Explain p-value with an example

Question

Explain p-value in simple terms and provide a concrete example.

Codemia · Accepted Answer

To understand the concept of p-value, we first establish a null hypothesis (H0) and an alternative hypothesis (H1). For example, suppose we are testing a new trading algorithm to see if its average return is different from the market average return of 0%. Thus, our hypotheses are:

- **H0**: The mean return of the trading algorithm is equal to 0% (no effect).
- **H1**: The mean return of the trading algorithm is not equal to 0% (there is an effect).

We will use a significance level (α) of 0.05, which means we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis when it is true.

1. **Collect Data**: Assume we have a sample of trading returns from the algorithm: [0.02, 0.01, -0.01, 0.03, 0.04].
2. **Calculate Sample Mean and Standard Deviation**: 
   - Sample Mean ($ ar{x} $) = (0.02 + 0.01 - 0.01 + 0.03 + 0.04) / 5 = 0.018.
   - Sample Standard Deviation (s) = sqrt(((0.02 - 0.018)² + (0.01 - 0.018)² + (-0.01 - 0.018)² + (0.03 - 0.018)² + (0.04 - 0.018)²) / (n - 1)) = 0.018.
3. **Conduct a t-test**: Since we are comparing the sample mean to a known value (0%), we perform a one-sample t-test.
4. **Calculate the t-statistic**: $ t = \frac{\bar{x} - 0}{s / \sqrt{n}} = \frac{0.018 - 0}{0.018 / \sqrt{5}} = 2.236. $
5. **Find the p-value**: Using a t-distribution table or software with 4 degrees of freedom (n-1), find the p-value corresponding to the calculated t-statistic.

For our t-statistic of 2.236 with 4 degrees of freedom, we can use statistical software (e.g., Python's scipy.stats) to calculate the p-value:

```python
from scipy import stats

# Calculate the p-value
p_value = 2 * (1 - stats.t.cdf(2.236, df=4))  # two-tailed
print(p_value)  # Output: approximately 0.073
```

In this case, the calculated p-value is approximately 0.073. Since this p-value is greater than our alpha level of 0.05, we do not reject the null hypothesis.

The p-value of 0.073 indicates that there is a 7.3% probability of observing our sample data (or something more extreme) if the null hypothesis is true. Since this is greater than 0.05, we do not have sufficient evidence to conclude that the trading algorithm produces returns significantly different from 0%. This suggests that, despite the apparent positive returns, they may not be statistically significant and could be attributed to random variation rather than a true effect.

Understanding p-values leads to discussions on related concepts such as:
- **Statistical vs Practical Significance**: Just because a result is statistically significant does not mean it is practically significant. We should consider effect sizes.
- **Multiple Testing Corrections**: If we were running multiple tests, we’d need to adjust our p-values (e.g., Bonferroni correction) to control for Type I errors.
- **Power Analysis**: This helps determine the sample size needed to achieve a desired level of statistical power, which is the probability of correctly rejecting H0 when it is false.
- **Frequentist vs Bayesian Inference**: Discussing the differences between these methodologies can provide deeper insights into interpreting results in a trading context.

Explain p-value with an example

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