Hello,
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
If you want to apply a function (such as var) across a factor such as Run or Rep, you can use tapply:
> with(variance, tapply(Value, Run, var))
1 2 3 4
0.005833333 0.310000000 0.610000000 0.043333333
> with(variance, tapply(Value, Rep, var))
1 2 3
0.48562500 0.88729167 0.05583333
You have four groups of three observations:
> run1 = c(9.85, 9.95, 10.00)
> run2 = c(9.90, 8.80, 9.50)
> run3 = c(11.20, 11.10, 9.80)
> run4 = c(9.70, 10.10, 10.00)
> runs = c(run1, run2, run3, run4)
> runs
[1] 9.85 9.95 10.00 9.90 8.80 9.50 11.20 11.10 9.80 9.70 10.10 10.00
Make some labels:
> n = rep(3, 4)
> group = rep(1:4, n)
> group
[1] 1 1 1 2 2 2 3 3 3 4 4 4
Calculate within-run stats:
> withinRunStats = function(x) c(sum = sum(x), mean = mean(x), var = var(x), n = length(x))
> tapply(runs, group, withinRunStats)
$`1`
sum mean var n
29.800000000 9.933333333 0.005833333 3.000000000
$`2`
sum mean var n
28.20 9.40 0.31 3.00
$`3`
sum mean var n
32.10 10.70 0.61 3.00
$`4`
sum mean var n
29.80000000 9.93333333 0.04333333 3.00000000
You can do some ANOVA here:
> data = data.frame(y = runs, group = factor(group))
> data
y group
1 9.85 1
2 9.95 1
3 10.00 1
4 9.90 2
5 8.80 2
6 9.50 2
7 11.20 3
8 11.10 3
9 9.80 3
10 9.70 4
11 10.10 4
12 10.00 4
> fit = lm(runs ~ group, data)
> fit
Call:
lm(formula = runs ~ group, data = data)
Coefficients:
(Intercept) group2 group3 group4
9.933e+00 -5.333e-01 7.667e-01 -2.448e-15
> anova(fit)
Analysis of Variance Table
Response: runs
Df Sum Sq Mean Sq F value Pr(>F)
group 3 2.57583 0.85861 3.5437 0.06769 .
Residuals 8 1.93833 0.24229
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> degreesOfFreedom = anova(fit)[, "Df"]
> names(degreesOfFreedom) = c("treatment", "error")
> degreesOfFreedom
treatment error
3 8
Error or within-group variance:
> anova(fit)["Residuals", "Mean Sq"]
[1] 0.2422917
Treatment or between-group variance:
> anova(fit)["group", "Mean Sq"]
[1] 0.8586111
This should give you enough to do confidence intervals.
I'm going to take a crack at this when I have more time, but meanwhile, here's the dput() for Kiar's data structure:
structure(list(Run = c(1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4), Rep = c(1,
2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3), Value = c(9.85, 9.95, 10, 9.9,
8.8, 9.5, 11.2, 11.1, 9.8, 9.7, 10.1, 10)), .Names = c("Run",
"Rep", "Value"), row.names = c(NA, -12L), class = "data.frame")
... in case you'd like to take a quick shot at it.
I've been looking at a similar problem. I've found reference to caluclating confidence intervals by Burdick and Graybill (Burdick, R. and Graybill, F. 1992, Confidence Intervals on variance components, CRC Press)
Using some code I've been trying I get these values
> kiaraov = aov(Value~Run+Error(Run),data=kiar)
> summary(kiaraov)
Error: Run
Df Sum Sq Mean Sq
Run 3 2.57583 0.85861
Error: Within
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 8 1.93833 0.24229
> confint = 95
> a = (1-(confint/100))/2
> grandmean = as.vector(kiaraov$"(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.
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