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Background

Sometimes we might want to compare three or four tube types for a particular analyte on a group of patients or we might want to see if a particular analyte is stable over time in aliqioted samples. In these experiments are essentially doing the multivariable analogue of the paired t-test. In the tube-type experiment, the factor that is differing between the ('paired') groups is the container: serum separator tubes (SST), EDTA plasma tubes, plasma separator tubes (PST) etc. In a stability experiment, the factor that is differing is storage duration.

Since this is a fairly common clinical lab experiment, I thought I would just jot down how this is accomplished in R – though I must confess I know just about $$\lim_{x\to0}x$$ about statistics. In any case, the statistical test is a repeated-measures ANOVA and this is one way to do it (there are many) including an approach to the post-hoc testing.

Some Fake Data to Work With

I'm going to make some fake data. I tried to dig up the data from an experiment I did as a resident but alas, I think the raw data died on an old laptop. But fake data will do for demonstration purposes. Let's suppose we are looking at parathyroid hormone (PTH) in three different vacutainer tubes: SST, EDTA and PST. For the sake of argument, let's say that we collect samples from 20 patients simultaneously and we anlayze them all as per our usual process. This means that each patient has three samples of material that should be otherwise identical outside of the effects of the collection contained.

library(magrittr)
set.seed(100) #to force the same pseudo-random each time
#data in pmol/L
#induce some heteroscedastic error
SST <- runif(20,3,50)
PST <- 1.03*SST + rnorm(20,0,0.1)*SST #set the data up to show no difference
EDTA <- 1.15*SST + rnorm(20,0,0.1)*SST  #set the data up to show a difference
tube.data <- data.frame(SST,PST,EDTA) %>% round(.,1)
tube.data <- data.frame(Subject = factor(1:20), tube.data)

This is the way we usually express (and receive) data like this in an Excel spreadsheet:

Subject SST PST EDTA
1 17.5 18.1 19.9
2 15.1 15.7 20.0
3 29.0 29.2 32.9
4 5.7 6.2 6.4
5 25.0 26.1 27.0
6 25.7 26.4 29.0
7 41.2 40.8 48.1
8 20.4 22.1 24.3
9 28.7 26.9 36.0
10 11.0 13.9 13.7
11 32.4 31.9 36.9
12 44.5 49.2 57.4
13 16.2 17.1 15.7
14 21.7 24.1 26.3
15 38.8 36.8 42.6
16 34.4 34.0 44.2
17 12.6 12.1 14.1
18 19.8 20.9 25.4
19 19.9 18.2 23.0
20 35.4 37.4 34.1

This Excel-ish way of storing the data is referred to as the “datawide” format for obvious reasons.

Gather the Grain

As it turns out this is not the way that we want to store data to do the statistical analyses of interest. What we want to do is have the tube type in a single column because this is the factor that is different within the subjects. We want to gather() or melt() the data (depending on your package of choice) to be like so:

library(tidyr)
tube.data.2 <- gather(tube.data, key = "Subject")
tube.data.2 %>% kable()

Subject Subject value
1 SST 17.5
2 SST 15.1
3 SST 29.0
4 SST 5.7
5 SST 25.0
6 SST 25.7
7 SST 41.2
8 SST 20.4
9 SST 28.7
10 SST 11.0
11 SST 32.4
12 SST 44.5
13 SST 16.2
14 SST 21.7
15 SST 38.8
16 SST 34.4
17 SST 12.6
18 SST 19.8
19 SST 19.9
20 SST 35.4
1 PST 18.1
2 PST 15.7
3 PST 29.2
4 PST 6.2
5 PST 26.1
6 PST 26.4
7 PST 40.8
8 PST 22.1
9 PST 26.9
10 PST 13.9
11 PST 31.9
12 PST 49.2
13 PST 17.1
14 PST 24.1
15 PST 36.8
16 PST 34.0
17 PST 12.1
18 PST 20.9
19 PST 18.2
20 PST 37.4
1 EDTA 19.9
2 EDTA 20.0
3 EDTA 32.9
4 EDTA 6.4
5 EDTA 27.0
6 EDTA 29.0
7 EDTA 48.1
8 EDTA 24.3
9 EDTA 36.0
10 EDTA 13.7
11 EDTA 36.9
12 EDTA 57.4
13 EDTA 15.7
14 EDTA 26.3
15 EDTA 42.6
16 EDTA 44.2
17 EDTA 14.1
18 EDTA 25.4
19 EDTA 23.0
20 EDTA 34.1

Now we see that there is a column for tube-type and a column for the PTH results which we can name accordingly. You can see why this called the “datalong” format.

names(tube.data.2) <- c("Subject", "Tube.Type", "PTH")
tube.data.2$Tube.Type <- as.factor(tube.data.2$Tube.Type) #turns tube type into factor

Visualize

Summarize the data:

summary(tube.data)

##     Subject        SST             PST             EDTA
##  1      : 1   Min.   : 5.70   Min.   : 6.20   Min.   : 6.40
##  2      : 1   1st Qu.:17.18   1st Qu.:17.85   1st Qu.:19.98
##  3      : 1   Median :23.35   Median :25.10   Median :26.65
##  4      : 1   Mean   :24.75   Mean   :25.36   Mean   :28.85
##  5      : 1   3rd Qu.:32.90   3rd Qu.:32.42   3rd Qu.:36.23
##  6      : 1   Max.   :44.50   Max.   :49.20   Max.   :57.40
##  (Other):14

Let's just have a quick look graphically:

library(mcr)
plot(mcreg(SST, EDTA,
method.reg = "PaBa",
mref.name = "SST",
mtest.name = "EDTA"))

plot(mcreg(SST, PST,
method.reg = "PaBa",
mref.name = "SST",
mtest.name = "PST"))

And as a boxplot with the points overtop:

boxplot(PTH ~ Tube.Type,
data = tube.data.2,
col = c("purple", "lightgreen", "gold"))
stripchart(PTH ~ Tube.Type,
vertical = TRUE,
data = tube.data.2,
method = "jitter",
pch = 20,
col = rgb(0,0,0,0.5))

Separate the Wheat from the Chaff

Now we want to make comparisons to see if these are different. To accomplish this, we will use the aov() function. This requires us to have data formatted “datalong” as it is in the tube.data.2 dataframe.

fit <- aov(PTH ~ Tube.Type + Error(Subject/Tube.Type), data=tube.data.2)

If you are like me, this syntax is confusing. But it goes like this. PTH is a function of Tube.Type which is straight forward–hence the PTH ~ Tube.Type bit. The error term has the Subject in front of the / and the factor that is different within the subjects (Tube.Type) after the /. That's my grade 2 explanation from reading this and this and this.

summary(fit)

##
## Error: Subject
##           Df Sum Sq Mean Sq F value Pr(>F)
## Residuals 19   7307   384.6
##
## Error: Subject:Tube.Type
##           Df Sum Sq Mean Sq F value   Pr(>F)
## Tube.Type  2  195.9   97.97   22.47 3.63e-07 ***
## Residuals 38  165.7    4.36
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

This tells us that there is a difference between the groups but it does not specify where the difference is.

I can't see the difference. Can you see the difference?

Sorry – I just had to make a pop-culture reference to this. We want to be specific about where the differences are without making a Type I error which might arise if we blindly charge ahead and do multiple paired t-tests. One easy way to accomplish this is to use the pairwise.t.test() function which does corrections for multiple comparisons. You can choose from a number of approaches for adjustment for pairwise comparison. This requires the “response vector” which is PTH and the “grouping factor” which is the tube type.

# choices for p.adjust.method are: c("holm", "hochberg", "hommel", "bonferroni", "BH", "BY", "fdr", "none")
pwt <- pairwise.t.test(tube.data.2$PTH, tube.data.2$Tube.Type, p.adj = "bonferroni", paired = TRUE)
pwt

##
##  Pairwise comparisons using paired t tests
##
## data:  tube.data.2$PTH and tube.data.2$Tube.Type
##
##     EDTA    PST
## PST 0.00083 -
## SST 7.9e-05 0.35033
##
## P value adjustment method: bonferroni

This is pretty easy to understand. There are statistically significant differences found between the EDTA and PST (p = 0.00083) and the EDTA and PST (p = 0.00008) but none between SST and PST (p = 0.35033).

Conclusion

Non-statistician's approach to tube-type comparisons, which is also applicable to analyte stability studies. This is a one-way repeated measures ANOVA with one within-subjects factor. There is a great deal more to say on the matter by people who know much more in the citations in the links provided above.

God probably uses datawide format

All the nations will be gathered before him, and he will separate the people one from another as a shepherd separates the sheep from the goats. He will put the sheep on his right and the goats on his left.

(Matt 25:32-33)