Calculate variance for click-through rates

Question

Given the following scenario about user conversion, calculate the the p-value.

Codemia · Accepted Answer

To calculate the p-value for click-through rates (CTR), we first need to set up our hypotheses. Let’s denote the click-through rates for two different user groups as follows:

- $ p_1 $: click-through rate of Group 1 (e.g., users exposed to a new feature)
- $ p_2 $: click-through rate of Group 2 (e.g., users with the standard experience)

We will test the following hypotheses:

- Null Hypothesis $ H_0: p_1 = p_2 $ (there is no difference in click-through rates)
- Alternative Hypothesis $ H_1: p_1 
eq p_2 $ (there is a difference in click-through rates)

We will use a two-proportion z-test to compare the CTRs of the two groups and calculate the p-value accordingly.

1. **Collect Data**: Assume we have the following data:
   - Group 1 (New feature): 200 clicks out of 1000 impressions
   - Group 2 (Standard): 150 clicks out of 1000 impressions

2. **Calculate Sample Proportions**:
   - $ \hat{p}_1 = rac{200}{1000} = 0.20 $
   - $ \hat{p}_2 = rac{150}{1000} = 0.15 $

3. **Calculate the Combined Proportion**: 
   - $ \hat{p} = rac{200 + 150}{1000 + 1000} = rac{350}{2000} = 0.175 $

4. **Calculate the Standard Error (SE)**:
   - $ SE = \sqrt{\hat{p}(1 - \hat{p}) \left( rac{1}{n_1} + rac{1}{n_2} ight)} $
   - $ SE = \sqrt{0.175 	imes 0.825 \left( rac{1}{1000} + rac{1}{1000} ight)} \approx 0.021 $

5. **Calculate the Z-Statistic**:
   - $ Z = rac{\hat{p}_1 - \hat{p}_2}{SE} = rac{0.20 - 0.15}{0.021} \approx 2.38 $

6. **Calculate the p-value**: Using the z-table, the two-tailed p-value for $ Z = 2.38 $ is approximately 0.017.

Given our previous calculations:
- Group 1 has a CTR of 20% and Group 2 has a CTR of 15%.
- The Z-Statistic we calculated is 2.38, leading to a p-value of approximately 0.017.

**Interpretation of the p-value**: Since the p-value (0.017) is less than the conventional alpha level of 0.05, we reject the null hypothesis $ H_0 $. This indicates that there is statistically significant evidence to suggest that the click-through rates differ between the two groups.

The result informs us that the new feature implemented in Group 1 significantly increases user engagement, as evidenced by a higher click-through rate compared to Group 2. However, it is crucial to note that while the result is statistically significant, one should consider practical significance—does the increase in CTR translate into higher conversions or revenue? Furthermore, other factors such as user demographics, time on site, or external marketing influences should be accounted for as potential confounders.

This analysis connects to several related concepts:
- **Power Analysis**: If we were planning a future experiment, we would want to conduct a power analysis to determine the required sample size to detect a meaningful difference in CTRs with a specified level of confidence.
- **Multiple Testing Corrections**: If we were testing multiple features simultaneously, we would need to apply corrections such as the Bonferroni correction to control the family-wise error rate.
- **Bayesian Inference**: An alternative approach could involve using Bayesian methods to calculate the probability of the hypotheses given the observed data, providing a different perspective on the significance of the results.

Calculate variance for click-through rates

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