Linearity studies are often not done right

Seeing how linearity studies are typically made, we have identified several issues that need attention. More than 10% error in estimated Concentration can easily happen if you are not aware of these.

Even though there are excellent guidance’s on how to do it right, like CLSI EP-06, they are clearly not always being followed.

Issues

The main issues we see are:

1.???Linear regression assumes that Precision is the same at all levels, requiring the Assay to have constant standard deviation. However, most Assays are closer to having a constant Coefficient of Variation (CV), i.e., constant relative standard deviation.

2.???R-Squared is being used as criteria for linearity of Assay. Although linearity influences R-Squared in a linear regression, it depends as much on Assay Precision and how much the concentration is varied.

3.???If Assay is found non-linear within the approved range of concentrations, this non-linearity is added to the measurement uncertainty for the Assay, thereby increasing it.

Solutions

The solutions to the issues above are straight forward:

1.???Build a regression model with the measured area as response and concentration as discrete predictor. Thereby the residual variance for replicates can be studied versus level. A Box-Cox transformation will give you the optimal power transformation λ. Then variance depends on level as shown in bullet a. below. If λ≠1, variance depends on level and weighted regression is needed with inverse variance, i.e., the formula in bullet b.:

?

2.???Fit data with a polynomial instead and use Information Criteria′s as stopping rule (we recommend Akaike Information Criteria corrected; AICc) to see if a polynomial fit is better than a linear fit. Polynomial fits will typically get a lower loglikelihood than a linear fit, as it has more fitting parameters, whereas AICc is loglikelihood corrected for the number of fitting parameters.

3.???If the Assay turns out to be non-linear, there is no reason to add the non-linearity to the measurement uncertainty of the assay. Instead use the polynomial fit to convert Area to Concentration.

Below is given an example on how to do this in the statistical software JMP from SAS.

Example

An Assay was tested by 5 times diluting a reference sample of known concentration a factor of 2. At each dilution the Area was measured in triplicate as shown in Figure 1.

An example using unweighted linear regression as is shown in Figure 2. If using a success criterion for linearity of R-Squared > 0.98, one would (erroneously) have concluded that the test for linearity was passed.

However, looking at the regression 2 things related to the issues mentioned above can be seen:

1.???The triplicate variation clearly increases as expected with level.

2.???There is a clear curvature on the residuals

1 Triplicate variation depends on level

To investigate triplicate variation, a model is built with Area as response and discrete Concentration as predictor. The only thing this model cannot predict perfectly is the triplicate variation.

Figure 3 shows that the best Box-Cox λ is close to 0. Since the Assay is expected to have a constant CV, a λ of 0 is chosen, and according to the proposed solution previously described, a weighted regression will be adopted using Area-2.

2 R-Square should not be the acceptance criteria for linearity

According to CLSI EP06, a linearity study starts with making a 3rd order polynomial fit, to see if the 2nd and/or 3rd order terms are statistically significant, by looking at P-values. However, P-values are very dependent on sample size and noise level on top of non-linearity. We recommend using AICc instead, as mentioned in the solutions section above.

Figure 4 shows 3rd, 2nd and 1st order polynomial fits weighted with Area-2. The 2nd order fit has the lowest AICc (minus loglikelihood corrected for number of fitting parameters) and is therefore the fit to go for. The first order fit has by far the highest AICc. i.e., by far the worst fit.

In the modeling above observations are assumed independent, but there is a risk that the triplicates for each concentration might not be e.g., independent dilutions. In that case DF will be over estimated. When you have multiple observations on experimental unit and there is a risk of dependencies, you have 2 options to get DF right:

1. Make the model on the means of the duplicates
2. Enter experimental unit as random factor in the model

We prefer the last method using all observations. This is especially important for detection of outliers, that might be hidden by a mean. The result of entering Concentration as attribute (triplicate group) random factor, is shown in figure 4.a.

It is seen that the triplicate groups can be assumed the same (variance component 0 for Triplicate Group) and thereby observations can be assumed independent. P-Values are exactly the same as in Figure 4.

3 Conversion of Area to Concentration with non-linear fi

In the use of the Assay, an Area is being measured and the goal is to get the Concentration. A model can then be fitted with Concentration as response and Area as predictor, again weighted with Area-2

Figure 5 shows this model including the equation converting Area to Concentration taking non-linearity into consideration.

4 What difference does it make?

Figure 6 shows that the difference between the wrong linear unweighted fit and the more proper weighted curved fit is 1.77 vs. 1.60, at the nominal concentration. This is a difference of more than 10%.?