Problem 8 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

An examination of 1000 people showed that 41 were carriers (heterozygotic) of the gene for cystic fibrosis. Let \(p\) be the proportion of all people who are carriers of cystic fibrosis. We can not say with certainty that \(p=41 / 1000\). For any number \(p\) in [0,1] , let \(L(p)\) be the likelihood of the event that 41 of 1000 people are carriers of cystic fibrosis given that the probability of being a carrier is \(p\). Then $$ L(p)=\left(\begin{array}{c} 1000 \\ 41 \end{array}\right) p^{41} \times(1-p)^{959} $$ where \(\left(\begin{array}{c}1000 \\ 41\end{array}\right)\) is a constant \(^{1}\) approximately equal to \(1.3 \times 10^{73}\). a. Compute \(L^{\prime}(p)\). b. Find the value \(\hat{p}\) of \(p\) for which \(L^{\prime}(p)=0\) and compute \(L(\hat{p})\). The value \(L(\hat{p})\) is the maximum value of \(L(p)\) and \(\hat{p}\) is called the maximum likelihood estimator of \(p\).

Step-by-Step Solution

Answer

\(\hat{p} = 0.041\); \(L(\hat{p})\) is the max likelihood.

Step 1: Understand the likelihood function

The likelihood function given is \[ L(p) = \binom{1000}{41} p^{41} (1-p)^{959} \] This function represents the probability of observing 41 carriers in a sample of 1000, given the probability of being a carrier is \( p \).

Step 2: Differentiate the likelihood function

To find \( L'(p) \), we need to differentiate \[ L(p) = \binom{1000}{41} p^{41} (1-p)^{959} \] using the product and chain rules. This can be expressed as follows:\[ L'(p) = \binom{1000}{41} \left( 41p^{40}(1-p)^{959} - 959p^{41}(1-p)^{958} \right) \]Simplify to:\[ L'(p) = \binom{1000}{41} p^{40}(1-p)^{958} (41(1-p) - 959p) \].

Step 3: Solve for p when L'(p)=0

Set \( L'(p) = 0 \) to find the maximum likelihood estimator \( \hat{p} \).\[ 41(1-p) = 959p \]\[ 41 - 41p = 959p \]\[ 41 = 1000p \]\[ \hat{p} = \frac{41}{1000} \]

Step 4: Compute the maximum likelihood L(\hat{p})

Substitute \( \hat{p} = 0.041 \) into the likelihood function:\[ L(\hat{p}) = \binom{1000}{41} (0.041)^{41} (1-0.041)^{959} \]Substitute the values to compute \[ L(\hat{p}) = 1.3 \times 10^{73} \times (0.041)^{41} \times (0.959)^{959} \].This is the maximum likelihood value of observing 41 carriers.

Key Concepts

Likelihood Function

In statistics, the likelihood function is a fundamental concept. It's used to quantify how likely a given set of parameters is in explaining the observed data. In other words, it tells us how probable our observed data is, given our assumption about the model parameters. For the case of cystic fibrosis carriers, our estimated likelihood function is \[ L(p) = \binom{1000}{41} p^{41} (1-p)^{959} \] This function measures the probability of observing exactly 41 carriers in a sample of 1000, assuming that the probability of any single individual being a carrier is \( p \).

• The term \( \binom{1000}{41} \) is a binomial coefficient representing the number of ways to choose 41 carriers from 1000 people.
• \( p^{41} \) indicates that 41 people are carriers, each with probability \( p \).
• \((1-p)^{959}\) represents the probability that the remaining 959 people are not carriers.

Understanding the likelihood function helps us determine the parameter value \( p \) that makes the observed data most probable.

Probability Theory

Probability theory forms the backbone of many statistical methods, including maximum likelihood estimation. It provides a framework for quantifying the likelihood of various outcomes. Using probability theory, we model real-world random processes mathematically.

• In our context, the problem deals with binomial probability, as each individual in the sample either is or isn't a carrier of the gene.
• The probability of being a carrier is represented by \( p \), while \( 1-p \) is the probability of not being a carrier.

The main task is to find a value for \( p \) that makes our observed data—the number of carriers—most probable. This is where the maximum likelihood estimation comes in. It's an application of probability theory that guides you to the most plausible parameter values based on the data you have.

Differentiation in Calculus

Differentiation is a core technique in calculus, used to find the rate at which a function is changing. In maximum likelihood estimation, differentiation allows us to find the parameter value that maximizes the likelihood function. To solve our exercise, we'd find the derivative of \( L(p) \) and set it to zero, helping us discover the point where the likelihood function reaches its maximum. Here’s how it’s done:

• The derivative of the likelihood function, \( L'(p) \), reflects the change in likelihood with respect to \( p \).
• By setting \( L'(p) = 0 \), we identify critical points where the change of the likelihood function could be significant.

Solving the resulting equation allows us to find \( \hat{p} \), the maximum likelihood estimator for \( p \). The end goal is to pinpoint where the function \( L(p) \) is at its highest, signifying the most likely parameter value based on the data provided.