Lecture 1 - BIOL 260 - Biostatistics Complete Study Guide

Principles of Population and Sample Design

Most scientific studies rely on sampling because time and financial resources are limited, making it impossible to measure every individual unit within a population.

Defining Populations and Samples

• Population: The complete set of all units or individuals in a specified group of interest. Populations are typically large and defined by spatial or conceptual boundaries.

  • Examples: All people in the U.S., the CSULB student population, bobcats residing in Joshua Tree Park, or all individuals diagnosed with diabetes.

• Sample: A subset of individuals drawn from the target population that should accurately represent the population.

  • Examples: A sample of registered voters, students enrolled in BIOL 260, bobcats counted via aerial survey, or diabetes patients treated at Long Beach Medical Center.

Properties of a Good Sample

To be representative of the population, a sample must possess three key properties:

1. Random Selection: Selection of individuals must be random to avoid bias.

2. Sufficient Sample Size: The sample must be large enough to increase precision.

3. Independent Selection: The choice of one individual should not influence the choice of another, aiding in avoiding bias and increasing precision.

Sampling Variation, Estimates, and Parameters

Quantifying Sampling Variation

Due to chance, estimated values vary across different samples drawn from the same underlying population.

Consider a hypothetical population containing $27.3\%$ gold turkeys (true parameter = $0.273$). Drawing repeated samples of size $n = 9$ yields varying estimated proportions:

• Sample 1: $\text{Proportion of gold} = \frac{1}{9}$, giving $\text{Pr}[\text{gold}] = 0.11$

• Sample 2: $\text{Proportion of gold} = \frac{2}{9}$, giving $\text{Pr}[\text{gold}] = 0.22$

• Sample 3: $\text{Proportion of gold} = \frac{3}{9}$, giving $\text{Pr}[\text{gold}] = 0.33$

Probability math allows the quantification of expected variation under repeated sampling. Every sample estimate contains an inherent measure of uncertainty. For instance, an estimated proportion of $0.33$ may have an associated uncertainty interval (such as a $95\%$ Confidence Interval) ranging from $0.06$ to $0.62$.

Estimates vs. Parameters

• Parameter: A fixed numerical quantity describing an entire population. Parameters represent exact true values and contain no uncertainty (e.g., population proportion of gold turkeys = $0.273$).

• Estimate (or Statistic): A numerical quantity calculated from sample data used to infer an unknown population parameter. Estimates always carry uncertainty (e.g., sample estimated proportion = $0.33$ with a $95\%$ confidence interval of $0.06 - 0.62$).

Accuracy, Precision, Bias, and Sampling Error

Accuracy and Precision

• Accuracy: The degree to which an estimated value conforms to the true population parameter.

• Precision: The degree of closeness or repeatability among independent measurements of the same quantity.

Accuracy and precision operate independently. High performance in both attributes is necessary to eliminate errors:

• A sample can have high precision but low accuracy if measurements are tightly grouped together far from the true parameter value (e.g., sample mean = $8.3\,\text{units}$ when the true population parameter = $3$).

Sampling Error vs. Bias

• Sampling Error:

  • The chance difference between a sample statistic and the true population parameter.

  • Arises because a sample is, purely by random chance, not perfectly representative of the population.

  • Precision describes the expected spread of estimates resulting from sampling error (quantifying typical sampling error).

  • Increasing sample size directly increases precision and repeatability by reducing the spread of sample statistics.

• Bias:

  • Systematic inaccuracy caused by sampling methods that systematically misrepresent the population.

  • Results in consistent overestimation or underestimation of the true population parameter.

  • Example: Examining returning combat aircraft to decide where to add armor, which misrepresents planes that were shot down and failed to return (survivorship bias).

Sample Size Effects on Distribution and Precision

Consider a larval population with a true mean length of $\text{Mean} = 7.094$ and standard deviation of $SD = 0.374$:

• Sample size $n = 5$: Distribution of sample means shows a mean of $\text{Mean} = 7.094$ and standard deviation of $SD = 0.156$. The wider distribution indicates lower precision.

• Sample size $n = 20$: Distribution of sample means shows a mean of $\text{Mean} = 7.097$ and standard deviation of $SD = 0.084$. The narrower distribution demonstrates higher precision and lower sampling error.

Core Goals of Statistical Inference

Statistical analysis serves two primary objectives: Estimation and Hypothesis Testing.

Estimation

• Definition: The process of inferring an unknown population parameter value from sample statistics and establishing the level of certainty around that estimate.

• Procedure:

  1. Collect sample data.

  2. Calculate a sample statistic (e.g., mean blood pressure).

  3. Apply probability math to establish a range likely to encompass the true parameter (e.g., mean blood pressure = $71.2 \pm 13.1\,\text{mmHg}$).

• Conceptual Basis: Imagines drawing infinite repeated samples of identical size to model expected statistic variability.

• Logical Framework: Employs inductive reasoning (inferring general population rules from specific sample observations).

Hypothesis Testing

• Definition: The process of evaluating whether observed sample data are consistent or inconsistent with a proposed null explanation of how the system functions.

• Procedure:

  1. Formulate a hypothesis explaining a biological phenomenon (e.g., smoking causes cancer).

  2. Collect sample data across comparison groups (e.g., smokers vs. non-smokers).

  3. Establish a baseline test value expected if the hypothesis is false (e.g., equal cancer rates between groups).

  4. Compute the mathematical probability of observing the empirical results if the hypothesis were false.

  5. If the observed data are extremely improbable under the assumption that the hypothesis is false, the hypothesis is supported.

• Logical Framework: Employs deductive reasoning (testing specific predictions derived from hypotheses).

Summary of Core Competencies

When reviewing this foundational material, ensure mastery of the following skills:

• Explaining why sampling variation is critical for statistical inference.

• Differentiating between bias, sampling error, and estimate precision.

• Identifying key sampling design attributes (randomness, sample size, independence) that minimize bias and maximize precision.

• Evaluating study descriptions to identify sources of bias and determine whether the primary objective is estimation or hypothesis testing.