Detailed Notes on Paired Samples T-test and Related Statistical Concepts

PH102: Biostatistics in Public Health - Paired Samples T-test

One Sample T Test Review

  • Definition: The One Sample T Test is utilized to assess whether the mean of a continuous random variable (R.V.) from one population is equal to a hypothesized value, denoted as $$.

Example of One Sample T Test

  • Question: Is the mean juice consumption in 1997 the same as the known mean juice consumption of 1975 with $ = 4.5$ oz/day?

Comparing Mean Between Two Populations

  • Test Selection Criteria: The choice of statistical test depends on:

    1. Relationship between the populations:

    • Are the two populations related (dependent)?

    • Are the two populations unrelated (independent)?

    1. Nature of the variable:

    • Is the question regarding a continuous variable (means) or a binary variable (proportions)?

    1. Population Standard Deviation Status:

    • Independent means lead to the use of:

      • Z Test for a population mean if standard deviation is known.

      • One Sample t-Test if standard deviation is not known.

    • Dependent means lead to:

      • Paired Sample t-Test if samples are paired.

      • Two Independent Samples t-Test if samples are independent.

Paired Samples Design

  • Characteristics: Pre- and post-intervention studies involve measuring a random variable before and after treatment on the same individuals, creating effective paired designs.

  • Example:

    • Measure plasma cholesterol of the same random sample of individuals before and after a specified treatment, such as eating avocados for six months.

  • Statistical Power: Paired samples are considered statistically powerful since utilizing the same participants minimizes variability caused by extraneous factors.

Paired Samples Design Hypothesis Set-Up

  • Example of Pre- vs. Post-Treatment Studies: The hypothesis posits that treatment alters a continuous variable outcome:

    • Sample 1: Pre-Treatment

    • Sample 2: Post-Treatment

  • Null Hypothesis Overview: Can be stated in three equivalent ways:

    1. $H0: {pre-treatment} = _{post-treatment}$

    2. $H0: {post-treatment} - _{pre-treatment} = 0$

    3. $H0: d = 0$ where $d = {post-treatment} - _{pre-treatment}$

  • Alternative Hypothesis (Ha): Opposite statements to the null hypothesis based on the defined parameters.

Paired Samples t Test Procedure

  • Utilizes the same operational approach as a One Sample t Test:

    • Null Hypothesis: $H0: d = 0$ vs $Ha: d
      eq 0$

  • A new variable, $d$, can be created as $( ext{post-treatment} – ext{pre-treatment})$, upon which the One Sample t Test is run.

Practical Example: Hypertension Study

  • Study Overview:

    • Dataset: Hypertension.sav with $n=409$ subjects diagnosed and treated for hypertension.

    • Objective: To compare systolic blood pressure before and after treatment with baseline SBP and follow-up SBP representing Sample 1 and Sample 2 respectively.

  • Null Hypothesis: No change in mean SBP between baseline and follow-up:

    • $H0: {baseline} = {follow-up}$ vs $Ha: {baseline} eq {follow-up}$

    • Equivalently: $H0: d = 0$ vs. $Ha: d
      eq 0$ with $d = {follow-up} - _{baseline}$

Data Presentation: Summary Statistics for Hypertension Study

  • Descriptive Statistics Output Overview:

    • Variables:

    • SBP.Baseline (409 samples, Mean: 144.70, SD: 23.467)

    • SBP.Followup (409 samples, Mean: 135.74, SD: 20.240)

    • SBP.Change (Mean: -8.95, SD: 23.23938)

    • Mean SBP Change Calculation: ($ ext{follow-up} - ext{baseline} = -8.95$ mmHg).

Testing Procedures

  • Null Hypothesis: Mean change in SBP equals zero:

    • $H0: d = 0$ vs $Ha: d
      eq 0$, where $d = {baseline}-_{follow-up}$.

Confidence Interval Calculation

  • 95% Confidence Interval for $_d$: Based on our analysis, the 95% CI for mean change in SBP was calculated as: $(-11.21, -6.69)$ mmHg, indicating that since zero is not included in the interval, the Null Hypothesis is rejected, concluding a significant decrease in SBP from baseline to follow-up.

Calculating the T-Statistic

  • T-statistic Formula: $ ext{t}_{stat}$:

    • Formula: $ ext{t}{stat} = rac{ar{x} - 0}{ rac{s_d}{ ext{sqrt}(n)}}$ where:

    • $ar{x}$ is the mean difference.

    • $s_d$ is standard deviation of the differences.

    • $n$ is the number of observations.

Example Calculation with Data

  • For Hypertension Study:

    • $ar{x} = -8.95$ mmHg, $s_d = 23.24$ mmHg, $n = 409$

    • Calculate: $ ext{t}_{stat} = rac{-8.95 - 0}{ rac{23.24}{ ext{sqrt}(409)}} = -7.79$.

Critical Value Determination

  • Determining Critical Values: Determine critical t-values from the t-distribution table for your specific $ ext{df}$ (degrees of freedom) and p-value criteria.

P-value Assessment

  • After obtaining the t-statistic, the associated p-value must be determined to conclude whether to reject the Null Hypothesis or not, based on standard thresholds (e.g., $ ext{p} < 0.05$ denotes significance).

Summary of Assumptions for Paired Samples t-Test

  1. The random variable under evaluation must be continuous.

  2. Normality: The distribution of differences between pairs should be approximately normal; alternatively, if the sample size is large enough (n > 30), the Central Limit Theorem allows for normality.

  3. Related Groups: Samples must consist of matched pairs or be related in some way.

Using SPSS for Paired Samples T-Test

  • SPSS Process Overview:

    1. One-sample t-test for differences or paired sample t-test selections.

    2. Calculating differences via the Transform function to create a new variable linking SBP changes.

    3. Analyzing results through Analyze > Compare Means for either perspective.

Exercise Proposal

  • Task: Assume normality in a dataset; conduct Paired Sample t-Test using SPSS and perform hand calculation verifications for hypothesis testing in comparable drug efficacy for sleep induction.