Question

I need to calculate the within and between run variances from some data as part of developing a new analytical chemistry method. I also need confidence intervals from this data using the R language

I assume I need to use anova or something ?

My data is like

> variance
   Run Rep Value
1    1   1  9.85
2    1   2  9.95
3    1   3 10.00
4    2   1  9.90
5    2   2  8.80
6    2   3  9.50
7    3   1 11.20
8    3   2 11.10
9    3   3  9.80
10   4   1  9.70
11   4   2 10.10
12   4   3 10.00

Answer 1

I’ve been looking at a similar problem. I’ve found reference to caluclating confidence intervals by Burdick and Graybill (Burdick, R. and Graybill, F. 1992, Confidence Intervals on variance components, CRC Press)

Using some code I’ve been trying I get these values

> kiaraov = aov(Value~Run+Error(Run),data=kiar)

> summary(kiaraov)

Error: Run
    Df  Sum Sq Mean Sq
Run  3 2.57583 0.85861

Error: Within
          Df  Sum Sq Mean Sq F value Pr(>F)
Residuals  8 1.93833 0.24229               
> confint = 95

> a = (1-(confint/100))/2

> grandmean = as.vector(kiaraov$"(Intercept)"[[1]][1]) # Grand Mean (I think)

> within = summary(kiaraov)$"Error: Within"[[1]]$"Mean Sq"  # S2^2Mean Square Value for Within Run

> dfRun = summary(kiaraov)$"Error: Run"[[1]]$"Df"

> dfWithin = summary(kiaraov)$"Error: Within"[[1]]$"Df"

> Run = summary(kiaraov)$"Error: Run"[[1]]$"Mean Sq" # S1^2Mean Square for between Run

> between = (Run-within)/((dfWithin/(dfRun+1))+1) # (S1^2-S2^2)/J

> total = between+within

> between # Between Run Variance
[1] 0.2054398

> within # Within Run Variance
[1] 0.2422917

> total # Total Variance
[1] 0.4477315

> betweenCV = sqrt(between)/grandmean * 100 # Between Run CV%

> withinCV = sqrt(within)/grandmean * 100 # Within Run CV%

> totalCV = sqrt(total)/grandmean * 100 # Total CV%

> #within confidence intervals

> withinLCB = within/qf(1-a,8,Inf) # Within LCB

> withinUCB = within/qf(a,8,Inf) # Within UCB

> #Between Confidence Intervals

> n1 = dfRun

> n2 = dfWithin

> G1 = 1-(1/qf(1-a,n1,Inf)) # According to Burdick and Graybill this should be a

> G2 = 1-(1/qf(1-a,n2,Inf))

> H1 = (1/qf(a,n1,Inf))-1  # and this should be 1-a, but my results don't agree

> H2 = (1/qf(a,n2,Inf))-1

> G12 = ((qf(1-a,n1,n2)-1)^2-(G1^2*qf(1-a,n1,n2)^2)-(H2^2))/qf(1-a,n1,n2) # again, should be a, not 1-a

> H12 = ((1-qf(a,n1,n2))^2-H1^2*qf(a,n1,n2)^2-G2^2)/qf(a,n1,n2) # again, should be 1-a, not a

> Vu = H1^2*Run^2+G2^2*within^2+H12*Run*within

> Vl = G1^2*Run^2+H2^2*within^2+G12*within*Run

> betweenLCB = (Run-within-sqrt(Vl))/J # Betwen LCB

> betweenUCB = (Run-within+sqrt(Vu))/J # Between UCB

> #Total Confidence Intervals

> y = (Run+(J-1)*within)/J

> totalLCB = y-(sqrt(G1^2*Run^2+G2^2*(J-1)^2*within^2)/J) # Total LCB

> totalUCB = y+(sqrt(H1^2*Run^2+H2^2*(J-1)^2*within^2)/J) # Total UCB

> result = data.frame(Name=c("within", "between", "total"),CV=c(withinCV,betweenCV,totalCV),LCB=c(sqrt(withinLCB)/grandmean*100,sqrt(betweenLCB)/grandmean*100,sqrt(totalLCB)/grandmean*100),UCB=c(sqrt(withinUCB)/grandmean*100,sqrt(betweenUCB)/grandmean*100,sqrt(totalUCB)/grandmean*100))

> result
     Name       CV      LCB      UCB
1  within 4.926418 3.327584  9.43789
2 between 4.536327      NaN 19.73568
3   total 6.696855 4.846030 20.42647

Here the lower confidence interval for between run CV is less than zero, so reported as NaN.

I’d love to have a better way to do this. If I get time I might try to create a function to do this.

Paul.

—

Edit: I did eventually write a function, here it is (caveat emptor)

#' avar Function
#' 
#' Calculate thewithin, between and total %CV of a dataset by ANOVA, and the
#' associated confidence intervals
#' 
#' @param dataf - The data frame to use, in long format 
#' @param afactor Character string representing the column in dataf that contains the factor
#' @param aresponse  Charactyer string representing the column in dataf that contains the response value
#' @param aconfidence What Confidence limits to use, default = 95%
#' @param digits  Significant Digits to report to, default = 3
#' @param debug Boolean, Should debug messages be displayed, default=FALSE
#' @returnType dataframe containing the Mean, Within, Between and Total %CV and LCB and UCB for each
#' @return 
#' @author Paul Hurley
#' @export
#' @examples 
#' #Using the BGBottles data from Burdick and Graybill Page 62
#' assayvar(dataf=BGBottles, afactor="Machine", aresponse="weight")
avar<-function(dataf, afactor, aresponse, aconfidence=95, digits=3, debug=FALSE){
    dataf<-subset(dataf,!is.na(with(dataf,get(aresponse))))
    nmissing<-function(x) sum(!is.na(x))
    n<-nrow(subset(dataf,is.numeric(with(dataf,get(aresponse)))))
    datadesc<-ddply(dataf, afactor, colwise(nmissing,aresponse))
    I<-nrow(datadesc)
    if(debug){print(datadesc)}
    if(min(datadesc[,2])==max(datadesc[,2])){
        balance<-TRUE
        J<-min(datadesc[,2])
        if(debug){message(paste("Dataset is balanced, J=",J,"I is ",I,sep=""))}
    } else {
        balance<-FALSE
        Jh<-I/(sum(1/datadesc[,2], na.rm = TRUE))
        J<-Jh
        m<-min(datadesc[,2])
        M<-max(datadesc[,2])
        if(debug){message(paste("Dataset is unbalanced, like me, I is ",I,sep=""))}
        if(debug){message(paste("Jh is ",Jh, ", m is ",m, ", M is ",M, sep=""))}
    }
    if(debug){message(paste("Call afactor=",afactor,", aresponse=",aresponse,sep=""))}
    formulatext<-paste(as.character(aresponse)," ~ 1 + Error(",as.character(afactor),")",sep="")
    if(debug){message(paste("formula text is ",formulatext,sep=""))}
    aovformula<-formula(formulatext)
    if(debug){message(paste("Formula is ",as.character(aovformula),sep=""))}
    assayaov<-aov(formula=aovformula,data=dataf)
    if(debug){
        print(assayaov)
        print(summary(assayaov))
    }
    a<-1-((1-(aconfidence/100))/2)
    if(debug){message(paste("confidence is ",aconfidence,", alpha is ",a,sep=""))}
    grandmean<-as.vector(assayaov$"(Intercept)"[[1]][1]) # Grand Mean (I think)
    if(debug){message(paste("n is",n,sep=""))}

    #This line commented out, seems to choke with an aov object built from an external formula
    #grandmean<-as.vector(model.tables(assayaov,type="means")[[1]]$`Grand mean`) # Grand Mean (I think)
    within<-summary(assayaov)[[2]][[1]]$"Mean Sq"  # d2e, S2^2 Mean Square Value for Within Machine = 0.1819
    dfRun<-summary(assayaov)[[1]][[1]]$"Df"  # DF for within = 3
    dfWithin<-summary(assayaov)[[2]][[1]]$"Df"  # DF for within = 8
    Run<-summary(assayaov)[[1]][[1]]$"Mean Sq" # S1^2Mean Square for Machine
    if(debug){message(paste("mean square for Run ?",Run,sep=""))}
    #Was between<-(Run-within)/((dfWithin/(dfRun+1))+1) but my comment suggests this should be just J, so I'll use J !
    between<-(Run-within)/J # d2a (S1^2-S2^2)/J
    if(debug){message(paste("S1^2 mean square machine is ",Run,", S2^2 mean square within is ",within))}
    total<-between+within
    between # Between Run Variance
    within # Within Run Variance
    total # Total Variance
    if(debug){message(paste("between is ",between,", within is ",within,", Total is ",total,sep=""))}

    betweenCV<-sqrt(between)/grandmean * 100 # Between Run CV%
    withinCV<-sqrt(within)/grandmean * 100 # Within Run CV%
    totalCV<-sqrt(total)/grandmean * 100 # Total CV%
    n1<-dfRun
    n2<-dfWithin
    if(debug){message(paste("n1 is ",n1,", n2 is ",n2,sep=""))}
    #within confidence intervals
    if(balance){
        withinLCB<-within/qf(a,n2,Inf) # Within LCB
        withinUCB<-within/qf(1-a,n2,Inf) # Within UCB
    } else {
        withinLCB<-within/qf(a,n2,Inf) # Within LCB
        withinUCB<-within/qf(1-a,n2,Inf) # Within UCB
    }
#Mean Confidence Intervals
    if(debug){message(paste(grandmean,"+/-(sqrt(",Run,"/",n,")*qt(",a,",df=",I-1,"))",sep=""))} 
    meanLCB<-grandmean+(sqrt(Run/n)*qt(1-a,df=I-1)) # wrong
    meanUCB<-grandmean-(sqrt(Run/n)*qt(1-a,df=I-1)) # wrong
    if(debug){message(paste("Grandmean is ",grandmean,", meanLCB = ",meanLCB,", meanUCB = ",meanUCB,aresponse,sep=""))}
    if(debug){print(summary(assayaov))}
#Between Confidence Intervals
    G1<-1-(1/qf(a,n1,Inf)) 
    G2<-1-(1/qf(a,n2,Inf))
    H1<-(1/qf(1-a,n1,Inf))-1  
    H2<-(1/qf(1-a,n2,Inf))-1
    G12<-((qf(a,n1,n2)-1)^2-(G1^2*qf(a,n1,n2)^2)-(H2^2))/qf(a,n1,n2) 
    H12<-((1-qf(1-a,n1,n2))^2-H1^2*qf(1-a,n1,n2)^2-G2^2)/qf(1-a,n1,n2) 
    if(debug){message(paste("G1 is ",G1,", G2 is ",G2,sep=""))
        message(paste("H1 is ",H1,", H2 is ",H2,sep=""))
        message(paste("G12 is ",G12,", H12 is ",H12,sep=""))
    }
    if(balance){
        Vu<-H1^2*Run^2+G2^2*within^2+H12*Run*within
        Vl<-G1^2*Run^2+H2^2*within^2+G12*within*Run
        betweenLCB<-(Run-within-sqrt(Vl))/J # Betwen LCB
        betweenUCB<-(Run-within+sqrt(Vu))/J # Between UCB
    } else {
        #Burdick and Graybill seem to suggest calculating anova of mean values to find n1S12u/Jh
        meandataf<-ddply(.data=dataf,.variable=afactor, .fun=function(df){mean(with(df, get(aresponse)), na.rm=TRUE)})
        meandataaov<-aov(formula(paste("V1~",afactor,sep="")), data=meandataf)
        sumsquare<-summary(meandataaov)[[1]]$`Sum Sq`
        #so maybe S12u is just that bit ?
        Runu<-(sumsquare*Jh)/n1
        if(debug){message(paste("n1S12u/Jh is ",sumsquare,", so S12u is ",Runu,sep=""))}
        Vu<-H1^2*Runu^2+G2^2*within^2+H12*Runu*within
        Vl<-G1^2*Runu^2+H2^2*within^2+G12*within*Runu
        betweenLCB<-(Runu-within-sqrt(Vl))/Jh # Betwen LCB
        betweenUCB<-(Runu-within+sqrt(Vu))/Jh # Between UCB
        if(debug){message(paste("betweenLCB is ",betweenLCB,", between UCB is ",betweenUCB,sep=""))}
    }
#Total Confidence Intervals
    if(balance){
        y<-(Run+(J-1)*within)/J
        if(debug){message(paste("y is ",y,sep=""))}
        totalLCB<-y-(sqrt(G1^2*Run^2+G2^2*(J-1)^2*within^2)/J) # Total LCB
        totalUCB<-y+(sqrt(H1^2*Run^2+H2^2*(J-1)^2*within^2)/J) # Total UCB
    } else {
        y<-(Runu+(Jh-1)*within)/Jh
        if(debug){message(paste("y is ",y,sep=""))}
        totalLCB<-y-(sqrt(G1^2*Runu^2+G2^2*(Jh-1)^2*within^2)/Jh) # Total LCB
        totalUCB<-y+(sqrt(H1^2*Runu^2+H2^2*(Jh-1)^2*within^2)/Jh) # Total UCB
    }
    if(debug){message(paste("totalLCB is ",totalLCB,", total UCB is ",totalUCB,sep=""))}
#   result<-data.frame(Name=c("within", "between", "total"),CV=c(withinCV,betweenCV,totalCV),
#           LCB=c(sqrt(withinLCB)/grandmean*100,sqrt(betweenLCB)/grandmean*100,sqrt(totalLCB)/grandmean*100),
#           UCB=c(sqrt(withinUCB)/grandmean*100,sqrt(betweenUCB)/grandmean*100,sqrt(totalUCB)/grandmean*100))
    result<-data.frame(Mean=grandmean,MeanLCB=meanLCB, MeanUCB=meanUCB, Within=withinCV,WithinLCB=sqrt(withinLCB)/grandmean*100, WithinUCB=sqrt(withinUCB)/grandmean*100,
            Between=betweenCV, BetweenLCB=sqrt(betweenLCB)/grandmean*100, BetweenUCB=sqrt(betweenUCB)/grandmean*100,
            Total=totalCV, TotalLCB=sqrt(totalLCB)/grandmean*100, TotalUCB=sqrt(totalUCB)/grandmean*100)
    if(!digits=="NA"){
        result$Mean<-signif(result$Mean,digits=digits)
        result$MeanLCB<-signif(result$MeanLCB,digits=digits)
        result$MeanUCB<-signif(result$MeanUCB,digits=digits)
        result$Within<-signif(result$Within,digits=digits)
        result$WithinLCB<-signif(result$WithinLCB,digits=digits)
        result$WithinUCB<-signif(result$WithinUCB,digits=digits)
        result$Between<-signif(result$Between,digits=digits)
        result$BetweenLCB<-signif(result$BetweenLCB,digits=digits)
        result$BetweenUCB<-signif(result$BetweenUCB,digits=digits)
        result$Total<-signif(result$Total,digits=digits)
        result$TotalLCB<-signif(result$TotalLCB,digits=digits)
        result$TotalUCB<-signif(result$TotalUCB,digits=digits)
    }
    return(result)
}

assayvar<-function(adata, aresponse, afactor, anominal, aconfidence=95, digits=3, debug=FALSE){
    result<-ddply(adata,anominal,function(df){
                resul<-avar(dataf=df,afactor=afactor,aresponse=aresponse,aconfidence=aconfidence, digits=digits, debug=debug)
                resul$n<-nrow(subset(df, !is.na(with(df, get(aresponse)))))
                return(resul)
            })
    return(result)
}