A few weeks ago we discussed how to estimate the average bias over the whole measuring range. We also discussed how to recognize situations where the average bias gives enough information that you don’t need to examine bias in more detail. In method comparisons this is typically not the case. Instead, you need to evaluate bias as a function of concentration.
So how is that done? That’s the subject for today.
What is linear regression all about, and what does it have to do with bias?
In the previous blog post about bias we discussed fitting a horizontal line to a difference plot to describe the average bias. When we examine our data using a linear regression model, we can basically turn to the regression plot to do a similar examination.
On a regression plot, the y-axis represents the candidate method and x-axis represents the comparative method. Results of all samples are drawn on the plot. A line is then fitted to these results to represent the difference between the methods. The distance between the regression line and the y=x line represents bias.
Images 1, 2 and 3 visualize this for three different data sets. In Image 1, bias seems quite constant on absolute scale. The results given by the regression model are consistent with the calculated average bias. The slope of the regression line is near to 1 (1 falling within the 95% CI of the slope) and the effect it has on bias estimation is very small compared to the effect related to the intercept. Average bias (calculated as absolute mean difference) gives a reasonable estimation to describe the whole data set.
Also in Image 2, the results given by the regression model are consistent with the calculated average bias. Now bias seems nearly constant on proportional scale. The intercept of the regression line is near zero (95% CI of intercept containing zero) and the effect it has on bias estimation is very small compared to the effect related to the slope. Average bias (calculated as relative mean difference) gives a reasonable estimation to describe the whole data set.
In Image 3 on the other hand, mean difference (nor median difference) does not describe the behavior of the data set throughout the whole measuring interval. Yet it is possible to fit the linear regression line to the regression plot. In this case the regression line has a rather large positive intercept, while the slope is significantly smaller than 1. This leads to positive bias on small concentrations and negative bias on large concentrations.
Linear regression analysis is used to find the regression line that best describes the data set. Bias at a specific concentration is obtained by calculating the distance between the regression line and x=y line at that specific concentration.
You may remember that we have mean and median values as options for finding an estimate for average bias. Both of them have their own benefits in estimating bias. Whether one of them is better than another depends on the data set.
Similarly, there are multiple models that can be used in finding the regression line. We will introduce ordinary linear regression model, weighted least squares model, Deming model, weighted Deming model and Passing-Pablok model, as those are the established regression models used in method comparisons. All of them are available in the measurement procedure comparison study in Validation Manager.
Ordinary linear regression – the simplest model there is
Probably the most commonly known regression model is ordinary linear regression (OLR, also known as ordinary least squares OLS). Because it’s rather simple, it gives useful examples when you try to understand the basic idea of linear regression. And back in the old days, when we didn’t have such fancy computers and softwares as we do now, it also had the benefit that it could be calculated with reasonable effort.
Ordinary least squares is actually meant for finding out how a dependant variable y depends on an independent variable x (e.g. how the price of an apartment in specific type of locations depends on it’s size). The idea is to minimize the vertical distance of all points to the fitted line. Values on x-axis are used as they are, assuming that they do not contain error.
In method comparisons, our variable on y-axis is not really dependant on our variable on x-axis. Instead, both y and x are dependant on true sample concentrations that are unknown for us. This makes the case a little different than what ordinary linear regression has been designed for. The reasoning for using ordinary linear regression anyway is that the values on x-axis are kind of our best existing knowledge about the true concentrations. If the error related to each data point is small enough and other assumptions of the model are met, use of ordinary linear regression is justified.
When can you use ordinary linear regression and why you shouldn’t
When we use linear regression in measurement procedure comparisons, the underlying assumption of ordinary linear regression that there’s no error in results given by our comparative method is never really true. However there may be cases when this error is negligible compared to the range of measured concentrations, and therefore will have little effect on our bias estimation. In that case this assumption is close enough for the purposes of linear regression. This is checked by calculating the correlation coefficient. If r is at least 0.975 (or equivalently r2 is at least 0.95), we can consider this assumption to be met.
Ordinary linear regression also assumes that the error related to the candidate method follows a normal distribution. Basically this is required to justify estimating bias with a similar logic as when you calculate mean value. Assumption of data being normally distributed also enables us to calculate confidence intervals.
In ordinary linear regression, standard deviation (SD) of the data is assumed to be constant on absolute scale throughout the measuring range. When talking about clinical laboratory measurement data, this assumption is often unrealistic. Analytes with a very small measuring range do tend to show constant SD, but with a small measuring range the first assumption of r>0.975 can be difficult to achieve. You can use the constant difference plot and the scatter on regression plot to visually evaluate whether the data set shows constant SD, but it’s not easy to be sure that this assumption would hold.
Image 5 shows an example data set where the assumptions of ordinary linear regression may hold. In the middle we have a difference plot on absolute scale using direct comparison, i.e. comparative method is on x-axis. On the left we have a constant Bland-Altman difference plot, where the mean of both methods is on x-axis. (For more information about difference plots, see the earlier blog post about average bias.) Looking at the difference plots and the scatter on the regression plot on the right, the assumptions of constant SD and normal distribution of errors are reasonable. Pearson correlation r = 0.981. Therefore ordinary linear regression can be used.
Now let us compare different regression models to represent this data. In Image 6 we can see regression plots using three different regression models. On the left the regression line is created using ordinary linear regression model. In the middle we can see that Deming model gives very similar results as OLR. On the left, Passing-Bablok has a little bit larger confidence intervals, and we can easily see that the linear fit is a little different than with the other models. If we are confident that SD is constant and normally distributed throughout the measuring range, ordinary linear regression and Deming models can both be used.
The only benefit of using ordinary linear regression instead of Deming model or Passing-Bablok is that it’s easier to calculate. Deming regression model works just as well in cases where ordinary linear regression model would be justified, and it doesn’t require us to check the correlation.
When using Validation Manager, you don’t need to worry about how complicated the calculations are. Therefore the only valid reasoning to use ordinary linear regression instead of Deming regression is if considering the comparative measurement procedure as a reference happens to suit your purposes.
How about weighted least squares?
If the variability of the data does not show constant SD, but instead constant %CV (i.e. the spread of the results on the regression plot grows wider as a function of concentration), the simplest regression model to use is weighted least squares (WLS).
In weighted least squares, each point is given a weight inversely proportional to the square of the assumed concentration (i.e. value on x-axis). After that the linear fit is created by minimizing the vertical distance between data points and the regression line. Thanks to the weights, values at low concentrations have a higher influence on where the regression line will be drawn than high concentrations. Otherwise the high variance at high concentrations could sort of pull the regression line to one or the other direction depending on what kind of a scatter the samples happen to show. This would deteriorate the bias estimation.
Otherwise ordinary linear regression model and weighted least squares model make practically the same assumptions of the statistical qualities of the data set. Error is assumed to be found only in the vertical direction and this error is assumed to be normally distributed. The difference is whether variability is expected to show constant SD or constant %CV.
The constant SD assumption of ordinary linear regression model made it possible to use correlation as a measure of whether the error related to the results is small enough that we can omit the error related to the comparative method in our calculations. With constant %CV there’s no similar rule of thumb that could be used in evaluating whether using the weighted least squares model is justified.
One way to examine the reliability is to switch the comparison direction (i.e. which method is the candidate method and which is comparative) and see how much it affects the bias estimation.
Image 7 shows why switching the comparison direction can have a significant effect on the bias estimation given by weighted least squares. If we compare the methods the other way round, it’s a different method whose results are considered to contain error that explains the distance of the data point from the regression line. The assumed true concentrations of samples may change significantly. E.g. on the left, S3 is interpreted to have smaller concentration than S4. On the right, S3 in interpreted to have larger concentration than S4. These kind of effects may cause bias estimation to be affected by which method is used as comparative method. In this example, the upper end of the regression line is clearly placed differently into the graphs. (Similar effects are also possible when using ordinary linear regression, though the correlation requirement keeps it rather small.)
This is the main problem with weighted least squares. Comparison direction affects the interpretation of sample concentrations. Unlike in ordinary linear regression, the assumed sample concentration affects how a sample is weighted in weighted least squares calculations. There is no way to evaluate how much the error related to the comparative method affects the results.
To get a better idea about what this means, take a look at Image 8 and Image 9. In Image 8 we have plotted the same data set to the regression plots. On the left, method A is our candidate method, while on the right, method B is our candidate method. Regression fit is done using weighted least squares to both of these graphs. Comparing the two graphs, we can clearly see that the regression line has been placed differently into the data set. On the left, the linear fit goes through the two data points of lowest concentrations (S1 and S2). On the right, these dots are clearly below the regression line. A little bit up the concentrations, sample S5 seems to touch the regression line on the left. On the right there’s clearly some distance between the dot and the regression line. When we continue up the concentrations to the next data point that seems to touch the regression line, we find sample S16 that is below the regression line in both graphs. That means that on the left it’s assumed that the random error related to method A is negative in that data point. On the right it is assumed that the random error related to method B is also negative in that data point. As we are calculating bias between methods A and B, these two interpretations are clearly in conflict with each other.
Image 9 shows us the regression plots of this same data set with method A as the candidate method. In the middle we see the linear regression fit made by weighted Deming model. As Deming model takes into account the errors in both methods, it’s not surprising that the linear fit goes through sample S16 that was placed below the regression line on both graphs in Image 8. The regression line created by Passing-Bablok on the right is quite concordant with the results of the Deming model, but the confidence interval is significantly wider. If we are confident that %CV is constant and errors are normally distributed throughout the measuring range as WLS requires, weighted Deming model is a better choice than weighted least squares model.
So basically weighted least squares is never a reasonable choice unless considering the comparative measurement procedure as a reference happens to suit your purposes.
Deming models handle error from both methods
Deming models take a slightly more complicated approach to linear regression. They make a more realistic assumption that both measurement procedures contain error, making them applicable for data sets with less correlation. Otherwise the reasoning behind these models is quite similar as the reasoning behind ordinary linear regression and weighted least squares.
That’s why we can think that Deming models improve our bias estimations compared to OLR and WLS similarly as Bland-Altman comparison improves average bias estimation compared to direct comparison. Both methods are now assumed to contain error, but we are still effectively calculating mean values to estimate bias.
To handle situations where one of the measurement procedures gives more accurate results than the other, both Deming models use an estimate of the ratio of measurement procedure imprecisions to give more weight to the more reliable method. For similar measurement procedures, this ratio is often estimated as 1.
Otherwise the assumptions are similar as for ordinary least squares and weighted least squares. Deming models also assume symmetrical distribution. Random errors that cause variance on candidate and comparative measurement procedures are assumed to be independent and normally distributed with zero averages.
Variance related to random error is assumed to be constant on absolute scale (constant SD when using Deming regression) or on proportional scale (constant %CV when using Weighted Deming regression) for each method throughout the measuring range.
Deming regression models are very sensitive to outliers. If you use either one of the Deming models, you will need to make sure that all real outliers are removed from the analysis. In Validation Manager this is rather easy, as on the difference plot Validation Manager shows you which results are potential outliers. After considering whether a highlighted data point really is an outlier you can remove it from the analysis with one click of a button.
Passing-Bablok survives stranger data sets
Clinical data often contains results that are aberrant. The distribution may not be symmetrical. Often variability (neither SD nor %CV) of the results is not constant throughout the measuring range. Instead it is mixed, e.g. showing absolute variability at low concentrations and proportional variability at high concentrations.
In average bias estimation, we had to use median instead of mean for skewed data sets. Similarly in linear regression, we need to use Passing-Bablok model.
Passing-Bablok regression model does not make any assumptions about the distribution of the data points (samples nor errors). It basically calculates the slope by taking a median of all possible slopes of the lines connecting data point pairs. Correspondingly, the intercept is calculated by taking the median of possible intercepts. As a result, there are approximately as many data points above the linear fit as there are below it.
Passing-Bablok regression also has the benefit of not being sensitive to outliers. That basically means that it’s ok not to remove them from the analysis.
Image 11 shows one example data set that is too difficult for the other regression models to interpret. Mere visual examination of the regression lines created with weighted least squares (upper left corner) and weighted Deming model (lower left corner) raises doubt that these regression lines do not describe the behavior of the data set very well. Also with weighted Deming model the 95% CI lines are so far from the regression line that the only way to interpret this result is to conclude that weighted Deming model doesn’t tell us anything about this data set. Prediction made by constant Deming model (lower right corner) seems more convincing, but as the data set doesn’t really seem to have constant SD and normally distributed errors, use of Deming regression model is questionable. Passing-Bablok (upper right corner) is the model to use in this case, though it is advisable to measure more samples to get more confidence on the linearity of the data and to reach a narrower confidence interval.
The challenge with Passing-Bablok is similar as in using median as the estimated average bias. As a nonparametric technique it requires higher sample size than a parametric technique such as Deming model. E.g. in Image 6 and Image 9 this could be seen in the wider confidence interval of the Passing-Bablok regression fit. So in cases where the assumptions of a Deming model hold and your data set is small, the appropriate Deming model gives more accurate results than Passing-Bablok.
However, when examining medical data, one can quite safely use a rule of thumb that the assumptions of Deming models (nor OLR or WLS) are not met and Passing-Bablok is a better choise. And the less data you have the more difficult it is to know whether the variability really is constant and whether the errors follow normal distribution or not.
How much samples are needed?
If you gather your verification samples following the CLSI EP09-A3 guideline and your data is linear, it’s pretty safe to say that you have enough samples. In many cases though, laboratories perform some verifications with a smaller scope, as there’s no reason to suspect significant differences.
With smaller data amounts, it is good to check the regression plot and the bias plot to see how the confidence interval behaves. With Passing-Bablok, the confidence intervals tend to be quite large with small data amounts. (Though as we can see in Image 11, it depends on the data set whether Deming models perform any better.) In that case you basically have three options. You can add samples to your study to make a more accurate bias estimation. Optionally you can evaluate whether your data set would justify using either of the Deming models instead. But you might also find average bias estimation sufficient for your purposes.
The importance of having enough samples is visualized in Image 12. Here we have measured sets of 10 samples that all represent the same population. All the three data sets have been created with identical parameters. The only difference in between them comes from the random error that is normally distributed with %CV 15%. What this means is that the same set of samples could lead to any of these data sets. For each data set, the upper graph shows weighted Deming regresion fit, while the lower graph shows Passing-Bablok regression fit. For all the data sets, Passing-Bablok shows wider confidence intervals than weighted Deming does. The data set in the middle shows significantly smaller bias than the others. So with small data sets, it depends on your luck how well the calculated bias really describes the methods.
When we combine these three data sets into a single verification data set of 30 samples, we get the regression plot shown in Image 13. The difference in the confidence intervals is big compared to Image 12 where none of the data points were left outside the area drawn by Passing-Bablok confidence intervals.
To really get an idea about the behavior of bias we recommend you to set goals for bias. After that the bias plot in Validation Manager shows a green area on concentration levels where your calculated bias is within your goals. The orange background shows where the goals were not met.
Image 13 shows bias plot for the same data set on proportional scale and on absolute scale. In our example data set, we have a goal of bias < 2 mg/dl for concentrations smaller than 30 mg/dl, and bias < 7% for concentrations larger than 30 mg/dl. If our medically relevant areas happen to be below 27 mg/dl and above 36 mg/dl, bias calculated from the regression line seems ok. But in this case the upper limit of our confidence interval is way above the goals which should worry us and give a reason for further investigation.
In addition to the amount of data, it’s also worthwhile to consider how well the data covers the measuring interval. Whichever regression model you use, all relevant concentration areas should show multiple data points. Otherwise mere chance can have a significant effect on the results.
Image 14 shows regression plots for three data sets of 19 samples that all represent the same population. All the three data sets have been created with identical parameters. The only difference in between them comes from the random error that is normally distributed with %CV 15%. For each data set, the upper graph shows weighted Deming regresion fit, while the lower graph shows Passing-Bablok regression fit. The data set on the left looks like there’s maybe no bias at all, whle the other data sets clearly show bias. As most of the data is on low concentrations, it’s difficult to say how the bias behaves at large concentrations. That’s why it is recommended to measure more samples of high concentrations to cover the measuring range more evenly.
What if it’s not linear?
There may be cases when no straight line would describe the behavior of the data set. Small peculiarities in the shape of the distribution may be ignored if it seems that their effect is within acceptable limits. But if the distribution is somehow strongly curved, a linear fit will not give meaningful information about the bias.
In these cases you can divide the measuring interval into ranges that will be analysed separately. This is done on Validation Manager report, where you can add new ranges. You can set the limits of the range by looking at the difference plot and regression plot to make sure that your range selection will be appropriate, as shown in Image 15. This way it is possible to get separately calculated results e.g. for low concentrations and high concentrations.
Another question, of course, is whether the nonlinearity between the methods indicates a bigger problem with the candidate method. Bias is supposed to be calculated within the range where the methods give quantitative values. Usually a method is supposed to behave linearly throughout this range. If both methods should be linear throughout the whole data set, nonlinearity between them may give reason to suspect that one of the methods is not linear after all. To investigate this, you can perform a linearity study.
If you want to dig deeper into regression models, here are some references for you to go through:
CLSI Approved Guideline EP09-A3 – Measurement Procedure Comparison and Bias Estimation Using Patient Samples
Deming regression in Wikipedia
W.E. Deming (1943), “Statistical adjustment of data”. Wiley, NY (Dover Publications edition, 1985). ISBN 0-486-64685-8
K. Linnet (1990), “Estimation of the linear relationship between the measurements of two methods with proportional errors”, Stat Med. Vol 9, Issue 12 p 1463-1473.
Passing-Pablok regression model in Wikipedia
H. Passing, W. Bablok (1983), “A new biometrical procedure for testing the equality of measurements from two different analytical methods. Application of linear regression procedures for method comparison studies in clinical chemistry, Part I.” J. Clin. Chem. Clin. Biochem. Vol 21, Issue 11 p 709-720.
W. Bablok, H. Passing, R. Bender, B. Schneider (1988),”A general regression procedure for method transformation. Application of linear regression procedures for method comparison studies in clinical chemistry, Part III” J. Clin. Chem. Clin. Biochem. Vol 26, Issue 11 p 783-790