Limit of detection is the sensitivity of the assay — how low a concentration can the test detect?

To test this, the lab did a dilution series over a range of concentrations. When the concentration is low enough, the fusion won’t be detected.

Using the data from the dilution series, we used 4 models to predict the limit of detection.

1. Linear regression using all data
2. Linear regression using just the concentrations where data was observed
3. Logit
4. Probit

1. Linear regression using all data

My data frame “chan” contains deltaCq values and the Percentage_Fusion.

First do the linear regression:


chan.lm <- lm ( deltaCq ~ log2 ( Final_Percentage_Fusion ), data=chan) summary (chan.lm)

with a prediction interval:

PI <- predict (chan.lm, newdata = new.dat, interval='prediction', level=0.90)

The fusion dilution where the top prediction limit crosses the delta Ct cut-off will be estimated and chosen to be the model's estimate of the C95.

My delta Ct cutoff is 5 (determined from the Accuracy study).

cutoff <- 5

Our linear model on the upper prediction interval is :


upper.lm <- lm (PI[,"upr"] ~ log2(conc) )

y = slope * x + intercept and hence x = (y- intercept)/slope. In our case x is the concentration, and it's what we want. y is delta Ct and we want to know what concentration is predicted to have a delta Ct of 5 (our cutoff).

slope <- upper.lm$coefficients[2]
intercept <- upper.lm$coefficients[1]


C95 = (cutoff - intercept)/slope


conc_LoD = 2^C95


conc_LoD is our limit of detection.

2. Linear regression using just the concentrations where data was observed

To do a linear regression on just some of the data, identify which concentrations to remove.

remove_because_partial_or_no_data <- c(0.0488, 0.0977, 0.1953, 0.3906, 0.781, 1.563)

Remove these concentrations from the data frame:

chan <- chan[!(chan$Final_Percentage_Fusion %in% remove_because_partial_or_no_data),]
and then follow the same steps as in #1.

3. Logit model

First, make a column of whether the PCR product is called as a positive or not. This column is called "CalledPositive".


chan["CalledPositive"] <- 1

If no signal is detected or deltaCq is > 5, then the fusion is not called and we have to assign CalledPositive as false or "0".

chan[is.na(chan$deltaCq), "CalledPositive"] <- 0

chan[!is.na(chan$deltaCq) & chan$deltaCq > 5, "CalledPositive"] <- 0

Run the logit analysis

mylogit <- glm (CalledPositive ~ log2Conc , family = binomial(link = "logit"), data = chan)

To find the sensitivity at 95% based on the logit model, I manually feed in ranges of values until I get something close to 95%.

For example, by inspecting the graph I know that my LoD is somewhere between 0 and 2 so I set


xlower <- 0

xupper <- 2

and then I make a series of points:

temp.data <- data.frame(log2Conc = seq(from = xlower, to = xupper, length.out = 10))

And I input temp.data to get predicted sensitivities:


predicted.data.logit <- predict(mylogit, temp.data, type = "response", se.fit=TRUE)

I then look at predicted.data.logit to see which concentration gives me close to 95%. This can take several iterations until I'm very close to 0.95

4. Probit model

We make use of the "CalledPositive" column defined from the section 3 logit model.

myprobit <- glm(CalledPositive ~ log2Conc , family = binomial(link = "probit"), data = chan)

To find the concentration with sensitivity of 95%, do the same steps as described in 3.Logit model, but use predicted.data.probit instead.


predicted.data.probit <- predict(myprobit, temp.data, type = "response", se.fit=TRUE)

You can find my complete linear regression R scripts and probit and logit scripts on github.