MAT MILESTONE 3

Application-Based Assessment: Two Population Hypothesis Testing Assessment

Scenario: MediCore Pharmaceuticals is conducting comprehensive clinical trials for a new cardiovascular medication called "HeartGuard." As the lead biostatistician, you must analyze multiple aspects of the trial data to determine the medication's effectiveness and safety profile compared to existing treatments and control groups.

The dataset contains clinical trial results with the following variables:

· PatientID: Unique identifier for each patient

· Treatment: Treatment group (HeartGuard, Placebo, StandardDrug)

· Gender: Patient gender (Male, Female)

· Age: Patient age in years

· BloodPressure_Before: Systolic blood pressure before treatment (mmHg)

· BloodPressure_After: Systolic blood pressure after 6 months (mmHg)

· SideEffects: Whether patient experienced side effects (Yes, No)

· Compliance: Treatment compliance rate (%)

· Population_SD_Known: Blood pressure population standard deviation for known variance studies

Submission: Submit your Word document with all graphs, charts, and outputs and your .mpx Minitab file.

Part A: Hypothesis Test Classification

Background

Before conducting any statistical analysis, you must properly classify the types of hypothesis tests needed for different research questions.

A1. Test Type Identification

For each of the following research questions, identify:

1. The type of hypothesis test needed

2. The parameter(s) being tested

3. Whether it involves one or two populations

4. The nature of the data (means, proportions, paired samples)

Research Questions: a) Is there a difference in average blood pressure reduction between HeartGuard and the standard drug?

b) Do patients taking HeartGuard have a different rate of side effects compared to those taking placebo?

c) Is HeartGuard more effective than placebo in reducing blood pressure?

d) Does HeartGuard reduce blood pressure more effectively than the standard drug, assuming population standard deviations are known from previous studies?

e) Is there a significant difference in blood pressure before and after HeartGuard treatment in the same patients?

f) Do male and female patients show different compliance rates with HeartGuard?

Part B: Two Population Means - Known Standard Deviations

Background

Previous large-scale studies have established that the population standard deviation for systolic blood pressure reduction is 8.5 mmHg for HeartGuard and 9.2 mmHg for standard drug treatments.

B1. Research Question Setup

Research Question: Is there a significant difference in mean blood pressure reduction between HeartGuard and the standard drug, assuming known population standard deviations?

Minitab: Using only HeartGuard and StandardDrug groups, calculate the blood pressure reduction for each patient (Before - After) in a new column.

B2. Hypothesis Test Execution

a) State your null and alternative hypotheses.

b) Use Minitab to perform the test (provide screenshots of your setup and results).

c) Report the test statistic and the p-value.

d) Make a decision using α = 0.05

B3. Results Interpretation

a) Interpret your statistical results in the context of the clinical trial

b) What are the limitations of assuming known population standard deviations in this context?

Part C: Two Population Means - Unknown Standard Deviations

Background

In reality, population standard deviations are typically unknown and must be estimated from sample data.

C1. Comparative Effectiveness Analysis

Research Question: Is HeartGuard significantly more effective than placebo in reducing blood pressure when population standard deviations are unknown?

a) Filter data for HeartGuard and Placebo groups only:

· Go to Data > Subset Worksheet.

· In the dialog box, select "Specify which rows to include".

· Click Condition button.

· Set condition: Treatment = "HeartGuard" OR Treatment = "Placebo".

· Click OK to create a new worksheet with only these two groups.

· Name the new worksheet "HeartGuard_vs_Placebo".

b) Calculate blood pressure reduction for each group:

· In the new worksheet, go to Calc > Calculator.

· Store result in: BP_Reduction.

· Expression: 'BloodPressure_Before' - 'BloodPressure_After'.

· Click OK.

· This creates a new column showing how much each patient's blood pressure decreased.

· Note: Positive values indicate blood pressure reduction (improvement).

c) Create separate columns for each treatment group:

· Go to Calc > Calculator again.

· Store result in: HeartGuard_Reduction.

· Expression: IF('Treatment'="HeartGuard",'BP_Reduction',"*").

· Click OK.

· Repeat for Placebo.

· Store result in: Placebo_Reduction.

· Expression: IF('Treatment'="Placebo",'BP_Reduction',"*").

· Click OK.

d) Check assumptions for the two-sample t-test using appropriate diagnostics:

· Normality Check:

· Go to Stat > Basic Statistics > Normality Test.

· Select HeartGuard_Reduction (ignore missing values).

· Choose Anderson-Darling test, click OK.

· Repeat for Placebo_Reduction.

· Equal Variances Check:

· Go to Stat > Basic Statistics > 2 Variances.

· First sample: HeartGuard_Reduction.

· Second sample: Placebo_Reduction.

· Click OK.

· Visual Inspection:

· Go to Graph > Boxplot > Multiple Y's Simple.

· Add both HeartGuard_Reduction and Placebo_Reduction.

· Click OK to compare distributions visually.

C2. Statistical Analysis

d) Conduct the appropriate two-sample t-test in Minitab .

1. Navigate to the test:

· Go to Stat > Basic Statistics > 2-Sample t.

2. Set up the data format:

· Select "Samples in different columns".

· First: Select first group's data column (e.g., HeartGuard_Reduction).

· Second: Select second group's data column (e.g., Placebo_Reduction).

3. Configure the test.

· Alternative hypothesis: Choose from:

· "Not equal" (two-tailed test: μ₁ ≠ μ₂).

· "Less than" (left-tailed test: μ₁ < μ₂).

· "Greater than" (right-tailed test: μ₁ > μ₂).

· Confidence level: Usually 95.0 (for α = 0.05).

4. Handle equal variances:

· Check "Assume equal variances" if your variance test showed equal variances.

· Leave unchecked for Welch's t-test (unequal variances).

5. Click OK to run the test.

e) Report all relevant statistics (means, standard deviations, test statistic, degrees of freedom, p-value).

C3. Comprehensive Comparison

g) Now compare HeartGuard vs. StandardDrug using the same methodology.

h) Create a summary table comparing HeartGuard against both Placebo and StandardDrug.

i) Which comparison shows the strongest evidence of HeartGuard's effectiveness?

C4. Effect Size Analysis

j) Calculate and interpret Cohen's d for both comparisons.

k) Classify the effect sizes as small, medium, or large.

l) Discuss the clinical significance of these effect sizes.

Part D: Two Population Proportions

Background

Side effects are a critical consideration in pharmaceutical approval. You need to compare side effect rates between treatment groups.

D1. Side Effects Analysis - Primary Comparison

Research Question: Is there a significant difference in the proportion of patients experiencing side effects between HeartGuard and Placebo groups?

Using Minitab:

a) Create a contingency table for Treatment (HeartGuard vs. Placebo) and SideEffects.

For creating a contingency table with counts and percentages:

1. Navigate to the tool:

· Stat > Tables > Cross Tabulation and Chi-Square

· For rows: Treatment

· For columns: SideEffects

· Check: Counts, Row percents, Column percents

· Click OK

b) Verify that conditions for the two-proportion z-test are met.

For Proportion Testing:

After creating the contingency table, you can:

1. Stat > Basic Statistics > 2 Proportions

2. Select "Summarized data"

3. Enter the counts from your contingency table

4. Enter sample sizes for each group

D2. Hypothesis Testing for Proportions

c) Set up appropriate hypotheses for this research question.

d) Conduct the two-proportion z-test in Minitab.

e) Report the test statistic, p-value, and confidence interval for the difference in proportions.

f) Make a statistical decision using α = 0.01.

Part E: Matched or Paired Samples

Background

The most direct way to assess HeartGuard's effectiveness is to compare each patient's blood pressure before and after treatment.

E1. Before-After Analysis Setup

Research Question: Does HeartGuard significantly reduce blood pressure in patients after 6 months of treatment?

Using Minitab:

a) Filter data to include only HeartGuard patients

b) Create a new variable for blood pressure differences (Before - After)

c) Examine the distribution of differences using appropriate plots.

E2. Paired Sample Analysis

d) Check assumptions for the paired t-test.

e) Conduct the paired t-test in Minitab.

f) Report the mean difference, standard deviation of differences, test statistic, and p-value.

g) Calculate and interpret the 95% confidence interval for the mean difference.