STOR 390: HOMEWORK 8 Solution

Abstract: This homework explores various aspects of data privacy and fairness in statistical analysis, integrating topics such as differential privacy and the ethical responsibilities surrounding data interpretation. By exploring the Randomized Response technique, we demonstrate its effectiveness in maintaining𝜀-differential privacy and protecting individual responses in sensitive surveys, where 𝜀 = ln(3). Additionally, we engage with regression models to control for confounding variables such as age and smoking status, illustrating how these factors can influence the outcomes

in studies related to lung capacity.
Question 1
Prove that Randomized Response Differential Privacy is
𝜀−differentially private. In particular, show that 𝜀 =
𝑙𝑛(3).
The Randomized Response technique is a method used to protect the privacy of individuals in surveys, particularly for sensitive questions. The method involves the following steps,
1. The respondent flips a fair coin.
2. If the coin lands heads, they flip another coin,
• If the second coin lands heads, they answer “Yes”.
• If the second coin lands tails, they answer “No”.
3. If the coin lands tails, they answer truthfully.
Let 𝑓 denote the true answer, where 𝑓 = 1 for “Yes” and 𝑓 = 0 for “No”. The probability of responding “Yes”
(𝑌 = 1) is calculated as follows,

The definition of 𝜀-differential privacy is given by,

If we let 𝑦 = 1,
𝜀 =
Therefore, the Randomized Response technique is 𝜀differentially private with 𝜀 = ln(3).
Question 2
Recall the example of regressing out a confounding variable we used in class. Here, we established that not only is ‘age‘ related to lung capacity, but so too is ‘smoke‘. Interpret the model coefficients produced below, and in particular explain why including both ‘age‘ and ‘smoke‘ helps to control for confounding. (Hint: This should involve interpreting your coefficients marginally.)
Correlation: 0.756459 Model Summary:
Call:
lm(formula = fev ~ age + smoke, data = fev)
Residuals:
Min 1Q Median 3Q Max -1.6653 -0.3564 -0.0508 0.3494 2.0894
Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.367373 0.081436 4.511 7.65e-06
***
age 0.230605 0.008184 28.176 < 2e-16 *** smoke -0.208995 0.080745 -2.588 0.00986 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*'
0.05 '.' 0.1 ' ' 1
Residual standard error: 0.5651 on 651 degrees of freedom Multiple R-squared: 0.5766, Adjusted
R-squared: 0.5753
F-statistic: 443.3 on 2 and 651 DF, p-value:
< 2.2e-16
The coefficients from our regression model give us the marginal effects of each variable,
• The coefficient of ‘age’ (0.230605) tells us lung capacity is expected to change by 0.230605 units with each additional unit change (in years) of age, holding smoking status constant.
• The coefficient of ‘smoke’ (-0.208995) tells us us the lung capacity is expected to change by 0.208995 units with each additional unit change in smoking, holding age constant.
Question 3
Describe one experimental design technique that could be used to mitigate known confounders or batch effects (as opposed to latent batch effects).
An effective experimental design technique to control for known confounders or batch effects is stratified random sampling. This method can be defined by dividing a population being observed or surveyed into smaller groups, known as strata, that are homogeneous with respect to one or more characteristics which are known to be confounders.
Implementation of Stratified Random Sampling (SRS):
• Strata Formation: The first step is to identify the known confounder(s) which might impact the study outcome. The entire population is then divided into subgroups or strata based on these confounders. Each stratum should be homogeneous regarding the confounder (a few examples of confounders could be age, sex, socioeconomic status).
• Random Sampling: From each stratum, participants are randomly selected to participate in the study. This ensures that each stratum is represented proportionally or equally, depending on the study design.
• Control Over Confounding: By stratifying the population by known confounders, the impact of these confounders is controlled across the treatment groups, reducing their effect on the outcome variable.
Question 4
I have described our responsibility to proper interpretation as an obligation or duty. How might a Kantian Deontologist defend such a claim?
From the perspective of Kantian Deontology, ethical behavior is grounded in adherence to universal maxims and the fulfillment of duties, irrespective of the consequences. According to Immanuel Kant, actions are morally right when they are performed out of duty and conform to universal law. This approach can be directly applied to the concept of the duty of proper interpretation. According to Kant’s principle of universalizability, an act is morally acceptable if it can be universalized, meaning it can consistently be willed as a law that everyone should follow. In terms of proper interpretation, this means that if the act of interpreting information accurately and responsibly could be universalized, it becomes not only a preferable practice but a moral duty. This stems from the desire for a society where truth and understanding are upheld as common values. Kant’s Categorical Imperative demands that we act only according to maxims that could be adopted universally. This imperative implies that misinterpreting or distorting information would be contrary to this principle because it would not be desirable for such practices to become universal law. Thus, a Kantian Deontologist would argue that we have a moral duty to interpret correctly to respect and uphold the rationality and autonomy of all individuals affected by the interpretation. Further, Kant’s notion of treating humanity, whether in oneself or in another, always as an end and never merely as a means to an end, reinforces the duty to interpret accurately. This ensures that individuals are not misled or harmed through misinterpretation, thus respecting their autonomy and dignity.
Therefore, from a Kantian Deontological perspective, proper interpretation is not merely a professional responsibility but a moral obligation. This statement aligns with Kant’s emphasis on actions that can be universally willed and that inherently respect the autonomy and dignity of others involved. Misinterpretation, by contrast, would undermine trust, dignity, and rational decision-making, thereby violating Kantian moral laws.
Question 5
(Free Points) What was the most informative/interesting unit in this course? The least?