Stats 2 Normal Distribution
Workshop 2 Overview
Presenter: Obi Ukoumunne, PenARC, UEMS
Contact: O.C.Ukoumunne@exeter.ac.uk
Outline of Workshop 2:
Normal Distribution
Importance of Statistical Methods in Medical and Health Research
Inferential Statistics
a. Estimation (Standard errors & Confidence intervals)
b. Hypothesis Testing (p-values)
Normal Distribution
Key Features
The Normal Distribution is a significant type of distribution for a quantitative variable, describing many commonly studied quantitative variables.
It influences the choice of statistical methods for summarizing (as discussed in Workshop 1) and analyzing data (to be covered in Workshops 3 and 5).
It forms the basis for statistical inference.
Definition and Characteristics
Mathematically, the Normal Distribution is a theoretical distribution characterized by a symmetrical bell-shaped curve.
The shape of this curve approximately describes the distribution of many commonly occurring quantitative variables, such as height and birth weight.
With knowledge of the mean and standard deviation (SD) of a normally distributed variable, percentages of scores can be calculated as follows:
68% of individuals score within 1 SD of the mean,
95% of individuals score within 1.96 SDs of the mean,
99.7% of individuals score within 3 SDs of the mean.
Probability Within Standard Deviations
68% Probability: A score lies within 1 standard deviation of the mean.
This is displayed by a shaded area in the Normal distribution curve.
95% Probability: A score lies within 1.96 standard deviations of the mean.
Shown through a shaded area in the Normal distribution curve.
99.7% Probability: A score lies within 3 standard deviations of the mean.
Again represented with a shaded area in the Normal distribution curve.
95% Range Calculation for Normally Distributed Variables
To calculate the 95% range for Normal variables, the following formula is used:
Lower Bound: mean - 1.96 × standard deviation
Upper Bound: mean + 1.96 × standard deviation
Example: IQ Scores
For a sample of 100 IQ scores with a mean of 98.8 and SD of 13.4, the 95% Normal range is computed as follows:
Lower Bound: (98.8 - (1.96 × 13.4)) = 72.5
Upper Bound: (98.8 + (1.96 × 13.4)) = 125.1
Thus, 95% of the sample has IQ scores between 72.5 and 125.1, leveraging the properties of the Normal distribution.
Statistical Tests for Normality
To verify Normality, the best practice is to draw a histogram, ideally requiring a sample size of around 50 individuals.
For smaller samples, statistical tests can be applied (e.g., Shapiro-Wilk test):
If p < 0.05, there is evidence of non-Normality.
If p > 0.05, data does not significantly deviate from Normality.
Importance of Statistical Methods in Medical and Health Research
Research Questions
Examples of health-related research questions include:
Does controlled crying resolve sleep problems in babies?
How common is ADHD?
Does mindfulness training in schools improve pupils' well-being?
Population vs Sample
Research questions involve collecting data from a sample of units (e.g., individuals selected from a population).
The population encompasses the entire set of units relevant to the study query. The sample is a smaller subset of this population, selected to participate in the research study.
Data refers to the information gathered from these sampled units.
Population Definition
The full set of units of interest, typically considered infinite in size, includes all existing and future units relevant to the research question.
Common characteristics of units depend on the study query, such as babies with sleep challenges or healthy rats.
Parameters and Estimates
In medical research, a sample of individuals is selected to derive estimates of true population parameters.
Examples of Study Parameters:
Mean systolic blood pressure among city traders.
Percentage of individuals with Type II diabetes.
Mean difference in heart rate post-exercise between athletes and non-athletes.
The Need for Statistical Methods
Answers derived from sample data often do not reflect the true population answer due to several uncertainties:
Individual variability in measurements.
Samples being mere subsets that are not entirely representative of the population.
Functions of Statistics
Summarizing data in the sample.
Quantifying uncertainty in the sample results, utilizing:
Descriptive statistics
Inferential statistics:
Standard errors (estimation)
Confidence intervals (estimation)
P-values (hypothesis testing)
Inferential Statistics
Overview
Research inquiries involve data collection from a sample representing a larger population.
Conclusions from the sample provide estimates concerning the population, but absolute certainty is unattainable.
Inferential statistics leverages standard errors, confidence intervals, and p-values to aid decision-making.
Standard Errors (Estimation)
Concept of Standard Error
The standard error (SE) reflects the accuracy of the sample estimate relative to the true value of the population parameter based on various samples.
Example: Infant Sleep Study
For the infant sleep study, a nurse-delivered intervention was randomized in which 328 children (aged 7 months) participated. Data were collected at age 12 months, measuring the mother's depression score from 0 to 30.
Sample Mean (Intervention): 5.9
Sample Mean (Control): 7.2
Mean Difference: 1.3 units lower for the intervention group.
Standard Error of the Mean Difference: 0.6
Interpretation of Standard Error
The standard error indicates how close sample estimates are to the true mean difference if the study were repeated multiple times. A smaller SE indicates greater precision.
Larger sample sizes yield smaller SEs due to increased information about the variables, enhancing estimation precision.
SE vs SD
Standard Error (SE) should not be mistaken for Standard Deviation (SD).
SE assesses the precision of an estimate;
SD measures variability in observations within the sample for quantitative variables.
Confidence Intervals (Estimation)
Concept
Confidence intervals (CIs) specify the range of values wherein we can ascertain the truth of the population parameter with confidence, often expressed as a percentage (e.g., 95% CI).
Example: Infant Sleep Study
Again examining the mean depression score from the infant sleep study:
Mean difference: 1.3
95% Confidence Interval: [0.1, 2.4]
The CI indicates that we can be 95% confident that the mean maternal depression score at 12 months is between 0.1 and 2.4 units lower in the intervention group compared to the control group.
Characteristics of Confidence Intervals
CIs encompass a range denoting where the true population parameter is likely to lie.
In repeated studies, a specified percentage of confidence intervals would encompass the true parameter value (e.g., in 100 studies, 95 of the 95% CIs should contain the true parameter value).
Confidence Intervals versus 95% Range
95% Confidence Interval (CI): A wide range providing insight about what the true parameter might be.
95% Range: Reflects variability of scores within the sample and does not convey precision about the parameter’s truth.
Levels of Confidence
90% CI indicates the range within which we are 90% certain of containing the true parameter.
99% CI provides a higher confidence, ensuring a greater likelihood that the true value falls within the range.
95% CI is preferred for its higher confidence level.
Sample Size Implications
Larger sample sizes yield narrower confidence intervals, providing more accurate estimates of the population parameters.
Hypothesis Testing (p-values)
Overview
Hypothesis Testing is crucial in examining whether the sample estimate aligns with the null hypothesis (H0).
The null hypothesis is a statement essentially proposing no relationship or difference between variables.
Procedure
State the null hypothesis (e.g., no difference between groups).
Analyze data to assess evidence against the null hypothesis, quantified as a p-value.
The p-value ranges from 0 to 1, with lower values indicating increased evidence against the null hypothesis.
Example: Infant Sleep Study
For the intervention, the p-value was calculated as 0.03, suggesting moderate evidence that suggests the mean maternal depression score differs between intervention and control groups (i.e., the intervention was beneficial).
Interpretation of P-values
The p-value helps quantify how much evidence exists against the null hypothesis. - Traditionally, a p-value of 0.05 is the threshold for rejecting/nullifying the null hypothesis, but it’s important to avoid overly strict adherence to this value as p-values are not a binary measure.
Broadening Understanding of P-values
Absence of evidence (high p-value) does not confirm the null hypothesis's truth; it may also reflect sample size issues.
The p-value does not measure the size of a difference or strength of association, rather it indicates only the existence of differences.
Clarification on Hypotheses
The null hypothesis is the baseline truth (e.g., no association).
The research hypothesis is what the researcher anticipates might be true and informs the direction of the study.
P-values and Confidence Intervals Relation
A 95% confidence interval can also test hypotheses:
If it includes the null hypothesis value (e.g., includes 0), p-value is > 0.05.
If it excludes it, p-value < 0.05; extends a nuanced understanding of hypothesis testing beyond mere significance.
Summary of Workshop 2
Understanding the Normal distribution is foundational to inferential statistical methods.
These methods include:
Standard Error: How close our sample estimate may be to the true answer.
Confidence Interval: What the true answer is.
Hypothesis Testing (p-value): What the true answer isn’t.
Preview of Workshop 3
Workshop 3 will focus on comparing quantitative (continuous) variables between various groups, examining independent versus paired groups, and exploring parametric as well as non-parametric methods to analyze data.