Lecture on Hypothesis Testing and Statistics

Overview

• Focus on testing hypotheses, statistics, and p-values.
• Statistical inference: Making conclusions about a population based on a sample.
• Population parameters (Greek letters) vs. Sample statistics (Roman letters).

Statistical Inference

• Process of making conclusions about a population parameter from a sample.
• Use of confidence intervals to specify a range of plausible parameter values.

Hypothesis Testing

• Involves forming a null hypothesis (H0) and an alternative hypothesis (H1 or HA).
• Null Hypothesis (H0): Assumes no effect or no change.
• Alternative Hypothesis (H1): Represents the research hypothesis. Can be one-sided (directional) or two-sided (non-directional).

Example 1: Drug Testing

• Aribulin in Metastatic Bladder Cancer:
  • Null Hypothesis: µ ≤ 30% shrinkage.
  • Alternative Hypothesis: µ > 30% shrinkage (one-sided).

Example 2: Coin Tossing

• PM510 Students' Coin Flipping Simulation:
  • Null Hypothesis: µ = 6.98 (same as theoretical value).
  • Alternative Hypothesis: µ ≠ 6.98 (two-sided).

Test Statistics and P-values

• Test Statistic: Summarizes the data for statistical inference.
• P-value: Measures consistency of observed data with null or alternative hypothesis.
  • Larger p-value: More consistent with null hypothesis.
  • Smaller p-value: More consistent with alternative hypothesis.
• Decision based on p-value:
  • p ≤ α (significance level): Reject the null.
  • p > α: Do not reject the null.

Decision Errors

• Type 1 Error (False Positive): Rejecting a true null hypothesis.
• Type 2 Error (False Negative): Failing to reject a false null hypothesis.
• Importance of avoiding Type 1 errors in the biomedical field.

Example Scenarios

• Aribulin and Tumor Shrinkage:
  • Analyzing decisions based on p-values under different scenarios.

Legal System Analogy

• Null Hypothesis: Accused is innocent.
• Alternative Hypothesis: Accused is guilty.
• Type 1 Error: Convicting an innocent person.
• Type 2 Error: Acquitting a guilty person.

Factors Affecting P-values

• Sample size and standard deviation impact p-values.
• The effect size affects statistical significance.

Procedure for Testing Hypotheses

1. Formulate research question and hypotheses.
2. Choose significance level (commonly α = 0.05).
3. Collect data and calculate test statistic and p-value.
4. Make decisions based on comparison of p-value and α.
5. Write conclusions based on statistical significance.