15.4 Biostatistics: Sampling, Central Tendency & Significance Testing

Biostatistics: Sampling, Central Tendency & Significance Testing

1. Classification of Variables / Data Types

Data collected in biomedical studies are broadly classified into two categories:

+---------------------------------------------------------------------------------------------------+
|                                      TYPES OF BIOMEDICAL DATA                                     |
+---------------------------------------------+-----------------------------------------------------+ 
|           QUALITATIVE (CATEGORICAL)         |               QUANTITATIVE (NUMERICAL)              |
+----------------------+----------------------+----------------------+------------------------------+
|       Nominal        |       Ordinal        |       Discrete       |          Continuous          |
+----------------------+----------------------+----------------------+------------------------------+
| Unordered categories | Ordered ranks/scales | Whole counts only    | Measured on real continuum   |
| • Blood Group (A/B/O)| • Cancer Stage (I-IV)| • Number of children | • Serum Bilirubin (mg/dL)    |
| • Sex (Male/Female)  | • Socioeconomic status| • Pulse rate/min    | • Hemoglobin (g/dL)          |
| • Eye color          | • Severity (Mild-Mod)| • Bed occupancy count| • Height (cm), Weight (kg)   |
+----------------------+----------------------+----------------------+------------------------------+

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2. Measures of Central Tendency & Skewness

A. Summary Measures

• Mean (Arithmetic Average): Sum of all observations divided by total count ($n$). Highly sensitive to extreme values (outliers). Best measure of central tendency for normally distributed continuous data.
• Median (50th Percentile): The middle value when data are arranged in ascending order. Unaffected by extreme outliers. Best measure for skewed numerical data or ordinal data.
• Mode: The most frequently occurring value in a dataset. Useful for nominal data or identifying bimodal distributions.

B. Relationships in Skewed Distributions

$\text{Normal (Symmetrical) Distribution: } \text{Mean} = \text{Median} = \text{Mode}$ $\text{Positive (Right-Tailed) Skew: } \text{Mean} > \text{Median} > \text{Mode}$ $\text{Negative (Left-Tailed) Skew: } \text{Mean} < \text{Median} < \text{Mode}$

Examples: Serum IgE levels and income distributions exhibit positive skew. Age at death in developed nations exhibits negative skew.

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3. Measures of Dispersion (Variability)

• Range: Difference between maximum and minimum values. Highly unstable.
• Interquartile Range (IQR): $Q3 - Q1$ (75th percentile minus 25th percentile). Accompanies Median for skewed data.
• Standard Deviation (SD): Measures the scatter of individual observations around the sample mean: $SD = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}}$
• Standard Error of Mean (SEM): Measures the sampling variability of sample means around the true population mean: $SEM = \frac{SD}{\sqrt{n}}$
• 95% Confidence Interval (95% CI): The range within which the true population parameter lies with 95% probability: $95\% \text{ CI for Mean} = \text{Mean} \pm (1.96 \times SEM)$
• Coefficient of Variation (CV): Relative measure of dispersion expressed as a percentage, enabling comparison between variables measured in different units (e.g., comparing height in cm vs weight in kg): $CV = \frac{SD}{\text{Mean}} \times 100$

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4. Normal (Gaussian) Distribution

The normal distribution is a symmetrical, bell-shaped continuous probability distribution defined by parameters $\mu$ (mean) and $\sigma$ (standard deviation).

+---------------------------------------------------------------------------------------------------+
|                                 GAUSSIAN NORMAL DISTRIBUTION CURVE                                |
+---------------------------------------------------------------------------------------------------+
|                                         |                                                         |
|                                    /    |    \                                                    |
|                                  /      |      \                                                  |
|                                /        |        \                                                |
|                              /          |          \                                              |
|                            /            |            \                                            |
|                          /              |              \                                          |
|                         |               |               |                                         |
|   ------------+---------+---------------+---------------+---------+------------                   |
|             μ-3σ      μ-2σ            μ-1σ     μ      μ+1σ    μ+2σ      μ+3σ                      |
|             |<------------ 68.27% ------------>|                                                  |
|             |<----------------------- 95.45% ----------------------->|                            |
|             |<--------------------------------- 99.73% -------------------------------->|          |
+---------------------------------------------------------------------------------------------------+

• Area Coverage Limits:
  • $\text{Mean} \pm 1 SD$ encompasses 68.27% of values.
  • $\text{Mean} \pm 2 SD$ encompasses 95.45% of values (specifically $\pm 1.96 SD$ encompasses 95.0%).
  • $\text{Mean} \pm 3 SD$ encompasses 99.73% of values.

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5. Sampling Techniques

A. Probability Sampling (Every element has known non-zero selection chance)

1. Simple Random Sampling: Every individual in the sampling frame has an equal chance of selection (lottery or random number generator).
2. Systematic Random Sampling: Every $k^{\text{th}}$ unit is selected after a random starting point ($k = N/n$).
3. Stratified Random Sampling: Population is divided into homogeneous strata (e.g., age groups, socioeconomic strata), and a random sample is drawn from each stratum. Reduces sampling error.
4. Cluster Sampling: Population is divided into heterogeneous clusters (e.g., villages, schools). A random selection of clusters is chosen, and all individuals within selected clusters are sampled.
   • WHO 30x7 Cluster Sampling: Used globally for immunization coverage surveys. Evaluates 30 clusters of 7 children each aged 12-23 months (total $N = 210$).
5. Multistage Sampling: Sampling is carried out in successive hierarchical stages (e.g., State $\rightarrow$ District $\rightarrow$ Village $\rightarrow$ Household).

B. Non-Probability Sampling

Includes Convenience, Purposive/Judgemental, Quota, and Snowball Sampling (used for hard-to-reach populations like IV drug users or sex workers).

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6. Hypothesis Testing & Choice of Statistical Tests

Error Matrix in Hypothesis Testing

• p-value: The probability of obtaining the observed result (or more extreme) purely by chance if $H_0$ is true. If $p < 0.05$, $H_0$ is rejected.

Statistical Test Selection Algorithm

$\chi^2 = \sum \frac{(O - E)^2}{E} \quad \text{with Degrees of Freedom } df = (r - 1)(c - 1)$