique points. These technical details become relevant when planning your standardization approach, as they affect how polynomial terms relate to your standardized predictors.

In practice, the choice between orthogonal and raw polynomials depends on your modeling goals. If your primary concern is model stability and avoiding multicollinearity, orthogonal polynomials may be preferable. If interpretation and consistency across datasets are more important, raw polynomials might be the better choice, despite their multicollinearity issues.

Proper Standardization Approach for Polynomial Terms

Now that we understand the challenges with polynomial terms in GLMMs and the distinction between orthogonal and raw polynomials, let’s explore a proper standardization approach that ensures comparable estimates across all predictors in your model. This approach requires careful consideration of when to standardize and how to handle the unique properties of polynomial terms.

Standardization Timing: Before or After Polynomial Transformation?

The first critical decision is whether to standardize your predictors before or after applying the polynomial transformation. Each approach has distinct implications for your GLMM:

Standardizing before polynomial transformation:

• Apply z-score standardization to raw predictors first
• Then use poly() on the standardized predictors
• This is the approach you’re currently using

Standardizing after polynomial transformation:

• Apply poly() to raw predictors first
• Then standardize each polynomial term individually
• This less common approach addresses scaling issues differently

When you standardize before polynomial transformation, you’re working with predictors that have mean 0 and standard deviation 1. The poly() function then creates polynomial terms from these standardized values. While this seems logical, it doesn’t fully solve the scaling problem because polynomial transformations inherently amplify scales at higher degrees.

For example, if you have a standardized predictor with values ranging from -3 to 3:

• Linear term ranges from -3 to 3
• Quadratic term ranges from 0 to 9
• Cubic term ranges from -27 to 27

Even though the input was standardized, the polynomial terms have dramatically different scales, leading to the coefficient size differences you’re observing in your GLMM.

The Two-Stage Standardization Approach

A more sophisticated approach involves a two-stage standardization process that accounts for the unique scaling properties of polynomial terms:

Stage 1: Standardize the raw predictor

• Apply z-score standardization to your original predictor
• This creates a standardized variable with mean 0 and SD 1

Stage 2: Apply polynomial transformation and re-standardize

• Use poly() on the standardized predictor (or raw=TRUE for raw polynomials)
• Standardize each resulting polynomial term individually

This two-stage approach ensures that each polynomial term is on the same scale, making coefficient estimates directly comparable. The key insight is that polynomial terms need their own standardization to account for their unique scaling properties.

Handling Orthogonal vs. Raw Polynomials

The standardization approach depends on whether you’re using orthogonal or raw polynomials:

For orthogonal polynomials (poly() default):

• The orthogonal transformation changes the scaling, so post-standardization is still necessary
• Each orthogonal polynomial term should be standardized individually
• Remember that orthogonal polynomials are dataset-specific, so standardization should be consistent across training and test data

For raw polynomials (raw=TRUE):

• Raw polynomials maintain the mathematical relationship to the original predictor
• Standardizing each polynomial term individually still provides comparable estimates
• The advantage is that raw polynomials are more interpretable and consistent across datasets

Practical Implementation Considerations

When implementing this standardization approach in your GLMM, consider these practical considerations:

1. Data splitting: If you’re using cross-validation or have separate training/test datasets, ensure consistent standardization parameters across all datasets. This means using the same mean and standard deviation for standardization across all data splits.

2. Centering vs. full standardization: For polynomial terms, you might consider centering (subtracting the mean) without dividing by the standard deviation. This preserves the relative scaling between polynomial terms while addressing the mean offset.

3. Interaction terms: If your model includes interaction terms between polynomial and other predictors, you’ll need to consider how standardization affects these interactions as well.

4. Random effects: In GLMMs, standardization primarily affects fixed effects, but it can indirectly influence random effects through changes in the model’s optimization landscape.

Addressing the Scaling Disparity

The scaling disparity between polynomial terms occurs because higher-order terms naturally have larger values. To address this, consider these strategies:

1. Rescaling polynomial coefficients: After fitting your GLMM, you can rescale polynomial coefficients to make them comparable. This involves dividing each coefficient by the standard deviation of its corresponding polynomial term.

2. Standardized effect sizes: Calculate standardized effect sizes (like Cohen’s f²) for each predictor, which accounts for differences in scaling and provides a more meaningful comparison.

3. Variance partitioning: Use variance partitioning techniques to understand the relative contribution of each polynomial term to the total variance explained by the model.

4. Visualization: Visualize the polynomial effects by plotting the predicted values across the range of each predictor. This helps interpret the practical significance of polynomial terms regardless of their coefficient sizes.

The Role of poly() Parameters

The poly() function offers several parameters that can affect standardization:

• coefs: By specifying coefficients, you can create consistent polynomial transformations across datasets. This is particularly useful when you want to apply the same polynomial transformation to new data.

• simple: When set to TRUE, this creates simple polynomials rather than orthogonal ones, which might be preferable for standardization purposes.

• degree: The degree of the polynomial affects how many terms you’ll need to standardize individually. Higher degrees require more careful standardization to ensure comparable estimates.

Alternative Approaches to Polynomial Standardization

If the two-stage standardization approach seems too complex, consider these alternatives:

1. Use raw polynomials with regularization: Instead of orthogonal polynomials, use raw polynomials with regularization techniques (like ridge regression) to handle multicollinearity.

2. B-spline basis functions: Instead of polynomial terms, consider using B-spline basis functions, which can be more stable and easier to standardize.

3. Effect coding: For categorical predictors used in polynomial relationships, consider effect coding instead of standard dummy coding.

4. Domain-specific transformations: Depending on your specific domain, there might be meaningful transformations that are more appropriate than polynomial terms.

Implementation Best Practices

When implementing polynomial standardization in your GLMM, follow these best practices:

1. Document your standardization approach: Clearly document how you standardized polynomial terms so others can replicate your analysis.

2. Check model diagnostics: After standardization, check model diagnostics to ensure that standardization didn’t introduce new issues like heteroscedasticity or non-normality.

3. Compare approaches: Try different standardization approaches and compare model performance and interpretability.

4. Consider the research question: Let your specific research question guide your standardization approach. Some questions might prioritize interpretability, while others prioritize predictive accuracy.

By implementing a proper standardization approach that accounts for the unique properties of polynomial terms, you can ensure that coefficient estimates in your GLMM are truly comparable across all predictors, regardless of their polynomial degree or original scale.

Implementation in R: Code Examples

Let’s translate the theoretical concepts into practical R code that demonstrates proper standardization of polynomial terms in GLMMs. These examples will show you step-by-step how to implement the standardization approaches we’ve discussed.

Example 1: Basic Two-Stage Standardization with Orthogonal Polynomials

Here’s how to implement the two-stage standardization approach with orthogonal polynomials:

# Load required libraries
library(lme4)
library(dplyr)

# Create example data
set.seed(123)
data <- data.frame(
 id = rep(1:50, each = 10),
 time = rep(1:10, 50),
 predictor = rnorm(500),
 outcome = rnorm(500)
)

# Stage 1: Standardize the raw predictor
data$predictor_std <- scale(data$predictor)

# Stage 2: Apply orthogonal polynomial transformation
# We'll create quadratic terms (degree = 2)
poly_terms <- poly(data$predictor_std, degree = 2, raw = FALSE)
data[, c("poly1", "poly2")] <- as.data.frame(poly_terms)

# Stage 3: Standardize each polynomial term individually
data$poly1_std <- scale(data$poly1)
data$poly2_std <- scale(data$poly2)

# Fit GLMM with standardized polynomial terms
model <- lmer(outcome ~ poly1_std + poly2_std + (1 | id), data = data)

# Check coefficient sizes
summary(model)

This approach ensures that both polynomial terms are on the same scale, making coefficient estimates directly comparable.

Example 2: Two-Stage Standardization with Raw Polynomials

If you prefer raw polynomials for easier interpretation:

# Stage 1: Standardize the raw predictor
data$predictor_std <- scale(data$predictor)

# Stage 2: Apply raw polynomial transformation
poly_terms_raw <- poly(data$predictor_std, degree = 2, raw = TRUE)
data[, c("poly1_raw", "poly2_raw")] <- as.data.frame(poly_terms_raw)

# Stage 3: Standardize each polynomial term individually
data$poly1_raw_std <- scale(data$poly1_raw)
data$poly2_raw_std <- scale(data$poly2_raw)

# Fit GLMM with standardized raw polynomial terms
model_raw <- lmer(outcome ~ poly1_raw_std + poly2_raw_std + (1 | id), data = data)

# Check coefficient sizes
summary(model_raw)

Notice that with raw polynomials, the interpretation is more straightforward—you’re directly seeing the effects of linear and quadratic terms on the standardized outcome.

Example 3: Function for Polynomial Standardization

Creating a reusable function makes the process more efficient:

# Function to standardize polynomial terms
standardize_polynomials <- function(predictor, degree = 2, raw = FALSE) {
 # Stage 1: Standardize the raw predictor
 predictor_std <- scale(predictor)
 
 # Stage 2: Apply polynomial transformation
 poly_terms <- poly(predictor_std, degree = degree, raw = raw)
 poly_df <- as.data.frame(poly_terms)
 
 # Stage 3: Standardize each polynomial term individually
 poly_std <- scale(poly_df)
 
 # Return standardized polynomial terms
 return(poly_std)
}

# Use the function
data$poly_std <- standardize_polynomials(data$predictor, degree = 2, raw = FALSE)
data[, c("poly1_std", "poly2_std")] <- data$poly_std

# Fit model
model_func <- lmer(outcome ~ poly1_std + poly2_std + (1 | id), data = data)
summary(model_func)

Example 4: Visualization of Standardized Polynomial Effects

Visualizing the effects helps interpret standardized polynomial terms:

library(ggplot2)

# Create a new dataset for prediction
new_data <- data.frame(
 predictor_std = seq(min(data$predictor_std), max(data$predictor_std), length.out = 100),
 id = 1
)

# Apply the same polynomial transformation
new_poly_terms <- poly(new_data$predictor_std, degree = 2, raw = FALSE)
new_data[, c("poly1", "poly2")] <- as.data.frame(new_poly_terms)

# Standardize using the same parameters as training data
new_data$poly1_std <- scale(new_data$poly1, 
 center = attr(data$poly1, "scaled:center"), 
 scale = attr(data$poly1, "scaled:scale"))
new_data$poly2_std <- scale(new_data$poly2, 
 center = attr(data$poly2, "scaled:center"), 
 scale = attr(data$poly2, "scaled:scale"))

# Add random effect (using mean)
new_data$pred <- predict(model, newdata = new_data, re.form = NA)

# Plot the polynomial effect
ggplot(new_data, aes(x = predictor_std, y = pred)) +
 geom_line() +
 labs(title = "Standardized Quadratic Effect",
 x = "Standardized Predictor",
 y = "Predicted Outcome") +
 theme_minimal()

Example 5: Comparing Standardization Approaches

Let’s compare different standardization approaches to see how they affect coefficient estimates:

# Approach 1: No standardization
model_no_std <- lmer(outcome ~ poly(predictor, degree = 2, raw = TRUE) + (1 | id), data = data)

# Approach 2: Standardize before poly()
model_std_before <- lmer(outcome ~ poly(predictor_std, degree = 2, raw = TRUE) + (1 | id), data = data)

# Approach 3: Two-stage standardization (our recommended approach)
model_two_stage <- lmer(outcome ~ poly1_std + poly2_std + (1 | id), data = data)

# Compare coefficient tables
coef_table_no_std <- coef(summary(model_no_std))
coef_table_std_before <- coef(summary(model_std_before))
coef_table_two_stage <- coef(summary(model_two_stage))

# Create comparison dataframe
comparison <- data.frame(
 Approach = c("No Standardization", "Std Before Poly", "Two-Stage Std"),
 Poly1 = c(coef_table_no_std["poly(predictor, degree = 2, raw = TRUE)1", "Estimate"],
 coef_table_std_before["poly(predictor_std, degree = 2, raw = TRUE)1", "Estimate"],
 coef_table_two_stage["poly1_std", "Estimate"]),
 Poly2 = c(coef_table_no_std["poly(predictor, degree = 2, raw = TRUE)2", "Estimate"],
 coef_table_std_before["poly(predictor_std, degree = 2, raw = TRUE)2", "Estimate"],
 coef_table_two_stage["poly2_std", "Estimate"])
)

print(comparison)

This comparison will show you how different standardization approaches affect coefficient estimates, helping you choose the most appropriate method for your analysis.

Example 6: Cross-Validation with Proper Standardization

When using cross-validation, it’s crucial to standardize within each fold to avoid data leakage:

# Create cross-validation folds
library(caret)
folds <- createFolds(data$outcome, k = 5)

# Function for CV with proper standardization
cv_with_std <- function(data, folds) {
 cv_results <- list()
 
 for (i in seq_along(folds)) {
 # Split data
 test_indices <- folds[[i]]
 train_data <- data[-test_indices, ]
 test_data <- data[test_indices, ]
 
 # Standardize based on training data
 train_mean <- mean(train_data$predictor)
 train_sd <- sd(train_data$predictor)
 
 # Apply standardization to both training and test data
 train_data$predictor_std <- (train_data$predictor - train_mean) / train_sd
 test_data$predictor_std <- (test_data$predictor - train_mean) / train_sd
 
 # Apply polynomial transformation
 train_poly <- poly(train_data$predictor_std, degree = 2, raw = TRUE)
 test_poly <- predict(train_poly, newdata = test_data$predictor_std)
 
 # Standardize polynomial terms based on training data
 train_poly_std <- scale(train_poly)
 test_poly_std <- scale(test_poly, 
 center = attr(train_poly_std, "scaled:center"), 
 scale = attr(train_poly_std, "scaled:scale"))
 
 # Add to data
 train_data[, c("poly1_std", "poly2_std")] <- train_poly_std
 test_data[, c("poly1_std", "poly2_std")] <- test_poly_std
 
 # Fit model
 model <- lmer(outcome ~ poly1_std + poly2_std + (1 | id), data = train_data)
 
 # Predict on test data
 test_data$pred <- predict(model, newdata = test_data, re.form = NA)
 
 # Store results
 cv_results[[i]] <- test_data[, c("outcome", "pred")]
 }
 
 # Combine results
 all_results <- do.call(rbind, cv_results)
 
 # Calculate RMSE
 rmse <- sqrt(mean((all_results$outcome - all_results$pred)^2))
 
 return(rmse)
}

# Run CV
cv_rmse <- cv_with_std(data, folds)
print(paste("Cross-validated RMSE:", cv_rmse))

This example demonstrates how to properly handle standardization in a cross-validation context, ensuring that data from the test set doesn’t influence the standardization parameters.

Example 7: Exporting Standardization Parameters

For reproducibility, save the standardization parameters:

# Save standardization parameters
std_params <- list(
 predictor_mean = mean(data$predictor),
 predictor_sd = sd(data$predictor),
 poly1_mean = attr(data$poly1_std, "scaled:center"),
 poly1_sd = attr(data$poly1_std, "scaled:scale"),
 poly2_mean = attr(data$poly2_std, "scaled:center"),
 poly2_sd = attr(data$poly2_std, "scaled:scale")
)

# Save to RDS file
saveRDS(std_params, "standardization_params.rds")

# Load when needed
loaded_params <- readRDS("standardization_params.rds")

These code examples provide a comprehensive toolkit for implementing proper standardization of polynomial terms in GLMMs. By following these approaches, you can ensure that coefficient estimates are comparable across all predictors in your model, regardless of their polynomial degree or original scale.

Best Practices for Comparable Estimates in GLMMs

Now that we’ve covered the technical aspects of standardizing polynomial terms in GLMMs, let’s discuss best practices that will help you ensure truly comparable estimates across all predictors in your model. These practices go beyond the technical implementation and address the broader context of mixed effects modeling.

Choose the Right Polynomial Degree

One of the most critical decisions in polynomial modeling is selecting the appropriate degree. Higher-degree polynomials can capture more complex relationships but also introduce more parameters to standardize and interpret:

• Start low: Begin with linear or quadratic terms unless you have strong theoretical reasons to use higher degrees.
• Consider the research question: Let your hypothesis guide the degree selection rather than trying to maximize model fit.
• Use model selection criteria: Compare models with different polynomial degrees using AIC, BIC, or cross-validation.
• Visualize relationships: Plot the relationship between your predictor and outcome to guide degree selection.

Remember that each additional polynomial degree requires another standardization step, increasing the complexity of ensuring comparable estimates.

Balance Between Orthogonal and Raw Polynomials

The choice between orthogonal and raw polynomials involves trade-offs between statistical properties and interpretability:

When to use orthogonal polynomials:

• When multicollinearity is a significant concern
• When you’re primarily interested in model fit and prediction
• When you have a large sample size relative to the number of parameters
• When you plan to use stepwise selection or automated model building

When to use raw polynomials:

• When interpretation of individual polynomial terms is important
• When you need consistent polynomial transformations across datasets
• When communicating results to non-statistical audiences
• When working with smaller sample sizes where orthogonality gains are less critical

In many cases, the best approach might be to use raw polynomials with additional techniques to handle multicollinearity, such as ridge regression or variance inflation factor (VIF) monitoring.

Document Your Standardization Approach

Thorough documentation is essential for reproducibility and transparency:

• Record standardization parameters: Save the means and standard deviations used for standardization.
• Describe your rationale: Explain why you chose a particular standardization approach.
• Version control: Use git or other version control systems to track changes to your standardization code.
• Create a data dictionary: Document how each variable in your model was transformed and standardized.

Documentation becomes particularly important when working with polynomial terms because the standardization process is more complex than with simple linear predictors.

Check Model Diagnostics After Standardization

Standardization can affect various aspects of your model, so thorough diagnostics are essential:

• Residual analysis: Check for patterns in residuals that might indicate issues with polynomial specification.
• Influence diagnostics: Identify influential observations that might be disproportionately affecting polynomial terms.
• Random effects distribution: Verify that random effects are approximately normally distributed.
• Convergence checks: Ensure that your model converged properly, especially with complex polynomial specifications.

Pay special attention to diagnostics related to the polynomial terms, as standardization might mask or reveal issues with the polynomial specification.

Consider Alternative Approaches to Polynomial Terms

In some cases, alternatives to polynomial terms might provide better results with simpler standardization:

B-splines:

• More flexible than polynomials for capturing complex relationships
• Easier to standardize because they’re bounded
• Can be constrained to be continuous and smooth

GAMs (Generalized Additive Models):

• Allow for more flexible functional forms
• Can incorporate random effects (GAMMs)
• Often provide better fit with fewer parameters

Piecewise linear models:

• Simpler to interpret and standardize
• Can capture non-linear relationships with breakpoints
• More transparent than polynomial models

Non-linear mixed models:

• Directly model non-linear relationships
• Avoid the need for polynomial transformations
• Can be more interpretable for specific theoretical relationships

Consider these alternatives, especially if polynomial standardization proves challenging or if your relationships have clear theoretical justifications for non-polynomial forms.

Handle Missing Data Appropriately

Missing data can affect polynomial standardization in complex ways:

• Impute before standardization: If you’re using imputation, apply it before standardizing predictors.
• Consider multiple imputation: For complex models with polynomial terms, multiple imputation might provide more robust results.
• Document missing data patterns: Report the amount and patterns of missing data to help interpret your results.

Be particularly careful with missing data in polynomial terms, as they can create complex patterns that affect model convergence and interpretation.

Communicate Results Effectively

Even with proper standardization, communicating polynomial effects can be challenging:

• Visualize effects: Plot predicted values across the range of each predictor.
• Report standardized coefficients: Include standardized effect sizes alongside unstandardized coefficients.
• Use confidence intervals: Report uncertainty around polynomial estimates.
• Provide practical interpretations: Translate statistical findings into meaningful practical effects.

When presenting polynomial effects, consider focusing on the overall shape of the relationship rather than individual coefficients, which can be difficult to interpret in isolation.

Validate Your Model

External validation is crucial for models with polynomial terms:

• Cross-validation: Use k-fold or leave-one-out cross-validation to assess model performance.
• External datasets: Test your model on independent datasets when possible.
• Sensitivity analysis: Test how sensitive your results are to changes in standardization approach.

Validation is particularly important for polynomial models because they can overfit to specific patterns in the data, especially with higher degrees.

Consider Computational Efficiency

Polynomial terms can affect model computational efficiency:

• Sparse matrices: For large datasets, consider using sparse matrix representations.
• Parallel processing: Use parallel computing for model fitting with complex polynomial specifications.
• Model simplification: Balance model complexity with computational constraints.

Computational considerations become more important with higher-degree polynomials and larger datasets.

Update Your Approach as Needed

Statistical modeling is an iterative process:

• Monitor new research: Stay updated on best practices for polynomial modeling.
• Revisit standardization: Periodically review whether your standardization approach is still optimal.
• Adapt to new data: As you collect more data, revisit your polynomial specifications and standardization.

The field of mixed modeling continues to evolve, with new methods and best practices emerging regularly.

By following these best practices, you can ensure that your polynomial terms in GLMMs provide meaningful, comparable estimates that accurately represent the relationships in your data while being interpretable and useful for decision-making.

Sources

1. R Documentation for poly() — Detailed documentation on polynomial functions in R statistics: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/poly.html
2. RDocumentation poly() Function — Comprehensive reference for poly() function parameters and behavior: https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/poly
3. R Core Team — Core developers of R statistical computing environment: https://www.r-project.org
4. Keith Jewell (Campden BRI Group, UK) — Contributor to R’s poly() function improvements: https://stat.ethz.ch/R-manual/R-devel/library/stats/html/poly.html

Conclusion

Properly standardizing predictors before using poly() in a GLMM requires a nuanced approach that accounts for the unique properties of polynomial transformations. While z-score standardization is generally beneficial for mixed models, polynomial terms need special handling because the poly() function returns orthogonal polynomials by default, which affects their scaling and interpretation.

The key to ensuring comparable estimates across predictors is implementing a two-stage standardization process: first standardizing the raw predictor, then applying the polynomial transformation, and finally standardizing each polynomial term individually. This approach addresses the fundamental issue that polynomial terms of different degrees naturally have different scales, even when derived from the same standardized predictor.

Your observation that polynomial terms have significantly larger estimates than other predictors is mathematically expected and can be resolved through proper standardization. By following the implementation strategies and best practices outlined in this guide, you can build GLMMs with polynomial terms that provide interpretable, comparable coefficient estimates while maintaining statistical rigor and model performance.

Remember that the choice between orthogonal and raw polynomials, the selection of polynomial degree, and the handling of interactions all influence how you should approach standardization. By carefully considering these factors and documenting your approach, you can create robust, interpretable models that accurately represent the complex relationships in your data.