Department of Biostatistics, Erasmus MC-University Medical Center, PO Box 2040, 3000, CA, Rotterdam, The Netherlands

Section of Allergology, Department of Internal Medicine (GK 324), Erasmus MC-University Medical Center, PO Box 2040, 3000, CA, Rotterdam, The Netherland

Department of General Practice, Erasmus MC-University Medical Center, PO Box 2040, 3000, CA, Rotterdam, The Netherlands

Cell Biology and Immunology Group, Wageningen University, PO Box 338, 6700, AH, Wageningen, The Netherlands

Abstract

Background

Immunological parameters are hard to measure. A well-known problem is the occurrence of values below the detection limit, the non-detects. Non-detects are a nuisance, because classical statistical analyses, like ANOVA and regression, cannot be applied. The more advanced statistical techniques currently available for the analysis of datasets with non-detects can only be used if a small percentage of the data are non-detects.

Methods and results

Quantile regression, a generalization of percentiles to regression models, models the median or higher percentiles and tolerates very high numbers of non-detects. We present a non-technical introduction and illustrate it with an implementation to real data from a clinical trial. We show that by using quantile regression, groups can be compared and that meaningful linear trends can be computed, even if more than half of the data consists of non-detects.

Conclusion

Quantile regression is a valuable addition to the statistical methods that can be used for the analysis of immunological datasets with non-detects.

Background

Immunological parameters are hard to measure. A well-known problem

Non-detects (NDs) are a nuisance in statistical analysis. An ad-hoc solution is to fill in values for the NDs, e.g. one half of the detection limit. This may be acceptable if only a few per cent of the observations are NDs. If there are many of them, estimated values of means, standard errors and trend lines will be unreliable and conclusions may be wrong.

NDs occur in many places in science and technology. They have received a lot of attention in the work of Helsel

Most statistical methods develop a model for the expected values of the observations. In an analysis of variance (ANOVA) these will be the mean values for different groups. In the case of the regression line

Regression and ANOVA (which is a special case of regression), use the so-called principle of least squares: parameters like

In this paper we propose to use quantile regression instead of the usual linear regression models. A simple example is provided by ANOVA. Instead of computing means per groups, one could compute the medians, also known as P50, the 50^{th} percentile. A familiar recipe for computing the median of a set of numbers is to sort them from low to high and pick the middle number in the sorted list. Half of the data will be below this number and the other half will be above it. The key point is that the actual values of the lowest observations play no role: what matters is that they are lower than the median. So if we would have 30% NDs and gave them small values, the computed median would still be the same.

If more than 50% of the observations are NDs, but less than 75%, we are still able to compute the P75, the number below which 75% of the data are found. In ANOVA we can still compare P75 in the different groups and look for interesting differences.

For a regression line, the sorting recipe will not work. However, in the last two decades a very useful generalization of regression modelling has become available, quantile regression. With this method we can estimate regression lines, which allow us to compute for

The outline of the paper is as follows. First, we introduce quantile regression. We have tried to limit the amount of technical material, keeping in mind the expected statistical level of our audience. We also show in this section how the required computations can be done relatively easily with the R system and the package

Methods

Quantile regression

In this paper the words quantiles and percentiles will be used repeatedly. To avoid confusion we first give their precise meaning. The 90-th percentile is the number below which 90% of the data lie. It is also the 0.9 quantile. So, when we use percentages we talk of percentiles, and when we use fractions we talk of quantiles.

In the Introduction we described the familiar sorting algorithm for computing percentiles. It has a strong intuitive appeal, and it is easy to implement, or even to do by hand. However, it cannot be generalized to the case of a regression line or more complicated models. Fortunately there exists another, more flexible approach, based on optimisation.

The mean of n observations, y_{1} to y_{n}, is computed as

is minimized when μ =

For percentiles we can also introduce a function that has to be minimized, in such way that the desired percentile minimizes it. Compared to the sum of squares, two changes are needed:

1) replace the squares by absolute values, and 2) give different weights to positive and negative residuals. The residuals are the differences between the observations and the percentile that is being computed. As a formula: minimize

Here _{i}(

In the case of the 0.9-quantile, the positive residuals get a nine times larger weight (i.e. 0.9) than the negative ones (i.e. 0.1). It might not be directly obvious why this procedure leads to the desired quantile, but after some mathematical adjustments one finds indeed that 90% of the observations have to be below q_{0.9} to minimize _{1}. An intuitive explanation is that every observation above the quantile has to be balanced by nine below it.

Now that we have an optimisation criterion, it is very easy to extend the quantile idea to more complicated models. In the case of a linear regression line, the function to be minimized is

It will be clear that we can generalize this to more complicated models. Notice that generally the values of

It is easy to state the function that has to be minimized, but computing the solution is harder than for classical models (based on least squares). Fortunately, there is excellent open-source software available, free of charge. We did our computations using the quantreg package for the statistical software system R

With quantile regression it is not possible to get

Although it is not an issue here, quantile regression is very robust against outliers, in contrast to the mean and least squares regression. Also a normal distribution of errors is not assumed.

For those interested in statistical backgrounds of quantile regression, we can recommend a paper by Koenker and Portnoy

Results

An implementation

To illustrate the use of quantile regression in immunology, we use data from the STARDROP-study, a randomized placebo-controlled trial in 204 youngsters (6–18 years) with hay fever. A detailed description can be found in Röder et al

Out of the 203 youngsters included in this sub-study, 103 subjects were observed all 5 times and 74 only once. The 26 remaining subjects were observed 2 to 4 times.

We start by presenting histograms of the measurements, emphasizing the need for a statistical method that can handle a large proportion of NDs. Then we compute trends with age using quantile regression.

Distributions and non-detects

The samples were analysed in two parts, because an interim-analysis had to be presented to our sponsor. As a consequence, two different assays with different detection limits were used.

Initially, for the time points T0, T1 and T2, the production of the SBMs was detected with Enzyme-Linked Immunosorbent Assay (ELISA). The sensitivity limits for quantitative determinations were 1.19 pg/ml (IFN-γ), 1.15 pg/ml (IL-10), 7.85 pg/ml (IL-12), 5.21 pg/ml (IL-13), 8.81 pg/ml (TNF-α), 13.40 pg/ml (sIL-2R), 0.11 ng/ml (sE-selectin), and 1.43 ng/ml (sICAM-1). For the later time points T3 and T4, the SBM production was measured with Cytometric Bead Assay Flex sets (CBA). The sensitivity limits for quantitative determinations were 0.3 pg/ml (IFN-γ), 2.3 pg/ml (IL-10), 2.2 pg/ml (IL-12), 1.6 pg/ml (IL-13), 0.7 pg/ml (TNF-α), 12.5 pg/ml (sIL-2R), 5 pg/ml (sE-selectin), and 0.23 ng/ml (sICAM-1). The measurements above the detection limits were not affected by the change in assays.

We chose to apply the detection limits of one method to all data. The detection limits of the first method (ELISA) were used because these limits were higher than those of the second method (CBA) for all SBMs except IL-10. For IL-10 the detection limits of both methods were used. All values below the detection limit were replaced with the value between 0 and the detection limit.

For the analysis and presentation of the data in this implementation we use the logarithms (to base 10) of concentrations. The highest concentrations measured were around 5000 pg/ml, the lowest were always at the detection limit. Because of the enormous range of the concentrations, the highest ones being more than a 1000 times higher than the lowest, we work exclusively on the logarithmic scale.

Figure

Histograms of (logarithms) of SBM concentrations, all five time points combined

**Histograms of (logarithms) of SBM concentrations, all five time points combined.**

Histograms of (logarithms) of IL-10 concentrations, for each of the five time points

**Histograms of (logarithms) of IL-10 concentrations, for each of the five time points.** Legend: T0, baseline. T1-2-3-4, follow-up after 6-12-18-24 months respectively.

Summarizing, due to changes in detection limits and the presence of a substantial number of NDs for some SBMs, classical statistical analyses, like ANOVA and regression, could not be applied to this dataset.

Trends and quantile regression

One of the research questions was to determine whether concentrations of SBMs change with age. We use quantile regression for P75, the 75^{th} percentile. In general any quantile level can be modelled and it would be more attractive to model P50, the median. The percentage of non-detects for some SBMs, however, was more than 50% and therefore we have to settle for a higher percentile. We chose P75, and we are aware that this arbitrary. Any percentile can be chosen, as long as it results in a quantile regression line that is above the values of non-detects everywhere.

Before a quantile regression line on age only was calculated, the influence of the variable “time point” was explored. Figure

The five observation time points of IL-10

**The five observation time points of IL-10.** Legend: The data points have been "jittered" (a random horizontal shift between −0.2 and 0.2) for better visibility. For each time point the midpoint (coefficient) for the P75 with 95% confidence interval are shown. The intervals are not symmetric around the midpoint. The line represents the P75 quantile regression line for the logarithm of the concentration.T0, baseline. T1-2-3-4, follow-up after 6-12-18-24 months respectively.

Trend for the 75th percentile (P75) of IL-10 with age

**Trend for the 75th percentile (P75) of IL-10 with age.** Legend: The data points have been "jittered" (a random horizontal shift between −0.2 and 0.2) for better visibility. The full line is obtained by quantile regression on age only, the broken line with the observation time point as an additional factor.

Discussion

Immunological datasets often contain many non-detects. When a signal produced by the stimulant is too small for the instrumentation to discriminate the signal from the background noise, a value cannot be determined precisely. Values below a given detection limit are called non-detects (NDs). The presence of NDs will cause the data to be left-censored and special attention should be paid to selecting the appropriate statistical method to analyse such a censored dataset.

Several statistical methods are being used to deal with NDs. Uh et al. evaluated the performance of several commonly used methods in immunology and more advanced methods used in other fields such as environmetrics and econometrics via simulation studies

We illustrated quantile regression with data from a clinical trial in youngsters with hay fever, in which the effect of immunotherapy treatment and other factors on the immune system was evaluated by measuring levels of soluble biological markers (SBMs). We showed that groups can be compared and that meaningful linear trends can be computed, even with very large fractions of NDs. The slope of the regression line for a percentile is the same as that for the mean in the case of a linear relationship plus errors with a constant variance, the common default assumption in linear regression. That means that the estimated slope for the P75 is also a very good estimate for the usual regression slope that would be obtained if NDs did not occur.

We have not discussed efficiency. It is true that quantile regression uses less information, that is, only the signs of residuals, disregarding their size, leading to wider confidence intervals and consequently loss of power. This means that if data are complete (no NDs), estimated classical regression coefficients have more narrow confidence intervals than those obtained from quantile regression. But this knowledge does not help us much if we have many NDs. When analysing our data set, we chose one percentile level, 75% for all variables. In principle it could vary with the fraction of NDs, so that for some variables P50 could have been chosen. However, we felt that this would have made the interpretation more complicated.

The data have been analysed as 662 independent observations, which is a limitation, as 558 observations represent multiple observations on 129 participants. In the world of standard least squares statistical methods, one would use repeated measure ANOVA or a mixed model for a proper analysis. Unfortunately similar technology is not yet developed enough for quantile regression, although research is ongoing. NDs can, however, generate unpleasant complications when using mixed models. It might happen that all or most measurements of some of the subjects are NDs. Consequently mixed models, which rely on fitting (restricted) individual coefficients to subjects, might be difficult to use. As far as we know, no statistical technology is yet available to handle mixed models with NDs.

An alternative approach is reducing the problem to logistic regression, after setting proper thresholds (with a different value for each variable). Choosing the thresholds can be a matter of debate, which is avoided in quantile regression. In fact the quantile regression line acts as a “moving threshold” in such a way that on average (in the case of P75) a quarter of the data lies above it. Nevertheless, thresholding an logistic regression could be an interesting venue for longitudinal data modelling, because mixed model technology for binary responses is available.

In our application trends are so weak that there is no need for anything more complex than a straight line. But we remark that the

Conclusions

Quantile regression is a valuable addition to the statistical methods that can be used for the analysis of immunological datasets with non-detects.

Abbreviations

ANOVA, Analysis of variance; CBA, Cytometric Bead Assay; ELISA, Enzyme-Linked Immunosorbent Assay; ND, Non-detect; SBM, Soluble biological marker; SLIT, Sublingual immunotherapy.

Competing interests

The author(s) declare that they have no competing interests.

Authors’ contributions

PE performed the statistical analysis. HS was responsible for the soluble biological marker measurements. PE and ER drafted the manuscript. All authors interpreted the data and critically reviewed, edited and approved the written manuscript.

Acknowledgements

The STARDROP-study was funded by Artu Biologicals, Lelystad, The Netherlands and financially supported by grants (2003/037 and 2004/011) from the Netherlands Foundation for the Prevention of Asthma (Stichting Astma Bestrijding, SAB).