Explain power analysis with an example

Question

Explain power analysis in simple terms and provide a concrete example.

Codemia · Accepted Answer

In the context of SpaceX, let's say we want to determine whether a new rocket design has improved launch reliability compared to the previous design. The analytical question we are addressing is: "What is the minimum sample size needed to detect a significant difference in launch reliability between the two designs with a given level of confidence?"

To perform this power analysis, we need historical data on launch reliability (success rates) of both the old and new rocket designs. Specifically, we need:
- The historical success rate of the old rocket design (e.g., 85% success rate).
- An estimated success rate for the new rocket design (e.g., 95%).
- The desired significance level (commonly alpha = 0.05).
- The power level we aim for (commonly set at 0.80 or 80%).

To calculate the required sample size for our power analysis, we can use the formula for comparing two proportions. The formula to calculate the required sample size (n) for each group is:

$$ n = \frac{(Z_{\alpha/2} + Z_{\beta})^2 \cdot (p_1(1-p_1) + p_2(1-p_2))}{(p_1 - p_2)^2} $$

Where:
- $Z_{\alpha/2}$ is the Z-value for the significance level (for alpha = 0.05, Z = 1.96).
- $Z_{\beta}$ is the Z-value for the power (for 80% power, Z = 0.84).
- $p_1$ is the success rate for the old design (0.85).
- $p_2$ is the success rate for the new design (0.95).

Using this formula, we can compute the necessary sample size for each design.

Let’s plug in the values into the formula:
- $p_1 = 0.85$
- $p_2 = 0.95$
- $Z_{\alpha/2} = 1.96$
- $Z_{\beta} = 0.84$

Calculating the variances:
- $p_1(1-p_1) = 0.85(0.15) = 0.1275$
- $p_2(1-p_2) = 0.95(0.05) = 0.0475$

Now substituting back into the sample size formula:
$$ n = \frac{(1.96 + 0.84)^2 \cdot (0.1275 + 0.0475)}{(0.85 - 0.95)^2} $$
$$ n = \frac{(2.8)^2 \cdot (0.175)}{(-0.1)^2} $$
$$ n = \frac{7.84 \cdot 0.175}{0.01} $$
$$ n = \frac{1.372}{0.01} = 137.2 $$

Thus, we would need approximately 138 launches for each rocket design to ensure we have sufficient power to detect a significant difference.

The result from our power analysis indicates that we need 138 launches for both the old and new rocket designs to confidently determine if the new design has a statistically significant improvement in launch reliability. If we find a significant difference, we can infer that the new design likely leads to better performance. Conversely, if we don't find a significant difference with our sample size, we cannot conclude that the two designs are equivalent, as we may not have had enough power to detect a difference if one exists.

Common pitfalls in power analysis include:
- **Underestimating the effect size**: If the estimated success rate of the new design is overly optimistic, you may calculate an insufficient sample size.
- **Ignoring variability**: Launch reliability can vary due to external factors (e.g., weather conditions), which should be accounted for in your model.
- **Not considering cohort segmentation**: Different groups (e.g., different launch sites) may have varying success rates, so stratifying the sample may be necessary.
- **Misinterpreting power**: A high power level does not guarantee finding a significant result; it merely indicates the probability of detecting an effect if it exists.

Explain power analysis with an example

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