Hypothesis Testing

Inference from Sample to Populations

• The lecture focuses on statistical inference, which involves drawing conclusions about a population based on data from a sample.

Hypothesis Testing

• Definition: Hypothesis testing is a method used to make decisions or inferences about a population based on sample data.
• Null Hypothesis: A statement of no effect or no difference. It's the hypothesis that researchers try to disprove.
• Alternative Hypothesis: A statement that contradicts the null hypothesis, suggesting there is a difference or effect.
• Type-I Error: Rejecting the null hypothesis when it is actually true (false positive). Denoted by $\\alpha$.
• Type-II Error: Failing to reject the null hypothesis when it is actually false (false negative). Denoted by $\\beta$.
• P-values: The probability of obtaining results as extreme as, or more extreme than, the observed results, assuming the null hypothesis is true. A small p-value suggests evidence against the null hypothesis.
• Confidence Intervals: A range of values within which the true population parameter is likely to fall.
• Single Mean Test: A statistical test used to determine whether a sample mean is significantly different from a known or hypothesized population mean.
• Single Proportion Test: A statistical test used to determine whether a sample proportion is significantly different from a known or hypothesized population proportion.

STATA Demonstration

• Demonstration of STATA commands for data analysis, including:
  • Upload “HospAdmNeu.dta” data in STATA.
  • summarize Ordinary1213 (Summary measures of ordinary admission).
  • summarize Ordinary1213, detail (Detailed summary of ordinary admission).
  • Graph box Ordinary1213 (Draw a boxplot to check for outliers).
  • scatter Ordinary1213 Daycase1213 (Scatter plot between two variables).
  • hist Ordinary1213 (Draw a histogram for ordinary admission).
  • hist Ordinary1213, bin(20) (Change the bin for better representation).
  • hist Ordinary1213, bin(20) normal (Check normality with a normal curve).
  • Homework: Repeat all commands for a day case hospital admission.

Statistical Inference: The Big Picture

• Real World: Data is collected from the real world.
• Theoretical World: Scientific and statistical models are used to represent the data.
• Sample: A subset of the population from which data is collected.
• Population: The entire group of individuals or objects of interest.
• Conclusion: Inferences are made from the sample to the population.

Populations and Samples

• A population is a collection of objects, people, or events of interest.
• Data collection on the entire population is often impractical.
• A sample is a subset of the population used to infer information about the population.
• Samples are not of interest in their own right but for what they reveal about the population.

Example: Babies Birth Weight and Mum’s Smoking

• Study in Newham, London, investigating causes of low birth weight in babies born in 2016.
• Data collected from 1000 babies in Newham hospital database.
• Dataset includes variables like baby ID, birth weight (bwt), gestational age (gest), mother’s age (mat_age), and number of cigarettes smoked per week (cigs).

Statistical Inference

• Summary statistics (mean, percentiles, SD) are calculated from the sample.
• These statistics are used to infer characteristics about the total population.
• The goal is to understand what sample statistics tell us about the theoretical distributions of the population.

Random Samples

• Research Question: Average weight of babies and reasons for underweight babies.
• Theoretical Population: Defined before data collection to understand the generalizability of inferences.
• Examples: all babies (current and future), babies born in the UK between 1990 and 2000.
• Average Baby Weight: Around 7 pounds (3.2 kg) for females and 7 pounds 5 ounces (3.3 kg) for males.
• Reasons for Underweight Babies: Premature birth, Intrauterine Growth Restriction (IUGR), infections during pregnancy, inadequate weight gain, smoking, alcohol or drug use, and maternal age.

Random Samples (contd.)

• The sample is a subset of the population and needs to be representative.
• Random Sampling: Each individual in the population has an equal chance of being included, and the inclusion of one individual does not affect the inclusion of another.
• Opportunistic Sampling: Also known as convenience sampling, involves selecting participants based on availability (e.g., recruiting participants from a local support group for a rare neurological disorder like Stiff Person Syndrome).

Stratified Random Sampling

• Example: Global Adult Tobacco Survey (GATS) in Bangladesh.
• Overall prevalence of tobacco smoking:
  • 2009: 23.00% (95% CI 22.98 to 23.00)
  • 2017: 16.44% (95% CI 16.43 to 16.45)
• Methodology: Two-stage stratified sampling.
  • First stage: Eight administrative divisions were created, stratified by urban and rural Enumeration Areas (EAs).
  • Second stage: 30 households were systematically selected from each sampled PSU (EA).
  • One participant was randomly picked from all eligible men and women in a participating household.

Statistical Inference: From Sample to Populations

• Assessing the accuracy of sample statistics in estimating population parameters.
• Repeatedly choosing samples from the same population results in different values for a statistic (e.g., the mean).
• Uncertainty associated with the estimate needs to be assessed.

Sampling Variability and Standard Errors

• The standard deviation of the sampling distribution of the mean ($s / \sqrt{n}$) measures the typical error between the sample mean and the population mean.
• $s / \sqrt{n}$ quantifies the accuracy of the sample mean as an estimate of the population mean and is known as the standard error (SE) of the mean.
• Since the theoretical standard deviation (s) is unknown, the sample standard deviation (SD) is used in its place.

Standard Deviation (SD) vs Standard Error (SE)

• Standard Deviation (SD): Measures how a typical observation in the sample differs from the sample mean.
• Standard Error (SE): Quantifies the typical error between the mean measured in a sample and the theoretical mean in the population.
• SD measures variability in the population or sample.
• SE ($= SD/\sqrt{n}$) measures variability in the sample means.

Importance of Normal Distribution in Medical Research

• Central Tendency and Variability: Helps in understanding the mean and standard deviation of medical data.
• Statistical Inference: Many tests and confidence intervals assume normality (e.g., t-tests and ANOVA).
• Predictive Modeling: Used in risk assessment and predictive modeling.
• Quality Control and Standardization: Used to monitor and maintain consistency of medical tests and procedures.

Sampling Distributions

• The frequency distribution of the sample means is called the sampling distribution of the mean.
• If the population distribution is Normal, the distribution of the sample mean over repeated samples is also Normal.
• The variation in the sample means depends on the variance of the population ($s^2$) and the sample size n.
• For large samples (n>30), the distribution of the sample mean is approximately normal, regardless of the population distribution.

Hypothesis Testing

• Null and alternative hypotheses depend on the type of investigation.
  • To see if there is a difference between two procedures:
    • Null Hypothesis: No difference.
    • Alternative Hypothesis: There is a difference.
  • To find out if a bold claim is true:
    • Null Hypothesis: There is no difference.
    • Alternative Hypothesis: The claim is true (drug A is better/worse than drug B).

Task-1: State Null Hypothesis

• In a criminal court, the accused is assumed innocent unless proven guilty.
• Null Hypothesis: The accused is innocent.

Assumptions for Hypothesis Testing

• My sample(s):
  1. Is representative
  2. Is independent
  3. Has homogeneous variance
  4. Is normal
• Assumptions 1 and 2 are usually considered automatically met.
• Assumptions 3 and 4 need to be tested using appropriate tools & techniques.

Types of Data

• Quantitative:
  • Continuous (e.g., blood pressure, age).
  • Discrete (e.g., number of children, number of cigarettes per day).
• Categorical:
  • Ordinal (ordered categories, e.g., grade of breast cancer, disease severity).
  • Nominal (unordered categories, e.g., sex, ethnicity).

Statistical Tests for Continuous Data

1. If sample size >= 30 and assumptions are met, use normal distribution.
2. If sample size < 30 and assumptions are met, use t-distribution.
3. If assumptions are not met, transform variables and repeat steps 1 or 2.
4. If none of the assumptions are met, use non-parametric tests.

• Parametric tests are generally more powerful.

Appropriate Tests for Continuous Data

• One-Sample:
  • t-test
  • Sign test
• Paired (2 groups):
  • Paired t-test
  • Wilcoxon signed rank test
  • Sign test
• Independent (2 groups):
  • Unpaired t-test
  • Wilcoxon rank sum test
• Independent (>2 groups):
  • One-way ANOVA
  • Kruskal-Wallis ANOVA

Categorical Data

• Categorical covariate data are often called factors
• Categorical data that take on only two distinct values are said to be dichotomous or binary
• Categorical data are often coded using numerical values (e.g. 0 = NO, 1 = YES) –
  statistical packages usually treat numeric data as quantitative unless you explicitly declare it to be categorical
• Limiting factor for any continuous observation is the accuracy of the measurement instrument

Appropriate Tests for Categorical Data

• 1 group:
  • z test for a proportion
  • Sign test
• Paired (2 categories):
  • McNemar's test
• Independent (2 groups):
  • Chi-squared test
  • Fisher's exact test
• Independent (>2 groups):
  • Chi-squared test
  • Chi-squared trend test
• >2 Categories:
  • Chi-squared test

Type I and Type II Errors

P-value and Confidence Interval (CI)

• P-value: Measures the strength of evidence against the null hypothesis.
  • P > 0.10: No evidence against null hypothesis.
  • 0.05 < P < 0.10: Weak evidence against null hypothesis.
  • 0.01 < P < 0.05: Moderate evidence against null hypothesis.
  • 0.001 < P < 0.01: Strong evidence against null hypothesis.
  • P < 0.001: Very strong evidence against null hypothesis.
• Confidence Interval (CI): Range of values within which the true population value is likely to be found.
  • If the CI for the difference in mean scores excludes ‘0’, there is strong evidence against the null hypothesis.

Confidence Interval for a Single Population Mean

• A range of likely values for the parameter.
• To construct a confidence interval for the population mean, make use of the following pieces of information:
  • Sample mean
  • Standard Error of the mean
• 95% of sample means lie within 1.96 SE above or below the population mean

Confidence Interval (contd.)

• Approximated 1.96~2
• There is 95% probability that this interval contains the unknown but true value of the population mean

Example: Finding Single Mean and 95% CI

• Baby weight study with 1000 babies in Newham hospital.
• Random sample of 1000 babies chosen, 997 considered after deleting unusual weights.
• Mean weight (x) = 3305 gm; SD = 505 gm.
• SE (mean) = $505/ \sqrt{997} = 16$ gm.
• 95% CI for mean = $x - z(5\%) \times SE(x)$ to $x + z(5\%) \times SE(x)$
  • = 3305 - 2 × 16 to 3305 + 2 × 16
  • = 3273 to 3337 gm
• We are 95% confident that the true mean birth weight is between 3273 and 3337 gm.

Single Proportion

• If out of total sample of size n only d individuals from our sample experience some event:
  $p = \frac{d}{n}$

Example (vaccination):

• In a trial of a new vaccine, 20 out of 1000 children vaccinated showed signs of adverse reaction:

$p = \frac{20}{1000} = 0.020 = 2.0\%$, thus advising parents that the vaccine is associated with an estimated 2% risk of adverse reaction.

Single Proportion and its 95% CI – Vaccination Example

• Step 1: Calculate proportion:$p = \frac{20}{1000} = 0.020$
• Step 2: Calculate Standard error:
  $se(p) = \sqrt{\frac{0.02(1-0.02)}{1000}} = 0.004$
• Step 3: Calculate 95% Confidence interval:
  • $0.02-1.96 \times 0.004$ to $0.02 +1.96 \times 0.004$
  • = 0.012 to 0.027 = 1.2% to 2.7%

Practice Example: Single Proportion and its 95% CI

• Smoking habits survey in Birmingham, UK, among 1000 teenagers aged 15-16 in 2001.
• 123 reported being current smokers.
• Find the proportion of teenagers who smoked and the 95% confidence interval.

Practice Example : Single proportion and its 95% CI – Birmingham teenage smoking habit - solution of class exercise

• Step 1 : calculate proportion, $p = \frac{123}{1000} = 0.123$
• Step 2: Calculate the standard error of the proportion, $se(p) = \sqrt{\frac{0.123(1−0.123)}{1000}} = 0.0104$
• Step 3: Calculate 95% CI using: $p −1.96×se(p)$, $p + 1.96×se(p)$
• This gives, 0.123−1.96×0.0104 and 0.123 + 1.96×0.0104 = 0.103 to 0.143 = 10.3% to 14.3%

Homework

• Upload babies data from Moodle using STATA.
• Using STATA command work out i) mean & sd of babies weight
• Using STATA command work out 95% CI for mean baby weight & comment on your findings.

Recommended Reading

1. Practical Statistics for medical research by Douglas Altman : Chapter 10, page 232-234.
2. Medical Statistics by B. Kirkwood & J. Sterne : Chapter-6
3. Statistics notes: The normal distribution BMJ 1995; 310 doi: https://doi.org/10.1136/bmj.310.6975.298
4. Statistics Notes: Standard deviations and standard errors. BMJ. 2005 Oct 15; 331(7521): 903. doi: 10.1136/bmj.331.7521.903
5. Comparison between t and normal distribution – separate file uploaded in Moodle (3 slides only)