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Chapter 13: Problem 52

Cancer prediction A breast cancer study at a city hospital in New York used logistic regression to predict the probability that a female has breast cancer. One explanatory variable was \(x=\) radius of the tumor (in \(\mathrm{cm}\) ). The results are as follows: Term zf Constant -2.165 radius 2.585 The quartiles for the radius were \(\mathrm{Q} 1=1.00, \mathrm{Q} 2=1.35\), and \(Q 3=1.85\) a. Find the probability that a female has breast cancer at \(\mathrm{Q} 1\) and \(\mathrm{Q} 3 .\) b. Interpret the effect of radius by estimating how much the probability increases over the middle half of the sampled radii, between \(\mathrm{Q} 1\) and \(\mathrm{Q}_{3}\).

Short Answer

Expert verified
a. Probability at \(Q1\) is 0.603 and at \(Q3\) is 0.932. b. Probability increases by approximately 32.9% from \(Q1\) to \(Q3\).

Step by step solution

01

Set Up the Logistic Regression Equation

The logistic regression model is used to predict the probability of an event occurring, such as having breast cancer. The equation is given by \[\log\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 x\]where \(p\) is the probability of having cancer, \(\beta_0 = -2.165\) is the constant term, and \(\beta_1 = 2.585\) is the coefficient for the explanatory variable \(x\), which is the radius of the tumor.
02

Calculate Probability at Q1

To find the probability at \(Q1 = 1.00\), substitute \(x = 1.00\) into the logistic regression equation:\[\log\left(\frac{p}{1-p}\right) = -2.165 + 2.585 \times 1.00\]Solving this gives:\[\log\left(\frac{p}{1-p}\right) = 0.42\]To find \(p\), use the logistic transformation:\[p = \frac{e^{0.42}}{1 + e^{0.42}} \approx 0.603\]Thus, the probability at \(Q1\) is approximately 0.603.
03

Calculate Probability at Q3

Next, find the probability at \(Q3 = 1.85\) by substituting \(x = 1.85\) into the equation:\[\log\left(\frac{p}{1-p}\right) = -2.165 + 2.585 \times 1.85\]This gives:\[\log\left(\frac{p}{1-p}\right) = 2.62525\]Again, apply the logistic transformation:\[p = \frac{e^{2.62525}}{1 + e^{2.62525}} \approx 0.932\]The probability at \(Q3\) is approximately 0.932.
04

Interpret the Effect of Radius

The increase in probability between the first and third quartile (from \(Q1\) to \(Q3\)) is the difference in probabilities: \[0.932 - 0.603 = 0.329\]This means that the probability of having cancer increases by approximately 32.9% over the middle half of the sampled radii, indicating that larger tumor radii significantly increase the cancer probability.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cancer Prediction
In the field of medicine, predicting the likelihood of a disease, such as breast cancer, is crucial for early intervention and treatment. This is where logistic regression plays an important role. Logistic regression is a statistical method used to model and predict outcomes based on one or more predictor variables. In the context of cancer prediction, one of the variables used could be the size of a tumor.
Logistic regression specifically is utilized for binary outcomes, like the presence or absence of cancer. It helps in understanding which variables contribute most significantly to the outcome. In our exercise, the constant and the radius of the tumor are key components in forming our logistic regression equation. By substituting specific values, such as quartiles, into the equation, one can predict the probability of breast cancer occurrence.
Tumor Radius
The tumor radius is a significant factor in determining the probability of breast cancer. Larger radii often suggest a higher likelihood of malignancy. In the example, the logistic regression model uses the radius as the explanatory variable to predict cancer probability.
  • For instance, a small change in the tumor radius may lead to a significant change in the predicted probability.
  • At the first quartile ( Q1 = 1.00 cm ), the probability of having breast cancer is approximately 60.3%.
  • At the third quartile ( Q3 = 1.85 cm ), this probability rises to about 93.2%.
This dramatic increase highlights how sensitive the cancer probability is to changes in the tumor radius. Such insights are invaluable for healthcare professionals who need to make informed decisions regarding patients' health interventions.
Quartiles
The use of quartiles is essential in statistical analysis for summarizing data and interpreting results. Quartiles divide data into four equal parts, making it easier to grasp how data points relate to the overall distribution.
In our example with tumor radius for cancer prediction, the first quartile ( Q1 ) and the third quartile ( Q3 ) represent the spread of radius sizes among patients. By examining the probability of cancer at these quartiles, the analysis shows that there is an approximate 32.9% increase in cancer probability from the first to third quartile.
Understanding these quartiles assists in providing a clearer picture of how variations among patients impact cancer risk, allowing healthcare providers to prioritize monitoring and treatment based on specific, measurable tumor characteristics.

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Most popular questions from this chapter

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