Statistical Analysis with R

Statistical analysis is a cornerstone of research, and R provides a powerful, flexible environment for conducting a wide range of statistical analyses. This comprehensive guide will walk you through the essential concepts, methodologies, and practical applications of statistical analysis using R, perfect for students seeking to deepen their understanding and master this critical skill. We'll cover everything from fundamental model formulae to advanced diagnostic checks, ensuring you have a solid grasp of how to effectively analyze data with R.

Understanding Statistical Models in R: Core Concepts

Every statistical model in R is composed of two main parts: a systematic component and a stochastic component. The systematic component describes the deterministic relationships between variables, while the stochastic component accounts for random variation and uncertainty.

R Model Formulae and Operators for Statistical Analysis

R's model formulae are a concise way to specify the relationships between your response and explanatory variables. They use specific operators to define these relationships:

• +: Adds terms (main effects).
• -: Deletes terms.
• :: Specifies an interaction between terms.
• *: Includes all main effects and their interactions (e.g., A*B is equivalent to A + B + A:B).
• ^: Includes all main effects and interactions up to a specified order (e.g., (A+B+C)^2 includes all main effects and two-way interactions).
• I(): "Interpreter" that treats operators inside as mathematical, not formulaic (e.g., I(x^2) for a quadratic term).
• poly(x, n): Generates orthogonal polynomial terms for x up to degree n.
• 1: Represents the intercept term (e.g., y ~ 1 for a null model).

Here are some common model formulae examples and their descriptions:

• y ~ 1: Null model ($f(\mu_i) = \alpha$)
• y ~ x: Linear model with one explanatory variable ($f(\mu_i) = \alpha + \beta x_i$)
• log(y) ~ x - 1: Linear model without an intercept, with log-transformed response ($\log(\mu_i) = \beta x_i$)
• y ~ x + I(x^2) or y ~ poly(x, 2): Quadratic model with one explanatory variable ($f(\mu_i) = \alpha + \beta x_i + \gamma x_i^2$)
• y ~ x1 + x2: Linear model with two explanatory variables ($f(\mu_i) = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i}$)
• y ~ A*B*C: 3-way ANOVA with all main effects and interactions.
• y ~ (A+B+C)^2: 3-way ANOVA with main effects and two-way interactions.
• y ~ x*A: 1-way ANCOVA, including an interaction between a continuous covariate and a factor.

Stochastic Component: Choosing Distributions in GLM

The stochastic component of a model defines the probability distribution of the response variable and its relationship with the mean. For Generalized Linear Models (GLM), selecting the correct distribution is crucial. This decision is typically based on theoretical models or prior experience with similar data.

Response variables can be:

• Continuous measurements: Values that can be made with infinite precision (e.g., length, pH).
• Gauss (normal) distribution: Bell-shaped, symmetric, mean = median = mode. Variance is independent of the mean. Suitable for many natural phenomena. R uses family=gaussian with link=identity by default for glm().
• Lognormal distribution: Positive real values, asymmetric, skewed right. Variance increases quadratically with the mean. Logarithmic transformation can stabilize variance.
• Gamma distribution: Positive real values, asymmetric, skewed right. Variance increases quadratically with the mean.
• Inverse Gaussian distribution: Positive real values, used for diffusion processes. Variance increases steeply with the mean.
• Counts: Discrete, integer values (e.g., number of events).
• Poisson distribution: Asymmetric, skewed right. Variance is equal to the expected value, and increases with the mean.
• Negative-binomial distribution: Discrete, integer values, strongly skewed right. Variance is larger than the expected value, increasing parabolically with the mean (useful for overdispersed count data).
• Proportions: Relative frequencies (y/n) representing the qualitative character of an event.
• Binomial & Binary distributions: Measurements are integers from n independent trials, with a single parameter $\pi$ (probability of occurrence). Variance is maximal at 0.5. Often modeled using a logit transformation: $\log(\frac{p}{1-p})$.

Quasi "distribution" refers to cases where an exact distribution isn't used, but rather an expected value and a relationship between the expected value and variance are specified. These are not true distributions but rather mixtures of available settings.

Model Criticism and Assumptions for Robust Statistical Analysis with R

After fitting a model, model criticism is essential to assess its quality and validate assumptions. We can never prove a model is adequate, but we can identify potential issues. This involves studying both the systematic and stochastic components.

Residuals ($\varepsilon_i$) are key to checking assumptions. Ideally, they should be normally distributed with a mean of zero and constant variance (homoscedasticity), and be independent of each other. Violations suggest issues.

Checking Model Assumptions in R: Diagnostic Plots

Diagnostic plots, typically generated by plot() on an lm or glm object, are crucial for informal assessment:

• Predictor's adequacy (Residuals vs. Fitted values): Raw (LM) or deviance (GLM) residuals plotted against fitted values. A curved pattern suggests a missing polynomial term (e.g., quadratic).
• Normality (Q-Q Plot): A Q-Q plot of standardized (LM) or standardized deviance (GLM) residuals. Deviations from a straight line (e.g., "J" or "S" shaped) indicate non-normal residuals, possibly requiring a link function change or variable transformation.
• Variance homogeneity (Scale-Location plot): A plot of standardized residuals against fitted/predicted values. If variance increases with the mean, consider using Poisson or Gamma distributions or a log transformation.
• Influence (Cook's distance, Residuals vs. Leverage): Cook's distance shows the influence of individual observations. Values close to 1 or higher suggest influential points. Such observations might need to be omitted or explanatory variables transformed.
• Independence: Standardized (LM) or Pearson residuals (GLM) plotted against a continuous explanatory variable, or checking for serial dependence if the explanatory variable is time or space.

Formal tests can also be used, such as the Bartlett test for homogeneity of variances or the Shapiro-Wilk test for normality of residuals, but plots are often more informative for identifying the nature of the violation.

Types of Statistical Analyses with R: Practical Applications

Simple Regression with R

Simple regression analyzes the relationship between one response variable and one continuous explanatory variable. For instance, to study if the number of seeds per oat ear predicts yield:

• Hypothesis: Is the number of seeds related to the yield? What is the predictive model?
• Variables: grain (explanatory, continuous), yield (response, continuous).
• Model: yield_i = α + β grain_i + ε_i, where ε_i ~ N(0, σ^2).
• R function: lm(yield ~ grain).

Adding a quadratic term (yield ~ poly(grain, 2) or yield ~ grain + I(grain^2)) allows checking for non-monotonous trends. If the quadratic term is not significant, a simpler linear model might be preferred.

1-way ANOVA for Comparing Means in R

1-way ANOVA is used to compare means across different levels of a single categorical explanatory variable (factor). For example, testing the effect of five diet types on cockroach body weight:

• Hypothesis: Is nutritional quality of the diet affecting the size of organisms? Is weight similar among diet groups?
• Variables: DIET (explanatory, categorical), weight (response, continuous).
• Model: weight_ij = DIET_j + ε_ij, where ε_ij ~ N(0, σ^2).
• R function: lm(weight ~ diet) followed by anova().

Comparisons and Contrasts

After a significant ANOVA, individual differences between factor levels are explored using comparisons.

• Contrasts: Specified a priori (before analysis) to test specific hypotheses about group differences. In R, contrasts() function can be used to set specific contrasts (e.g., contr.treatment, contr.helmert, contr.sum). Orthogonal contrasts ensure each comparison is unique.
• Simplification: Grouping similar factor levels and testing the new model against the original using anova().
• Multiple Comparisons (Post-hoc tests): Performed a posteriori (after analysis) using methods like Tukey's HSD, Bonferroni correction, Scheffe test, or Dunn test (library(multcomp) and glht(model, linfct = mcp(factor = "Tukey"))). These adjust p-values for multiple testing.

Multiple Regression for Complex Relationships in R

Multiple regression extends simple regression to include two or more explanatory variables (continuous, categorical, or both). For predicting cereal yield with multiple factors:

• Hypothesis: Which of several variables affect yield? What is the predictive model?
• Variables: yield (response), winter, ears, pH, P, K, Mg (explanatory).
• General Linear Predictor: $\alpha + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_k x_k$.
• Model: yield_i = α + β_1 winter_i + β_2 ears_i +... + ε_i.
• R function: lm(yield ~ winter + ears + pH + P + K + Mg). Interactions and polynomial terms can be added (e.g., (winter+ears+PpH+K+Mg)^2 + I(winter^2) +...).

Simplification of multiple regression models often involves a top-down (backward selection) approach: start with a maximal model (all main effects, interactions, and quadratic terms) and progressively remove insignificant terms based on p-values or AIC. A common rule of thumb is to have less than n/3 parameters in the model at any time to avoid overfitting.

ANCOVA: Combining Regression and ANOVA in R

ANCOVA (Analysis of Covariance) combines categorical factors and continuous covariates. For example, a model with one factor A and one covariate x would have a linear predictor: $\alpha + A_j + \beta x + \delta_j x$.

• y ~ x*A: This formula directly specifies the main effects of x and A, plus their interaction x:A.
• Model interpretation involves looking at how the slope of x might differ across levels of A (if the interaction A:x is significant), or if only main effects are significant, how A shifts the intercept and x contributes a common slope.

2-way ANOVA for Two Factors and Their Interaction in R

2-way ANOVA examines the effects of two categorical factors and their interaction on a continuous response. For instance, studying toxin production by bacteria species under different disease diagnoses:

• Hypothesis: Is the amount of toxin similar for four bacteria species and four diseases? If not, what are the differences? Which species can be used as an indicator?
• Variables: toxin (response), SPECIES, DIAGNOSIS (factors).
• Model: toxin_ijk = α + SPECIES_j + DIAGNOSIS_k + SPECIES:DIAGNOSIS_jk + ε_ijk.
• R function: lm(toxin ~ species*diagnosis).

The anova() output for lm uses Type I Sum of Squares, which is sequential. For unbalanced data (unequal observations per treatment), it's more appropriate to consider Type III Sum of Squares, which assesses effects independently.

Weighted Regression for Varying Precision in R

Weighted regression is used when observations have different levels of precision or influence. By assigning weights to measurements, we can increase or decrease their effect on the model fit. Only positive values are allowed for weights.

• Scenario: Sexual size dimorphism (ratio) varying with temperature and influenced by the number of individuals sampled (number).
• Model: ratio_i = α + β temp_i + ε_i, where ε_i ~ N(0, σ^2 / number_i).
• R function: lm(ratio ~ temp, weights=number). This explicitly models the variance of residuals as inversely proportional to the number of individuals.

Practical R Operations and Exploratory Data Analysis (EDA)

Essential R Functions and Data Structures

• Functions: sin, log, sqrt, sum, mean, var, sd, length, c, seq, matrix, cbind, data.frame, factor, relevel, scale (for centering and scaling variables).
• Variables: R is case-sensitive. Variable names should not contain spaces or special characters (except _).
• Vectors: Numeric, character, logical (T/F).
• Data Frames: Rectangular data structure (columns = variables, rows = observations). Can be created with data.frame() or imported using read.delim("clipboard") from Excel or text files.
• Missing data are NA. is.na() checks for missing values. $, attach(), names() are used for accessing variables.

Exploratory Data Analysis (EDA) with R

EDA is a crucial first step, involving visual and tabular analysis to understand data characteristics, check for errors, suggest models, and assess assumptions.

Expected Value and Variation

• Expected value ($E(y), \mu$): Theoretical long-term average. Estimated by:
• Mean: mean(y)
• Median: median(y) (robust for asymmetric distributions)
• Trimmed mean: mean(y, trim=0.1) (removes a percentage of observations from tails)
• Variance ($\operatorname{Var}(y), \sigma^2$): Theoretical measure of variability. Estimated by:
• Variance: var(y) ($s^2$)
• Standard deviation: sd(y) ($s$)
• Standard error of the mean (SEM): $s / \sqrt{n}$
• Range: min(y), max(y)
• Quantiles: quantile(y) (0%, 25%, 50%, 75%, 100%)
• Confidence Intervals (CI): For mean, typically based on t-distribution: $\overline{y} \pm t_{0.975,\nu} \times SEM$. For asymmetric distributions, CIs are estimated on transformed values.

Tabular Analysis

• Basic summaries: summary() provides min, max, quantiles, median, mean for all variables.
• Summary tables for explanatory variables: tapply(X=response, INDEX=list(factor1, factor2), FUN=mean).
• Frequencies: table().

Graphics for Data Visualization

Graphical displays are powerful for EDA. Key functions include:

• plot(): Basic scatterplots. Arguments: type, las, xlab, ylab, cex.lab, xlim, ylim, cex.axis, log, main, main.cex, pch (symbol type), cex (symbol size), col (color), font.
• hist(): Histograms for studying distribution.
• stem(): Stem-and-leaf plots.
• qqnorm(), qqline(), qqplot(): Q-Q plots for checking normality or comparing distributions.
• boxplot(): Box plots for categorical explanatory variables, with notch for CI95 of median.
• xyplot() (from lattice library): Advanced panel plots for continuous and categorical variables.
• interaction.plot(): For two categorical variables, plots means connected by lines.
• barplot(): For counts or proportions.
• pairs(): Matrix of all possible scatterplots for several continuous variables.
• wireframe() (from lattice library): 3D plots for two continuous explanatory variables.
• ggplot2: For elegant and customizable plots.
• visreg: For plotting estimated models.

Collinearity

Collinearity occurs when two or more explanatory variables are correlated. This can complicate the interpretation of effects. Principal Component Analysis (PCA) can be used to reduce the dimensionality of correlated variables by using PCA scores instead of original variables.

General Linear Models (GLM) and Modeling Procedure

Understanding GLM: Beyond Linear Models

General Linear Models (GLM) extend the traditional Linear Model (LM) by allowing for non-normal response variables and non-linear relationships between the mean of the response and the linear predictor.

LM form: y = α + β_1 x_1 +... + β_k x_k + ε, where ε ~ N(0, σ^2). It assumes the response variable y itself is normally distributed and the relationship is linear in parameters. Many non-linear relationships can be linearized through transformations (e.g., log(y) = a + bx + ε).

GLM extends this with three components:

1. Random Component: Specifies the probability distribution of the response variable (y ~ distribution).
2. Systematic Component: The linear predictor (α + β_1 x_1 +... + β_k x_k).
3. Link Function ($f(\mu)$): Connects the mean of the response variable ($\mu$) to the linear predictor. For example:

• Gaussian distribution uses identity link: $f(\mu) = \mu$.
• Poisson distribution uses log link: $f(\mu) = \log(\mu)$.
• Binomial distribution uses logit link: $f(\mu) = \log(\frac{\mu}{1-\mu})$.

GLMs use deviance as a measure of fit (analogous to sum of squares in LM). Null deviance is like SST, and residual deviance is like SSE.

Modeling Procedure: Building and Refining Models

There are two main approaches to building a statistical model:

• Bottom-up (forward selection): Start with a simple model and progressively add variables based on significance.
• Top-down (backward selection): Start with a maximal (saturated) model including all main effects and interactions, and then remove insignificant terms.

Recommended top-down procedure for robust statistical analysis with R:

1. Fit maximal model: Include all main effects and interactions.
2. Remove insignificant terms: Start with higher-order interactions, then main effects. Intercept is marginal to slope, slope is marginal to quadratic term.
3. Group similar factor levels: If a factor has multiple levels, group those with statistically similar effects.
4. Check diagnostic plots: Critically evaluate model assumptions (normality, homogeneity of variance, independence, influence).
5. Alter model if necessary: Based on diagnostic checks, transform variables, change distribution, or remove influential observations.
6. Achieve minimal adequate model: The final model should contain only terms where all parameters are significantly different from zero. Criteria for removal include p-values (from F-tests or $\chi^2$-tests) or Akaike Information Criterion (AIC), where a lower AIC indicates a better balance between fit and complexity.

FAQ: Statistical Analysis with R for Students

What is a minimal adequate model in R statistical analysis?

A minimal adequate model is the simplest model that adequately explains the phenomenon under study. It contains only terms whose parameters are statistically significant. The goal is to simplify reality while retaining explanatory power, adhering to the principle of parsimony.

Why are diagnostic plots important in R statistical analysis?

Diagnostic plots are crucial for informally checking the assumptions of your statistical model, such as normality of residuals, homogeneity of variance, and independence of errors. They help identify potential issues like omitted polynomial terms, non-normal distributions, or influential data points that could invalidate your model's conclusions.

What is the difference between an LM and a GLM in R?

A Linear Model (LM) assumes a normally distributed response variable and a linear relationship between the response and predictors. A Generalized Linear Model (GLM) is an extension that allows for non-normal response variables (e.g., counts, proportions) by using a specified probability distribution and a "link function" to connect the mean of the response to the linear predictor.

How does R handle collinearity between explanatory variables?

R doesn't automatically "handle" collinearity in standard lm() or glm() functions, but it provides tools to address it. You can identify collinearity by looking at correlation matrices. Solutions often involve removing one of the correlated variables, combining them, or using techniques like Principal Component Analysis (PCA) to create uncorrelated composite variables.

When should I use weights in my R regression model?

You should use weights in your regression model when observations have different levels of precision or reliability. For example, if some measurements are based on a larger sample size, they might be more reliable and should be given more weight. The weights argument in lm() allows you to incorporate this differential precision into your analysis, leading to more accurate parameter estimates. This is often crucial in situations where the error variance is not constant across observations.`, 0, 50,