Statistical analysis and regression in R are powerful tools for understanding data. This guide provides a comprehensive overview for students, covering everything from fundamental concepts and model formulae to specific distributions and diagnostic checks, using practical R examples.
Introduction to Statistical Analysis and Regression in R
R is a versatile environment for data manipulation, calculation, and graphical display. It's a high-level programming language that makes you think critically about your analysis, offering extensive packages and dynamic development. While known for its command-line interface, R is a robust and free statistical system.
Why Learn R for Statistical Analysis?
- Freeware: Accessible to everyone without cost.
- Extensive: One of the largest statistical systems available.
- Dynamic: Open environment with continuous development and thousands of packages.
- Analytical Depth: Encourages a deeper understanding of the analysis process by requiring explicit commands.
Core Concepts in Statistical Modeling
Statistical modeling helps us understand the relationship between variables. It consists of two main parts: the systematic component (the predictable part of the model) and the stochastic component (the random, unexplained part).
Variables in Statistical Models
Understanding variable types is crucial for choosing the right statistical method:
- Response Variable (Dependent): The outcome we want to understand or predict (e.g., yield, weight, toxin amount). It's typically plotted on the ordinate (Y-axis) and can be continuous, counts, or proportions.
- Explanatory Variable (Independent): Variables that influence the response. They are manipulated or observed and plotted on the abscissa (X-axis). Explanatory variables can be numeric (continuous or discrete measurements, called a covariate) or categorical (a factor with two or more levels).
Model Formulae in R for Regression Analysis
R uses a specific syntax for defining statistical models. The general structure is response variable ~ explanatory variable(s).
Key Operators in R Model Formulae
+: Adds terms (main effects).-: Deletes terms.:: Specifies an interaction between terms.*: Includes all main effects and interactions (e.g.,A*B*Cexpands toA + B + C + A:B + A:C + B:C + A:B:C).^: Includes main effects and interactions up to a specified order (e.g.,(A+B+C)^2includes main effects and all 2-way interactions).1: Represents the intercept term (e.g.,y ~ 1for a null model).I(): Interprets mathematical operators within the formula (e.g.,y ~ x + I(x^2)for a quadratic term)./: Denotes nested relationships.|: Indicates conditioned relationships.
Common R Model Formulae Examples
Here are some essential model formulae and their descriptions, vital for R statistical analysis:
| Model formula | Description | Mathematical Expression |
|---|---|---|
y ~ 1 | Null model (intercept only) | $f(\mu_i) = \alpha$ |
y ~ x | Simple linear model | $f(\mu_i) = \alpha + \beta x_i$ |
log(y) ~ x - 1 | Linear model with log-transformed response, no intercept | $\log(\mu_i) = \beta x_i$ |
y ~ x + I(x^2) | Quadratic model with one explanatory variable | $f(\mu_i) = \alpha + \beta x_i + \gamma x_i^2$ |
y ~ poly(x, 2) | Quadratic model using orthogonal polynomials | (Similar to x + I(x^2) but reparameterized) |
y ~ x1 + x2 | Linear model with two explanatory variables | $f(\mu_i) = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i}$ |
y ~ x*A | One-way ANCOVA (covariate x, factor A) | $f(\mu_{ij}) = \alpha + A_j + \beta x_i + \delta_j x_k$ |
Stochastic Components and Distributions
The stochastic component accounts for random variation. In Generalized Linear Models (GLMs), you choose a distribution for your response variable based on its nature and theoretical considerations.
Response Variable Types and Distributions
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Continuous Measurements: Values with infinite precision (e.g., length, pH, concentration).
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Gauss (Normal) Distribution: Bell-shaped, symmetric, mean = median = mode. Parameters: $\mu, \sigma^2$. Use
glm(formula, family=Gaussian)for LM in GLM framework. -
Lognormal Distribution: Positive real values, right-skewed. Variance increases quadratically with mean. Often normal after log transformation.
-
Gamma Distribution: Positive real values, right-skewed. Variance increases quadratically with mean.
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Inverse Gaussian Distribution: Positive real values, used for diffusion processes. Variance increases steeply with mean.
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Counts: Discrete integer values (e.g., number of seeds, individuals).
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Poisson Distribution: Discrete, right-skewed. Variance equals expected value.
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Negative-binomial Distribution: Discrete, strongly right-skewed. Variance is larger than expected value, increasing parabolically with mean.
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Proportions: Relative frequencies ($y/n$) from a whole population.
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Binomial & Binary Distributions: For integer counts of $n$ independent trials. Parameter $\pi$ (probability of event) from 0 to 1. Variance maximal at 0.5.
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Quasi “Distribution”: Not a true distribution but specifies the relationship between expected value and variance, useful for overdispersion.
Flashcards
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Analytical Methods: Regression, ANOVA, and ANCOVA
Choosing the right analytical method depends on your explanatory variables:
| Explanatory Variable(s) | Method |
|---|---|
| Continuous | Regression |
| Categorical | ANOVA |
| Continuous and Categorical | ANCOVA |
Regression Analysis
- Simple Regression: One explanatory variable.
- Multiple Regression: Two or more explanatory variables.
The general linear predictor for multiple regression is $\alpha + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_k x_k$. The x terms can include polynomial functions (e.g., $x^2$, $x^3$) and interactions (e.g., $x_1:x_2$). A rule of thumb is to have less than $n/3$ parameters in the model at any time to avoid overfitting.
Example: Cereal Yield Prediction
A study on 100 plots aimed to predict wheat yield using six variables (winter plants, ears/m$^2$, pH, P, K, Mg). Initial analysis showed quadratic terms and interactions, leading to model simplification. Centering (subtracting mean) and scaling (dividing by SD) variables improved interpretability of coefficients in complex models.
m1 <- lm(yield ~ (winter+ears+PpH+K+Mg)^2 + I(winter^2) + I(ears^2) + I(PpH^2) + I(K^2) + I(Mg^2))
After stepwise simplification (m2 <- step(m1)), a final model might highlight significant factors like winter plants, quadratic pH, and potassium, as seen in the example output: $a + b_1 \frac{\text{winter} - \bar{y}\text{winter}}{s\text{winter}} + b_2 \frac{\text{pH} - \bar{y}\text{pH}}{s\text{pH}} + c_1 \left(\frac{\text{pH} - \bar{y}\text{pH}}{s\text{pH}}\right)^2 + b_3 \frac{K - \bar{y}_K}{s_K}$.
ANOVA (Analysis of Variance)
ANOVA is used when all explanatory variables are categorical factors (e.g., 1-way, 2-way, k-way ANOVA). For two categorical variables $A$ and $B$ with two levels, a model with treatment contrasts is $\alpha + A_i + B_j + A:B_{ij}$.
Example: Cockroach Diet Study (1-way ANOVA)
An experiment tested the effect of five diet types (control, lipid1, lipid2, protein1, protein2) on cockroach body weight. The model was weight_ij = DIET_j + epsilon_ij. The anova(m1) output revealed a highly significant effect of diet on weight, with a p-value < 2.2e-16.
m1 <- lm(weight~diet)
anova(m1)
ANCOVA (Analysis of Covariance)
ANCOVA combines regression and ANOVA, involving both continuous (covariates) and categorical (factors) explanatory variables. For one factor $A_j$ and one covariate $x$, a linear predictor might be $\alpha + A_j + \beta x + \delta_j x$.
Model Simplification and Comparisons
Simplifying models is crucial for finding the most parsimonious yet adequate model. This process involves removing insignificant terms and grouping similar factor levels.
Strategies for Model Simplification
- Top-down (Backward Selection): Start with a maximal (saturated) model including all main effects and interactions. Gradually remove insignificant terms, beginning with higher-order interactions.
- Bottom-up (Forward Selection): Build up a model by adding variables one by one.
Criteria for Term Removal
- P-value: Use F-tests or $\chi^2$-tests from
anova()to assess significance. Remove terms with p-values above a chosen threshold (e.g., 0.05). - Akaike Information Criterion (AIC):
AIC = -2 logLik + 2p(wherepis the number of parameters). Lower AIC values indicate a better balance between model fit and complexity.AIC(model1, model2)can compare models.
Contrasts and Multiple Comparisons
- Contrasts: Used to compare individual differences between factor levels. R's default is
contr.treatment(compares levels to a reference). Other options includecontr.helmertandcontr.sum. Contrasts should be specified prior to analysis. - Multiple Comparisons (Post-hoc Tests): Applied after analysis if a factor is significant. Examples include Bonferroni correction, Dunn test, Scheffe test, and Tukey HSD test (e.g., using
glhtfrom themultcomplibrary). These tests adjust p-values for multiple comparisons.
Example: Toxin Production and Disease
A study measured toxin amount produced by four bacteria species in patients with different diseases. The model toxin ~ species*diagnosis showed significant interaction, implying that the effect of species on toxin levels varied by diagnosis. Subsequent simplification involved grouping similar diagnoses and species.
m1 <- lm(toxin ~ species*diagnosis)
anova(m1)
Model Criticism and Diagnostics in R
Assessing model quality and assumptions is critical. You can never