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Antibody Titres and Two Types of Bias

  • Jozef Nauta
Chapter
Part of the Springer Series in Pharmaceutical Statistics book series (SSPS)

Abstract

In this chapter, two types of possible bias for antibody titres are explored. The first type of bias is due to how antibody titres are defined. An alternative definition is presented, the mid-value definition. With this definition, the bias is properly corrected. The second type of bias occurs when antibody titres above (or below) a certain level are not determined. If this bias is ignored, the geometric mean titre will be underestimated. It is demonstrated how the method of maximum likelihood estimation for censored observation can be applied to eliminate this bias.

4.1 Standard Antibody Titres Versus Mid-Value Titres

Statisticians have pointed out that when antibody titres are determined using the standard definition—the reciprocal of the highest dilution at which the assay read-out did occur (see Sect.  2.1.2)—, the true titre value is underestimated [21]. It is easy to see why. By definition, the true antibody titre \(\tau \) lies between the standard titre \(t_s\) and the reciprocal of the next dilution, \(r_n\):
$$\begin{aligned} t_s \le \tau < r_n. \end{aligned}$$
Thus, for most serum samples the standard antibody titre will be lower than the true titre. This means that if standard antibody titres are used, the geometric mean titre will underestimate the geometric mean of the distribution underlying the antibody values.

There are antibody assays that try to correct for this bias. An example is the interpolated serum bactericidal assay (SBA) to demonstrate humoral immune responses induced by meningococcal vaccines. Meningococcal disease is caused by the bacterium Neisseria meningitidis, also known as meningococcus bacteria. Attack rates of the disease are highest among infants aged younger than two years and adolescents between 11 and 19 years of age. The disease can cause substantial mortality. Five serogroups, A, B, C, Y and W135, are responsible for virtually all cases of the disease. The standard SBA titre is defined as the reciprocal of the highest dilution of serum immediately preceding the 50% survival/kill value for colony-forming units (50% cut-off). The interpolated SBA titre is calculated using a formula that determines the percentage kill in dilutions on either side of the 50% cut-off. The titre is the reciprocal of the dilution of serum at the point where the antibody curve intersects the 50% cut-off line.

Another approach to correct for the bias is changing the definition of the antibody titre to
$$\begin{aligned} \text {antibody titre} = \sqrt{t_s \times r_n}, \end{aligned}$$
the geometric mean of the standard antibody titre and the reciprocal of the next dilution. For titres based on serial twofold dilutions \(r_n\) equals \(2 \times t_s\), in which case the definition of the antibody titre becomes
$$\begin{aligned} \text {antibody titre} = \sqrt{2} \times t_s. \end{aligned}$$
This definition is called the mid-value definition of antibody titres [22]. Mid-value antibody titres are higher than standard titres. For example, if the predefined dilutions are 1:4, 1:8, 1:16, etc., and the standard antibody titre for a serum sample is 64, then the mid-value titre for the sample is the geometric mean of 64 and 128
$$\begin{aligned} \sqrt{64\times 128} = 90.5 = \sqrt{2} \times 64. \end{aligned}$$
On a logarithmic scale, the mid-value antibody titre \(t_{mv}\) is the mid-point between \(t_s\) and \(r_n\):
$$\begin{aligned} \log t_{mv} = (\log t_s + \log r_n)/2. \end{aligned}$$
Hence the name mid-value antibody titre.
In almost all practical situations the mid-value definition reduces the bias of standard antibody titres, meaning that on average the mid-value titre is closer to the true titre than the standard titre, i.e.
$$\begin{aligned} |t_{mv} - \tau | \le |t_s - \tau |. \end{aligned}$$
A sufficient set of conditions for the mid-value definition to reduce the bias is:
  1. 1.

    The log-transformed antibody titres are normally distributed.

     
  2. 2.

    The dilutions are predefined.

     
  3. 3.

    The distance between two consecutive log-transformed dilution factors is small compared to the range of observed log-transformed titre values.

     

For post-vaccination titres, this is usually the case. (For pre-vaccination titres, often condition 1 or 3 is not met.)

In case of serial twofold dilutions, when the antibody titres are determined using the standard definition and the primary outcome measure is the geometric mean titre, there is no need to calculate the individual mid-value titres. Let \(GMT_s\) be the geometric mean of a single group of standard antibody titres. Then the geometric mean of the corresponding mid-value titres is
$$\begin{aligned} GMT_{mv} = \sqrt{2} \times GMT_s. \end{aligned}$$
A similar expression holds for the confidence limits for \(GMT_{mv}\).
For the geometric mean fold increase, which can be expressed as the ratio of a post- and a pre-vaccination geometric mean titre (see Sect.  3.3.1), the bias correction is also not needed, because
$$\begin{aligned} gMFI_{mv}&= \frac{GMT_{mv}\ \mathrm{post}}{GMT_{mv}\ \mathrm{pre}} = \frac{\sqrt{2}GMT_{s}\ \mathrm{post}}{\sqrt{2}GMT_{s}\ \mathrm{pre}} \\&= \frac{GMT_{s}\ \mathrm{post}}{GMT_{s}\ \mathrm{pre}} = gMFI_s. \end{aligned}$$
When the summary statistic of interest is the geometric mean fold increase, both definition, the standard and the mid-value one, will produce the same result. The same holds true for the geometric mean ratio, the ratio of two independent geometric mean titres:
$$\begin{aligned} GMR_{mv}&= \frac{GMT_{mv\ 1}}{GMT_{mv\ 0}} = \frac{\sqrt{2} GMT_{s\ 1}}{\sqrt{2} GMT_{s\ 0}} \\&= \frac{GMT_{s\ 1}}{GMT_{s\ 0}} = { GMR}_s. \end{aligned}$$
Both definitions produce the same result.

4.2 Censored Antibody Titres and Maximum Likelihood Estimation

If the number of dilutions in an antibody assay is limited, then it may happen that at the highest tested dilution the assay read-out did not occur. In that case, often the titre is set to the reciprocal of the highest dilution. The result of this practice is bias. The geometric mean titre will be underestimated, and the assumption of normality for the distribution of the log-transformed titres may not hold.

Example 4.1

In Fig. 4.1, the histogram of the frequency distribution of log-transformed post-vaccination measles HI antibody titres of a hypothetical group of 300 children is displayed. The starting dilution was 1:4, and as log transformation the standard transformation (\(\log _2\)(titre/2)) was used. The arithmetic mean of the log-transformed titres is 5.80, with estimated standard deviation 2.49. This arithmetic mean corresponds to a geometric mean titre of
$$\begin{aligned} 2\times 2^{5.80} = 111.4. \end{aligned}$$
The median and the maximum of the titres are 128 and 16,384, respectively. Next, assume that the highest tested dilution would have been 1:512 rather than a much higher dilution, and that titres above 512 would have been set to 512. The result of this censoring is also shown in Fig. 4.1. The bars above 9, 10, 11, 12 and 13 are added to the bar above 8. Due to the censoring, the frequency distribution is no longer symmetrical and thus no longer normally shaped. The arithmetic mean of the censored log-transformed titres is 5.53 (\(GMT =\) 92.4), with estimated standard deviation 2.05. Both estimates are smaller than the estimates based on the uncensored data.
In the following sections, it is explained how this bias due to censoring can be eliminated [23].
Fig. 4.1

Uncensored and censored frequency distribution of log-transformed measles HI antibody titres

4.2.1 ML Estimation for Censored Normal Data

A censored observation is an observation for which a lower or an upper limit is known but not the exact value. An example of a censored observation is a value below the detection limit of a laboratory test. The upper limit of the test result is known, the detection limit, but not the test result itself. If for an observation only an upper limit is known, the observation is called left-censored. Observation for which only a lower limit is known is called right-censored. Censored observations are often assigned the value of the limit. If this is not taken into account in the statistical analysis, estimates will be biased. A powerful statistical method to eliminate this bias is maximum likelihood (ML) estimation for censored data. As an introduction to ML estimation for censored antibody titres, in this section ML estimation for censored normal data is discussed.

Let \(x_1,\ldots , x_n\) be a group of non-censored \(N(\mu ,\sigma ^2)\) distributed observations. The log-likelihood function is
$$\begin{aligned} LL(\mu ,\sigma ) = \sum _{i=1}^n \log {f(x_i;\mu ,\sigma )}, \end{aligned}$$
where \(f(x;\mu ,\sigma )\) is the normal density function. The ML estimates of \(\mu \) and \(\sigma \) are those values that maximize the log-likelihood function. For normal data, the ML estimates are the arithmetic mean and the estimated standard deviation (but with the \((n-1)\) in the denominator replaced by n).
Next, assume that r of the observations are right-censored. Let \(x_1,\ldots , x_{n-r}\) be the non-censored observations and \(x_{n-r+1}, \ldots , x_n\) the right-censored observations. Then the log-likelihood function becomes
$$\begin{aligned} LL(\mu ,\sigma ) = \sum _{i=1}^{n-r} \log {f(x_i;\mu ,\sigma )} + \sum _{i=n-r+1}^n \log {[1-F(x_i;\mu ,\sigma )]}, \end{aligned}$$
where \(F(x;\mu ,\sigma )\) is the normal distribution function. Thus, for a censored observation x, the density for x is replaced with the probability of an observation beyond x. Again, the ML estimates of \(\mu \) and \(\sigma \) are those values that maximize the log-likelihood function, and they are unbiased estimates of these parameters.
Finally, assume that there are l left-censored observations: \(x_1,\ldots , x_l\). The log-likelihood function for normal data with both left- and right-censored observations is
$$\begin{aligned} LL(\mu ,\sigma )&= \sum _{i=1}^l \log {F(x_i;\mu ,\sigma )} + \sum _{i=l+1}^{n-r} \log {f(x_i;\mu ,\sigma )} \\&\quad +\sum _{i=n-r+1}^n \log {[1-F(x_i;\mu ,\sigma )]}. \end{aligned}$$
Maximum likelihood estimation for censored normal data can be intuitively understood as follows. For a series of values for \(\mu \) and \(\sigma \) a normal curve is fitted to the histogram of the frequency distribution of the observation, and it is checked how well the curve fits to the data. This includes a comparison of the areas under the left and the right tail of the fitted curve with the areas of the histogram bars below or above the censored values. The censored tails are reconstructed, and correct estimates of the mean and standard deviation of the distribution are obtained. The ML estimates of \(\mu \) and \(\sigma \) are those values that give the best fit, and they are found by iteration.

Maximum likelihood estimation for censored observations was introduced for the analysis of survival data, where it is used to handle censored survival times [24].

4.2.2 ML Estimation for Censored Antibody Titres

To obtain ML estimates for censored, log-transformed antibody titres, SAS procedure PROC LIFEREG can be used. This is a procedure to fit probability distributions to data sets with censored observations. A wide variety of distributions can be fitted, including the normal distribution.

Before the log-likelihood function for censored observations can be applied to log-transformed antibody titres, a modification is needed. In the previous section, it was assumed that the data were censored normal observations. Log-transformed antibody titres, however, are not continuous observations; they are so-called interval-censored observations. An interval-censored observation is an observation for which the lower and the upper value is known but not the exact value. If this is not taken into account, i.e. if the values are treated as if continuous, the ML estimates PROC LIFEREG returns will be invalid.

In Sect. 4.1, it was explained that the true titre value \(\tau \) lies between the standard titre \(t_s\) and the reciprocal \(r_n\) of the next dilution:
$$\begin{aligned} t_s \le \tau < r_n. \end{aligned}$$
Thus, let \(t_i\) be an interval-censored standard titre, with \(r_i\) the reciprocal of the next dilution. The second term of the log-likelihood function, the term for the non-censored observations becomes
$$\begin{aligned} \sum _{i=l+1}^{n-r} \log {[F(\log {r_i};\mu ,\sigma ) - F(\log {t_i};\mu ,\sigma )]}. \end{aligned}$$
The first term of the log-likelihood function, the term for the left-censored observations, becomes
$$\begin{aligned} \sum _{i=1}^{l} \log {F(\log {r_L};\mu ,\sigma )}, \end{aligned}$$
where \(r_L\) is the reciprocal of the starting dilation. The third term of the log-likelihood function, the term for the right-censored observations, becomes
$$\begin{aligned} \sum _{i=n-r+1}^{n} \log {[1-F(\log {r_H};\mu ,\sigma )]}, \end{aligned}$$
where \(r_H\) is the reciprocal of the highest dilution tested.
In PROC LIFEREG this modification can be handled by the MODEL statement with the LOWER and UPPER syntax; Lower and Upper are two variables containing the lower and the upper ranges for the observations. For an interval-censored standard titre
$$\begin{aligned} \texttt {Lower}\, = \log t_i \end{aligned}$$
and
$$\begin{aligned} \texttt {Upper}\, = \log r_i \end{aligned}$$
For left-censored standard titres Lower has to be set to missing (interpreted by PROC LIFEREG as minus infinity) and Upper to \(\log r_L\); for right-censored standard titres Lower \(= \log r_H\) and Upper has to be set to missing (interpreted as plus infinity). By definition, all observations below the detection limit are left-censored.

Example 4.2

(continued) The starting dilution was 1:4, and thus antibody titres less or equal to 4 are to be considered as left-censored. Below a SAS code to fit a normal distribution to the censored log-transformed antibody titres in Fig. 4.1 is given.

Two parameters are estimated, an intercept, which is the ML estimate of \(\mu \), and a scale parameter, which is the ML estimate of \(\sigma \). For both parameters, two-sided 95% confidence limits are given. Note that the \(\mu \) estimated is the mean of the distribution underlying the log-transformed mid-value titres (Sect. 4.1), and not the mean of the distribution underlying the log-transformed standard titres. This is due to the values assigned to the SAS variables Lower and Upper, which are consistent with the mid-value definition. To estimate the \(\mu \) consistent with the definition of standard titres, in the above SAS code Lower should be set to Logtitre0.5 and Upper to Logtitre \(+\) 0.5. Then SAS Output 4.1B is obtained.

SAS Code 4.1  Fitting a normal distribution to the censored antibody titres of Fig. 4.1

SAS Output 4.1A

The ML estimate of \(\mu \) is now consistent with the standard titre definition. (This value could have of course also been obtained by subtracting 0.5 from the ML estimates in SAS Output 4.1A: \(6.227-0.5 = 5.727\).) Note that the correction does not have an effect on the ML estimate of \(\sigma \).

SAS Output 4.1B

The ML estimates 5.73 and 2.39 are in good agreement with the estimates based on the uncensored data, 5.80 and 2.49. This demonstrates the powerful tool ML estimation for censored observations is.

Above as log transformation, the standard transformation \(\log t = \log _2 [t/(D/2)]\) was used, with D the starting dilution factor. A general SAS code to fit a normal distribution to the censored \(\log _e\) transformed serial twofold antibody titres is presented below.

SAS Code 4.2  Fitting a normal distribution to censored serial twofold antibody titres

The approach discussed above can be readily extended to the case of two vaccine groups. Let Group be the SAS variable for the groups, taking the value 1 for the experimental group and 0 for the control group. Then the MODEL statement in SAS Code 4.1 must be changed to

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Jozef Nauta
    • 1
  1. 1.AmsterdamThe Netherlands

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