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}</p></li><li><p></p></li><li><p>H0: {post-treatment} - _{pre-treatment} = 0</p></li><li><p></p></li><li><p>H0: d = 0wherewhered = {post-treatment} - _{pre-treatment}</p></li></ol></li><li><p><strong>AlternativeHypothesis(Ha):</strong>Oppositestatementstothenullhypothesisbasedonthedefinedparameters.</p></li></ul><h4id="254d3ab4−9405−4c2e−9024−f9c7b38d462b"data−toc−id="254d3ab4−9405−4c2e−9024−f9c7b38d462b"collapsed="false"seolevelmigrated="true">PairedSamplestTestProcedure</h4><ul><li><p>UtilizesthesameoperationalapproachasaOneSampletTest:</p><ul><li><p><strong>NullHypothesis:</strong></p></li></ol></li><li><p><strong>Alternative Hypothesis (Ha):</strong> Opposite statements to the null hypothesis based on the defined parameters.</p></li></ul><h4 id="254d3ab4-9405-4c2e-9024-f9c7b38d462b" data-toc-id="254d3ab4-9405-4c2e-9024-f9c7b38d462b" collapsed="false" seolevelmigrated="true">Paired Samples t Test Procedure</h4><ul><li><p>Utilizes the same operational approach as a One Sample t Test:</p><ul><li><p><strong>Null Hypothesis:</strong>H0: d = 0vsvsHa: d
      eq 0</p></li></ul></li><li><p>Anewvariable,</p></li></ul></li><li><p>A new variable,d,canbecreatedas, can be created as( ext{post-treatment} – ext{pre-treatment}),uponwhichtheOneSampletTestisrun.</p></li></ul><h4id="d80bccd3−2155−4c5e−a1b1−ef4052eea553"data−toc−id="d80bccd3−2155−4c5e−a1b1−ef4052eea553"collapsed="false"seolevelmigrated="true">PracticalExample:HypertensionStudy</h4><ul><li><p><strong>StudyOverview:</strong></p><ul><li><p>Dataset:Hypertension.savwith, upon which the One Sample t Test is run.</p></li></ul><h4 id="d80bccd3-2155-4c5e-a1b1-ef4052eea553" data-toc-id="d80bccd3-2155-4c5e-a1b1-ef4052eea553" collapsed="false" seolevelmigrated="true">Practical Example: Hypertension Study</h4><ul><li><p><strong>Study Overview:</strong></p><ul><li><p>Dataset: Hypertension.sav withn=409subjectsdiagnosedandtreatedforhypertension.</p></li><li><p>Objective:TocomparesystolicbloodpressurebeforeandaftertreatmentwithbaselineSBPandfollow−upSBPrepresentingSample1andSample2respectively.</p></li></ul></li><li><p><strong>NullHypothesis:</strong>NochangeinmeanSBPbetweenbaselineandfollow−up:</p><ul><li><p>subjects diagnosed and treated for hypertension.</p></li><li><p>Objective: To compare systolic blood pressure before and after treatment with baseline SBP and follow-up SBP representing Sample 1 and Sample 2 respectively.</p></li></ul></li><li><p><strong>Null Hypothesis:</strong> No change in mean SBP between baseline and follow-up:</p><ul><li><p>H0: {baseline} = {follow-up}vsvsHa: {baseline} eq {follow-up}</p></li><li><p>Equivalently:</p></li><li><p>Equivalently:H0: d = 0vs.vs.Ha: d
      eq 0withwithd = {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 = 0vsvsHa: d
      eq 0,where, whered = {baseline}-_{follow-up}.</p></li></ul></li></ul><h4id="3824b84a−c78c−49c4−bdbc−35eac4bc9ef9"data−toc−id="3824b84a−c78c−49c4−bdbc−35eac4bc9ef9"collapsed="false"seolevelmigrated="true">ConfidenceIntervalCalculation</h4><ul><li><p><strong>95.</p></li></ul></li></ul><h4 id="3824b84a-c78c-49c4-bdbc-35eac4bc9ef9" data-toc-id="3824b84a-c78c-49c4-bdbc-35eac4bc9ef9" collapsed="false" seolevelmigrated="true">Confidence Interval Calculation</h4><ul><li><p><strong>95% Confidence Interval for_d</strong>:Basedonouranalysis,the95</strong>: 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}isthemeandifference.</p></li><li><p>is the mean difference.</p></li><li><p>s_disstandarddeviationofthedifferences.</p></li><li><p>is standard deviation of the differences.</p></li><li><p>nisthenumberofobservations.</p></li></ul></li></ul><h4id="0abfa39e−d930−4bf0−9122−a872d8da7126"data−toc−id="0abfa39e−d930−4bf0−9122−a872d8da7126"collapsed="false"seolevelmigrated="true">ExampleCalculationwithData</h4><ul><li><p>ForHypertensionStudy:</p><ul><li><p>is the number of observations.</p></li></ul></li></ul><h4 id="0abfa39e-d930-4bf0-9122-a872d8da7126" data-toc-id="0abfa39e-d930-4bf0-9122-a872d8da7126" collapsed="false" seolevelmigrated="true">Example Calculation with Data</h4><ul><li><p>For Hypertension Study:</p><ul><li><p>ar{x} = -8.95mmHg,mmHg,s_d = 23.24mmHg,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.