Comparing Q values to T values in statistical analysis is essential for accurate interpretation, and compare.edu.vn can guide you through the process. Understanding their roles in hypothesis testing helps in making informed decisions by controlling the false discovery rate and assessing statistical significance, enhancing research validity. Explore comparative analytics and decision-making tools with us.
1. Understanding P-Values, Q-Values, and T-Values
To understand how to compare Q values to T values, we must first define each and how they are used in statistical analysis.
1.1. What is a P-Value?
A p-value (probability value) assesses the statistical significance of a hypothesis test. It measures the probability of observing a test statistic as extreme as, or more extreme than, the value calculated from the sample data, assuming the null hypothesis is true. In simpler terms, it indicates the strength of evidence against the null hypothesis.
- Definition: The probability that the observed result (or a more extreme result) occurred by chance if the null hypothesis is true.
- Range: Varies between 0 and 1.
- Interpretation:
- Small P-Value (typically ≤ 0.05): Strong evidence against the null hypothesis, leading to its rejection. The result is considered statistically significant.
- Large P-Value (typically > 0.05): Weak evidence against the null hypothesis, failing to reject it. The result is considered not statistically significant.
- Use: Determining whether the results of a study support the research hypothesis or if they are likely due to random chance.
- Example: If a study finds a p-value of 0.03 for a new drug’s effectiveness, there is a 3% chance of observing such results if the drug has no effect. This usually leads to rejecting the null hypothesis and concluding the drug is effective.
1.2. What is a T-Value?
A T-value is a statistical measure that indicates the difference between group means, relative to the variability within the groups. It is used in a T-test to determine if the differences between two sets of data are statistically significant.
- Definition: The ratio of the difference between the group means and the standard error of the difference.
- Formula: ( t = frac{bar{x}_1 – bar{x}_2}{s_p sqrt{frac{1}{n_1} + frac{1}{n_2}}} )
- (bar{x}_1) and (bar{x}_2) are the sample means of the two groups.
- (s_p) is the pooled standard deviation.
- (n_1) and (n_2) are the sample sizes of the two groups.
- Interpretation:
- Large Absolute T-Value: Indicates a large difference between group means relative to the variability, suggesting a statistically significant difference.
- Small Absolute T-Value: Indicates a small difference between group means relative to the variability, suggesting a non-statistically significant difference.
- Use: Comparing the means of two groups to determine if they are significantly different.
- Example: In a study comparing the test scores of two classes, a high T-value would suggest a significant difference in performance between the classes.
1.3. What is a Q-Value?
A Q-value is used in multiple hypothesis testing to control the false discovery rate (FDR). The FDR is the expected proportion of false positives among the rejected hypotheses. Q-values provide a way to adjust for the increased chance of false positives when performing multiple tests.
- Definition: The minimum FDR at which a test can be called significant.
- Calculation: Q-values are calculated after p-values have been obtained and adjusted for multiple comparisons, often using methods like the Benjamini-Hochberg procedure.
- Interpretation:
- Small Q-Value (typically ≤ 0.05): Indicates that the test is significant while controlling for the FDR at 5%. This means that among all tests called significant at this level, no more than 5% are expected to be false positives.
- Large Q-Value (typically > 0.05): Indicates that the test is not significant when controlling for the FDR.
- Use: Managing the false discovery rate in studies involving multiple comparisons, such as genomics, proteomics, and large-scale experiments.
- Example: In a gene expression study, a q-value of 0.01 for a particular gene suggests that if all genes with q-values ≤ 0.01 are considered significant, then approximately 1% of these are expected to be false positives.
1.4. Key Differences Summarized
To summarize the differences, here is a table for quick reference:
| Feature | P-Value | T-Value | Q-Value |
|---|---|---|---|
| Definition | Probability of observing the result by chance. | Measure of the difference between group means relative to variability. | Minimum FDR at which a test is significant. |
| Purpose | Assess statistical significance of a single test. | Determine if the means of two groups are significantly different. | Control the false discovery rate in multiple hypothesis testing. |
| Interpretation | Lower value indicates stronger evidence against the null. | Higher absolute value indicates a significant difference between the groups. | Lower value indicates higher confidence in the significance of the result after FDR control. |
| Multiple Tests | Requires adjustment. | Used in calculating p-values for comparison of means. | Adjusts p-values specifically for multiple comparisons. |
| Primary Use Case | Single hypothesis tests. | Comparison of two group means. | Large-scale studies like genomics and proteomics where many tests are performed simultaneously. |
2. When to Use P-Values, T-Values, and Q-Values
2.1. Use Cases for P-Values
P-values are used in a wide range of statistical tests to determine the significance of results.
- Single Hypothesis Testing: In simple experiments or studies, p-values help determine if the observed results are statistically significant. For example, when testing if a new drug is effective compared to a placebo, a p-value helps determine if the observed difference is due to the drug or random chance.
- Regression Analysis: In regression models, p-values are used to assess the significance of individual predictors. A small p-value for a predictor suggests it has a significant impact on the outcome variable.
- Chi-Square Tests: In chi-square tests, p-values help determine if there is a significant association between categorical variables. For instance, testing if there is an association between smoking and lung cancer.
- Example:
- Scenario: A researcher is studying the effect of a new teaching method on student test scores.
- Null Hypothesis: The new teaching method has no effect on student test scores.
- Result: After conducting the study, the p-value is found to be 0.04.
- Interpretation: Since the p-value (0.04) is less than the significance level (0.05), the researcher rejects the null hypothesis and concludes that the new teaching method has a statistically significant effect on student test scores.
2.2. Use Cases for T-Values
T-values are specifically used for comparing the means of two groups.
- Independent Samples T-Test: Used to compare the means of two independent groups. For example, comparing the test scores of students in two different schools.
- Paired Samples T-Test: Used to compare the means of two related groups, such as before and after measurements on the same subjects. For example, measuring blood pressure before and after a medical intervention.
- Determining Group Differences: T-values quantify the difference between group means relative to the variability within the groups, providing a standardized measure for comparison.
- Example:
- Scenario: A company wants to know if a new training program improves employee productivity.
- Data: They measure the productivity of employees before and after the training program.
- Analysis: A paired samples t-test is used to compare the means of the “before” and “after” measurements.
- Result: The t-value is 3.5, and the corresponding p-value is 0.01.
- Interpretation: Since the p-value (0.01) is less than the significance level (0.05), the company concludes that the training program significantly improved employee productivity. The t-value of 3.5 indicates the magnitude of this improvement relative to the variability in the data.
2.3. Use Cases for Q-Values
Q-values are essential when dealing with multiple hypothesis tests to control the false discovery rate.
- Genomics and Proteomics: In gene expression studies or proteomics experiments, thousands of genes or proteins are tested simultaneously. Q-values help identify significant changes while controlling for the FDR.
- Genome-Wide Association Studies (GWAS): GWAS involve testing millions of genetic variants for association with a trait. Q-values are used to manage the large number of false positives.
- Exploratory Data Analysis: In large datasets, when many different hypotheses are tested, q-values provide a way to prioritize findings for further investigation.
- Example:
- Scenario: A genomic study investigates the expression levels of 20,000 genes to identify those that are differentially expressed between cancer cells and normal cells.
- Analysis: After performing 20,000 t-tests (one for each gene), the p-values are adjusted to q-values using the Benjamini-Hochberg procedure.
- Result: The researcher sets a significance level of q ≤ 0.05.
- Interpretation: Genes with q-values less than or equal to 0.05 are considered significant. This means that among all the genes identified as differentially expressed, the expected proportion of false positives is no more than 5%. This approach helps to focus on the most reliable findings for further validation.
2.4. Practical Scenarios
| Scenario | Statistical Test | Primary Metric | Additional Metrics |
|---|---|---|---|
| Drug Efficacy | T-Test | P-Value, T-Value | Confidence Interval, Effect Size |
| Gene Expression Analysis | Multiple T-Tests | Q-Value | P-Value, Fold Change |
| Marketing Campaign Performance | Chi-Square Test | P-Value | Odds Ratio, Confidence Interval |
| Employee Training Effectiveness | Paired Samples T-Test | P-Value, T-Value | Confidence Interval, Effect Size |
| A/B Testing | T-Test | P-Value, T-Value | Conversion Rate, Confidence Interval |
3. How to Compare Q-Values to T-Values
Comparing Q values to T values involves understanding their distinct roles and how they contribute to the interpretation of statistical results. Q values are used to control the false discovery rate (FDR) in multiple hypothesis testing, while T values are used to assess the significance of the difference between two group means. Here’s a step-by-step guide on how to compare them:
3.1. Understand the Context
- Type of Study: Determine if the study involves a single comparison or multiple comparisons. If it’s a single comparison between two groups, T values and their associated p-values are appropriate. If it involves multiple comparisons, Q values are necessary to control the FDR.
- Hypothesis Testing: Identify the null and alternative hypotheses being tested. Understand what a significant T value implies about the group means and what a significant Q value implies about the overall set of tests.
3.2. Calculate and Interpret T-Values
- Perform T-Test: Conduct the appropriate T-test (independent samples, paired samples, etc.) based on your study design.
- Obtain T-Value and P-Value: Note the calculated T value and its associated p-value.
- Interpret T-Value: A large absolute T value indicates a significant difference between the group means. Compare the p-value to your significance level (alpha, usually 0.05). If ( p leq alpha ), reject the null hypothesis.
3.3. Adjust for Multiple Comparisons if Necessary
- Identify Multiple Testing: If you are performing multiple T-tests (e.g., comparing multiple genes or proteins), you need to adjust for multiple comparisons.
- Choose an FDR Control Method: Common methods include the Benjamini-Hochberg procedure.
- Calculate Q-Values: Apply the chosen method to adjust the p-values and obtain Q values.
3.4. Interpret Q-Values
- Set FDR Level: Determine the acceptable FDR level (e.g., 0.05 or 0.10).
- Compare Q-Values to FDR Level: For each test, compare the Q value to the chosen FDR level. If ( q leq FDR ), the test is considered significant while controlling for the FDR.
- Interpret Significance: A small Q value indicates that the test is significant and that the proportion of false positives among all significant tests is controlled at the specified FDR level.
3.5. Compare and Contrast
- Single Test vs. Multiple Tests: If you performed a single T-test, you would only interpret the T value and its p-value. If you performed multiple T-tests, you would use the Q values to determine which results remain significant after controlling for the FDR.
- Assess Consistency: Check if the conclusions based on the T values and Q values are consistent. For example, a test with a significant T value and a small Q value indicates a robust finding. A test with a significant T value but a large Q value might be a false positive.
- Prioritize Findings: Use the Q values to prioritize findings for further investigation. Tests with smaller Q values are more likely to be true positives and should be prioritized.
3.6. Examples
3.6.1. Example 1: Gene Expression Analysis
- Scenario: A study compares gene expression levels between two groups (e.g., cancer vs. normal cells) for 10,000 genes.
- T-Test Results: After performing T-tests for each gene, many genes show significant p-values (e.g., ( p leq 0.05 )).
- Q-Value Adjustment: The p-values are adjusted to Q values using the Benjamini-Hochberg procedure.
- Comparison:
- Gene A has a T value of 3.2, p-value of 0.01, and Q value of 0.02. This gene is considered significant after FDR control.
- Gene B has a T value of 2.8, p-value of 0.03, and Q value of 0.07. This gene is not considered significant after FDR control.
- Interpretation: Gene A is a more reliable finding because it remains significant after controlling for the FDR.
3.6.2. Example 2: Drug Testing
- Scenario: A clinical trial compares the effectiveness of a new drug to a placebo in treating a specific condition.
- T-Test Results: A T-test is used to compare the mean improvement in the treatment group vs. the placebo group.
- Single Comparison: Since this is a single comparison, Q values are not needed.
- Interpretation:
- The T value is 4.5, and the p-value is 0.001.
- Conclusion: The new drug is significantly more effective than the placebo.
3.7. Key Considerations
- Sample Size: Ensure adequate sample sizes for reliable T-tests and FDR control. Small sample sizes can lead to unstable T values and unreliable Q value estimates.
- Assumptions: Verify that the assumptions of the T-tests are met (e.g., normality, equal variances). Violations of these assumptions can affect the validity of the T values and p-values.
- FDR Control Method: Choose an appropriate FDR control method based on the characteristics of your data and the dependencies between tests.
- Contextual Knowledge: Interpret the statistical results in the context of your research question and existing knowledge. Consider the biological or practical significance of the findings.
3.8. Summary Table: Comparing Q-Values and T-Values
| Feature | T-Value | Q-Value |
|---|---|---|
| Purpose | Assess significance of difference between two group means. | Control FDR in multiple hypothesis testing. |
| Context | Single or few comparisons. | Multiple comparisons. |
| Interpretation | Magnitude of difference relative to variability. | Minimum FDR at which a test is considered significant. |
| Use | Determining if a single comparison is statistically significant. | Identifying significant results while controlling for false positives. |
| Requires Adjustment | Not applicable for single tests. | Requires p-value adjustment for multiple comparisons. |
By following these steps, you can effectively compare Q values to T values, ensuring robust and reliable interpretations of your statistical results. Remember to consider the context of your study, the assumptions of the tests, and the importance of controlling for multiple comparisons when necessary.
4. Advantages and Disadvantages of Each Value
Each of the values (p-value, T-value, and q-value) has unique advantages and disadvantages in statistical analysis. Understanding these can help you choose the appropriate metric and interpret results more effectively.
4.1. P-Value
Advantages:
- Widespread Use: P-values are universally recognized and used across various scientific disciplines, making it easy to communicate results.
- Simplicity: They provide a straightforward measure of the statistical significance of a single hypothesis test.
- Ease of Calculation: P-values are readily calculated by most statistical software packages.
Disadvantages:
- Misinterpretation: Often misinterpreted as the probability that the null hypothesis is true or the probability that a significant result is practically important.
- Context Dependence: P-values do not provide information about the size or importance of an effect. A small p-value does not necessarily mean the effect is large or meaningful.
- Multiple Testing Issues: In multiple hypothesis testing, using p-values without adjustment leads to an increased risk of false positives.
4.2. T-Value
Advantages:
- Direct Comparison: T-values directly quantify the difference between group means relative to the variability within the groups.
- Standardized Measure: Provides a standardized measure that can be compared across different studies or datasets.
- Foundation for P-Value: T-values are used in the calculation of p-values for t-tests, linking the magnitude of the difference to its statistical significance.
Disadvantages:
- Limited Scope: Only applicable for comparing the means of two groups.
- Assumptions: T-tests have assumptions (normality, equal variances) that must be met for the results to be valid.
- Single Comparison: Does not address the issue of multiple comparisons; each T-test must be considered independently.
4.3. Q-Value
Advantages:
- FDR Control: Q-values directly address the problem of multiple comparisons by controlling the false discovery rate.
- Prioritization: Helps prioritize significant findings in large-scale studies by indicating the expected proportion of false positives.
- Increased Power: Compared to more conservative methods like Bonferroni correction, FDR control often provides increased statistical power, allowing for more true positives to be identified.
Disadvantages:
- Complexity: Requires understanding of multiple testing procedures and FDR control methods, which can be complex.
- Dependence on Method: Q-values depend on the specific method used for FDR control (e.g., Benjamini-Hochberg), and different methods can yield different results.
- Less Intuitive: Q-values are less intuitive than p-values and may be more difficult to communicate to a general audience.
4.4. Summary Table: Advantages and Disadvantages
| Value | Advantages | Disadvantages |
|---|---|---|
| P-Value | Widely used, simple to calculate, easy to interpret for single tests. | Often misinterpreted, does not indicate effect size, prone to false positives in multiple testing. |
| T-Value | Directly compares group means, provides a standardized measure, foundation for p-value calculation. | Limited to two-group comparisons, requires assumptions, does not address multiple comparisons. |
| Q-Value | Controls FDR in multiple testing, helps prioritize findings, often provides increased statistical power. | Complex, depends on the FDR control method, less intuitive than p-values. |
5. Step-by-Step Guide to Calculating Q-Values
Calculating Q-values involves adjusting p-values for multiple hypothesis testing to control the false discovery rate (FDR). Here’s a step-by-step guide using the Benjamini-Hochberg (BH) procedure, one of the most common methods for FDR control.
5.1. Step 1: Perform Hypothesis Tests
- Conduct Tests: Perform all individual hypothesis tests (e.g., T-tests, ANOVA, Chi-square tests) for each comparison you are making.
- Collect P-Values: Obtain the p-values for each test.
5.2. Step 2: Sort the P-Values
- Order P-Values: Sort the p-values from smallest to largest. Denote the sorted p-values as ( p{(1)}, p{(2)}, dots, p{(m)} ), where ( p{(1)} ) is the smallest p-value and ( p_{(m)} ) is the largest, and ( m ) is the total number of tests.
5.3. Step 3: Calculate the BH Critical Values
- Formula: For each sorted p-value ( p_{(i)} ), calculate the critical value using the formula:
[
text{Critical Value}_i = frac{i}{m} times alpha
]
where:- ( i ) is the rank of the p-value in the sorted list.
- ( m ) is the total number of tests.
- ( alpha ) is the desired FDR level (e.g., 0.05 for a 5% FDR).
5.4. Step 4: Determine Significant P-Values
- Compare P-Values to Critical Values: Starting from the smallest p-value, compare each ( p_{(i)} ) to its corresponding critical value.
- Find the Largest Significant P-Value: Find the largest ( i ) such that ( p{(i)} leq frac{i}{m} times alpha ). Denote this ( i ) as ( k ). All p-values ( p{(1)}, p{(2)}, dots, p{(k)} ) are considered significant.
5.5. Step 5: Calculate Q-Values
- Formula: Calculate the q-values for each p-value using the formula:
[
q{(i)} = minleft( p{(j)} times frac{m}{j} right) text{ for } j = i, i+1, dots, m
]
This means for each ( p{(i)} ), you calculate ( p{(i)} times frac{m}{i} ) and take the minimum of this value and all subsequent values. - Ensure Monotonicity: The q-values should be monotonically increasing. If they are not, adjust them to ensure this property.
5.6. Step 6: Interpret the Q-Values
- Compare to FDR Level: Compare each q-value to the desired FDR level ( alpha ).
- Significant Results: Any test with a q-value less than or equal to ( alpha ) is considered significant while controlling for the FDR at level ( alpha ).
5.7. Example Calculation
Suppose you have performed 10 hypothesis tests and obtained the following p-values:
| Test | P-Value |
|---|---|
| 1 | 0.005 |
| 2 | 0.015 |
| 3 | 0.025 |
| 4 | 0.035 |
| 5 | 0.045 |
| 6 | 0.055 |
| 7 | 0.065 |
| 8 | 0.075 |
| 9 | 0.085 |
| 10 | 0.095 |
Let’s calculate the q-values using the Benjamini-Hochberg procedure with ( alpha = 0.05 ).
- Sort P-Values:
| Rank ((i)) | P-Value ((p_{(i)})) |
|---|---|
| 1 | 0.005 |
| 2 | 0.015 |
| 3 | 0.025 |
| 4 | 0.035 |
| 5 | 0.045 |
| 6 | 0.055 |
| 7 | 0.065 |
| 8 | 0.075 |
| 9 | 0.085 |
| 10 | 0.095 |
- Calculate Critical Values:
| Rank ((i)) | P-Value ((p_{(i)})) | Critical Value ((frac{i}{m} times alpha)) |
|---|---|---|
| 1 | 0.005 | (frac{1}{10} times 0.05 = 0.005) |
| 2 | 0.015 | (frac{2}{10} times 0.05 = 0.010) |
| 3 | 0.025 | (frac{3}{10} times 0.05 = 0.015) |
| 4 | 0.035 | (frac{4}{10} times 0.05 = 0.020) |
| 5 | 0.045 | (frac{5}{10} times 0.05 = 0.025) |
| 6 | 0.055 | (frac{6}{10} times 0.05 = 0.030) |
| 7 | 0.065 | (frac{7}{10} times 0.05 = 0.035) |
| 8 | 0.075 | (frac{8}{10} times 0.05 = 0.040) |
| 9 | 0.085 | (frac{9}{10} times 0.05 = 0.045) |
| 10 | 0.095 | (frac{10}{10} times 0.05 = 0.050) |
- Determine Significant P-Values:
The largest ( i ) such that ( p{(i)} leq frac{i}{m} times alpha ) is ( i = 1 ), since ( 0.005 leq 0.005 ). Therefore, only ( p{(1)} ) is initially considered significant.
- Calculate Q-Values:
| Rank ((i)) | P-Value ((p_{(i)})) | (p_{(i)} times frac{m}{i}) | Q-Value |
|---|---|---|---|
| 1 | 0.005 | (0.005 times frac{10}{1} = 0.050) | 0.050 |
| 2 | 0.015 | (0.015 times frac{10}{2} = 0.075) | 0.050 |
| 3 | 0.025 | (0.025 times frac{10}{3} = 0.083) | 0.050 |
| 4 | 0.035 | (0.035 times frac{10}{4} = 0.088) | 0.050 |
| 5 | 0.045 | (0.045 times frac{10}{5} = 0.090) | 0.050 |
| 6 | 0.055 | (0.055 times frac{10}{6} = 0.092) | 0.050 |
| 7 | 0.065 | (0.065 times frac{10}{7} = 0.093) | 0.050 |
| 8 | 0.075 | (0.075 times frac{10}{8} = 0.094) | 0.050 |
| 9 | 0.085 | (0.085 times frac{10}{9} = 0.094) | 0.050 |
| 10 | 0.095 | (0.095 times frac{10}{10} = 0.095) | 0.050 |
Since the q-values must be monotonically increasing, we adjust them:
| Rank ((i)) | P-Value ((p_{(i)})) | Q-Value |
|---|---|---|
| 1 | 0.005 | 0.050 |
| 2 | 0.015 | 0.050 |
| 3 | 0.025 | 0.050 |
| 4 | 0.035 | 0.050 |
| 5 | 0.045 | 0.050 |
| 6 | 0.055 | 0.050 |
| 7 | 0.065 | 0.050 |
| 8 | 0.075 | 0.050 |
| 9 | 0.085 | 0.050 |
| 10 | 0.095 | 0.095 |
- Interpret Q-Values:
With ( alpha = 0.05 ), only the first test with ( p_{(1)} = 0.005 ) has a q-value less than or equal to 0.05 (i.e., ( q leq 0.05 )). Therefore, only the first test is considered significant after controlling for the FDR at 5%.
5.8. Software Implementation
Most statistical software packages provide functions to calculate q-values. For example, in R:
p_values <- c(0.005, 0.015, 0.025, 0.035, 0.045, 0.055, 0.065, 0.075, 0.085, 0.095)
q_values <- p.adjust(p_values, method = "BH")
print(q_values)This code will output the q-values for each test, allowing you to determine which results remain significant after FDR control.
5.9. Key Considerations
- Choice of FDR Control Method: The Benjamini-Hochberg procedure is widely used, but other methods like Benjamini-Yekutieli may be more appropriate for certain datasets.
- Assumptions: Ensure the assumptions of the underlying hypothesis tests are met.
- Interpretation: Understand that controlling the FDR means controlling the expected proportion of false positives, not eliminating them entirely.
By following these steps, you can effectively calculate and interpret q-values, ensuring robust and reliable interpretations of your statistical results in multiple hypothesis testing scenarios.
6. Real-World Examples
6.1. Example 1: Analyzing Gene Expression Data
In a study examining gene expression changes in cancer cells compared to normal cells, researchers often measure the expression levels of thousands of genes. This results in a multitude of p-values from individual t-tests for each gene.
- Scenario: A researcher conducts a microarray experiment to identify genes that are differentially expressed between cancerous and normal cells. The experiment measures the expression levels of 20,000 genes.
- Statistical Analysis: The researcher performs a t-test for each of the 20,000 genes to compare the expression levels between the two groups. This results in 20,000 p-values.
- Multiple Testing Problem: With 20,000 tests, even if only 5% of the genes are truly differentially expressed, about 1,000 false positives (genes incorrectly identified as differentially expressed) would be expected if no correction for multiple testing is applied.
- Applying Q-Values: To control the false discovery rate, the researcher adjusts the p-values to q-values using the Benjamini-Hochberg (BH) procedure.
- Interpretation: The researcher sets a significance threshold of q ≤ 0.05. Genes with q-values below this threshold are considered significantly differentially expressed. This means that among all the genes identified as significant, the expected proportion of false positives is no more than 5%.
- Benefits: By using q-values, the researcher can confidently identify a set of genes that are likely to be truly differentially expressed, reducing the number of false positives and focusing on the most promising candidates for further investigation.
- Source: According to Storey and Tibshirani (2003), using q-values in genomic studies helps manage the false discovery rate, allowing for more reliable identification of significant genes.
6.2. Example 2: Evaluating the Effectiveness of Marketing Campaigns
A company runs multiple marketing campaigns across different channels and wants to determine which campaigns are truly effective in increasing sales.
- Scenario: A marketing team conducts A/B tests on ten different marketing campaigns to determine which ones lead to a significant increase in sales.
- Statistical Analysis: The team performs a t-test for each campaign to compare the sales generated by the new campaign (group A) versus the existing campaign (group B). This results in ten p-values.
- Multiple Testing Problem: Without correcting for multiple
