Goal of the training

The goal of this training is to:

• Provide an overview of fundamental statistical methods for the quantitative data analysis, including key theoretical results presented.
  • Equip staff with essential knowledge and skills to interpret data effectively.
  • Apply statistical methods that support organizational goals.
• Introduce Advanced Topics in Quantitative Data Analysis:
  • Regression analysis, Survival Analysis, Longitudinal data analysis
• Enable trainees to pose scientific questions within the context of appropriate statistical methods and carry out and interpret analyses effectively.

What is statistics and quantitative data analysis?

• Statistics is the science of:
  • Collecting, analyzing, summarizing, and interpreting data
  • Drawing conclusions to support effective decision-making.
• Transforms raw data into meaningful information.
• Helps us learn about the world through data-driven insights.
• Widely applied in fields like:
  • Biology, sociology, economics, public health, and business
• When applied to biological or health-related data, it is referred to as Biostatistics

Branches of Statistics

• The field of statistics consists of two major branches that help us describe data and make informed decisions based on evidence.

Purpose of Descriptive Statistics

Purpose of Descriptive Statistics

Types of variables

What is a Variable?

A variable is any characteristic, number, or quantity that can be measured or counted and differs across individuals or observations.

Measure of central tendency

Mean

Median

• The middle value in an ordered dataset (from smallest to largest).
• Represents the 50th percentile, half the values lie above and half below.
• Not affected by extreme values (outliers) → ideal for skewed distributions

Mean vs Median

• Median is much more robust against scattering.
• An outlier usually has no influence on the median, but it has a more or less large influence on the mean.

Mean vs Median (continued)

• When the data is symmetric and has no outliers, the mean is the best measure because it uses all values and reflects the overall dataset accurately.
• However, if the quantitative data is skewed or has outliers, the median is a better choice.
• It is often presented with histograms or summary tables showing the mean and standard deviation.

Mode (Modal Value)

Measure of Dispersion

• Dispersion or variability refers to how dispersed or deviated the values from one another in a distribution.
• Three most widely used measures of dispersion are:
  • Variance, Standard deviation
  • Interquartile and Range
• Variance
• Variance measures the amount of spread or variability of observation from mean.
• The sample variance (s²) is the average of the squares of each deviation from the sample mean.

\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}\left(x_i - \bar{x}\right)^2\]

Standard deviation

Example of variance and standard deviation calculation

\[s^2 = \frac{1}{n-1}\sum_{i=1}^{n}\left(x_i - \bar{x}\right)^2\]

Interquartile range (IQR)

• IQR is the distance between the 1st and 3rd quartile.
• It is not sensitive to extreme values (outliers). Thus, it is usually described together with the median in skewed distribution of observation.

\[\text{Formula: } IQR = 3rd\ quartile - 1st\ quartile = q_3 - q_1\]

Tabular Presentation

Charts

Bar chart

• Bar charts are one of the most common charts in scientific work and they are mostly used to show either frequencies or averages.
• Depending on how the bars are arranged, a distinction is made between horizontal and vertical bar charts.

Bar chart (continued)

Grouped bar charts

• If two categorical variables are present, grouped bar charts can be created.
• In grouped bar charts, either the frequency, the percent, or the percent in each group can be specified.

Overall, bar chart is

Histogram

Used for normality checking

Polygon

• A frequency polygon is a graph that displays the data using lines to connect points plotted for the frequencies

Box plot

How is a boxplot interpreted?

• The box itself indicates the range in which the middle 50% of all values lie.
• Thus, the lower end of the box is the 1st quartile, and the upper end is the 3rd quartile.

Boxplot illustration

Scatter Plots

Scatter Plots (continued)

• Helps identify the type of correlation:
• Positive correlation: Both variables increase together.
• Negative correlation: One variable increases while the other decreases.
• No correlation: Points appear randomly scattered.

Good practice in data presentation: tabular result

• When presenting descriptive statistics in tables, it is good practice to keep the table simple, organized, and easy to read.
• Use clear headings for each variable, and label columns properly (e.g., mean, median, SD).
• Align numbers neatly, usually to the right, and round them to a consistent number of decimal places.
• Always add a clear table title and footnotes if any explanations or abbreviations are used, so the reader can understand the table without extra help.

Tabular presentation or reporting

Graphical data presentation or reporting

Inferential Statistics

• Involves drawing conclusions about a population from a random sample. Educated guess rather that describing data.
• Useful when it is impractical to study the entire population.
• Allows generalization from sample data to a larger group.
• Example:
• Evaluating the effect of a cash transfer program by surveying a sample of recipients and inferring the results to the broader population.
• Key Methods: depends on data type, assumption, objective
• Group comparison tests:
  • t-test, chi-square test, ANOVA (Analysis of Variance)
• Relationship/correlation tests:
  • Correlation analysis, regression analysis

Random Sample and Sampling Error

• A random and representative sample approximates the population, but is rarely identical to it.
• The difference between a sample statistic and the true population value is called sampling error.
• Sampling error is:
  • Natural and expected
  • Unavoidable when using samples instead of full populations

Random Sample and Sampling Error (continued)

• Since population values are unknown, the exact sampling error is also unknown.
• We use statistical methods to estimate sampling error and evaluate reliability.

Role of Hypothesis Testing:

• Helps determine whether sample results reflect true population effects or are due to random chance.
• Accounts for expected variability caused by sampling error.

Hypothesis testing

• A hypothesis is an assumption about a relationship or effect that is neither proven nor disproven at the start of a study.
• It is developed based on a research question and typically justified through a literature review.
  • A hypothesis proposes an expected association (e.g., “Men earn more than women in the same job in Ethiopia”).
  • The goal is to reject the hypothesis based on data analysis.
  • Data from surveys or experiments are used, and a hypothesis test (e.g., t-test, correlation analysis) is applied.

Null and alternative hypothesis

• In hypothesis testing, we always define two opposing hypotheses:
• Null hypothesis (H₀): Assumes no difference or no effect between groups.
  • Example: The salaries of men and women in Ethiopia do not differ.
• Alternative hypothesis (H₁): Assumes a difference or an effect between groups.
  • Example: The salaries of men and women in Ethiopia differ.
• The hypothesis you want to prove (based on theory or research) is usually the alternative hypothesis.
• In a hypothesis test, you only test the null hypothesis and decide whether to reject it.

Level of significance or probability of error

• In hypothesis testing, we can never be 100% sure when rejecting the null hypothesis, there is always a small chance we are wrong.
• The significance level (α) is the allowed probability of wrongly rejecting a true null hypothesis.
  • If the p-value is smaller than α, we reject the null hypothesis.
  • If the p-value is greater than α, we do not reject the null hypothesis.
• Common significance levels are: 5% (α = 0.05) → 5% risk of wrongly rejecting a true null hypothesis.

Example: Two-Sample t-Test

• Used to compare the means of two independent groups.
• A larger difference between sample means suggests it’s less likely both groups come from the same population.

Key Concepts:

• If p-value < significance level (α) → Reject H₀ (null hypothesis)
  • Example: p = 0.04 < 0.05 → there’s a 4% chance the observed (or more extreme) mean difference occurred by random chance, assuming no true difference.
• The significance level (α):
  • Must be set before analysis (commonly 0.05)
  • Must not be adjusted afterward to influence results

Types of Errors in Hypothesis Testing

Hypothesis testing is based on sample data, so errors can occur due to random variation. No test is 100% foolproof - sample results naturally vary by chance.

Two Main Types of Errors:

Type I Error (\(\alpha\)) - False Positive

• Occurs when the null hypothesis is rejected even though it is actually true.
• Also called a false positive → concluding there is an effect when there isn’t one.
• The probability of committing a Type I error is the significance level \(\alpha\), set by the researcher (commonly \(\alpha = 0.05\)).

\[P(\text{Type I Error}) = \alpha\]

Example: A drug trial concludes the drug is effective, but in reality it has no effect. The result was due to random chance in the sample.

Type II Error (\(\beta\)) - False Negative

• Occurs when the null hypothesis is not rejected even though the alternative hypothesis is true.
• Also called a false negative → missing a real effect that actually exists.
• The probability of committing a Type II error is denoted \(\beta\).

\[P(\text{Type II Error}) = \beta\]

Example: A drug trial concludes the drug has no effect, but it actually does work. The study failed to detect the real effect.

Statistical Power

The power of a test is the probability of correctly rejecting \(H_0\) when it is false (the ability to detect a true effect).

\[\text{Power} = 1 - \beta\]

The Trade-off Between Type I and Type II Errors

• Decreasing \(\alpha\) (stricter threshold) → reduces Type I errors, but increases Type II errors (\(\beta\)).
• Increasing \(\alpha\) (lenient threshold) → reduces Type II errors, but increases Type I errors.
• The only way to reduce both simultaneously is to increase the sample size \(n\).

Why Errors Happen?

• Sample results vary by chance
  • no sample perfectly represents the population.
• No test is 100% foolproof
  • a decision based on probabilities will occasionally be wrong.
• The goal is not to eliminate errors entirely, but to control and minimise them through
  • appropriate study design,
  • sample size, and
  • significance thresholds.

Choosing the Right Statistical Test

Selecting the appropriate test depends on three key factors:

• Type of variables - categorical (nominal/ordinal) or numeric (continuous/discrete)
• Number of groups or samples - one, two, or more groups
• Relationship between samples - independent or related (paired)

Common Statistical Tests

a) T-test

Used to compare mean differences between groups. Three variants:

• \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \quad \text{(one-sample)}\)

• \(t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{s^2_p\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \quad \text{(independent two-sample)}\)

• \(t = \frac{\bar{d}}{s_d / \sqrt{n}} \quad \text{(paired)}\)

b) Chi-Square Test

Tests the association between two categorical variables.

\[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]

• \(O_i\) = observed frequency, \(E_i\) = expected frequency
• Used for nominal or ordinal data - never for continuous variables
• See the Chi-Square Test notes for full breakdown.

c) One-way / Two-way ANOVA

• One-way ANOVA: Compares means across three or more independent groups on one factor.
• Two-way ANOVA: Examines two independent factors simultaneously, including their interaction effect on the outcome.

\[F = \frac{\text{Variance between groups}}{\text{Variance within groups}}\]

If \(F\) is large, the between-group differences are unlikely due to chance alone.

Checking Assumptions of Statistical Tests

• Most statistical tests require certain assumptions to be valid.
• Assumptions should always be checked before running tests - violating them can lead to invalid conclusions.

How to Check Assumptions

Normality Assumption

Many tests including the t-test and ANOVA assume that the data (or residuals) follow a normal distribution.

Graphical Checks

Histogram

• Visually inspect the shape of the distribution.
• A bell-shaped, symmetric histogram suggests normality.
• Skewed or multi-modal shapes suggest non-normality.

Q-Q Plot (Quantile-Quantile Plot)

• Plots observed quantiles against theoretically expected quantiles under normality.
• If points fall along the diagonal line → normality is supported.
• Systematic deviations from the line → normality is violated.

Formal Normality Tests

Interpretation: If \(p < 0.05\), reject \(H_0\) → the data significantly deviate from normality.

What to Do When Normality Is Violated

Confidence Interval

• A CI defines a range where the true population parameter (e.g., mean) is likely to lie.
• Sample estimates (mean, variance) are only approximations of the true population values.
• Based on the sample mean, sample size (n), and sample standard deviation (s) and assumes a normal distribution of the parameter, CI is given as

Confidence Interval (continued)

Statistical tests for differences

• One-Sample t-Test
• Purpose: Tests if the sample mean differs significantly from a known or hypothesized population mean.
• Used when: You have one sample and a fixed reference value (e.g., target, population mean).

One-Sample t-Test (continued)

• Requirements
• Random sample and approx. normal distribution of the data
• Numeric (continuous) data

Types of Questions

• Two-tailed: Is the sample mean different from the reference value?
• One-tailed: Is the sample mean greater than or less than the reference value?

Hypotheses

• H₀ (Null): μ = μ₀ (Sample mean equals the reference mean)
• H₁ (Alternative): μ ≠ μ₀ (Sample mean differs from the reference)

Example: for pre specified average µ = 28

Two-Sample t-Test (Independent Samples)

• Purpose: Tests if two independent groups differ significantly in their means.
• Used when: Comparing two unrelated groups (e.g., treatment vs placebo, male vs female).
• Requirements
• Two independent samples & Numerical (continuous) data
• Normal distribution in each group (or large enough samples for approximation)
• Homogeneity of variances (can be tested with Levene’s test)

Examples

• Does Drug XY reduce weight compared to a placebo?
• Is there a health difference between people with and without a degree?

Hypotheses

• H₀ (Null): μ₁ = μ₂ (No difference between group means)
• H₁ (Alternative): μ₁ ≠ μ₂ (There is a difference between group means)

What is your conclusion?

Conclusion of the independent t-test

• In this example, the p-value is 31%, which is higher than the 5% significance level, so there is no significant difference between the two groups.
• The confidence interval (-6.328 to 18.118) crosses zero, also showing no significant difference.

Report a t-test for independent samples:

• An independent samples t-test was conducted to compare exam results in summer and winter. There was not a significant difference in the scores, p =0.31. The magnitude of the differences in the means (mean difference =5.9, 95% CI: [-6.33, 18.12]) was large.

Paired-Samples t-test (Dependent t-test)

• The paired-samples t-test is used to compare two related groups to determine whether their mean difference is statistically significant.
• Pairs of observations are needed:
  • Repeated measurements on the same individuals (before vs after treatment).
  • Matched subjects across two groups (e.g., twins, matched cases).
• Controls for individual variability because comparisons are within the same subjects.
• Greater chance to detect a real difference if one exists.

Look at before and after weight measure

ANOVA (Analysis of Variance)

• ANOVA is used to test whether statistically significant differences exist between three or more group means.
• Why not use multiple t-tests?
  • Every hypothesis test carries a risk of Type I error (commonly set at 5%).
  • Running multiple t-tests increases the probability of making at least one false-positive conclusion.
  • ANOVA helps to avoid inflated error rates when comparing several groups simultaneously.
• Independent variables (factors): Categorical.
• Dependent variable: Continuous variable
• F-ratio: The test statistic in ANOVA.
  • It compares between-group variability to within-group variability.

Types of Anova

Most common used ANOVA for non repeated measures are:

Example for One-factor ANOVA

• With the help of the dependent variable, e.g. “highest educational qualification” with the three characteristics group 1, group 2 and group 3 should be explained as much variance of the dependent variable “salary” as possible.

ANOVA hypotheses

• Null hypothesis H0: The mean of all groups is equal.
• Alternative hypothesis H1: There are differences in the means of the groups.
• You want to check whether there is a difference in coffee consumption between students in different subjects. To do this, ask 10 students from each field of study.

Two-Way ANOVA (Two-Factor ANOVA)

• Two-way ANOVA tests whether two independent categorical variables (factors) influence a continuous dependent variable.
• It also tests whether there is an interaction effect between the two factors.
• You want to test the effects of two factors on one dependent variable.
• Example questions:
  • Does gender and education affect salary?
  • Does therapy type and gender affect blood pressure?

Two-Way ANOVA (continued)

Three Questions Answered by Two-Way ANOVA

• Does Gender affect the dependent variable?
• Does Education affect the dependent variable?
• Is there an interaction effect between Factor 1 and Factor 2?

Hypothesis:

Steps in Two-Way ANOVA Calculation

• Calculate group means (e.g., mean attitude scores for male & studied, male & not studied, etc.).
• Calculate overall mean of all values.
• Calculate Sums of Squares:
  • SStotal (total variation from overall mean)
  • SSfactorA (variation explained by Factor 1)
  • SSfactorB (variation explained by Factor 2)
  • SSinteraction (variation explained by interaction of A and B)
  • SSerror (unexplained variation)
• Calculate degrees of freedom for each component.
• Calculate mean squares (variance estimates).
• Calculate F-values:
  • F = Variance of Factor / Error Variance
• Interpret p-values to accept or reject the null hypotheses.

Two-Way ANOVA example

• Example: Does gender (male or female) and education status (studied or not) influence a person’s attitude towards retirement planning?
• Dependent Variable: Attitude towards retirement planning (rated from 1 = not important to 10 = very important).

Chi-Square Test

The Chi-square test is a statistical hypothesis test used for categorical variables (i.e., nominal or ordinal scales, e.g., types of assistance received, satisfaction levels).

• It is used to determine whether there is a significant association between two or more categorical variables.

The Chi-square test can answer three types of questions:

Test of Independence

Used to test whether two categorical variables are independent of each other.

Hypotheses

\(H_0: \text{The two variables are independent (no association)}\)

\(H_1: \text{The two variables are not independent (association exists)}\)

\[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]

where the expected frequency for each cell is:

Decision rule

• If \(\chi^2_{\text{calculated}} > \chi^2_{\text{critical}}\) at significance level \(\alpha\) → Reject \(H_0\)
• If \(\chi^2_{\text{calculated}} \leq \chi^2_{\text{critical}}\) → Fail to reject \(H_0\)

Assumptions

• Observations are independent of each other.
• Each observation falls into exactly one cell.
• Expected frequency in each cell is \(\geq 5\) (if violated, consider Fisher’s Exact Test).
• Data are counts (frequencies), not proportions or percentages.

Test of Goodness-of-Fit

Used to determine whether observed frequencies match a theoretically expected distribution.

Formula

\[\chi^2 = \sum_{i=1}^{k} \frac{(O_i - E_i)^2}{E_i}\]

\[df = k - 1\]

where \(k\) is the number of categories.

Test of Homogeneity

Used to determine whether two or more independent groups share the same distribution of a categorical variable.

The Formula is the same as the Test of Independence:

\[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\]

\(df = (r - 1)(c - 1)\)

Note: The key distinction from the Test of Independence is the study design - in homogeneity testing, group membership (rows) is fixed by the researcher, whereas in independence testing, both variables are observed freely.

Comparison of the three Chi-Square tests

Example for Test of Independence

• To test whether two categorical, Gender and Education level variables are independent.

Test of Independence (Cont’ue)

The chi-square value is calculated via:

Research Question: Is umbrella use dependent on gender?

Statistical Methods for Testing Correlations

• What is Correlation?
• Correlation analysis is a statistical method used to examine the relationship between two continuous or ordinal variables.
• It measures:
  • The direction of the relationship (positive or negative)
  • The strength of the relationship (weak or strong)
• The measure of this relationship is the correlation coefficient, which ranges from -1 to +1.

Correlation vs Causation

• Correlation shows association, not causation.
  • A strong correlation does not prove that changes in one variable cause changes in the other.
• Example: Childhood speech and school success:
  • There may be a correlation, but correlation alone doesn’t prove that speaking earlier causes better school success.

Pearson Correlation Analysis

• The Pearson correlation coefficient measures the linear relationship between two continuous (interval/ratio) variables.

Step 1: Covariance

• Covariance measures how two variables change together.
  • Positive covariance → Positive relationship
  • Negative covariance → Negative relationship
• Covariance formula:

\[Cov(x,y) = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{N-1}\]

Pearson Correlation coefficient

Step 2: Correlation

• Covariance is not standardized, making it hard to compare across different datasets.
• So, we normalize it to get the correlation coefficient:

\[r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2 (y_i - \bar{y})^2}}\]

• Called Pearson correlation coefficient & can take values between -1 and +1.
• Before calculating the Pearson correlation, it’s important to visually check the relationship:
  • Scatterplots help detect if the relationship is linear or non-linear.

Pearson correlation: linearity

• Pearson correlation only captures linear relationships. If the data form a curve or other non-linear pattern, Pearson’s r may not be appropriate.

What is Regression?

Regression is a statistical method to model the relationship between a dependent variable and one or more independent variables.

• It is used to measure the influence of predictors and predict outcomes.

Example: Predicting a person’s salary based on education level, weekly working hours, and age.

• Variables:

Dependent Variable: The variable being predicted (e.g., salary), and Independent Variables: Variables used for prediction (e.g., education, working hours, age).

Types of Regression

Regression helps understand how the dependent variable changes when one unit of an independent variable changes, holding others constant.

Simple vs. Multiple Regression

• Simple Regression: Use when one independent variable (IV) predicts the dependent variable (DV).
  • Example: Does long work time (IV) affect a person’s income (DV)?
• Multiple Regression: Use when two or more independent variables predict the DV.
  • Example: Do work hours (IV₁), age (IV₂), and education (IV₃) affect a person’s income (DV)?

Simple Linear Regression

• Predicts the value of a dependent variable (DV) based on one independent variable (IV).
• The stronger the linear relationship between IV and DV, the more accurate the prediction.
• A higher proportion of explained variance in the DV leads to better prediction quality.
• A scatter plot can illustrate the relationship — when the relationship is strong, data points cluster closely along a straight line.

Method of Least Squares

Linear regression uses the Method of Least Squares to find the best-fit line:

\[\hat{y} = b \cdot x + a\]

Concept of Residual

The residual (error) is the difference between the actual and predicted \(y\)-values:

\[e = y - \hat{y}\]

• Goal: Minimize the sum of squared residuals (Ordinary Least Squares — OLS).

Multiple Linear Regression

In multiple linear regression, more than one independent variable is used to predict a single dependent variable. It allows for more accurate and complex prediction by considering multiple influencing factors.

Example

• Investigating cholesterol levels in patients:
  • Independent variables: age, hours of exercise per week, dietary habits, etc.
  • Dependent variable: cholesterol level.

Interpretation

An increase in an independent variable \(x_i\) by one unit will change the dependent variable \(y\) by \(b_i\) units, holding all other variables constant.

Coefficient of Determination (R²)

R² (also known as variance explained) shows how much of the variance in the dependent variable can be explained by the independent variables in the model.

\[R^2 = \frac{S^2_{\hat{y}}}{S^2_{y}} = \frac{\text{Variance of the Predicted values}}{\text{Variance of the Observed values}}\]

• \(R^2 = 1\): Perfect fit — all variance explained.
• \(R^2 = 0\): No variance explained — independent variables do not help predict the dependent variable.

The higher the \(R^2\), the better the regression model fits the data.

Adjusted R²

R² increases as more independent variables are added to the model, even if those variables don’t contribute meaningfully. Adjusted R² compensates for this by penalising for the number of predictors.

\[R^2_{adj} = 1 - \left(1 - R^2\right)\cdot\frac{n-1}{n-p-1}\]

Why the penalty works

When a new predictor is added, \((n - p - 1)\) decreases. Two scenarios arise:

• Useful predictor: \((1 - R^2)\) drops meaningfully, offsetting the smaller denominator \(\Rightarrow\) \(R^2_{adj}\) increases or stays stable.
• Useless predictor: \((1 - R^2)\) barely changes, but the ratio \(\frac{n-1}{n-p-1}\) grows \(\Rightarrow\) \(R^2_{adj}\) falls, signalling the predictor was not worth including.

Key properties

• \(R^2_{adj} \leq R^2\) always, with equality only when \(p = 0\).
• \(R^2_{adj}\) can be negative if the model fits worse than a horizontal line (intercept only).
• As \(n \to \infty\), the penalty vanishes and \(R^2_{adj} \approx R^2\), because large samples make overfitting less of a concern.
• Use \(R^2_{adj}\) when comparing models with different numbers of predictors — it provides a fair, penalised basis for comparison.

Assumptions of Linear Regression

For the results of regression analysis to be valid, the following assumptions must be met:

Linearity

Linear regression assumes a linear relationship between the dependent and independent variables. The goal is to draw a straight line that best represents the data points.

• Left Graph: A linear relationship is visible - data points align closely to the straight line, meaning a regression model will work effectively.
• Right Graph: A non-linear relationship is visible - a straight line cannot accurately represent the data, which may lead to incorrect predictions and conclusions.

Consequences of Non-Linearity

• Non-linearity can produce invalid regression coefficients and misleading predictions, leading to substantial errors and poor decision-making.

Homoscedasticity

In a regression model, there is always error (residuals) in predicting the dependent variable. Homoscedasticity means that the variance of the residuals is constant across all predicted values.

To test for homoscedasticity, plot: Dependent variable (DV) on the \(x\)-axis vs Residuals (errors) on the \(y\)-axis.

• If homoscedasticity exists → residuals scatter evenly across all values.
• If heteroscedasticity exists → residuals show varying spread depending on the range of the DV.

Heteroscedasticity causes inaccurate regression estimates and unreliable predictions, which may lead to incorrect conclusions.

Example

Results

Interpretation of Results

Model Summary

• \(R^2 = 75.4\%\) → 75.4% of the variation in weight is explained by the independent variables (height, age, and gender).
• \(R^2_{adj} = 63\%\) → After adjusting for the number of predictors and degrees of freedom, about 63% of the variance is truly explained by the model. Adjusted R² provides a more realistic measure of model fit, penalising the addition of variables that don’t improve the model.
• Average prediction error (residual standard error) = 6.587 kg.

Regression Equation

\[\text{Weight} = 47.379 \times \text{Height} + 0.297 \times \text{Age} + 8.922 \times \text{is\_male} - 24.41\]

Interpreting Coefficients

• Age: Each additional year increases weight by 0.297 kg (holding other variables constant).
• Gender (is_male): Being male adds 8.922 kg to weight compared to females.

Hypothesis Testing

• \(H_0\): Coefficient \(= 0\) (no effect)
• \(H_1\): Coefficient \(\neq 0\) (has effect)

In this model, only Age has \(p < 0.05\), making it the only statistically significant predictor.

What is **Survival Analysis?

• Survival analysis is a statistical method used to examine the time until a specific event occurs, such as: Death, Disease onset, Relapse, Recovery, Equipment failure
• It focuses on time-based variables, measuring the duration between a start event and an end event.
• Time is typically recorded in days, weeks, or months.

Key Components

• Start time: The point when observation begins (e.g., diagnosis date).
• Event: The point when the outcome occurs (e.g., death, failure).
• Survival time: The time between the start and the occurrence of the event.

• Start = End of withdrawal process, Event = Relapse; Survival time = Number of days or weeks until relapse
• Time from disease diagnosis to death: Start = Diagnosis date, Event = Death

What is Censoring?

• Censoring occurs when:
• The event of interest has not happened by the study’s end.
• The subject leaves the study before the event occurs.

Censoring (continued)

• Ignoring censoring leads to biased results and incorrect survival estimates.
• Censoring is a natural and common part of survival studies and a critical feature that distinguishes survival analysis from other types of statistical modeling.
• Imagine you are a dental technician analyzing the lifespan of tooth fillings:
  • Start time: Day the filling is placed.
  • Event: Filling breaks or falls out.
• You record the survival time for each patient’s filling. However, two important scenarios arise:
• A patient’s filling has not failed by the end of the study.
• A patient drops out (moves away, changes dentist) before the filling fails.
• These situations introduce censoring.

Basic Concepts of Survival analysis

• Survival time analysis uses specialized statistical methods designed to handle time-to-event data and account for censoring.

The three most common methods are:

• Kaplan-Meier Survival Curves
• Log-Rank Test
• Cox Proportional Hazards Regression

Kaplan-Meier Curve

Comparing different groups

• When studying survival time, researchers often want to compare two or more groups. For example:
  • Comparing patients receiving two different treatments.
  • Comparing male vs. female patients.
  • Comparing different age groups or exposure levels.
• In such cases, the Kaplan-Meier curve is drawn separately for each group. Each line on the plot represents the estimated survival rate over time for a particular group.

Comparing different groups (continued)

Hypotheses in the Log-Rank Test

• Null Hypothesis (H₀): The survival distributions of the groups are identical. (There is no difference in survival times between groups.)
• Alternative Hypothesis (H₁): The survival distributions of the groups are different. (There is a difference in survival times between groups.)

Result

Interpreting the Log-Rank Test

• If the p-value is small (typically < 0.05):
  • Reject the null hypothesis.
  • Conclude that there is a statistically significant difference between the groups’ survival experiences.
• If the p-value is large (typically ≥ 0.05):
  • Fail to reject the null hypothesis.
  • Conclude that there is no significant difference between the groups.

Cox Regression

• What is Cox Regression?
• Cox Regression (also called the Cox Proportional Hazards Model) is a method used in survival analysis to:
  • Examine the influence of several variables (continuous, binary, or categorical) on survival time.
  • Adjust for multiple variables at the same time.
  • Predict how changes in the variables affect the risk (hazard) of the event occurring.
• allows us to determine the effects of multiple independent variables on a time-to-event outcome,
  • to test hypotheses about which factors impact survival,
  • to build predictive models based on those factors.

Cox Regression Model (Cox Proportional Hazards Model)

The Cox model describes the hazard (risk) of an event over time:

\[H(t) = H_0(t) \times \exp\left(b_1 x_1 + b_2 x_2 + \cdots + b_k x_k\right)\]

• \(H_0(t)\): Baseline hazard when all predictors are zero.
• \(x_1,\ldots,x_k\): Predictor variables, \(b_1,\ldots,b_k\): Regression coefficients.
• The exponential of a coefficient exp(b) gives the Hazard Ratio (HR):
  • For binary variables (e.g., exposed vs. unexposed), HR shows how much more (or less) likely the event is in the exposed group.
  • For continuous variables (e.g., age), HR shows the change in hazard per one-unit increase:
    • Example: HR = 1.03 for age → each year increases the hazard by 3%.
    • Example: HR = 0.85 for albumin → each 1 g/dL increase reduces the hazard by 15%.
• Interpretation assumes other covariates are held constant.

Cox Regression: data

• Let’s assume that we have the following data and we want to evaluate them.

Cox Regression: results

• The following results of a Cox regression is generated from the above data

Interpretation

• Coefficient (β): Reflects the direction and strength of the variable’s association with survival.
  • Negative β → Lower risk (longer survival).
  • Positive β → Higher risk (shorter survival).
• P-value: Tests whether the coefficient is significantly different from zero.

Hypothesis Testing in Cox Regression

• Null Hypothesis (H₀): The coefficient is zero (no effect on survival).
• Alternative Hypothesis (H₁): The coefficient is not zero (there is an effect on survival).
• Decision rule:
  • If p-value < 0.05 → Reject H₀ → Variable has a significant effect.

Basic concepts of Longitudinal Data analysis

• In survey research and data collection, two major methods are:
  • Longitudinal studies and Cross-sectional studies
• Both are widely used across health, social science, and humanitarian fields.
• Knowing their differences is crucial for designing effective research and managing data collection workflows

Key Differences

When to Use a Longitudinal Study

• When the research question involves changes, trends, or trajectories over time.
• When exploring the effect of time-dependent exposures.
• When within-subject comparisons are critical.
• When aiming to establish temporal sequences for potential causal inferences.

Example: Tracking changes in maternal nutritional status from early pregnancy to postpartum to examine impacts on birth outcomes.

Key Features of Longitudinal Data

• Statistical techniques like ANOVA and regression usually assume independent and identically distributed (iid) residuals.
• In practice, data often violate this assumption due to correlations among observations.
• Correlated data structures include:
  • Clustered data, Repeated measurements, Spatially correlated data

Examples for clustered data: Families, Schools, Hospitals, Towns.

• When repeated measurements are collected over time, the resulting data form a longitudinal (or panel) study.
  • allows researchers to examine how individuals change over time and what factors influence those changes.
• Therefore, these correlated data have correlation among observations, which violates assumption of iid.

Objectives of Longitudinal Studies

Exploratory Data Analysis (EDA) in Longitudinal Studies

• What is Exploratory Data Analysis (EDA)?
• Exploratory analysis comprises techniques to visualize patterns in the data.
• Data analysis must begin by making displays that expose patterns relevant to the scientific question.
• Therefore EDA helps:
  • Helps uncover expected and unexpected patterns.
  • Graphical displays are crucial to highlight relationships and trends.
  • Indicates which model will be appropriate for analysis.

Example: Jimma Infant Survival Data

A follow-up study of newborn infants in Southwest Ethiopia with measurements at 7 time points (every 2 months, 0-12 months).

Long Format vs. Wide Format

Long Format: one row per observation per time point:

Wide Format: one row per individual:

Most longitudinal models require long format.

individual Vs Mean profiles

Conclusions from the profile:

Exploratory analysis conclusion

Exploring the random effects

• Choosing Random Effects:
  • Decide which parameters need random effects to account for between-group variation.

Linear Mixed Model

• Correctly modeling correlation is essential for valid inference about regression coefficients.
• Extend general linear models to account for correlated errors.
• Combine:
  • Fixed effects (e.g., sex, age group)
  • Random effects (e.g., individual subjects)
• LMMs make assumptions about:
• Mean structure: linear or nonlinear
• Variance function: constant or changing (e.g., quadratic)
• Correlation structure: independent, serial, etc.
• Subject-specific profiles: linear, quadratic, etc.

The model is given by

\[Y_i = \underbrace{X_i \beta}_{\text{fixed effect}} + \underbrace{Z_i b_i}_{\text{random effect}} + \varepsilon_i\]

Where:

LMM result for Jimma infant data

Intraclass Correlation Coefficient (ICC)

The ICC quantifies what proportion of the total variability is attributable to between-individual differences.

\[ICC = \frac{\sigma^2_{\text{between}}}{\sigma^2_{\text{between}} + \sigma^2_{\text{within}}} = \frac{345.302}{345.302 + 656.924} \approx 0.345\]

• An ICC of 34.5% indicates substantial individual variation in both initial weight and growth rate.
• This confirms that ignoring random effects would lead to invalid standard errors and misleading inference.