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predict.pointwise(results.predict, coverage = 0.99,
simultaneous = FALSE, individual = FALSE)predict with se.fit=TRUE.coverage=0.99.simultaneous=FALSE.individual=FALSE) or
prediction limits for an individual observation (individual=TRUEfit of the argument
results.predict.predict. The limits are computed at those points specified by the argument
newdata of predict.
The predict function is a generic function with methods for several
different classes. The funciton pointwise was part of the S language.
The modifications to pointwise in the package "lm").
Confidence Limits for a Predicted Mean Value (individual=FALSE).
Consider a standard linear model with $p$ predictor variables.
Often, one of the major goals of regression analysis is to predict a future
value of the response variable given known values of the predictor variables.
The equations for the predicted mean value of the response given
fixed values of the predictor variables as well as the equation for a
two-sided (1-$\alpha$)100% confidence interval for the mean value of the
response can be found in Draper and Smith (1998, p.80) and
Millard and Neerchal (2001, p.547).
Technically, this formula is a confidence interval for the mean of
the response for one set of fixed values of the predictor variables and
corresponds to the case when simultaneous=FALSE. To create simultaneous
confidence intervals over the range of of the predictor variables,
the critical t-value in the equation has to be replaced with a critical
F-value and the modified formula is given in Draper and Smith (1998, p. 83),
Miller (1981a, p. 111), and Millard and Neerchal (2001, p. 547).
This formula is used in the case when simultaneous=TRUE.
Confidence Limits for a Predicted Individual Value (individual=TRUE).
In the above section we discussed how to create a confidence interval for
the mean of the response given fixed values for the predictor variables.
If instead we want to create a prediction interval for a single
future observation of the response variable, the fomula is given in
Miller (1981a, p. 115) and Millard and Neerchal (2001, p. 551).
Technically, this formula is a prediction interval for a single future
observation for one set of fixed values of the predictor variables and
corresponds to the case when simultaneous=FALSE. Miller (1981a, p. 115)
gives a formula for simultaneous prediction intervals for $k$ future
observations. If we are interested in creating an interval that will
encompass all possible future observations over the range of the
preictor variables with some specified probability however, we need to
create simultaneous tolerance intervals. A formula for such an interval
was developed by Lieberman and Miller (1963) and is given in
Miller (1981a, p. 124). This formula is used in the case when
simultaneous=TRUE.predict, predict.lm,
lm, calibrate,
inversePredictCalibrate, detectionLimitCalibrate.# Using the data in the built-in data frame Air.df,
# fit the cube root of ozone as a function of temperature.
# Then compute predicted values for ozone at 70 and 90
# degrees F, and compute 95\% confidence intervals for the
# mean value of ozone at these temperatures.
# First create the lm object
#---------------------------
ozone.fit <- lm(ozone ~ temperature, data = Air.df)
# Now get predicted values and CIs at 70 and 90 degrees
#------------------------------------------------------
predict.list <- predict(ozone.fit,
newdata = data.frame(temperature = c(70, 90)), se.fit = TRUE)
pointwise(predict.list, coverage = 0.95)
# $upper
# 1 2
# 2.839145 4.278533
# $fit
# 1 2
# 2.697810 4.101808
# $lower
# 1 2
# 2.556475 3.925082
#--------------------------------------------------------------------
# Continuing with the above example, create a scatterplot of ozone
# vs. temperature, and add the fitted line along with simultaneous
# 95% confidence bands.
x <- Air.df$temperature
y <- Air.df$ozone
dev.new()
plot(x, y, xlab="Temperature (degrees F)",
ylab = expression(sqrt("Ozone (ppb)", 3)))
abline(ozone.fit, lwd = 2)
new.x <- seq(min(x), max(x), length=100)
predict.ozone <- predict(ozone.fit,
newdata = data.frame(temperature = new.x), se.fit = TRUE)
ci.ozone <- pointwise(predict.ozone, coverage=0.95,
simultaneous=TRUE)
lines(new.x, ci.ozone$lower, lty=2, lwd = 2, col = 2)
lines(new.x, ci.ozone$upper, lty=2, lwd = 2, col = 2)
title(main=paste("Cube Root Ozone vs. Temperature with Fitted Line",
"and Simultaneous 95% Confidence Bands",
sep=""))
#--------------------------------------------------------------------
# Redo the last example by creating non-simultaneous
# confidence bounds and prediction bounds as well.
dev.new()
plot(x, y, xlab = "Temperature (degrees F)",
ylab = expression(sqrt("Ozone (ppb)", 3)))
abline(ozone.fit, lwd = 2)
new.x <- seq(min(x), max(x), length=100)
predict.ozone <- predict(ozone.fit,
newdata = data.frame(temperature = new.x), se.fit = TRUE)
ci.ozone <- pointwise(predict.ozone, coverage=0.95)
lines(new.x, ci.ozone$lower, lty=2, col = 2, lwd = 2)
lines(new.x, ci.ozone$upper, lty=2, col = 2, lwd = 2)
pi.ozone <- pointwise(predict.ozone, coverage = 0.95,
individual = TRUE)
lines(new.x, pi.ozone$lower, lty=4, col = 4, lwd = 2)
lines(new.x, pi.ozone$upper, lty=4, col = 4, lwd = 2)
title(main=paste("Cube Root Ozone vs. Temperature with Fitted Line",
"and 95% Confidence and Prediction Bands",
sep=""))
#--------------------------------------------------------------------
# Clean up
rm(predict.list, ozone.fit, x, y, new.x, predict.ozone, ci.ozone,
pi.ozone)Run the code above in your browser using DataLab