residual volume

Examples of residual volume in the following topics:

• Lung Volumes and Capacities

  • The volume in the lung can be divided into four units: tidal volume, expiratory reserve volume, inspiratory reserve volume, and residual volume.
  • The residual volume (RV) is the amount of air that is left after expiratory reserve volume is exhaled.
  • Residual volume is also important for preventing large fluctuations in respiratory gases (O2 and CO2).
  • The functional residual capacity (FRC) includes the expiratory reserve volume and the residual volume.
  • It is the sum of the residual volume, expiratory reserve volume, tidal volume, and inspiratory reserve volume. .

• Lung Capacity and Volume

  • Even when we exhale deeply some air is still in the lungs (about 1,000 ml) and is called residual volume.
  • There are certain types of diseases of the lung where residual volume builds up because the person cannot fully empty the lungs.
  • Determination of the residual volume is more difficult as it is impossible to "completely" breathe out.
  • Therefore measurement of the residual volume has to be done via indirect methods such as radiographic planimetry, body plethysmography, closed circuit dilution (including the helium dilution technique), and nitrogen washout.
  • Standard errors in prediction equations for residual volume have been measured at 579 ml for men and 355 ml for women, while the use of 0.24*FVC gave a standard error of 318 ml.

• Residuals

  • Observations below the line have negative residuals.
  • The observation marked by an " has a small, negative residual of about -1; the observation marked by "+" has a large residual of about +7; and the observation marked by "$\Delta$" has a moderate residual of about -4.
  • The residuals are plotted at their original horizontal locations but with the vertical coordinate as the residual.
  • For instance, the point (85.0, 98.6)+ had a residual of 7.45, so in the residual plot it is placed at (85.0, 7.45).
  • The second data set shows a pattern in the residuals.

• Plotting the Residuals

  • Note that the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent.
  • To create a residual plot, we simply plot an $x$-value and a residual value.
  • The average of the residuals is always equal to zero; therefore, the standard deviation of the residuals is equal to the RMS error of the regression line.
  • Residual plots can allow some aspects of data to be seen more easily.
  • Differentiate between scatter and residual plots, and between errors and residuals

• Line fitting, residuals, and correlation exercises

  • If we were to construct a residual plot (residuals versus x) for each, describe what those plots would look like.
  • (a) Describe the relationship between volume and height of these trees.
  • (b) Describe the relationship between volume and diameter of these trees.
  • 7.1: (a) The residual plot will show randomly distributed residuals around 0.
  • (b) The residuals will show a fan shape, with higher variability for smaller x.

• Introduction to multiple regression exercises

  • (d) Calculate the residual for the first observation in the data set.
  • (e) The variance of the residuals is 249.28, and the variance of the birth weights of all babies in the data set is 332.57.
  • Instead, other variables, such as height and diameter, may be used to predict a tree's volume and yield.
  • Height is measured in feet, diameter in inches (at 54 inches above ground), and volume in cubic feet.
  • Determine if the model overestimates or underestimates the volume of this tree, and by how much.

• Productivity

  • The other 17.00 units came from the production volume growth.
  • The first is called "increasing returns" and occurs when productivity and production volume increase or when productivity and production volume decrease.
  • The second, "diminishing returns", occurs when productivity decreases and volume increases or when productivity increases and volume decreases.
  • It measures the residual growth that cannot be explained by the rate of change in the services of labour, capital and intermediate outputs, and is often interpreted as the contribution to economic growth made by factors such as technical and organizational innovation.
  • The volume measure of output reflects the goods and services produced by the workforce.

• Checking model assumptions using graphs exercises

  • 8.11: Nearly normal residuals: The normal probability plot shows a nearly normal distribution of the residuals, however, there are some minor irregularities at the tails.
  • Constant variability of residuals: The scatter-plot of the residuals versus the fitted values does not show any overall structure.
  • In addition, the residuals do appear to have constant variability between the two parity and smoking status groups, though these items are relatively minor.
  • Independent residuals: The scatterplot of residuals versus the order of data collection shows a random scatter, suggesting that there is no apparent structures related to the order the data were collected.
  • The rest of the residuals do appear to be randomly distributed around 0.

• Checking model assumptions using graphs

  • A normal probability plot of the residuals is shown in Figure 8.9.
  • In a normal probability plot for residuals, we tend to be most worried about residuals that appear to be outliers, since these indicate long tails in the distribution of residuals.
  • We consider a plot of the residuals against the cond new variable and the residuals against the wheels variable.
  • There appears to be curvature in the residuals, indicating the relationship is probably not linear.
  • We see some slight bowing in the residuals against the wheels variable.

• An objective measure for finding the best line

  • Mathematically, we want a line that has small residuals.
  • Perhaps our criterion could minimize the sum of the residual magnitudes:
  • However, a more common practice is to choose the line that minimizes the sum of the squared residuals:
  • In many applications, a residual twice as large as another residual is more than twice as bad.
  • Squaring the residuals accounts for this discrepancy.