Biostatistics: Basic Statistical Concepts, Sampling Methods, and Variables

Basic Concepts of Statistics

• Definition of Statistics (الإحصاء):

  • Statistics is the scientific method that specializes in collecting data and facts about a specific phenomenon or a specific group of phenomena.

  • It encompasses organizing and classifying this data and facts in a manner that facilitates their analysis and interpretation.

  • It provides the framework for drawing conclusions and making sound decisions in light of the analyzed results.

• Main Branches of Statistics:

  • Descriptive Statistics (الإحصاء الوصفي):

  • Concerned with summarizing and describing data using a set of statistical tools.

  • Goal: Organize and summarize data in a way that helps to understand it quickly and effectively.

  • Inferential Statistics (الإحصاء الاستدلالي):

  • Used to interpret data and draw general conclusions from a sample of data.

  • Used specifically for hypothesis testing and predicting future trends.

  • Key elements include Hypothesis testing and Data analysis.

• Scope of Application of Statistics (مجال تطبيق علم الإحصاء):

  • Used in every field that involves scientific research.

  • Applications encompass:

  • Agricultural

  • Industrial

  • Psychological

  • Social

  • Sports and Youth

  • Medical

  • Economic

  • Administrative

  • Engineering

Stages of the Statistical Method

• Stages of the Statistical Method (مراحل الطريقة الإحصائية):

  • Stage 1: Defining a problem or research hypothesis.

  • Stage 2: Collecting data and information about the phenomenon or phenomena related to the research.

  • Stage 3: Classifying, tabulating, and displaying data (جمع وتصنيف وتبويب البيانات).

  • Stage 4: Calculating statistical indicators, such as estimating the parameters of the research community (population).

  • Stage 5: Analyzing the study data and reaching the results.

  • Stage 6: Interpreting the results and making a decision regarding the research hypotheses.

Statistical Population and Statistical Term

• Statistical Population / Community (المجتمع الإحصائي):

  • Comprises all components of the phenomenon that is the subject of research or study.

  • These components are supposed to share a certain characteristic or set of characteristics.

  • Components may be a living organism or anything else.

• Classification of Populations:

  • Finite Population (المجتمع المحدد):

  • A population that consists of a limited and known number of individuals or elements.

  • All of its elements can be counted accurately.

  • Infinite Population (المجتمع غير المحدد):

  • A society/population that consists of an unlimited or unknown number of individuals or elements.

  • Completely counting all elements is impossible due to the sheer size or unlimited nature of the population.

• Statistical Term / Unit / Item (المفردة الإحصائية):

  • The smallest unit in the statistical community.

Data Collection Sources and Methods

• Sources of Research Data (مصادر البيانات):

  • Historical Sources (المصادر التاريخية):

  • Data and information stored and collected by various state agencies and institutions as a result of surveys conducted by these agencies or bodies for their own purposes.

  • Examples include:

    • Population census data

    • Agricultural and industrial production statistics

    • Foreign and domestic trade statistics

    • Statistics on students graduating from Iraqi universities

  • Field Sources (المصادر الميدانية):

  • Used when it is not possible to obtain data from historical sources.

  • Involves obtaining data directly from its original sources in the field.

• Methods of Collecting Data from the Field (أساليب جمع البيانات من الميدان):

  • Comprehensive Inventory Method (أسلوب الحصر الشامل):

  • Data is collected on all elements of the statistical community.

  • The statistical community must be specific and finite (every element in it can be observed).

  • Examples:

    • General population census

    • Inventory of establishments and industrial units in Iraq

  • Sample Method (أسلوب العينات):

  • Means collecting data and information about a specific group of statistical community items.

  • This chosen group of items is called a sample.

  • Chosen in a way that ensures it accurately represents the statistical community.

Random Sampling Methods

• Random Sample (العينة العشوائية):

  • A group of individuals selected from the statistical community in which the researcher has no interference in the selection process.

• Types of Random Samples:

  • 1. Simple Random Sample (العينة العشوائية البسيطة):

  • Chosen randomly to ensure that every single item in the statistical community has the same probability of appearance in the sample.

  • Condition: The statistical community must be homogeneous with respect to the characteristic(s) related to the research.

  • Mathematical Procedure:

    • Assume a homogeneous, finite statistical population with $N$ items.

    • When choosing a simple random sample of size $n$, each item has an equal probability of appearance equal to $\frac{1}{N}$.

    • The total number of unique simple random samples that can be selected from this population is calculated using the combinations formula:         $C_n^N = \frac{N!}{n!(N-n)!}$

  • Example:

    • A homogeneous statistical population consists of $N = 4$ items: $A$, $B$, $C$, $D$.

    • Goal: Select a simple random sample of $n = 3$ items.

    • Number of simple random samples:         $C_3^4 = \frac{4!}{3!(4-3)!} = \frac{24}{6} = 4$

    • Factorial expansion: $4! = 4 \times 3 \times 2 \times 1 = 24$

    • Probability of drawing any individual item: $\frac{1}{4} = \frac{1}{N}$

    • The $4$ possible simple random samples are:

    • $A, B, C$

    • $A, B, D$

    • $A, C, D$

    • $B, C, D$

  • 2. Stratified Random Sample (العينة الطبقية العشوائية):

  • Considered the best and most accurate sampling type for representing a heterogeneous statistical population.

  • In many cases, population items are non-homogeneous regarding the studied trait (e.g., studying family income where families are high-income, medium-income, or low-income).

  • Drawing a simple random sample from a heterogeneous population is invalid because it leads to biased estimates toward one class.

  • Procedure:

    • Partition the heterogeneous population $N$ into $L$ strata based on common characteristics among items within each stratum.

    • Stratum sizes satisfy: $N_1 + N_2 + \dots + N_L = N$.

    • Draw a simple random sample from each stratum proportional to its size in the population (Proportional Distribution Method / طريقة التوزيع المتناسب).

  • Proportional Distribution Formulas:

    • Weight of stratum $h$ in the population:         $W_h = \frac{N_h}{N}$         for $h = 1, 2, 3, \dots, L$, where $W_1 + W_2 + \dots + W_L = 1$

    • Size of simple random sample drawn from stratum $h$:         $n_h = n \times W_h = n \times \frac{N_h}{N}$

    • Total sample size $n$:         $n = n_1 + n_2 + \dots + n_L$

    • Note that ratio of contribution is preserved:         $\frac{n_h}{n} = \frac{N_h}{N}$

  • Example:

    • Population of $N = 2200$ families.

    • High-income families ($N_1$): $700$

    • Medium-income families ($N_2$): $900$

    • Low-income families ($N_3$): $600$

    • Required: Draw a stratified random sample of $n = 110$ families using proportional distribution.

    • Calculation of weights:         $W_1 = \frac{700}{2200} = \frac{7}{22}$         $W_2 = \frac{900}{2200} = \frac{9}{22}$         $W_3 = \frac{600}{2200} = \frac{6}{22}$

    • Calculation of sample sizes:         $n_1 = \frac{7}{22} \times 110 = 35$         $n_2 = \frac{9}{22} \times 110 = 45$         $n_3 = \frac{6}{22} \times 110 = 30$

    • Proportionality confirmation:         $W_1 = \frac{7}{22} = \frac{35}{110}$         $W_2 = \frac{9}{22} = \frac{45}{110}$         $W_3 = \frac{6}{22} = \frac{30}{110}$

  • Homework Exercise:

    • A community consists of $N = 1000$ people distributed by income into three classes: $200$ low-income, $550$ middle-income, and $250$ high-income people. Draw a random sample of $n = 100$ people representing all classes of society.

  • 3. Systematic Random Sample (العينة العشوائية المنتظمة):

  • Population of $N$ items is divided into $n$ groups, each group containing $k$ items, where:       $k = \frac{N}{n}$

  • Items must be arranged according to a specific system (e.g., ascending, descending, or address order along a street).

  • Selection Method:

    1. Select one item randomly from the first group of $k$ items (let its sequence number be $a$).

    2. Select subsequent items by adding interval $k$ sequentially:

    • First item: $a$

    • Second item: $a + k$

    • Third item: $a + 2k$

    • Continuing up to the last item.

    1. All items after the first are separated by equal intervals $k$

  • Number of Possible Samples:

    • The number of systematic random samples that can be selected from the population equals the number of items in each group, $k$.

  • Example:

    • $N = 24$ students arranged in descending order of grades.

    • Desired sample size $n = 6$ students.

    • Step 1: Calculate interval $k$:         $k = \frac{N}{n} = \frac{24}{6} = 4$

    • Students divided into $6$ groups of $4$ students each:

    • Group 1: Items $1, 2, 3, 4$

    • Group 2: Items $5, 6, 7, 8$

    • Group 3: Items $9, 10, 11, 12$

    • Group 4: Items $13, 14, 15, 16$

    • Group 5: Items $17, 18, 19, 20$

    • Group 6: Items $21, 22, 23, 24$

    • Step 2: Select a student randomly from Group 1 (e.g., sequence number $3$).

    • Step 3: Add $k = 4$ sequentially to obtain remaining sequences:

    • Student 1: Sequence $3$

    • Student 2: $3 + 4 = 7$

    • Student 3: $7 + 4 = 11$

    • Student 4: $11 + 4 = 15$

    • Student 5: $15 + 4 = 19$

    • Student 6: $19 + 4 = 23$

    • Final systematic random sample sequence: $(3, 7, 11, 15, 19, 23)$.

  • Homework Exercises:

    • H.W. 1: How do we select a regular random sample of $n = 50$ students from a population of $N = 500$ students?

    • H.W. 2: How do we choose a regular random sample from a population of $N = 3000$ teachers, where the desired sample size is $n = 500$ teachers?

  • 4. Multistage Sampling (المعاينة متعددة المراحل):

  • The population is divided into hierarchical sampling units across multiple stages:

    • Stage 1: Divide population into primary units; select a simple random sample of primary units.

    • Stage 2: Divide selected primary units into smaller secondary units; select a simple random sample of secondary units.

    • Stage 3: Divide selected secondary units into smaller units; select a simple random sample from them.

    • Continue subdivision and random selection until reaching the final units from which data is collected.

  • Example: Estimating average sugar consumption of an Iraqi family (statistical unit = family):

    • Stage 1: Divide Iraq into governorates (primary units); randomly select a sample of governorates.

    • Stage 2: Divide selected governorates into districts (secondary units); randomly select a sample of districts.

    • Stage 3: Divide selected districts into sub-districts (nawahis); randomly select a sample of sub-districts.

    • Stage 4: Divide selected sub-districts into residential neighborhoods (mahallat); randomly select a sample of neighborhoods.

    • Stage 5: Divide selected neighborhoods into alleys (azqah); randomly select a sample of alleys.

    • Stage 6: Divide selected alleys into residential houses; randomly select houses to reach the families for data collection.

Non-Random Sampling Methods

• Non-Random Samples (العينات غير العشوائية):

  • A group of items selected from the statistical population in a way where the researcher interferes in the selection process based on research considerations.

• Types of Non-Random Samples:

  • 1. Intentional / Purposive Sample (العينة العمدية):

  • Selected deliberately based on prior belief that items are best suited to represent the study population.

  • Example: Studying ways to develop football sports — deliberately selecting a sample of football specialists due to their experience in sports management.

  • 2. Quota Sample (العينة الحصصية):

  • Statistical population is divided into several strata based on criteria relevant to the study.

  • An intentional (non-random) sample is chosen from each stratum proportional to the stratum's size in the population.

  • The total sum of these intentional sample sizes forms the quota sample size.

Variables and Their Classification

• Definition of Variable (المتغيرات):

  • Any characteristics, number, or quantity capable of being measured or counted.

  • Also referred to as a data element (e.g., age, gender, expenses, country of birth).

  • Called a variable because its value varies across units in the population or over time.

  • Represented by symbols such as $X$, $Y$, $Z$, etc.

• Classification of Variables:

  • 1. Quantitative Variables (المتغيرات الكمية):

  • Variables that take numerical values and can be measured.

  • Discrete Variables (المتغيرات المنفصلة):

    • Take specific numerical values, often integers (صحيحة).

    • Examples: Number of students in a classroom, number of cars in a parking lot.

  • Continuous Variables (المتغيرات المتصلة):

    • Can take any numerical value within a given range, often including decimals.

    • Examples: Weight of people, height of buildings.

  • 2. Qualitative / Descriptive Variables (المتغيرات النوعية / الوصفية):

  • Variables describing categories or attributes that cannot be measured numerically.

  • Examples: Gender (male/female), eye color (blue, brown, green), social status (rich, middle class, poor).

• Classification Identification Exercises:

  • Intelligence score (درجة الذكاء): Descriptive / qualitative variable ($\text{متغير وصفي}$

  • Type of college (نوع الكلية): Descriptive / qualitative variable ($\text{متغير وصفي}$

  • Phone number (رقم الهاتف): Discrete quantitative variable ($\text{متغير كمي منفصل}$

  • Family monthly income (الدخل الشهري للأسرة): Continuous quantitative variable ($\text{متغير كمي متصل}$

  • Blood type (فصيلة الدم): Descriptive / qualitative variable ($\text{وصفي}$

  • Number of children in the family (عدد أطفال الأسر): Discrete quantitative variable ($\text{كمي منفصل}$

  • Skin color (لون البشرة): Descriptive / qualitative variable ($\text{وصفي}$

  • Gender (الجنس): Descriptive / qualitative variable ($\text{وصفي}$

  • Cotton yield (محصول القطن): Continuous quantitative variable ($\text{كمي متصل}$

  • Student grade (درجة الطالب): Discrete quantitative variable ($\text{كمي منفصل}$

  • Number of cars in the parking lot (عدد السيارات في موقف السيارات): Discrete quantitative variable ($\text{كمي منفصل}$

  • Number of students in class (عدد الطلاب في الفصل): Discrete quantitative variable ($\text{كمي منفصل}$