9.3 Likelihood ratio tests

Likelihood ratio tests are a powerful tool in hypothesis testing, comparing the probability of observed data under different hypotheses. They help assess which hypothesis makes the data most probable, providing a flexible approach for various statistical models and data types.

These tests are widely used in fields like genetics, medicine, and economics. By following specific steps to construct and interpret likelihood ratio tests, researchers can make informed decisions about hypotheses, balancing statistical significance with practical implications.

Likelihood Ratio in Hypothesis Testing

Concept and Principles

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• Likelihood ratio compares probability of observed data under two different hypotheses
• Assesses strength of evidence favoring one hypothesis over another
• Based on maximum likelihood estimation principle selects hypothesis making observed data most probable
• General form $λ = L(θ0) / L(θ1)$, where $L(θ0)$ represents likelihood under null hypothesis and $L(θ1)$ under alternative hypothesis
• Test statistic often expressed as $-2ln(λ)$, asymptotically follows chi-square distribution under certain conditions
• Considered powerful and flexible approach to hypothesis testing, particularly for complex statistical models
  • Adapts well to various types of data and hypotheses
  • Provides consistent framework for comparing nested models

Applications in Different Distributions

• Normal distributions use likelihood ratio tests to compare means, variances, or both simultaneously
  • Example comparing means: $H_0: μ = μ_0$ vs $H_1: μ ≠ μ_0$
• Binomial distributions construct tests for hypotheses about probability of success
  • Example testing coin fairness: $H_0: p = 0.5$ vs $H_1: p ≠ 0.5$
• Poisson distributions employ tests to compare rate parameters between populations
  • Example comparing event rates: $H_0: λ_1 = λ_2$ vs $H_1: λ_1 ≠ λ_2$
• Exponential distribution tests often applied in reliability analysis and survival studies
  • Example testing mean lifetime: $H_0: θ = θ_0$ vs $H_1: θ ≠ θ_0$
• Multivariate distributions involve considering joint probability density function and appropriate parameter spaces
  • Example testing correlation in bivariate normal: $H_0: ρ = 0$ vs $H_1: ρ ≠ 0$

Constructing Likelihood Ratio Tests

General Steps

• Specify null and alternative hypotheses clearly defining parameter space
• Write likelihood functions for both hypotheses based on probability model
• Calculate likelihood ratio by dividing null hypothesis likelihood by alternative hypothesis likelihood
• Determine distribution of test statistic under null hypothesis often using asymptotic properties
• Derive critical region based on chosen significance level and test statistic distribution
• Compute observed test statistic value using sample data
• Compare observed test statistic to critical region to make decision

Specific Examples

• Normal distribution mean test
  • Null hypothesis: $H_0: μ = μ_0$
  • Alternative hypothesis: $H_1: μ ≠ μ_0$
  • Likelihood ratio: $λ = \frac{L(μ_0, σ^2)}{L(μ, σ^2)}$
• Binomial distribution proportion test
  • Null hypothesis: $H_0: p = p_0$
  • Alternative hypothesis: $H_1: p ≠ p_0$
  • Likelihood ratio: $λ = \frac{L(p_0)}{L(p)}$
• Poisson distribution rate test
  • Null hypothesis: $H_0: λ = λ_0$
  • Alternative hypothesis: $H_1: λ ≠ λ_0$
  • Likelihood ratio: $λ = \frac{L(λ_0)}{L(λ)}$

Critical Region for Likelihood Ratio Tests

Determining Critical Values

• Critical region represents set of values for test statistic leading to null hypothesis rejection
• Typically based on asymptotic distribution of $-2ln(λ)$, often chi-square
• Degrees of freedom for chi-square distribution depend on difference in free parameters between null and alternative hypotheses
• Significance level (α) determines critical value from appropriate chi-square distribution
• Critical region usually takes form $\{-2ln(λ) > c\}$, where c represents critical value from chi-square distribution
• Power analysis assesses probability of correctly rejecting false null hypothesis helping determine appropriate sample size
  • Involves calculating Type II error rate (β) and power (1 - β)
  • Helps balance between Type I and Type II errors

Practical Considerations

• Choose significance level based on research context and consequences of errors
• Consider trade-offs between Type I and Type II errors when setting critical region
• Adjust critical values for multiple comparisons using methods (Bonferroni correction)
• Evaluate assumptions of asymptotic distributions for small sample sizes
• Use simulation or bootstrapping techniques for complex models lacking closed-form distributions
• Interpret p-values in conjunction with effect sizes for comprehensive analysis

Likelihood Ratio Tests in Practice

Real-World Applications

• Genetics uses tests to compare inheritance models or gene expression patterns
  • Example comparing dominant vs recessive inheritance models
• Medical research applies tests to assess treatment effectiveness or diagnostic accuracy
  • Example evaluating new drug efficacy compared to placebo
• Economics and finance utilize tests for model selection and hypothesis testing in time series analysis
  • Example testing for structural breaks in economic indicators
• Quality control employs tests to detect changes in manufacturing processes
  • Example monitoring production line for shifts in mean or variance
• Signal processing and communication theory use tests for detecting and classifying signals in noisy environments
  • Example distinguishing between different modulation schemes in wireless communications

Implementation Steps

• Define research question and corresponding hypotheses clearly stating parameters of interest
• Select appropriate probability model for data considering underlying assumptions
• Estimate model parameters using maximum likelihood methods
  • Numerical optimization techniques often required for complex models
• Calculate likelihood ratio and test statistic using estimated parameters
• Compare test statistic to critical value or compute p-value
• Interpret results in context of original problem considering practical significance
• Perform sensitivity analysis to assess robustness of conclusions
• Communicate findings clearly including limitations and potential sources of bias