residuals

(noun)

The difference between the observed value and the estimated function value.

Related Terms

Examples of residuals in the following topics:

  • 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

  • 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.

  • Inferences of Correlation and Regression

    • These differences are referred to as residuals, and they can be standardized and adjusted to follow a normal distribution with mean $0$ and standard deviation $1$.
    • The adjusted standardized residuals, $d_{ij}$, are given by:
    • Subclavian site/no infectious complication has the largest residual at 6.2.
    • As these residuals follow a Normal distribution with mean 0 and standard deviation 1, all absolute values over 2 are significant.
    • The association between femoral site/no infectious complication is also significant, but because the residual is negative, there are fewer individuals than expected in this cell.

  • Influential Observations

    • The most commonly used measure of distance is the studentized residual.
    • Observation B has small leverage and a relatively small residual.
    • Observation C has small leverage and a relatively high residual.
    • Observation D has the lowest leverage and the second highest residual.
    • Observation E has by far the largest leverage and the largest residual.

  • Multiple and logistic regression solutions

    • 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.

  • Homogeneity and Heterogeneity

    • This new data set can also be used to construct a histogram, which can subsequently be used to assess the assumption that the residuals are normally distributed.
    • To the extent that the histogram matches the normal distribution, the residuals are normally distributed.
    • A residual plot displaying homoscedasticity should appear to resemble a horizontal football.
    • To the extent that a residual histogram matches the normal distribution, the residuals are normally distributed.

  • Conditions for the least squares line

    • Generally the residuals must be nearly normal.
    • An example of non-normal residuals is shown in the second panel of Figure 7.13.