Fundamentals of Biostatistics - Unit 1 Notes

Origin and Basic Meanings of Statistics

• The word statistics is derived from various European languages, each sharing a core meaning related to a "political state":

  • Latin word: STATUS.

  • Italian word: STATISTA.

  • German word: STATISTIK.

• In common parlance, the term "Statistics" is often used synonymously with "data."

• The fundamental purpose of data in this context extends beyond simple collection; it encompasses a comprehensive methodology for handling, analyzing, and drawing valid inferences (interpretation) from numerical information.

Dual Senses of Statistics: Plural and Singular

• The term "statistics" is generally employed in two distinct senses: the plural sense and the singular sense.

• Plural Sense: This refers to statistics as numerical data or statistical data itself. It constitutes the raw facts and figures collected for study.

• Singular Sense: This refers to statistics as a science or a branch of knowledge. It involves the techniques or methods used for collecting, classifying, presenting, analyzing, and interpreting data.

Definitions and Features in the Plural Sense

• Definition by Bowley: "Statistics are numerical statements of facts in any department of enquiry placed in relation to each other."

• Definition by Yule and Kendall: "By Statistics we mean quantitative data affected to a market extent by multiplicity of causes."

• Key Features of Statistics in the Plural Sense:

  • Aggregate of Facts: Statistics do not refer to a single isolated figure; they represent a collection of many facts.

  • Numerically Expressed: Qualitative observations must be converted into numerical form to be considered statistics.

  • Affected by Multiplicity of Causes: Statistical data are influenced by numerous factors rather than a single cause.

  • Standard of Accuracy: Data must be enumerated or estimated according to a reasonable and predetermined standard of accuracy.

  • Systematic Collection: Data should be gathered in a planned and orderly manner.

  • Predetermined Purpose: Statistics are collected with a specific goal or objective in mind.

  • Relational Placement: Data points are placed in relation to one another to facilitate comparison.

Definitions and Features in the Singular Sense (Statistics as a Science)

• Definition by Croxton and Cowdon: "Statistics may be defined as a collection, presentation, analysis and interpretation of numerical data."

• Definition by Lovitt: "Statistics is the science which deals with the collection, classification and tabulation of numerical facts as a basis for the explanation, description and comparison of phenomena."

• Key Features of Statistics as a Science:

  • Collection of data: The initial gathering of facts.

  • Organization of data: Arranging data into logical patterns or classifications.

  • Presentation of data: Using tables, charts, or graphs for clarity.

  • Analysis of data: Applying statistical tools and tests to examine the data.

  • Interpretation of data: Drawing conclusions and explaining findings.

Applications and Importance of Biostatistics

• Broad Applications:

  • Clinical medicine and Medical science.

  • Community medicine and Public health.

  • Pharmacy and Nursing.

  • Genetical statistics.

  • Agriculture.

  • Demography.

  • Psychology.

• Major Functions and Importance:

  • Factual Presentation: Biostatistics presents complex facts through figures. For example, comparing the population of Nepal in 2017 to 2007 is more logical when stated as a growth from $1$ billion to $1.2$ billion rather than just stating it "increased."

  • Simplification of Complexity: It uses tools like averages, diagrams, and graphs to make complex, non-comprehensive data readily understandable.

  • Facilitation of Comparison: Tools such as ratios, percentages, and coefficients allow researchers to draw valid conclusions. For instance, stating a drug's recovery rate is $3$ days is only useful when compared against a standard drug's rate.

  • Forecasting Future Trends: Time series analysis allows hospital administrators and health officers to predict variables like the number of outpatients expected next year.

  • Policy Formulation: Vital statistics like birth rates, maternal mortality rates, and child death rates are used to create national health policies and family planning programs.

  • Understanding Relationships: Statistical tools like correlation, regression, chi-square tests, relative risk, and odds ratios determine links between different characteristics.

  • Hypothesis Testing: Inferential statistics (e.g., $Z$ test, $t$ test, $F$ test, Chi-square test) are essential for testing scientific hypotheses.

  • Quality Control: Use of graphical control methods such as $P$ charts, $C$ charts, $S$ charts, and $U$ charts ensures consistency and lack of error in laboratory and production activities.

  • Handling Uncertainty: Probability tools help determine the likelihood of events, such as the chance of a wrong diagnosis or the risk of heart disease in smokers.

  • Managing Biological Variation: Statistics provide tools like Range and Standard Deviation ($SD$) to study variations in human characteristics like height, weight, blood pressure, and urea levels.

Data Collection and Classification in Pharmacy

• Data Collection: The process of gathering info for research, which in pharmacy includes drug efficacy, patient responses, or epidemiological studies.

• Best Practices:

  • Units of measurement must be clearly defined.

  • Records should be correct, complete, clear, concise, and easy to comprehend.

• Types of Data:

  • Primary Data: Collected directly from experiments or surveys. Example: Measuring the blood pressure of patients on a new antihypertensive drug.

  • Secondary Data: Collected from existing sources like journals, databases, or hospital records. Example: Meta-analysis of previous clinical trials.

• Classification of Data: The process of grouping data based on shared characteristics to make it easier to interpret.

  • Qualitative (Categorical): Descriptive and non-numerical (e.g., Gender, Drug Type, Side Effect Type).

  • Quantitative (Numerical): Measurable in numbers (e.g., Blood pressure, drug concentration).

• Methods of Classification:

  • Discrete Classification: Data in distinct categories. Example: Grouping drugs as antibiotics, analgesics, or antihypertensives.

  • Continuous Classification: Data grouped into class intervals or ranges. Example: Blood glucose levels categorized as $70\text{--}100$, $101\text{--}120$, and $121\text{--}140\,mg/dL$.

Presentation of Data: Tabulation

• Principles of Presentation:

  • Should arouse reader interest and be concise without losing detail.

  • Must be in a simple form for quick impressions and suggest solutions to problems.

• Methods of Presentation: Tabulation and Graphical Presentation.

• Tabulation of Data: The arrangement of data in rows and columns.

  • Simple/One-way Table: Contains data for a single characteristic (e.g., a Frequency distribution).

  • Two-way Table: Contains data based on two characteristics.

  • Manifold Table: Contains data for three or more characteristics.

• Standard Principles for Tables:

  • Must be numbered and contain a brief, top-aligned title.

  • Rows and columns must be clearly defined with specified units of measurement.

  • The number of class intervals should be balanced (not too many or too few) and must not overlap.

  • Standard codes and symbols should be explained in footnotes.

  • Sources for secondary data must be cited at the bottom.

Constructing a Frequency Distribution

• Step-by-Step Procedure:

  1. Sort Data: Arrange in ascending order.

  2. Calculate Range: Identify the minimum and maximum values.

  3. Decide Intervals: Choose a number of intervals that summarizes without being too broad or too detailed.

  4. Determine Intervals: Create non-overlapping, all-encompassing intervals.

  5. Tally: Count observations falling into each interval.

• Example Construction:

  • Given $20$ observations ranging from $1.2$ to $9.8$.

  • Intervals chosen: $5$ intervals with a width of $2$ ($0\text{--}2$, $2\text{--}4$, $4\text{--}6$, $6\text{--}8$, $8\text{--}10$).

Graphical Presentation of Data

• Introduction: Graphics complement tables to summarize data attractively. They should be self-explanatory with clear captions and indices.

• Bar Diagram: Mutually exclusive discrete data. The length of the bar indicates frequency. Categories on one axis, frequency on the other.

  • Simple Bar Diagram: Single variable for different categories.

  • Multiple/Compound Bar Chart: Each observation has multiple values (e.g., Percentage of males vs. females across different countries).

  • Component/Proportional Bar Chart: Subdivides a single bar into sections representing the relative proportion of parts within a total (e.g., energy intake divided into Protein, Fat, and Carbohydrate).

• Histogram: Similar to bar charts but bars are adherent (no gaps), used for continuous data and class frequency tables.

• Frequency Polygon: Created by connecting the midpoints of the tops of histogram rectangles. A smoothed frequency polygon is a Frequency Curve (Normal Curve).

• Pie Diagram: A circle representing $100\%$ of frequency, divided into segments representing proportional compositions.

• Stem-and-Leaf Plot: Data visualization based on place value.

  • Stem: Represents leading digits.

  • Leaf: Represents the last digit.

  • Key: Essential for reading (e.g., $2 \mid 5 = 25$).

• Whiskers Box-Plot: Displays a five-number summary: Minimum score, First Quartile ($Q_1$), Median ($Q_2$), Third Quartile ($Q_3$), and Maximum score.

• General Graph Principles:

  • Variables usually on the X-axis; frequencies on the Y-axis.

  • Lines should not be extrapolated beyond the range of actual values.

  • Graphs are aids for visualization, not final statistical analysis.

Statistical Data Series and Class Intervals

• Individual Series: Data points given independently for each individual. No frequency. Example: Weight of $5$ children: $3, 3.5, 2.5, 4, 2$.

• Discrete Series (Ungrouped): Tabular frequency distribution for discrete variables ($X$ and $f$).

• Continuous Series (Grouped): Numerical variables taking any value within a range.

  • Exclusive (Overlapping): Upper limit of one class is the lower limit of the next (e.g., $10\text{--}20$, $20\text{--}30$).

  • Inclusive (Non-overlapping): Both limits are included in the interval; overlapping is avoided (e.g., $10\text{--}19$, $20\text{--}29$).

  • Open-end Classes: A limit is missing at the first or last interval.

Measures of Central Tendency

• Overview: These provide a "typical" value representing the center of the distribution where about $50\%$ of observations lie above and $50\%$ below.

• Arithmetic Mean ($\bar{x}$): The mathematical average.

  • Individual Series: $\bar{x} = \frac{\sum X}{n}$.

  • Discrete Series: $\bar{x} = \frac{\sum fX}{N}$.

  • Continuous Series: $\bar{x} = \frac{\sum fx}{N}$ (where $x$ is the midpoint).

  • Assumed Mean Method: $\bar{x} = A + \frac{\sum fd}{N}$ (where $A$ is assumed mean and $d = x - A$).

  • Advantages: Rigidly defined, based on all observations, suitable for further math analysis.

  • Disadvantages: Highly affected by extreme values, cannot be computed for open-ended classes or qualitative data.

• Median ($Md$): The middle value in an ordered set; a positional average.

  • Individual Series: If $n$ is odd, $\frac{n+1}{2}$th item; if even, average of the two middle terms.

  • Discrete Series: Value corresponding to cumulative frequency ($c.f.$) equal to or just greater than $\frac{N+1}{2}$.

  • Continuous Series: $Md = L + \frac{\frac{N}{2} - c.f.}{f} \times i$.

  • Advantages: Robust (not affected by extreme values), appropriate for qualitative data (intelligence, honesty) and open-ended classes.

• Mode ($Mo$): The value occuring most frequently.

  • Analysis: Determined by inspection. If multiple, it is "bimodal" or "multimodal."

  • Continuous Formula: $Mo = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i$.

  • Relationship: For moderately skewed distributions: $Mode = 3\,Median - 2\,Mean$.

Partition Values: Quartiles, Deciles, and Percentiles

• Quartiles ($Q_i, i = 1, 2, 3$): Divide data into 4 equal parts.

  • $Q_1$ (Lower): $25\%$ below it.

  • $Q_2$: Median ($50\%$ below it).

  • $Q_3$ (Upper): $75\%$ below it.

• Deciles ($D_j, j = 1 \dots 9$): Divide data into 10 equal parts.

• Percentiles ($P_k, k = 1 \dots 99$): Divide data into 100 equal parts.

• Advantages: Useful for dispersion and skewness; Easy to determine for individual/discrete series.

• Disadvantages: Not based on all observations; Requires data rearrangement.

Measures of Dispersion

• Definition: The state of data being spread or the extent to which numerical data vary around an average. It indicates homogeneity vs. heterogeneity.

• Absolute Measures (Same units as original data):

  • Range: Difference between Largest ($L$) and Smallest ($S$) value ($Range = L - S$).

  • Mean Deviation ($MD$): Arithmetic mean of absolute deviations from a central tendency ($MD = \frac{\sum |X - \bar{x}|}{N}$).

  • Variance ($\sigma^2$): The average of squared differences from the mean.

  • Standard Deviation ($\sigma$): The positive square root of variance ($\sigma = \sqrt{\sigma^2}$).

  • Quartile Deviation ($QD$): Half the distance between the third and first quartile ($QD = \frac{Q_3 - Q_1}{2}$).

• Relative Measures (Unitless ratios, often percentages):

  • Coefficient of Range: $\frac{L - S}{L + S}$.

  • Coefficient of Variation ($CV$): $\frac{\sigma}{\bar{x}} \times 100$.

  • Coefficient of SD: $\frac{\sigma}{\bar{x}}$.

  • Coefficient of QD: $\frac{Q_3 - Q_1}{Q_3 + Q_1}$.

Standard Deviation and Variance Details

• Standard Deviation ($SD$): Often called the "root mean square deviation." It is the most robust measure for inferential statistics.

  • Population SD: $\sigma = \sqrt{\frac{\sum(X - \mu)^2}{N}}$.

  • Sample SD: $S = \sqrt{\frac{\sum(X - \bar{x})^2}{n - 1}}$.

• Merits of SD: Summarizes deviation in one figure; differentiates between real differences vs. chance; helps calculate standard error.

• Demerits of SD: Lengthy calculations; gives extra weight to extreme values.

Standard Error (SE)

• Overview: Measures the dispersion of sample means around the true population mean. It indicates the precision of a sample estimate.

• Standard Error of the Mean: $SE_{\bar{x}} = \frac{s}{\sqrt{n}}$.

• Standard Error of a Proportion: $SE_p = \sqrt{\frac{pq}{n}}$.

• Comparison: Standard deviation describes variability within one sample, whereas standard error describes variability of means across multiple samples.

Scales of Measurement

• Nominal Scale: Mutual exclusive categories without magnitude. Arithmetics restricted to counting (freq, percentage, chi-square). Example: Gender, Blood group, Urban/Rural.

• Ordinal Scale: Categories with a meaningful natural order/rank but unequal intervals. Appropriate central tendency: Median. Example: Pain (Mild, Moderate, Severe), Cancer stages (I, II, III, IV).

• Interval Scale: Continuous scale with defined distances between measurements. No absolute zero (zero is arbitrary). Addition/subtraction allowed. Example: Temperature ($^{\circ}C$ or $^{\circ}F$).

• Ratio Scale: Has all properties of interval scale plus a true absolute zero. All mathematical and statistical operations are usable. Example: Blood pressure, serum cholesterol, height, weight, pulse rate.