--- title: "Confidence intervals for linear combinations of variances" date: "2016-03-19" output: html_document --- \newcommand{\EE}{\mathbb{E}} {r setup, include=FALSE} knitr::opts_chunk$set(echo = TRUE)  Let$x_i$'s be some observations of independent random variables$X_i \sim \theta_i \dfrac{\chi^2_{d_i}}{d_i}$. In this article we will take a look at some methods to get a confidence interval about a linear combination of the$\theta_i$'s. This situation occurs when we are interested in the variances of interest in an ANOVA model with random effects. ## Satterthwaite method In order to get a confidence interval about a linear combination$\sum a_i\theta_i$with nonnegative coefficents$a_i$, the Satterthwaite approximation consists in doing as if $$\sum a_i X_i \sim \left(\sum a_i\theta_i\right) \frac{\chi^2_\nu}{\nu} \quad \textrm{with }\; \nu = \frac{{\left(\sum a_ix_i\right)}^2}{\sum\dfrac{{(a_i x_i)}^2}{d_i}}.$$ Thus, taking a$100(1-\alpha)\%$-dispersion interval$[b^-, b^+]$of the$\chi^2_\nu$distribution, one gets the approximate$100(1-\alpha)\%$-confidence interval about$\sum a_i\theta_i$: $$\left[\nu\frac{\sum a_ix_i}{b^+}, \nu\frac{\sum a_ix_i}{b^-}\right].$$ This interval is returned by the ciSatt function below, taking the quantiles$\chi^2_\nu\bigl(\frac{\alpha}{2}\bigr)$and$\chi^2_\nu\bigl(1-\frac{\alpha}{2}\bigr)$for$b^-$and$b^+$respectively. {r} ciSatt <- function(x, dofs, a, alpha=0.05){ s <- sum(a*x) nu <- s^2/sum((a*x)^2/dofs) lwr <- s*nu/qchisq(1-alpha/2, nu) upr <- s*nu/qchisq(alpha/2,nu) return(c("lwr"=lwr, "upr"=upr)) }  ## Graybill & Wang's method [Graybill & Wang][GW] provided another method for nonnegative linear combinations. Their approximate$100(1-\alpha)\%$-confidence interval about$\sum a_i\theta_i$is $$\left[\sum a_i x_i - \sqrt{\sum{(G_ia_ix_i)}^2}, \sum a_i x_i + \sqrt{\sum{(H_ia_ix_i)}^2}\right]$$ where $$G_i = 1 - \frac{d_i}{\chi^2_{d_i}\bigl(1-\frac{\alpha}{2}\bigr)} \quad \text{and }\; H_i = \frac{d_i}{\chi^2_{d_i}\bigl(\frac{\alpha}{2}\bigr)} - 1.$$ ## Ting & al's generalization Graybill & Wang's confidence interval has been generalized to the case when some$a_i$are negative by [Ting & al][TBGJL] (see also [Burdick & al][BBM]). It is returned by the function ciTing given below. {r} ciTing <- function(x, dofs, a, alpha=0.05){ G <- 1 - sapply(dofs, function(d){ d/qchisq(1-alpha/2,d) }) H <- sapply(dofs, function(d){ d/qchisq(alpha/2,d) }) - 1 s <- sum(a*x) if(all(a>0)){ # this is Graybill & Wang's confidence interval lwr <- s - sqrt(sum((G*a*x)^2)) upr <- s + sqrt(sum((H*a*x)^2)) return(c("lwr"=lwr, "upr"=upr)) } pos <- which(a>0); neg <- which(a<0) A12 <- sum(sapply(pos, function(q){ sapply(neg, function(r){ Fqr <- qf(1-alpha/2, dofs[q], dofs[r]) # alpha/2 in the article is an error Lqr <- ((Fqr-1)^2 - G[q]^2*Fqr^2 - H[r]^2)/Fqr return(Lqr*a[q]*x[q]*a[r]*x[r]) }) })) B12 <- sum(sapply(pos, function(q){ sapply(neg, function(r){ Fqr <- qf(alpha/2, dofs[q], dofs[r]) # 1-alpha/2 in the article is an error Lqr <- ((Fqr-1)^2 - H[q]^2*Fqr^2 - G[r]^2)/Fqr return(Lqr*a[q]*x[q]*a[r]*x[r]) }) })) lwr <- s - sqrt(sum((G[pos]*a[pos]*x[pos])^2) + sum((H[neg]*a[neg]*x[neg])^2) - A12) upr <- s + sqrt(sum((H[pos]*a[pos]*x[pos])^2) + sum((G[neg]*a[neg]*x[neg])^2) - B12) return(c("lwr"=lwr, "upr"=upr)) }  We will study the performance of this confidence interval on an example. Some improvements of this interval are given in [Ting & al][TBGJL]'s paper, but we do not provide them here. ## Example Consider a balanced three-way ANOVA model with one fixed factor and two random factors: $$y_{hijk} = \underset{\mu_h}{\underbrace{\mu + A_h}} + B_i + C_j + {(AB)}_{hi} + {(AC)}_{hj} + {(BC)}_{ij} + {(ABC)}_{hij} + \epsilon_{hijk}, \\ \quad h = 1,\ldots, H, \quad i = 1,\ldots, I, \quad j = 1, \ldots, J, \quad k = 1, \ldots, K.$$ $$\scriptsize \begin{array}{lccc} \text{Source} & \text{dof} & \text{Mean square} & \text{Expected mean square} \\ B & I-1 & S^2_B & \theta_B = \sigma^2_E + K \sigma^2_{ABC} + HK \sigma^2_{BC} + JK \sigma^2_{AB} + HJK \sigma^2_B \\ C & J-1 & S^2_C & \theta_C = \sigma^2_E + K \sigma^2_{ABC} + HK \sigma^2_{BC} + IK \sigma^2_{AC} + HIK \sigma^2_C \\ A \times B & (H-1)(I-1) & S^2_{AB} & \theta_{AB}= \sigma^2_E + K \sigma^2_{ABC} + JK \sigma^2_{AB} \\ B \times C & (I-1)(J-1) & S^2_{BC} & \theta_{BC} = \sigma^2_E + K \sigma^2_{ABC} + HK \sigma^2_{BC} \\ A \times C & (H-1)(J-1) & S^2_{AC} & \theta_{AC} = \sigma^2_E + K \sigma^2_{ABC} + IK \sigma^2_{AC} \\ A \times B \times C & (H-1)(J-1)(K-1) & S^2_{ABC} & \theta_{ABC} = \sigma^2_E + K \sigma^2_{ABC} \\ E & n - HIJ & S^2_E & \theta_E = \sigma^2_E \end{array}$$ In matricial form, the variance components and the expected mean squares are related by $$\small \begin{pmatrix} \theta_B \\ \theta_C \\ \theta_{AB} \\ \theta_{BC} \\ \theta_{AC} \\ \theta_{ABC} \\ \theta_E \end{pmatrix} = \begin{pmatrix} HJK & 0 & JK & HK & 0 & K & 1 \\ 0 & HIK & 0 & HK & IK & K & 1 \\ 0 & 0 & JK & 0 & 0 & K & 1 \\ 0 & 0 & 0 & HK & 0 & K & 1 \\ 0 & 0 & 0 & 0 & IK & K & 1 \\ 0 & 0 & 0 & 0 & 0 & K & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \sigma^2_B \\ \sigma^2_C \\ \sigma^2_{AB} \\ \sigma^2_{BC} \\ \sigma^2_{AC} \\ \sigma^2_{ABC} \\ \sigma^2_E \end{pmatrix}$$ and conversely, $$\small \begin{pmatrix} \sigma^2_B \\ \sigma^2_C \\ \sigma^2_{AB} \\ \sigma^2_{BC} \\ \sigma^2_{AC} \\ \sigma^2_{ABC} \\ \sigma^2_E \end{pmatrix} = \begin{pmatrix} \frac{1}{HJK} & 0 & -\frac{1}{HJK} & -\frac{1}{HJK} & 0 & \frac{1}{HJK} & 0 \\ 0 & \frac{1}{HIK} & 0 & -\frac{1}{HIK} & -\frac{1}{HIK} & \frac{1}{HIK} & 0 \\ 0 & 0 & \frac{1}{JK} & 0 & 0 & -\frac{1}{JK} & 0 \\ 0 & 0 & 0 & \frac{1}{HK} & 0 & -\frac{1}{HK} & 0 \\ 0 & 0 & 0 & 0 & \frac{1}{IK} & -\frac{1}{IK} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{1}{K} & -\frac{1}{K} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} \theta_B \\ \theta_C \\ \theta_{AB} \\ \theta_{BC} \\ \theta_{AC} \\ \theta_{ABC} \\ \theta_E \end{pmatrix}$$ We look for a confidence interval about the reproductibily variance of factor$B$: $$\sigma^2_{\textrm{repro}, B} = \sigma^2_B + \sigma^2_{AB} + \sigma^2_{BC} + \sigma^2_{ABC}$$ which is the linear combination of the expected mean squares $$\small \begin{multline} \frac{1}{HJK} \theta_B + \left(\frac{1}{JK}-\frac{1}{HJK}\right)\theta_{AB} + \left(\frac{1}{HK}-\frac{1}{HJK}\right)\theta_{BC} \\ + \left(\frac{1}{HJK}-\frac{1}{JK} + \frac{1}{K}-\frac{1}{HK}\right)\theta_{ABC} - \frac{1}{K} \theta_E \\ = \frac{1}{HJK}\left(\theta_B + (H-1)\theta_{AB} + (J-1)\theta_{BC} + (1-H-HJ-J)\theta_{ABC} - HJ\theta_E \right) \quad (\ast) \end{multline}$$ and is estimated by $$\frac{1}{HJK}\left(S^2_B + (H-1)S^2_{AB} + (J-1)S^2_{BC} + (1-H-HJ-J)S^2_{ABC} - HJS^2_E \right).$$ ### Checking the coverage Let us check the frequentist coverage of the confidence interval. We firstly write a function to sample the mean squares: {r} rMeanSquares <- function(nsims, H, I, J, K, sigma2B=1, sigma2C=1, sigma2AB=1, sigma2BC=1, sigma2AC=1, sigma2ABC=1, sigma2E=1){ VCnames <- c("B", "C", "AB", "BC", "AC", "ABC", "E") VC <- setNames(c(sigma2B, sigma2C, sigma2AB, sigma2BC, sigma2AC, sigma2ABC, sigma2E), VCnames) dofs <- setNames(c(I-1, J-1, (H-1)*(I-1), (I-1)*(J-1), (H-1)*(J-1), (H-1)*(J-1)*(K-1), H*I*J*(K-1)), VCnames) M <- rbind( c(H*J*K, 0, J*K, H*K, 0, K, 1), c(0, H*I*K, 0, H*K, I*K, K, 1), c(0, 0, J*K, 0, 0, K, 1), c(0, 0, 0, H*K, 0, K, 1), c(0, 0, 0, 0, I*K, K, 1), c(0, 0, 0, 0, 0,K, 1), c(0, 0, 0, 0, 0, 0, 1) ) thetas <- setNames(as.vector(M %*% VC), VCnames) sims <- matrix(numeric(1), nrow=nsims, ncol=7) colnames(sims) <- VCnames for(j in VCnames){ sims[,j] <- thetas[j]/dofs[j]*rchisq(nsims, dofs[j]) } attr(sims, "VC") <- VC attr(sims, "dofs") <- dofs return(sims) }  Here we simulate the mean squares using not too small values of the degress of freedom. {r simulations1, collapse=TRUE, cache=TRUE} # simulations H <- 10; I <- 15; J <- 10; K <- 5 nsims <- 10000 sims <- rMeanSquares(nsims, H=H, I=I, J=J, K=K) dofs <- attr(sims, "dofs") VC <- attr(sims, "VC") sigma2Brepro <- sum(VC[c("B", "AB", "BC", "ABC")]) # linear combination a <- 1/K*c(1/H/J, 1/J-1/H/J, 1/H-1/H/J, 1/H/J-1/J+1-1/H, -1) # confidence intervals Bounds <- matrix(numeric(1), nrow=nsims, ncol=2) colnames(Bounds) <- c("lwr", "upr") for(i in 1:nsims){ Bounds[i,] <- ciTing(sims[i,c("B", "AB", "BC", "ABC", "E")], dofs[c("B", "AB", "BC", "ABC", "E")], a=a) } # coverage of the upper one-sided interval: mean(Bounds[,"lwr"] < sigma2Brepro) # coverage of the lower one-sided interval: mean(Bounds[,"upr"] > sigma2Brepro) # coverage of the two-sided interval: mean(Bounds[,"lwr"] < sigma2Brepro & Bounds[,"upr"] > sigma2Brepro)  As we observe, the difference between the coverage obtained from the simulations and the nominal coverage does not exceed$1\%$for each of the three confidence intervals (upper one-sided, lower one-sided and two-sided). ### A small degrees of freedom example Now let us assess the frequentist coverage with$H=3$,$I=3$,$J=3$and$K=5$. {r simulations2, collapse=TRUE, cache=TRUE} # simulations H <- 3; I <- 3; J <- 3; K <- 5 nsims <- 10000 set.seed(666) sims <- rMeanSquares(nsims, H=H, I=I, J=J, K=K) dofs <- attr(sims, "dofs") VC <- attr(sims, "VC") sigma2Brepro <- sum(VC[c("B", "AB", "BC", "ABC")]) # linear combination a <- 1/K*c(1/H/J, 1/J-1/H/J, 1/H-1/H/J, 1/H/J-1/J+1-1/H, -1) # confidence intervals Bounds <- matrix(numeric(1), nrow=nsims, ncol=2) colnames(Bounds) <- c("lwr", "upr") for(i in 1:nsims){ Bounds[i,] <- ciTing(sims[i,c("B", "AB", "BC", "ABC", "E")], dofs[c("B", "AB", "BC", "ABC", "E")], a=a) } # coverage of the upper one-sided interval: mean(Bounds[,"lwr"] < sigma2Brepro) # coverage of the lower one-sided interval: mean(Bounds[,"upr"] > sigma2Brepro) # coverage of the two-sided interval: mean(Bounds[,"lwr"] < sigma2Brepro & Bounds[,"upr"] > sigma2Brepro)  This time, the coverage is not so close to the nominal level. The upper one-sided confidence interval ([lwr, Inf[) is too short, and the lower one-sided confidence interval (]-Inf, upr]) is too long. In other words, the lower bound and the upper bound are higher than desired. Let's have a look to the bounds: {r, collapse=TRUE} head(Bounds, 10)  The upper bound is quite big ($\sigma_{\textrm{repro},B}=4here). ### Shortening the intervals with the Satterthwaite approximation Recall our linear combination of the mean squares: \begin{align*} & a_1 S^2_B + a_2 S^2_{AB} + a_3 S^2_{BC} + a_4 S^2_{ABC} + a_5 S^2_E \\ = & a_1 x_1 + a_2 x_2 + a_3 x_3 + a_4 x_4 + a_5 x_5 \end{align*} with coefficientsa_1,a_2,a_3,a_4>0$,$a_5<0$, and degrees of freedom$2$,$4$,$4$,$16$and$108$. A degree of freedom of$2$is pretty small, and it could be the cause of the large upper bound. To circumvent this problem, we could try to replace$a_1x_1 + a_2x_2$with its Satterthwaite approximation: $$\underset{y}{\underbrace{a_1 x_1 + a_2 x_2}} + a_3 x_3 + a_4 x_4 + a_5 x_5$$ and then apply the Ting & al interval to the new linear combination$y+a_3 x_3 + a_4 x_4 + a_5 x_5$. Let's look what it gives for the second row of simulations: {r, collapse=TRUE} x <- sims[2, c("B", "AB", "BC", "ABC", "E")] dofs <- c(2, 4, 4, 16, 108) y <- sum(a[1:2]*x[1:2]) nu <- y^2/sum((a[1:2]*x[1:2])^2/dofs[1:2]) x_new <- c(y, x[3], x[4], x[5]) dofs_new <- c(nu, dofs[3], dofs[4], dofs[5]) a_new <- c(1, a[3], a[4], a[5]) # original interval: ciTing(x, dofs, a) # new interval: ciTing(x_new, dofs_new, a_new)  The upper bound is considerably smaller. Now let's have a look at the coverage when we apply this method to the previous simulations: {r simulations3, collapse=TRUE, cache=TRUE} # confidence intervals Bounds_new <- matrix(numeric(1), nrow=nsims, ncol=2) colnames(Bounds_new) <- c("lwr", "upr") for(i in 1:nsims){ x <- sims[i, c("B", "AB", "BC", "ABC", "E")] y <- sum(a[1:2]*x[1:2]) nu <- y^2/sum((a[1:2]*x[1:2])^2/dofs[1:2]) x_new <- c(y, x[3], x[4], x[5]) dofs_new <- c(nu, dofs[3], dofs[4], dofs[5]) Bounds_new[i,] <- ciTing(x_new, dofs_new, a_new) } # coverage of the upper one-sided interval: mean(Bounds_new[,"lwr"] < sigma2Brepro) # coverage of the lower one-sided interval: mean(Bounds_new[,"upr"] > sigma2Brepro) # coverage of the two-sided interval: mean(Bounds_new[,"lwr"] < sigma2Brepro & Bounds_new[,"upr"] > sigma2Brepro)  This time, the upper one-sided interval achieves a coverage close to the nominal value. The lower one-sided interval still have a too large coverage, but the upper bounds we get are generally pretty shorter: {r, collapse=TRUE} head(Bounds) head(Bounds_new)  Note that the method we proposed here is not intended to be a general one. The only thing we propose to the user is to explore the performance of the confidence intervals with the help of simulations for a specific design (values of$H$,$I$,$J$and$K\$) and the expected values of the variance components. We also recall that [Ting & al][TBGJL]'s paper provides some improvements of the confidence intervals that we did not consider here. ## References Graybill & Wang: *Confidence Intervals on Nonnegative Linear Combinations of Variances*. Journal of the American Statistical Association 75 (1980), 869-873. Ting, Burdick, Graybill, Jeyaratman & Lu: *Confidence intervals on linear combinations of variance components that are unrestricted in sign*. Journal of Statistical Computation and Simulation 35 (1990), 135-143. Burdick, Borror, Montgomery: *Design and Analysis of Gauge R&R Studies*. SIAM 2005. [GW]: http://www.jstor.org/stable/pdf/2287174.pdf "Confidence Intervals on Nonnegative Linear Combinations of Variances" [TBGJL]: http://www.tandfonline.com/doi/abs/10.1080/00949659008811240?journalCode=gscs20 "Confidence intervals on linear combinations of variance components that are unrestricted in sign" [BBM]: https://books.google.be/books?id=yTMQRkuYCswC "Design and Analysis of Gauge R&R Studies"