Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/Hg) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:
| S. No. | Cholesterol (mg/dL) | SBP (mm/Hg) |
| 1 | 300 | 150 |
| 2 | 410 | 270 |
| 3 | 380 | 210 |
| 4 | 530 | 310 |
| 5 | 570 | 350 |
| 6 | 490 | 310 |
| 7 | 340 | 210 |
| 8 | 320 | 150 |
| 9 | 280 | 110 |
| 10 | 550 | 320 |
| 11 | 340 | 220 |
| 12 | 350 | 170 |
| 13 | 410 | 260 |
| 14 | 390 | 230 |
| 15 | 450 | 270 |
(i) Fit a linear regression model using the method of least squares.
(ii) Construct the normal probability plot for the data on serum cholesterol and systolic blood pressure.
(iii) Test the significance of the fitted regression model.
To address the given questions, the analysis of the relationship between serum cholesterol (independent variable) and systolic blood pressure (SBP, dependent variable) will be performed using biostatistical techniques. This involves fitting a linear regression model, constructing a normal probability plot, and testing the significance of the fitted model.
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### (i) Fitting a Linear Regression Model
The general form of a linear regression model is:
\[
Y = \beta_0 + \beta_1 X + \epsilon
\]
Where:
- \( Y \): Dependent variable (SBP in this case)
- \( X \): Independent variable (Cholesterol in this case)
- \( \beta_0 \): Intercept of the regression line
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Polytomous logistic models
State whether the following statements are True or False. Give reason in support of your answer:
(a) The value of sensitivity of the following results of a diagnostic test is 0.85.
| Disease | Test result | Total | |
| + | - | ||
| Present | 170 | 30 | 200 |
| Absent | 20 | 280 | 300 |
| Lung Cancer | No Lung Cancer | Totel | |
| Smokers | 100 | 1220 | 1320 |
| Non-smokers | 50 | 2260 | 2310 |
(c) The logit link function is log
(d) We define three indicator/dummy variables for a regressor variable with three categories.
(e) Left censoring occurs whenever the exact time of occurrence of an event is not known.
Poisson regression
The following data on diagnosis of coronary heart disease (where 0 indicating absence and 1 indicating presence), serum cholesterol (in mg/dl), resting blood pressure (in mmHg) and weight (in kg) were obtained for 80 patients to explore the relationship of coronary heart disease with cholesterol and weight.
| S. No. | Serum Cholesterol (mg/dl) | Weight (kg) | Number of Patients having CHD | Total Number of Patients | |||
| 1 | 420 | 60 | 10 | 20 | |||
| 2 | 450 | 68 | 15 | 30 | |||
| 3 | 400 | 54 | 4 | 15 | |||
| 4 | 510 | 74 | 2 | 10 | |||
| 5 | 480 | 62 | 1 | 5 | |||
(i) Fit a multiple logistic model for the dependence of coronary heart disease on the average serum cholesterol and weight considering and
the initial values of the parameters (solve only for one Iteration).
(ii) Test the significance of the fitted model using Hosmer-Lemeshow test at 5% level of significance.
A random sample of 250 patients was selected and their workout timing and diabetes status were recorded. The following table shows the workout timing and severity of diabetes:
| Workout (in minutes) | Severity of diabetes status | ||
| Low | Moderate | High | |
| 0 −15 | 06 | 27 | 19 |
| 15 to 30 | 08 | 38 | 17 |
| 30 to 45 | 21 | 45 | 33 |
| ≥ 45 | 14 | 18 | 06 |
Kaplan and Meier method
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