Designing a Limma Design Formula

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Introduction

The limma package is a powerful tool for analyzing gene expression data, particularly in the context of microarrays and RNA-seq. A critical part of any limma analysis is the design formula, which specifies the experimental conditions and contrasts that you are interested in. This vignette provides a guide on how to construct a limma design formula correctly, with examples and best practices.

Understanding the Design Matrix

The design matrix is a crucial component in any differential expression analysis using limma. It defines the relationships between the samples and the experimental conditions (factors) under investigation. A well-constructed design matrix allows limma to correctly model the effects of these factors and estimate the differential expression.

Basic Design Formula

In its simplest form, the design formula includes only one factor, such as treatment vs. control. If your experiment involves comparing two conditions (e.g., treated vs. untreated), you can create a design formula like this:

design <- model.matrix(~ 0 + condition, data = meta)

Here, condition is a factor variable in your metadata (meta) that represents the experimental groups.

Important Points: The ~ 0 + condition syntax tells R to create a design matrix without an intercept (i.e., a matrix where each level of the factor condition is represented by its own column). This approach is helpful when you want to make direct comparisons between conditions. Including Multiple Factors If your experiment includes more than one factor, such as time points and treatments, you can include both in the design formula:

design <- model.matrix(~ 0 + treatment + time, data = meta)

This formula assumes that the effects of treatment and time are additive (no interaction). If you suspect that the interaction between treatment and time might be important, you can include an interaction term:

design <- model.matrix(~ 0 + treatment * time, data = meta)

Interaction Term: The treatment * time term includes both the main effects of treatment and time and their interaction. Blocking Factors In some experiments, you might have technical or biological replicates, or other blocking factors (e.g., batch effects). You can include these blocking factors in your design formula:

design <- model.matrix(~ 0 + treatment + batch, data = meta)

This formula accounts for both treatment and batch effects, ensuring that your analysis is not confounded by batch effects.

Creating Contrasts

After defining your design matrix, you will likely want to make specific comparisons between conditions. This is where contrasts come in. For example, to compare treated vs. untreated, you can define a contrast matrix:

contrast <- makeContrasts(
  treated_vs_untreated = treatmenttreated - treatmentuntreated,
  levels = design
)

Practical Example

Let’s say you have an experiment with two treatments (A and B) and two time points (early and late). Your metadata might look like this:

meta <- data.frame(
  sample = c("S1", "S2", "S3", "S4"),
  treatment = factor(c("A", "A", "B", "B")),
  time = factor(c("early", "late", "early", "late"))
)

Your design formula would be:

design <- model.matrix(~ 0 + treatment * time, data = meta)

And a contrast to compare treatment A at the early time point against treatment B at the late time point would be:

contrast <- makeContrasts(
  A_early_vs_B_late = (treatmentA:timeearly) - (treatmentB:timelate),
  levels = design
)

Summary

1. Intercept or No Intercept:
   • Starting with ~ 0 means no intercept (i.e., not including a baseline group in the model).
   • Starting with ~ 1 (or just ~) includes an intercept (baseline group).
2. Additive Factors:
   • Factors separated by + indicate additive effects. For example, ~ 0 + factor1 + factor2 means you are modeling the effects of factor1 and factor2 additively, without considering interactions.
3. Interaction Terms:
   • The * symbol is used to model interactions between factors. For example, ~ 0 + factor1 * factor2 will include factor1, factor2, and their interaction (factor1:factor2).
   • Alternatively, you can specify the interaction explicitly with :. For example, ~ 0 + factor1 + factor2 + factor1:factor2 is equivalent to ~ 0 + factor1 * factor2.

Some examples:

• ~ 0 + factor1 + factor2: Additive model without an intercept.
• ~ 1 + factor1 + factor2: Additive model with an intercept.
• ~ 0 + factor1 * factor2: Model with main effects and their interaction, no intercept.
• ~ 1 + factor1 * factor2: Model with intercept, main effects, and their interaction.