Question

Refer to data of Hossack et al. [1980, 1981]. Ten normal men and 11 normal women were studied during a maximal exercise treadmill test. While being exercised they had a catheter (tube) inserted into the pulmonary (lung) artery and a short tube into the left radial or brachial artery. This allowed sampling and observation of arterial pressures and the oxygen content of the blood. From this, several parameters as described below were measured or calculated. The data for the 11 women are given in Table 11.17; the data for the 10 normal men are displayed in Table 11.18. Descriptions of the variables follow. • Activity: a subject who routinely exercises three or more times per week until perspiring was active (Act); otherwise, the subject was sedentary (Sed). • Wt: weight in kilograms. • $H t$ : height in centimeters. • VO2MaX: oxygen (in millimeters per kilogram of body weight) used in $1 \mathrm{~min}$ at maximum exercise. • FAI: functional aerobic impairment. For a patient's age and activity level (active or sedentary) the expected treadmill duration (ED) is estimated from a regression equation. The excess of observed duration (OD) to expected duration (ED) as a percentage of $\mathrm{ED}$ is the FAI. $\mathrm{FAI}=100 \times(\mathrm{OD}-\mathrm{ED}) / \mathrm{ED}$. •$\dot{Q}_{M A X}:$ output of the heart in liters of blood per minute at maximum. •$H R_{M A X}$ : heart rate in beats per minute at maximum exercise. • $S V_{M A X}$ : volume of blood pumped out of the heart in milliliters during each stroke (at maximum cardial output). • $\mathrm{CaO}_2$ : oxygen content of the arterial system in milliliters of oxygen per liter of blood. • $\mathrm{C} \overline{\mathrm{v}} \mathrm{O}_2$ : oxygen content of the venous (vein) system in milliliters of oxygen per liter of blood. • $a \bar{v} O_2 D_{M A X}$ : difference in the oxygen content (in milliliters of oxygen per liter of blood) between the arterial system and the venous system (at maximum exercise); thus, $\mathrm{avVO}_2 \mathrm{D}_{\mathrm{MAX}}=\mathrm{CaO}_2-\mathrm{C}_2$. • $\bar{P}_{\mathrm{SA}}, \mathrm{mAX}$ : average pressure in the arterial system at the end of exercise in milliliters of mercury $(\mathrm{mmHg})$. •$\bar{P}_{\mathrm{PA}}, \mathrm{MAX}$ : average pressure in the pulmonary artery at the end of exercise in $\mathrm{mmHg}$. After examining the normal probability plot of residuals, the regression of Problem 11.4 was rerun omitting cases 2 and 8 . In this case we find: TABLE CANT COPY Do tasks (a), (b-i), (c), (d), and (f). Comment: The very small residual (high $R^2$ ) indicates that the data are very likely highly "over fit." Compute $R^2$.

Name: Problem 5, Chapter 11: Biostatistics A Methodology For the Health Sciences
Uploaded: 2022-01-20T22:56:33-08:00
Duration: 6 min 14 s
Channel: Raymond Matshanda

   Refer to data of Hossack et al. [1980, 1981]. Ten normal men and 11 normal women were studied during a maximal exercise treadmill test. While being exercised they had a catheter (tube) inserted into the pulmonary (lung) artery and a short tube into the left radial or brachial artery. This allowed sampling and observation of arterial pressures and the oxygen content of the blood. From this, several parameters as described below were measured or calculated. The data for the 11 women are given in Table 11.17; the data for the 10 normal men are displayed in Table 11.18. Descriptions of the variables follow.
• Activity: a subject who routinely exercises three or more times per week until perspiring was active (Act); otherwise, the subject was sedentary (Sed).
• Wt: weight in kilograms.
• $H t$ : height in centimeters.
• VO2MaX: oxygen (in millimeters per kilogram of body weight) used in $1 \mathrm{~min}$ at maximum exercise.
• FAI: functional aerobic impairment. For a patient's age and activity level (active or sedentary) the expected treadmill duration (ED) is estimated from a regression equation. The excess of observed duration (OD) to expected duration (ED) as a percentage of $\mathrm{ED}$ is the FAI. $\mathrm{FAI}=100 \times(\mathrm{OD}-\mathrm{ED}) / \mathrm{ED}$.
•$\dot{Q}_{M A X}:$ output of the heart in liters of blood per minute at maximum.
•$H R_{M A X}$ : heart rate in beats per minute at maximum exercise.
• $S V_{M A X}$ : volume of blood pumped out of the heart in milliliters during each stroke (at maximum cardial output).
• $\mathrm{CaO}_2$ : oxygen content of the arterial system in milliliters of oxygen per liter of blood.
• $\mathrm{C} \overline{\mathrm{v}} \mathrm{O}_2$ : oxygen content of the venous (vein) system in milliliters of oxygen per liter of blood.
• $a \bar{v} O_2 D_{M A X}$ : difference in the oxygen content (in milliliters of oxygen per liter of blood) between the arterial system and the venous system (at maximum exercise); thus, $\mathrm{avVO}_2 \mathrm{D}_{\mathrm{MAX}}=\mathrm{CaO}_2-\mathrm{C}_2$.
• $\bar{P}_{\mathrm{SA}}, \mathrm{mAX}$ : average pressure in the arterial system at the end of exercise in milliliters of mercury $(\mathrm{mmHg})$.
•$\bar{P}_{\mathrm{PA}}, \mathrm{MAX}$ : average pressure in the pulmonary artery at the end of exercise in $\mathrm{mmHg}$.
After examining the normal probability plot of residuals, the regression of Problem 11.4 was rerun omitting cases 2 and 8 . In this case we find:
TABLE CANT COPY
Do tasks (a), (b-i), (c), (d), and (f). Comment: The very small residual (high $R^2$ ) indicates that the data are very likely highly "over fit." Compute $R^2$.

Biostatistics A Methodology For the Health Sciences

Gerald van Belle,… 2nd Edition

Chapter 11, Problem 5

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Solved by Expert Raymond Matshanda

01/20/2022

Step 1

The data includes measurements from 10 normal men and 11 normal women during a maximal exercise treadmill test. The variables of interest are Activity, Weight (Wt), Height (Ht), VO2Max, Functional Aerobic Impairment (FAI), Maximum Cardiac Output (Q_MAX), Maximum Show more…

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Refer to data of Hossack et al. [1980, 1981]. Ten normal men and 11 normal women were studied during a maximal exercise treadmill test. While being exercised they had a catheter (tube) inserted into the pulmonary (lung) artery and a short tube into the left radial or brachial artery. This allowed sampling and observation of arterial pressures and the oxygen content of the blood. From this, several parameters as described below were measured or calculated. The data for the 11 women are given in Table 11.17; the data for the 10 normal men are displayed in Table 11.18. Descriptions of the variables follow. • Activity: a subject who routinely exercises three or more times per week until perspiring was active (Act); otherwise, the subject was sedentary (Sed). • Wt: weight in kilograms. • $H t$ : height in centimeters. • VO2MaX: oxygen (in millimeters per kilogram of body weight) used in $1 \mathrm{~min}$ at maximum exercise. • FAI: functional aerobic impairment. For a patient's age and activity level (active or sedentary) the expected treadmill duration (ED) is estimated from a regression equation. The excess of observed duration (OD) to expected duration (ED) as a percentage of $\mathrm{ED}$ is the FAI. $\mathrm{FAI}=100 \times(\mathrm{OD}-\mathrm{ED}) / \mathrm{ED}$. •$\dot{Q}_{M A X}:$ output of the heart in liters of blood per minute at maximum. •$H R_{M A X}$ : heart rate in beats per minute at maximum exercise. • $S V_{M A X}$ : volume of blood pumped out of the heart in milliliters during each stroke (at maximum cardial output). • $\mathrm{CaO}_2$ : oxygen content of the arterial system in milliliters of oxygen per liter of blood. • $\mathrm{C} \overline{\mathrm{v}} \mathrm{O}_2$ : oxygen content of the venous (vein) system in milliliters of oxygen per liter of blood. • $a \bar{v} O_2 D_{M A X}$ : difference in the oxygen content (in milliliters of oxygen per liter of blood) between the arterial system and the venous system (at maximum exercise); thus, $\mathrm{avVO}_2 \mathrm{D}_{\mathrm{MAX}}=\mathrm{CaO}_2-\mathrm{C}_2$. • $\bar{P}_{\mathrm{SA}}, \mathrm{mAX}$ : average pressure in the arterial system at the end of exercise in milliliters of mercury $(\mathrm{mmHg})$. •$\bar{P}_{\mathrm{PA}}, \mathrm{MAX}$ : average pressure in the pulmonary artery at the end of exercise in $\mathrm{mmHg}$. After examining the normal probability plot of residuals, the regression of Problem 11.4 was rerun omitting cases 2 and 8 . In this case we find: TABLE CANT COPY Do tasks (a), (b-i), (c), (d), and (f). Comment: The very small residual (high $R^2$ ) indicates that the data are very likely highly "over fit." Compute $R^2$.

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Key Concepts

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. It helps in understanding how changes in predictor(s) are associated with changes in an outcome, and it involves fitting a model to empirical data to make predictions or infer relationships.

Residual Analysis

Residual analysis involves examining the differences between observed and predicted values from a regression model. It is a key diagnostic tool used to assess the adequacy of the model, check for violations of model assumptions, and identify patterns indicating potential outliers or model misspecification.

Coefficient of Determination (R²)

The coefficient of determination, denoted as R², measures the proportion of the variance in the dependent variable that is predictable from the independent variables. A high R² indicates that the model explains a large portion of the variation, but it can also be a sign of potential overfitting if the model is overly complex.

Overfitting

Overfitting occurs when a statistical model captures not only the underlying relationship in the data but also the random noise. This typically happens when the model is excessively complex relative to the amount of data available, leading to excellent performance on the sample data but poor generalizability to new data.

Outlier Detection and Management

Outlier detection and management involve identifying unusual observations that do not conform to the expected pattern of the data. Removing or mitigating the influence of outliers can lead to a more robust and reliable regression model by preventing these data points from disproportionately affecting the results.

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Transcript

0:00 Right.

00:01 This question says, fiscal fitness testing is an important aspect of athletic training.

00:08 A common measures of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during a strainer's exercise.

00:20 A study was conducted on 24 mega age meant to study the influence of the time it takes to complete a two -mile run.

00:27 The oxygen uptake measure was accomplished with standard laboratory methods as the subjects performed on a treadmill.

00:38 The work was published in the maximum oxygen intake prediction in young and middle -aged males.

00:45 Journal of sports medicine write 9, 1969, 17 to 22.

00:56 The data right the data are recorded are presented as the figures that you see in these questions okay this is an extract from the journal that was published in 1969 so let's see the questions let's use the data to answer the questions right they say the first question says estimate the parameters in a simple linear regression model so we have to estimate the parameters as in a simple linear regression model.

01:31 So in other ways, we have to come up with an estimate of a simple linear recreation model.

01:37 They're using all the formulas that we know for calculating the recreation, for coming up with the regression line there.

01:48 And bsa is does the time it takes to run two miles have a significant influence on maximum oxygen take up, use the now hypothesis right, beta is equal to zero or beta is not equal to zero.

02:09 Then you have to in the question, see, is plot the residuals on a graph against x and comment on the appropriateness of the simple linear model.

02:22 Okay.

02:23 So now let's look at the questions and my possible answers that i gave here.

02:30 I'm just presenting the answers that that i gave, i did the calculation and come up.

02:35 So my estimate of the linear model there, estimate of the parameters in a linear regression model there, this is what i came up with.

02:48 Okay, you don't just go to on the moon and come up with this because so what i did is i took calculations so you have to find the sum of x, you have to find the sum of y, you have to find the sum of y the sum of x squared right the sum of x y and then the sum of right you have to come up with these figures from the table and then from the table you have to find your the generally remember the general equation for the linear equation model is like this a right k plus right where our a is is the slope and our p is the so you have to use these formulas to then calculate the slope the slope is negative 0 .05 -13 there and my intercept was 90 .80904 there so you have to find use the methods to calculate these then you come up with a linear regression line there so this is what i came up with is it right then the second question says does the time it takes to run two miles have a significant influence on maximum oxygen take up there.

04:12 I say the t value in testing the nile hypothesis is the t value that i found is negative 6 .533, which results in a p value.

04:26 This t value, it results in a p value which is less than 0 .001 there.

04:33 Hence, the time it takes to run two miles has a significant influence...

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