module 3 stat section B 1

Module 3 Overview

  • Welcome to Module 3, focusing on statistical concepts from Module 2.

Module Structure

  • Section A: Analysis of categorical data

    • Part 1: Univariate analysis of categorical data

    • Part 2: Analysis of relationships between categorical variables

    • Part 3: Multivariate analysis of categorical data

  • Section B: Analysis of continuous data

Objectives of the Module

  1. Determine the most appropriate statistical test for different data types and comparisons.

  2. Understand the rationale and assumptions behind each statistical test.

  3. Interpret and report the results of the statistical analyses.

Categorical Data Analysis

  • Summarizing Categorical Data:

    • Use frequencies, proportions, or percentages for single variables.

    • Employ a two-way contingency table (cross tabulation) to summarize relationships between two categorical variables.

Key Questions in Categorical Data Analysis

  1. Is the observed proportion the same as a hypothesized proportion?

  2. Does our sample proportion align with another study's population proportion?

  3. Is there a difference in proportions between two or more groups?

  4. Does an intervention change the outcome proportions?

Univariate Analysis of Categorical Data

  • Example Question:

    • "Is the proportion of first-year QUT students who are female the same as in the Brisbane population aged 18-21?"

    • Analysis Type: Compare sample proportion to an external population.

  • Another Example Question:

    • "Is the distribution of individuals presenting to Royal Brisbane Hospital's emergency ward evenly spread across age groups?"

    • Analysis Type: Compare sample results to a hypothesized distribution.

Statistical Tests for Categorical Data

Chi-Square Goodness of Fit Test

  • Purpose: Used to compare observed and expected frequencies for a single categorical variable.

  • Hypotheses:

    • Null Hypothesis (H0): Observed frequencies are evenly distributed across categories.

    • Alternative Hypothesis (H1): Observed frequencies are not evenly distributed.

Data Example: Accident Emergency Ward

  • Observed Frequency Data for Age Groups:

    • 0-15: 40

    • 16-30: 62

    • 31-60: 50

    • 60+: 42

  • Total Sample Size: 200

Expected Frequencies (Assuming H0 is true)

  • Each age group should have 50 individuals if evenly distributed.

Calculation of Chi-Square Statistic (χ²)

  • Formula: χ² = Σ[(Observed - Expected)² / Expected] for each group.

  • Example Calculation:

    • For age group 0-15:

      • Observed: 40, Expected: 50

      • (40 - 50)² / 50 = 2

  • Continue for all groups and sum results for total chi-square value.

Test Statistic and P-Value Calculation

  • A test statistic is computed to determine whether observed values support the null hypothesis.

  • Finding Significance: If the calculated p-value is greater than 0.05, we do not reject the null hypothesis.

  • In this example:

    • χ² value = 6.88, P-value = 0.076

    • Conclusion: No evidence to suggest that the age distribution is uneven; frequencies are in line with H0.

Chi-Square Distribution

  • Non-symmetric, shape depends on degrees of freedom.

  • Degrees of Freedom Formula: df = number of groups - 1.

  • For our example, df = 4 - 1 = 3.

  • Chi-square values greater than a certain threshold suggest significant results.

Why Use Software?

  • Calculating χ² values traditionally requires referencing tables, which can be tedious.

  • Using statistical software like R Commander automates calculations and improves accuracy.

Recap

  • Chi-square goodness of fit is used to test proportions across categories.

  • Hypothesized distribution example: age groups presenting evenly in emergency studies.

Next Steps

  • Future topics will cover bivariate categorical data analysis.