How to compute the confidence interval of the ratio of two normal means

Question

I want to derive the limits for the $100(1-\alpha)\%$ confidence interval for the ratio of two means.
Suppose, $X_1 \sim N(\theta_1, \sigma^2)$ and $X_2 \sim N(\theta_2, \sigma^2)$ being independent, the mean ratio $\Gamma = \theta_1/\theta_2$. I tried to solve: $$\text{Pr}(-z(\alpha/2)) \leq X_1 - \Gamma X_2 / \sigma \sqrt {1 + \gamma^2} \leq z(\alpha/2)) = 1 - \alpha$$ but that equation couldn't be solved for many cases (no roots). Am I doing something wrong? Is there a better approach? Thanks

Answer 1

Fieller's method does what you want -- compute a confidence interval for the quotient of two means, both assumed to be sampled from Gaussian distributions.

• The original citation is: Fieller EC: The biological standardization of Insulin. Suppl to J R Statist Soc 1940, 7:1-64.

• The Wikipedia article does a good job of summarizing.

• I've created an online calculator that does the computation.

• Here is a page summarizing the math from the first edition of my Intuitive Biostatistics

Answer 2

R has the package mratios with the function t.test.ratio.

Gemechis Dilba Djira, Mario Hasler, Daniel Gerhard and Frank Schaarschmidt (2011). mratios: Inferences for ratios of coefficients in the general linear model. R package version 1.3.15. http://CRAN.R-project.org/package=mratios

See also http://www.r-project.org/user-2006/Slides/DilbaEtAl.pdf

Answer 3

Also if you want to compute Fieller's confidence interval not using mratios (typically because you don't want a simple lm fit but for example a glmer or glmer.nb fit), you can use the following FiellerRatioCI function, with model the output of the model, aname the name of the numerator parameter, bname the name of the denomiator parameter. You can also use directly the FiellerRatioCI_basic function giving, a, b and the covariance matrix between a and b.

FiellerRatioCI <- function (x, ...) { # generic Biomass Equilibrium Level
    UseMethod("FiellerRatioCI", x)
}
FiellerRatioCI_basic <- function(a,b,V,alpha=0.05){
    theta <- a/b
    v11 <- V[1,1]
    v12 <- V[1,2]
    v22 <- V[2,2]

    z <- qnorm(1-alpha/2)
    g <- (z^2)*v22/b^2
    C <- sqrt(v11 - 2*theta*v12 + theta^2 * v22 - g*(v11-v12^2/v22))
    minS <- (1/(1-g))*(theta- g*v12/v22 - z/b * C)
    maxS <- (1/(1-g))*(theta- g*v12/v22 + z/b * C)
    return(c(ratio=theta,min=minS,max=maxS))
}
FiellerRatioCI.glmerMod <- function(model,aname,bname){
    V <- vcov(model)
    a<-as.numeric(unique(coef(model)$culture[aname]))
    b<-as.numeric(unique(coef(model)$culture[bname]))
    return(FiellerRatioCI_basic(a,b,V[c(aname,bname),c(aname,bname)]))
}
FiellerRatioCI.glm <- function(model,aname,bname){
    V <- vcov(model)
    a <- coef(model)[aname]
    b <- coef(model)[bname]
    return(FiellerRatioCI_basic(a,b,V[c(aname,bname),c(aname,bname)]))
}

Example (based on standard glm basic example):

 counts <- c(18,17,15,20,10,20,25,13,12)
 outcome <- gl(3,1,9)
 treatment <- gl(3,3)
 glm.D93 <- glm(counts ~ outcome + treatment, family = poisson())

 FiellerRatioCI(glm.D93,"outcome2","outcome3")

ratio.outcome2            min            max 
      1.550427      -2.226870      17.880574

Answer 4

You can calculate it through:

• Fieller's method
• The Taylor method, also called Delta method: it's easier than Fieller's but will fail if the denominator approaches zero.
• The Hwang–bootstrap method, a bootstrap technique that does not result in unbounded confidence limits.

Here you can find a thorough description and comparison of these methods.