Applied Statistics for the Health Sciences

Overview of Statistical Tests for Mean Differences

This section introduces key statistical methods used to compare mean differences between groups or conditions, with a focus on applications in health sciences. Understanding these tests is essential for analyzing experimental data and drawing valid scientific conclusions.

• Mean differences between two groups or conditions: Statistical tests are used to determine if the average values of two samples differ significantly.

• Parametric and non-parametric tests: Selection depends on data distribution and sample characteristics.

• t-test for independent samples: Used when comparing two separate groups.

• t-test for repeated measures (paired samples): Used when the same subjects are measured under different conditions.

Course Review: Hypothesis Testing

Research Questions and Hypotheses

Statistical analysis begins with formulating a research question and corresponding hypotheses. Hypothesis testing is a structured process to determine if observed data support a specific claim.

• Null Hypothesis (H0): Assumes no effect or no difference between groups.

• Alternative Hypothesis (H1): Assumes there is an effect or a difference.

• Statistical significance: Results are considered statistically significant if the probability of observing the data under the null hypothesis is less than a chosen threshold (commonly ).

Mean Difference Between Two Groups or Conditions

Assessing Mean Differences

Researchers often compare the mean scores of two samples to determine if they differ statistically. This is common in clinical trials, laboratory experiments, and observational studies.

• Example: Comparing the benefits of intensive rehabilitation in patients with chronic obstructive pulmonary disease (COPD) versus standard care.

• Application: Used to evaluate treatment efficacy, behavioral interventions, or physiological responses.

Parametric and Non-Parametric Tests

Types of Statistical Tests for Mean Differences

Statistical tests for mean differences are classified as parametric or non-parametric, depending on the assumptions about the data.

• Parametric tests: Assume data are normally distributed and have equal variances. Examples include the independent samples t-test and paired samples t-test.

• Non-parametric tests: Do not require normality or equal variances. Examples include the Mann-Whitney U test and Wilcoxon signed-rank test.

• Assumption checking: Recent advances allow for more flexible use of parametric tests even when assumptions are mildly violated.

t-Test for Independent Groups

Comparing Two Independent Samples

The independent samples t-test is used to compare the means of two unrelated groups. It is widely used in experimental and clinical research.

• Example: Evaluating the effectiveness of a new drug in multiple sclerosis patients by comparing treated and untreated groups.

• Hypotheses:

  • Null hypothesis: (no difference in means)

  • Alternative hypothesis: (means are different)

• Formula: The t-statistic for independent samples is calculated as: where and are sample means, and are sample variances, and , are sample sizes.

Steps to Verify t-Test Results with SPSS

• Verify the result of the Levene's test for equality of variances.

• If p-value is significant (p < 0.05), use the row for unequal variances.

• Check the p-value of the t-test. If significant, conclude that the means differ.

• Calculate the effect size to assess the magnitude of the difference.

Effect Size Calculation

• Cohen's d: Measures the standardized difference between means. where is the pooled standard deviation.

• Interpretation:

  • Small effect:

  • Medium effect:

  • Large effect:

Interpretation of Results

• If the p-value is less than 0.05, the difference between group means is statistically significant.

• Effect size provides context for the practical significance of the result.

t-Test for Repeated Measures (Paired Samples)

Comparing Means Within the Same Subjects

The paired samples t-test is used when the same subjects are measured under two conditions, such as before and after treatment.

• Example: Determining whether patients with multiple sclerosis have lower scores on a verbal IQ task than on a performance IQ task.

• Hypotheses:

  • Null hypothesis: (no difference in means)

  • Alternative hypothesis: (means are different)

• Formula: The t-statistic for paired samples is calculated as: where is the mean of the differences, is the standard deviation of the differences, and is the number of pairs.

Steps to Verify t-Test Results with SPSS

• Verify the result of the t-test for paired samples.

• If p-value is significant, conclude that the means differ.

• Calculate the effect size to assess the magnitude of the difference.

Effect Size Calculation

• Cohen's d for paired samples:

• Interpretation:

  • Small effect:

  • Medium effect:

  • Large effect:

Interpretation of Results

• If the p-value is less than 0.05, the difference between condition means is statistically significant.

• Effect size provides context for the practical significance of the result.

Summary Table: Parametric vs. Non-Parametric Tests

The following table summarizes the main differences between parametric and non-parametric tests for mean differences.

Additional info: These notes are focused on statistical methods and do not cover biological mechanisms or processes directly. However, statistical analysis is a foundational skill for interpreting experimental results in biology and health sciences.