check.resid: Residual Diagnostics

Description

This function performs residual diagnostics for linear models estimated by using the lm() function for detecting nonlinearity (partial residual or component-plus-residual plots), nonconstant error variance (predicted values vs. residuals plot), and non-normality of residuals (Q-Q plot and histogram with density plot).

Usage

check.resid(model, type = c("linear", "homo", "normal"),
            resid = c("unstand", "stand", "student"),
            point.shape = 21, point.fill = "gray80", point.size = 1,
            line1 = TRUE, line2 = TRUE,
            line.type1 = "solid", line.type2 = "dashed",
            line.width1 = 1, line.width2 = 1,
            line.color1 = "#0072B2", line.color2 = "#D55E00",
            bar.width = NULL, bar.n = 30, bar.color = "black",
            bar.fill = "gray95", strip.size = 11,
            label.size = 10, axis.size = 10,
            xlimits = NULL, ylimits = NULL,
            xbreaks = ggplot2::waiver(), ybreaks = ggplot2::waiver(),
            check = TRUE, plot = TRUE)

Value

Returns an object of class misty.object, which is a list with following entries:

call: function call
type: type of analysis
model: model specified in model
plotdat: data frame used for the plot
args: specification of function arguments
plot: ggplot2 object for plotting the residuals

Arguments

model: a fitted model of class lm.
type: a character string specifying the type of the plot, i.e., "linear" for partial (component-plus-residual) plots, "homo" (default) for predicted values vs. residuals plot, and "normal" for Q-Q plot and histogram with a density plot. Note that partial plots are not available for models with interaction terms.
resid: a character string specifying the type of residual used for the partial (component-plus-residual) plots or Q-Q plot and histogram, i.e., "unstand" for unstandardized residuals "stand" for standardized residuals, and "student" for studentized residual. By default, studentized residuals are used for predicted values vs. residuals plot and unstandardized residuals are used for Q-Q plot and histogram.
point.shape: a numeric value for specifying the argument shape in the geom_point function.
point.fill: a numeric value for specifying the argument fill in the geom_point function.
point.size: a numeric value for specifying the argument size in the geom_point function.
line1: logical: if TRUE (default), regression line is drawn in the partial (component-plus-residual) plots, horizontal line is drawn in the predicted values vs. residuals plot, and t-distribution or normal distribution curve is drawn in the histogram.
line2: logical: if TRUE (default), Loess smooth line is drawn in the partial (component-plus-residual) plots, loess smooth lines are drawn in the predicted values vs. residuals plot, and density curve is drawn in the histogram.
line.type1: a character string or numeric value for specifying the argument linetype in the geom_smooth, geom_hline, or stat_function function.
line.type2: a character string or numeric value for specifying the argument linetype in the geom_smooth or geom_density function.
line.width1: a numeric value for specifying the argument linewidth in the geom_smooth, geom_hline, or stat_function function.
line.width2: a numeric value for specifying the argument linewidth in the geom_smooth or geom_density function.
line.color1: a character string or numeric value for specifying the argument color in the geom_smooth, geom_hline, or stat_function function.
line.color2: a character string or numeric value for specifying the argument color in the geom_smooth or geom_density function.
bar.width: a numeric value for specifying the argument bins in the geom_bar function.
bar.n: a numeric value for specifying the argument bins in the geom_bar function.
bar.color: a character string or numeric value for specifying the argument color in the geom_bar function.
bar.fill: a character string or numeric value for specifying the argument fill in the geom_bar function.
strip.size: a numeric value for specifying the argument size in the element_text function of the strip.text argument within the theme function.
label.size: a numeric value for specifying the argument size in the element_text function of the axis.title argument within the theme function.
axis.size: a numeric value for specifying the argument size in the element_text function of the axis.text argument within the theme function.
xlimits: a numeric value for specifying the argument limits in the scale_x_continuous function.
ylimits: a numeric value for specifying the argument limits in the scale_y_continuous function.
xbreaks: a numeric value for specifying the argument breaks in the scale_x_continuous function.
ybreaks: a numeric value for specifying the argument breaks in the scale_y_continuous function.
check: logical: if TRUE (default), argument specification is checked.
plot: logical: if TRUE (default), a plot is drawn.

Author

Takuya Yanagida takuya.yanagida@univie.ac.at

Details

Nonlinearity

The violation of the assumption of linearity implies that the model cannot accurately capture the systematic pattern of the relationship between the outcome and predictor variables. In other words, the specified regression surface does not accurately represent the relationship between the conditional mean values of $Y$ and the $X$s. That means the average error $E(\varepsilon)$ is not 0 at every point on the regression surface (Fox, 2015).

In multiple regression, plotting the outcome variable $Y$ against each predictor variable $X$ can be misleading because it does not reflect the partial relationship between $Y$ and $X$ (i.e., statistically controlling for the other $X$s), but rather the marginal relationship between $Y$ and $X$ (i.e., ignoring the other $X$s). Partial residual plots or component-plus-residual plots should be used to detect nonlinearity in multiple regression. The partial residual for the $j$th predictor variable is defined as

$$e_i^{(j)} = b_jX_{ij} + e_i$$

The linear component of the partial relationship between $Y$ and $X_j$ is added back to the least-squares residuals, which may include an unmodeled nonlinear component. Then, the partial residual $e_i^{(j)}$ is plotted against the predictor variable $X_j$. Nonlinearity may become apparent when a non-parametric regression smoother is applied.

By default, the function plots each predictor against the partial residuals, and draws the linear regression and the loess smooth line to the partial residual plots.

Nonconstant Error Variance

The violation of the assumption of constant error variance, often referred to as heteroscedasticity, implies that the variance of the outcome variable around the regression surface is not the same at every point on the regression surface (Fox, 2015).

Plotting residuals against the outcome variable $Y$ instead of the predicted values $\hat{Y}$ is not recommended because $Y = \hat{Y} + e$. Consequently, the linear correlation between the outcome variable $Y$ and the residuals $e$ is $\sqrt{1 - R^2}$ where $R$ is the multiple correlation coefficient. In contrast, plotting residuals against the predicted values $\hat{Y}$ is much easier to examine for evidence of nonconstant error variance as the correlation between $\hat{Y}$ and $e$ is 0. Note that the least-squares residuals generally have unequal variance $Var(e_i) = \sigma^2 / (1 - h_i)$ where $h$ is the leverage of observation $i$, even if errors have constant variance $\sigma^2$. The studentized residuals $e^*_i$, however, have a constant variance under the assumption of the regression model. Residuals are studentized by dividing them by $\sigma^2_i(\sqrt{(1 - h_i)}$ where $\sigma^2_i$ is the estimate of $\sigma^2$ obtained after deleting the $i$th observation, and $h_i$ is the leverage of observation $i$ (Meuleman et al, 2015).

By default, the function plots the predicted values against the studentized residuals. It also draws a horizontal line at 0, a loess smooth lines for all residuals as well as separate loess smooth lines for positive and negative residuals.

Non-normality of Residuals

Statistical inference under the violation of the assumption of normally distributed errors is approximately valid in all but small samples. However, the efficiency of least squares is not robust because the least-squares estimator is the most efficient and unbiased estimator only when the errors are normally distributed. For instance, when error distributions have heavy tails, the least-squares estimator becomes much less efficient compared to robust estimators. In addition, error distributions with heavy-tails result in outliers and compromise the interpretation of conditional means because the mean is not an accurate measure of central tendency in a highly skewed distribution. Moreover, a multimodal error distribution suggests the omission of one or more discrete explanatory variables that naturally divide the data into groups (Fox, 2016).

By default, the function plots a Q-Q plot of the unstandardized residuals, and a histogram of the unstandardized residuals and a density plot. Note that studentized residuals follow a $t$-distribution with $n - k - 2$ degrees of freedom where $n$ is the sample size and $k$ is the number of predictors. However, the normal and $t$-distribution are nearly identical unless the sample size is small. Moreover, even if the model is correct, the studentized residuals are not an independent random sample from $t_{n - k - 2}$. Residuals are correlated with each other depending on the configuration of the predictor values. The correlation is generally negligible unless the sample size is small.

References

Fox, J. (2016). Applied regression analysis and generalized linear models (3rd ed.). Sage Publications, Inc.

Meuleman, B., Loosveldt, G., & Emonds, V. (2015). Regression analysis: Assumptions and diagnostics. In H. Best & C. Wolf (Eds.), The SAGE handbook of regression analysis and causal inference (pp. 83-110). Sage.

Examples

Run this code

if (FALSE) {
#-------------------------------------------------------------------------------
# Residual diagnostics for a linear model

mod <- lm(Ozone ~ Solar.R + Wind + Temp, data = airquality)

# Example 1: Partial (component-plus-residual) plots
check.resid(mod, type = "linear")

# Example 2: Predicted values vs. residuals plot
check.resid(mod, type = "homo")

# Example 3: Q-Q plot and histogram with density plot
check.resid(mod, type = "normal")

#-------------------------------------------------------------------------------
# Extract data and ggplot2 object

object <- check.resid(mod, type = "linear", plot = FALSE)

# Data frame
object$plotdat

# ggplot object
object$plot
}

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