Chapter 2: Understanding Power Analysis

Experimental methods are the gold standard for uncovering relationships in our environment. However, careful planning is essential; a poorly designed experiment may yield results that are either insignificant or misleading. Power analysis plays a vital role in ensuring the integrity of experimental design.

Consider this essential question: Why is it crucial to estimate power prior to conducting an experiment? The answer lies in the fact that a lack of understanding regarding an experiment's power could lead to ineffective use of time and resources on experiments that fail to reveal meaningful outcomes.

In statistical terms, power is defined as the likelihood of correctly rejecting the null hypothesis. To simplify, I prefer to characterize the power of experiments as the probability of detecting a relationship, assuming one exists.

Power is fundamentally linked to two opposing forces: (1) signal and (2) noise. The balance between these variables dictates our ability to identify trends effectively.

To calculate power, we create two distributions: one with a mean of zero (indicating no relationship between our experimental variable and the response variable) and another with a non-zero mean (indicating a positive relationship). The first distribution corresponds to the null hypothesis, while the second represents the alternative hypothesis. In general, power is inversely related to the degree of overlap between these distributions—greater overlap equates to reduced power, while lesser overlap results in increased power.

Below is a visual representation to illustrate this concept:

In the illustration, the green distribution signifies the null distribution, representing possible relationship values if no connection exists between the experimental and response variables. Conversely, the blue distribution represents potential relationship estimates when a positive relationship does exist. The red line indicates the threshold for the top 5% of the null distribution. The area in the blue distribution that lies to the right of the red line depicts the power. If any of this seems unclear, remember that a quick online search can provide further clarity! Now, let’s delve into the primary focus of this article—simulation.

Chapter 3: Calculating Power Through Simulations

Having established a foundational understanding of power, let's explore how simulation can be utilized to calculate power, ultimately leading to enhanced experimental design.

I believe in the power of learning through examples, so we will walk through a simulation scenario together.

Imagine you are employed by an advertising agency tasked with estimating the power of a particular experiment. The goal of this experiment is to evaluate the effectiveness of an advertising strategy on sales. The design divides the country into several sub-groups, randomly assigning one group to receive the advertising campaign while another serves as the control.

To simplify, we will assume a difference-in-difference (DID) approach for our analysis (further details on this approach can be easily found online). The DID method divides our data into four groups: (1) pre/control, (2) post/control, (3) pre/test, and (4) post/test. Only the post/test dataset will be influenced by the campaign; the other three datasets will not receive any advertising and will serve as controls to account for confounding factors.

Note: While we have chosen the DID approach for this example, we can adjust our methodology to compute power for any analytical strategy.

To assess the power of our test, we will conduct two simulations. Simulation 1 assumes no impact from the program, while Simulation 2 posits a specific impact level. The first simulation corresponds to the null hypothesis (the campaign does not affect sales), whereas the second simulation represents the alternative hypothesis (the campaign has a positive effect on sales). Each simulation yields a single data point, specifically the DID calculation. For instance, running Simulation 1 once provides one DID value.

The next step involves executing each simulation multiple times to generate two distributions of our DID metrics. These distributions will allow us to estimate the power based on their overlap. A lower degree of overlap indicates higher power, while greater overlap implies reduced power.

Below is a breakdown of the datasets corresponding to the two simulations:

As we prepare to run our simulations, let's implement some Python code to generate the synthetic data and compute the power.

Here’s how the code aligns with the aforementioned setup:

import numpy as np

from matplotlib import pyplot as plt

import seaborn as sns

def calculate_diff_in_diff(pre_test, pre_control,

post_test, post_control):

# Function to calculate the difference-in-difference metric

test_diff = post_test - pre_test

control_diff = post_control - pre_control

diff_in_diff_ratio = test_diff - control_diff

return diff_in_diff_ratio

# Function definitions for simulating group data and multiple simulations follow...

The code above runs two simulations 1000 times. If we assume a 5% impact (indicating our program increased the probability of customer purchases by 5%) with a sample size of 1500 customers, we can evaluate the resulting power.

The red line in the visualization denotes the threshold for the top 5% of the green "no impact" distribution. At a significance level of 5%, with a sample size of 1500 and a 5% impact size, our calculated power is approximately 63%. This suggests a 63% chance of detecting a significant difference given the parameters of the simulation.

The true value of simulation in power analysis emerges here! A power of 63% is relatively low, indicating that we might struggle to discern significant effects in our test. Fortunately, we identified this issue prior to executing the experiment! With this framework in mind, we can explore strategies to address the low power.

There are two primary strategies for enhancing power: (1) increasing the sample size and (2) amplifying the impact of the test. Let’s examine both options.

By increasing the sample size from 1500 to 3000, we can easily adjust our code:

# Adjusting sample sizes for testing and control groups

test_n_customers = 3000

control_n_customers = 3000

As a result, our power rises to around 91%! We can achieve this increase by extending the duration of the experiment or broadening the geographical areas included in the study. Based on our power analysis, implementing one of these adjustments is advisable given the low initial power.

The second approach—boosting the impact size—is less straightforward, as we can only influence it to a certain extent. If we fully understood how to alter the impact size, we would not need to conduct a test in the first place! To enhance the impact size, we must increase the treatment level; in this case, that would entail launching more aggressive advertising campaigns. While we know that a stronger relationship between sales and the campaign should yield a more significant impact, the exact increase remains uncertain.

We can gain insights regarding the influence of varying impact levels on power through simulation. The following table illustrates how power corresponds to different simulated impact levels, aiding our understanding of confidence across various thresholds. If the campaign’s impact falls below 5%, our likelihood of detecting it diminishes. If this is unacceptable, we can either boost the sample size to mitigate noise or escalate the campaign efforts to enhance the likelihood of observing a substantial impact.

This table can serve as a valuable communication tool with stakeholders, clarifying the likelihood of detecting varying impact levels. Should everyone agree on a low probability of identifying impacts below 5%, we can proceed accordingly. If not, adjustments to the experimental design may be necessary.