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Testing of Hypothesis: Complete Guide

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1. What is a Hypothesis?

"In a statistical hypothesis, statements are made about the relationship among parameters of one or more populations."

• Barash, Cullen, and Stoelting's Clinical Anesthesia, 9e

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2. Basic Concepts in Hypothesis Testing

2.1 The Null Hypothesis (H₀)

• Example: "There is no difference in blood pressure between the treatment group and the control group."
• Algebraically: H₀: μ₁ = μ₂

"A comparative study commences with a fundamental presumption of 'no difference,' known as the null hypothesis. The null hypothesis posits that the difference between the two treatments being compared is zero."

• Rockwood and Green's Fractures in Adults, 10th ed.

2.2 The Alternative Hypothesis (H₁ or Hₐ)

• Example: "The treatment group has lower blood pressure than the control group."
• Algebraically: H₁: μ₁ ≠ μ₂ (two-tailed) or H₁: μ₁ < μ₂ (one-tailed)

• Two-tailed (non-directional): Tests for any difference, either greater or less (μ₁ ≠ μ₂). More conservative.
• One-tailed (directional): Tests for a difference in one specific direction only (μ₁ > μ₂ or μ₁ < μ₂).

2.3 Level of Significance (α)

"In the medical literature, this threshold is typically set at a value of less than 5% (i.e., p < .05)."

• Rockwood and Green's Fractures in Adults, 10th ed.

2.4 The P-Value

• If p < α: Reject H₀ - result is "statistically significant"
• If p ≥ α: Fail to reject H₀ - result is "not statistically significant"

"The level of significance is the probability level considered too low to warrant support of the null hypothesis being tested. If sample values are sufficiently unlikely to have occurred by chance (i.e., the probability of the sample test statistic is less than the chosen level of significance), the null hypothesis is rejected."

• Barash, Cullen, and Stoelting's Clinical Anesthesia, 9e

2.5 Test Statistic

• Z-statistic (large samples, known σ)
• t-statistic (small samples, unknown σ)
• Chi-square (χ²) (categorical data)
• F-statistic (comparing variances/ANOVA)
• Fisher's Exact Test (small sample categorical comparisons)

2.6 Type I and Type II Errors

• Type I Error (α): Rejecting a true null hypothesis (false positive). Probability = α (the significance level).
• Type II Error (β): Failing to reject a false null hypothesis (false negative).
• Power (1 - β): The probability of correctly rejecting a false H₀. A well-designed study aims for power ≥ 0.80.

"Type I errors are generally considered more serious."

• Statistics LibreTexts

2.7 Critical Region and Critical Value

2.8 Confidence Interval (CI)

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3. Procedure of Hypothesis Testing

Step 1: State the Hypotheses

• H₀ (Null): States no effect, no difference, or no relationship
• H₁ (Alternative): States the expected effect or difference

Step 2: Choose the Level of Significance (α)

Step 3: Select the Appropriate Statistical Test

• Type of data (continuous, categorical, ordinal)
• Number of groups (two groups → t-test; multiple groups → ANOVA)
• Sample size (large → Z-test; small → t-test)
• Distribution of data (parametric vs. non-parametric)

Step 4: Collect Data and Calculate the Test Statistic

Step 5: Determine the Critical Value or P-value

• Critical value method: Compare the test statistic against the critical value from standard tables (Z-table, t-table, etc.)
• P-value method: Calculate the probability of obtaining that test statistic under H₀

Step 6: Make a Decision

• If test statistic > critical value (or p < α): Reject H₀
• If test statistic ≤ critical value (or p ≥ α): Fail to reject H₀

Step 7: State the Conclusion

• Example: "At the 5% significance level, there is sufficient evidence to conclude that the new drug significantly reduces blood pressure (t = 3.2, p = 0.002)."

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4. Worked Example (Coin Toss)

• H₀: The coin is fair (P(heads) = 0.5)
• H₁: The coin is biased toward heads
• Test: Probability = 0.5^10 ≈ 0.001 → p < 0.001
• Decision: Since p < 0.05 (α), reject H₀
• Conclusion: The coin appears to be biased toward heads (statistically significant at p < 0.001)

This example is directly from Rockwood and Green's Fractures in Adults, 10th ed. to illustrate the intuitive nature of hypothesis testing.

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5. Summary Table

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ANOVA - Analysis of Variance

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1. What is ANOVA?

Why not just use multiple t-tests? If you compare 3 groups using separate t-tests (A vs B, B vs C, A vs C), you run 3 tests, each with a 5% chance of a false positive. Together, the cumulative Type I error rises to ~14%. With 6 groups, you'd have 15 comparisons and ~54% false positive risk. ANOVA controls this by performing a single "omnibus" test, keeping Type I error at exactly α (e.g., 5%).

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2. Core Logic of ANOVA

Total Variation = Between-Group Variation + Within-Group Variation
     (SST)             (SSB)                      (SSW)

• Between-Group Variation (SSB): How much the group means differ from the overall grand mean. This reflects the effect of the independent variable (treatment).
• Within-Group Variation (SSW): How much individual observations vary within each group (also called "error" or "residual"). This is natural random variation.

• F near 1: Group means are similar - evidence for H₀
• F much greater than 1: Group means differ more than expected by chance - evidence against H₀

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3. Hypotheses in ANOVA

• H₀ (Null): All group means are equal → μ₁ = μ₂ = μ₃ = ... = μₖ
• H₁ (Alternative): At least one group mean is different from the others (not all means are equal)

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4. Types of ANOVA

4.1 One-Way ANOVA

• Compares means of 3+ groups based on one independent variable (factor)
• Example: Comparing pain relief scores across 3 drug dosages (10 mg, 20 mg, 30 mg)
• One factor with multiple levels

4.2 Two-Way ANOVA

• Compares means based on two independent variables simultaneously
• Can detect:
  • Main effect of Factor A
  • Main effect of Factor B
  • Interaction effect between A and B (does the effect of A depend on the level of B?)
• Example: Comparing blood pressure across 3 drugs AND 2 genders simultaneously

4.3 Repeated Measures ANOVA

• Used when the same subjects are measured at multiple time points or under multiple conditions
• Controls for individual differences between subjects
• Example: Measuring anxiety scores in patients before, during, and after treatment
• Requires testing the assumption of sphericity (Mauchly's test)

4.4 MANOVA (Multivariate ANOVA)

• Extension of ANOVA for two or more dependent variables at once
• Example: Comparing both systolic AND diastolic blood pressure across groups simultaneously

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5. Assumptions of ANOVA

If assumptions are violated: Use Welch's ANOVA (for unequal variances) or Kruskal-Wallis test (non-parametric alternative to one-way ANOVA).

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6. Step-by-Step Procedure of ANOVA

Step 1: State Hypotheses

• H₀: μ₁ = μ₂ = μ₃ (all group means equal)
• H₁: At least one μᵢ ≠ μⱼ

Step 2: Set Significance Level

• Typically α = 0.05

Step 3: Calculate Sum of Squares

Step 4: Calculate Degrees of Freedom (df)

Step 5: Calculate Mean Squares (MS)

Step 6: Calculate the F-Statistic

Step 7: Build the ANOVA Table

Step 8: Compare F to Critical Value / Find p-value

• Look up the critical F value from the F-distribution table using df_B and df_W
• Or use software to find p-value
• If F_calculated > F_critical (or p < α): Reject H₀

Step 9: Post-Hoc Tests (if H₀ is rejected)

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7. Post-Hoc Tests

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8. Worked Example

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9. How to Report ANOVA Results

Format: F(df_between, df_within) = value, p = value, η² = effect size

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10. Effect Size in ANOVA

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11. Non-Parametric Alternative

• One-Way ANOVA → Kruskal-Wallis Test (uses ranks instead of raw values)
• Repeated Measures ANOVA → Friedman Test

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12. Quick Summary

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Measures of Central Tendency

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1. Definition

"Continuous variables are more typically described with a single summary statistic (i.e., measure of central tendency) as well as the associated uncertainty around the summary estimate (i.e., measure of spread)."

• Rockwood and Green's Fractures in Adults, 10th ed.

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2. THE MEAN (Arithmetic Mean)

Definition

Formula

Calculation Example

• Sum = 4+7+3+9+5+8+6 = 42
• n = 7
• Mean = 42 / 7 = 6

Types of Mean

For Grouped Data (Frequency Distribution)

Properties of Mean

• Uses all values in the dataset
• Affected by every observation
• Highly sensitive to outliers and skewed distributions
• Unique - only one mean per dataset
• Sum of deviations from the mean = 0: Σ(Xᵢ - X̄) = 0

Advantages

• Uses all data points - most mathematically complete
• Suitable for further algebraic and statistical calculations
• Basis for many inferential tests (t-test, ANOVA, etc.)

Disadvantages

• Distorted by extreme outliers
• Not suitable for skewed distributions
• Cannot be used for nominal (categorical) data
• May give a non-integer value that doesn't actually exist in the data

When to Use

• Data is normally distributed (symmetric)
• No extreme outliers
• Interval or ratio scale data
• When further statistical analysis is needed

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3. THE MEDIAN

Definition

"The median of a set of measurements is the number that falls in the middle."

• Rockwood and Green's Fractures in Adults, 10th ed.

How to Calculate

• If n is ODD: Median = value at position (n+1)/2
• If n is EVEN: Median = average of values at positions n/2 and n/2 + 1

Examples

For Grouped Data

• L = lower boundary of median class
• n = total frequency
• CF = cumulative frequency before median class
• f = frequency of median class
• h = class width

Properties of Median

• Divides distribution into two equal halves
• Not affected by extreme outliers (robust statistic)
• Only one median per dataset
• Can be used for ordinal data
• May not be an actual value in the dataset (for even n)

Advantages

• Unaffected by extreme values/outliers
• Ideal for skewed distributions
• Can be used for ordinal data
• Easy to understand and locate graphically

Disadvantages

• Does not use all values in the data
• Less suitable for further mathematical calculations
• Less efficient than mean for symmetric, normally distributed data

When to Use

• Data is skewed (not normally distributed)
• Ordinal data (e.g., pain scores: 0-5)
• Presence of extreme outliers
• Classic example: income data (right-skewed; median better represents the typical earner)

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4. THE MODE

Definition

"The mode is the most frequently occurring number in a set of measurements."

• Rockwood and Green's Fractures in Adults, 10th ed.

Examples

• Data: 2, 3, 3, 4, 4, 4, 5, 6 → Mode = 4 (appears 3 times)
• Data: 1, 2, 3, 4, 5 → No mode (all values appear once)
• Data: 2, 2, 3, 4, 4 → Bimodal: Mode = 2 and 4

Types of Mode

• Unimodal: One mode
• Bimodal: Two modes
• Multimodal: More than two modes
• No mode: All values occur with equal frequency

For Grouped Data

• L = lower boundary of modal class (class with highest frequency)
• f₁ = frequency of modal class
• f₀ = frequency of class before modal class
• f₂ = frequency of class after modal class
• h = class width

Properties of Mode

• Can be used for any type of data (nominal, ordinal, interval, ratio)
• May not be unique
• Not affected by extreme values
• May not exist in some datasets
• Easiest to identify visually

Advantages

• Works for all data types including categorical (nominal)
• Unaffected by outliers
• Represents the most "popular" value
• Easy to identify by inspection

Disadvantages

• May not be well-defined (bimodal or no mode)
• Ignores most of the data
• Least used for advanced calculations

When to Use

• Nominal (categorical) data - only measure that works (e.g., most common blood type)
• Bimodal distributions
• When you want the most typical/popular value
• Count data and probability distributions

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5. Relationship Between Mean, Median, and Mode

In a Normal (Symmetric) Distribution

In a Skewed Distribution

Karl Pearson's Empirical Relationship

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6. Choosing the Right Measure

"The appropriate choice often depends on the distribution of data. Most data can be assumed to be normally distributed... In this situation of normality, the mean is the best-used summary statistic. If the data are not normally distributed, then the median may be a better summary statistic."

• Rockwood and Green's Fractures in Adults, 10th ed.

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7. Summary Comparison Table

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8. Worked Example

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9. Connection to Dispersion

"Along with measures of central tendency, investigators will often (and should) include a measure of spread to indicate uncertainty around the estimate."

• Rockwood and Green's Fractures in Adults, 10th ed.

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Methods of Data Collection

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1. What is Data Collection?

Before collecting data, a researcher must clearly define:

• What data to collect
• Why it is needed (research objectives)
• Who it will be collected from (target population)
• How it will be collected (method)
• When and where it will be collected

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2. Types of Data

Based on Source

Based on Nature

Based on Scale

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3. PRIMARY DATA COLLECTION METHODS

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Method 1: Questionnaire

Types of Questions

• Open-ended: Free-form answers ("How do you feel about...?")
• Closed-ended: Fixed choices ("Yes/No", multiple choice)
• Likert Scale: Degree of agreement (Strongly Agree → Strongly Disagree)
• Ranking questions: Rank in order of preference

Types of Questionnaires

Advantages

• Can reach a large number of people
• Low cost, especially online
• Respondent anonymity encourages honest answers
• Easy to quantify and analyze
• No interviewer bias

Disadvantages

• Low response rate for mailed versions
• Cannot clarify ambiguous questions
• Limited to literate respondents
• Cannot capture non-verbal cues
• Risk of incomplete responses

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Method 2: Interview / Schedule Method

Types of Interviews

Difference: Questionnaire vs Schedule

Advantages of Interviews

• High response rate
• Can probe for deeper information
• Can clarify misunderstandings
• Works for illiterate respondents
• Captures non-verbal cues (body language, emotions)
• Flexible

Disadvantages of Interviews

• Time-consuming and expensive
• Interviewer bias possible
• Respondents may not be honest face-to-face
• Difficult to cover large samples
• Requires trained interviewers

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Method 3: Observation Method

Types of Observation

Advantages

• Captures actual behavior (not what people say they do)
• No reliance on memory or self-report
• Good for studying children, animals, or groups who can't respond verbally
• Natural setting = high ecological validity

Disadvantages

• Time-consuming
• Observer bias possible
• Hawthorne effect (people change behavior when observed)
• Limited to observable behavior (not feelings or thoughts)
• Ethical issues with covert observation

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Method 4: Experiment

Types

Key Features

• Independent variable (IV): What the researcher manipulates (e.g., drug dose)
• Dependent variable (DV): What is measured (e.g., blood pressure)
• Control group: Receives no treatment or a placebo
• Experimental group: Receives the treatment

Advantages

• Establishes cause-and-effect relationships
• High control over variables
• Replicable

Disadvantages

• Artificial (lab settings may not reflect real life)
• Ethical concerns (giving harmful treatments)
• Expensive and time-consuming
• Not possible for all research questions

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Method 5: Case Study

• Advantages: Rich, detailed data; explores rare or unusual cases
• Disadvantages: Cannot generalize findings; time-intensive; researcher bias

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Method 6: Focus Group

• Advantages: Group interaction generates ideas; time-efficient for qualitative data
• Disadvantages: Dominant members may influence others; not anonymous; moderator bias

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4. SECONDARY DATA COLLECTION METHODS

Sources of Secondary Data

A. Published Sources

B. Unpublished Sources

Advantages of Secondary Data

• Time-saving - data already exists
• Cost-effective - no fieldwork needed
• Large datasets available (e.g., national census)
• Useful for longitudinal analysis and trend identification
• Good starting point before primary research

Disadvantages of Secondary Data

• May not be specific to the research question
• Data may be outdated
• Quality and reliability may be uncertain
• Bias from the original collector
• May not match the exact variables needed
• Limited control over data collection process

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5. PRIMARY vs. SECONDARY DATA: Comparison

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6. QUANTITATIVE vs. QUALITATIVE Data Collection

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7. Choosing the Right Method

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8. Steps in Data Collection

1. Define the research problem - What are you trying to find out?
2. Set research objectives - What specific questions need answers?
3. Choose data type - Primary or secondary? Quantitative or qualitative?
4. Select the method - Survey, interview, observation, experiment, etc.
5. Design the instrument - Develop questionnaire, interview guide, observation checklist
6. Pilot test - Test on a small sample; revise any flaws
7. Collect data - Execute the chosen method with the target population
8. Record and organize - Code, enter, and store data securely
9. Validate and clean - Check for errors, missing values, inconsistencies
10. Analyze - Apply appropriate statistical or qualitative techniques

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9. Summary Table of All Methods

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