What Is the Meaning of the Family Confidence Coefficient

In a previous post on multiple regression with ii predictor variables, the human relationship between the number of products and the altitude traveled on total delivery time was examined in the delivery dataset. Often there is a need to form confidence intervals for the parameters of a model to estimate the range in which the bodily parameter supposedly lies (at a given level of conviction, which volition hereby default to 95%). Yet, forming individual intervals for each parameter as discussed in an earlier mail on simple regression conviction intervals does not lead to an overall 95 percent conviction that the estimates for each parameter are right. For one regression parameter and the intercept, the confidence would actually be \(.95^ii = .9025\). Therefore, to guess a family of coefficients, the demand for simulatenous confidence intervals arises. The essential difference between a family unit confidence coefficient and a statement conviction coefficient is the former indicates that the entire family of confidence intervals are right bold repeated sampling.

Simulatenous Confidence Intervals of Mean Response

There are two procedures for forming simultaneous conviction intervals, the Working-Hotelling and Bonferroni procedures. Each estimates intervals of the mean response using a family conviction coefficient. The Working-Hotelling coefficient is defined past \(Westward\) and Bonferroni \(B\). In practice, it is recommended to perform both procedures to determine which results in a tighter interval. The Bonferroni method volition be explored start.

The Bonferroni Procedure for Simultaneous Estimation of Mean Responses

The Bonferroni method is more than general and conservative than Working-Hotelling. Confidence intervals are formed by adjusting each conviction coefficient to be higher than \(1 - \alpha\) and so the overall family conviction coefficient stays at the desired level. The confidence limits of the Bonferroni procedure are defined as:

\[ \hat{Y}_h \pm Bs\{\lid{Y}_h\} \]

Where \(\hat{Y}_h\) is equal to the matrix of the fitted response values and B is defined equally:

\[ B = t_{1 - \alpha / 2g, due north - 2} \]

Working-Hotelling Procedure for Simultaneous Conviction Intervals

The Working-Hotelling procedure is reminiscent of the familiar confidence band around a regression line. The confidence band contains all of the regression line and thus all mean responses. Due to this holding, boundary values can be formed at various levels of the predictor variable in question. The confidence interval equation for the Working-Hotelling procedure is similar to the Bonferroni procedure with the exception of the former being F-distributed with \(2\) and \(n - 2\) degrees of freedom:

\[ \chapeau{Y}_h \pm Ws\{\lid{Y}_h\} \]

Where:

\[ W^2 = 2F_{one - \alpha, two, n - two} \]

The standard mistake in both simulatenous confidence interval procedures is defined as:

\[ southward^2\{\hat{Y}_h\} = MSE(Ten'_h (X'X)^{-1}X_h) = X'_h south^two\{b\}X_h \]

Forming Simultaneous Confidence Intervals in R

With the definitions and equations out of the way, nosotros tin can explore how to build the simulatenous conviction intervals in R. The investr is the only package I've constitute that performs the Bonferroni and Working-Hotelling procedures. Of course, non existence satisified with but using a package and calling it a solar day as I oftentimes am, nosotros will build a custom part that creates intervals using both procedures to verify our agreement.

Outset by loading the necessary packages and the delivery dataset.

            library(robustbase)          

            ## Alarm: bundle 'robustbase' was built under R version 3.three.1          

            library(investr)          

            ## Warning: package 'investr' was congenital nether R version iii.three.i          

            library(ggplot2) library(gridExtra) data("delivery")          

Fit linear models with each predictor variable.

            dist.lm <- lm(delTime ~ distance, data = commitment) prod.lm <- lm(delTime ~ northward.prod, data = delivery)          

Using the plotFit function from the investr parcel, plot the Bonferroni and Working-Hotelling confidence intervals. Setting the argument conform to Scheffe instructs the role to build Working-Hotelling intervals.

            par(mfrow=c(two,2))  plotFit(dist.lm, interval = 'confidence', adapt = 'Scheffe', master = 'Working-Hotelling DelTime ~ Altitude') plotFit(prod.lm, interval = 'confidence', arrange = 'Scheffe', main = 'Working-Hotelling DelTime ~ Products')  plotFit(dist.lm, interval = 'confidence', k = 0.95, adjust = 'Bonferroni', primary = 'Bonferroni DelTime ~ Distance') plotFit(prod.lm, interval = 'confidence', m = 0.95, suit = 'Bonferroni', main = 'Bonferroni DelTime ~ Products')          

It appears the Bonferroni intervals are tighter than the Working-Hotelling intervals, though there is no reported test statistic to ostend this. To verify the results of the function and our understanding, we can write a function that implements both the Working-Hotelling and Bonferroni simultaneous confidence intervals.

            working.hotelling.bonferroni.intervals <- function(ten, y) {   y <- equally.matrix(y)   x <- every bit.matrix(x)   northward <- length(y)    # Get the fitted values of the linear model   fit <- lm(y ~ ten)   fit <- fit$fitted.values      # Observe standard mistake as divers above   se <- sqrt(sum((y - fit)^2) / (n - ii)) *      sqrt(1 / n + (x - mean(x))^2 /             sum((ten - mean(10))^2))    # Calculate B and West statistics for both procedures.   W <- sqrt(two * qf(p = 0.95, df1 = 2, df2 = n - ii))   B <- one-qt(.95/(two * 3), n - 1)    # Compute the simultaneous conviction intervals      # Working-Hotelling   wh.upper <- fit + W * se   wh.lower <- fit - W * se      # Bonferroni   bon.upper <- fit + B * se   bon.lower <- fit - B * se      xy <- information.frame(cbind(x,y))      # Plot the Working-Hotelling intervals   wh <- ggplot(xy, aes(ten=ten, y=y)) +      geom_point(size=ii.5) +      geom_line(aes(y=fit, 10=x), size=ane) +      geom_line(aes(10=x, y=wh.upper), colour='blue', linetype='dashed', size=ane) +      geom_line(aes(x=x, wh.lower), colour='blue', linetype='dashed', size=1) +     labs(title='Working-Hotelling')      # Plot the Bonferroni intervals   bonn <- ggplot(xy, aes(ten=x, y=y)) +      geom_point(size=2.5) +      geom_line(aes(y=fit, ten=10), size=i) +      geom_line(aes(x=x, y=bon.upper), colour='blue', linetype='dashed', size=1) +      geom_line(aes(x=x, bon.lower), colour='blue', linetype='dashed', size=ane) +     labs(championship='Bonferroni')      filigree.adapt(wh, bonn, ncol = 2)      # Collect results of procedures into a data.frame and return   res <- data.frame(round(cbind(W, B), 3), row.names = c('Consequence'))   colnames(res) <- c('W', 'B')      return(res) }  working.hotelling.bonferroni.intervals(delivery$due north.prod, commitment$delTime)          

            ##            W     B ## Result two.616 2.023          

            working.hotelling.bonferroni.intervals(delivery$distance, commitment$delTime)          

            ##            W     B ## Issue 2.616 2.023          

The graphs from our function mirror those from the plotFit function. Equally we suspected, the Bonferroni intervals are indeed tighter as evidenced by a smaller \(B\) value compared to \(Due west\). Thus, the Bonferroni intervals should be used in this particular example. Notice the \(W\) and \(B\) values are the aforementioned regardless of the predictor variable beingness examined, this is due to the procedures using the family unit conviction coefficient rather than the argument confidence coefficient as mentioned previously.

Summary

Simultaneous confidence intervals were explored and computed with the Bonferroni and Working-Hotelling procedures using the investr package and our own part. In the multiple regression setting, simulatenous confidence intervals are recommended as they provide certainty entire family unit of confidence coefficients are right. Thus, the simulatenous intervals volition always be wider than the argument confidence intervals as the sometime must take into account the articulation conviction level of the coefficients. This reply on StackExchange goes into more item regarding why the simulatenous intervals are wider than intervals formed with the statement confidence coefficient.

References

Feng, Y. Simultaneous inferences and other topics in regression assay. Retrieved from http://www.stat.columbia.edu/~yangfeng/W4315/lectures/lecture-4/lecture_4.pdf

Kutner, One thousand. H., Nachtsheim, C. J., Neter, J., Li, Westward., & Wasserman, W. (2004). Applied linear statistical models (5th ed.). Boston, MA: McGraw-Hill Higher Education.

Why are simultaneous confidence intervals wider than the normal ones? (2016). Retrieved from http://stats.stackexchange.com/questions/188372/why-are-simultaneous-confidence-intervals-wider-than-the-normal-ones

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