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The effect of simple imputation on inferences about population means when data are missing in biomedical research due to detection limits.

1. Introduction

Detection limit is a long standing problem in experimental sciences. It refers to the limited ability of an instrument in measuring an outcome of interest in a certain range (typically small values close to 0). Many instruments cannot return meaningful measurements if signals fall below a certain threshold value. This problem is especially prevalent in biomedical sciences as signals are sometimes too weak to be detected in the presence of ambient noise. Although detection limits are often due to limitations of physical devices, the problem may also arise in psychosocial research with assessments based on instruments (questionnaires). For example, in alcohol and substance use research, alcohol or drug use may not be detected in a subject if the blood level is not sufficiently high. Also, if answers for all or most subjects to an item in a questionnaire fall below (or above) a certain score in the potential range of scores, the lack of variability in the outcome may prevent any useful analysis of the data.

Detection limit presents problems for statistical analysis since no data (or very little data) is observed in part of the potential range of the variable. It is not possible to gauge the variability of the outcome below the detection limit, but this information is needed to conduct standard statistical inference on the data in the whole range (for example, to estimate the geometric mean of the population from which the sample is drawn). A commonly used ad-hoc method to deal with this problem is to impute data below the detection limit and then apply standard statistical methods. [1] This practice is especially prevalent in biomedical research. Geometric means are the most popular method of imputing values below the detection limit because data are often log-transformed to reduce skewness before being analyzed (even though the log-transformation may not actually reduce the skewness [2]). The arithmetic mean of the log-transformed outcome is the logarithm of the sample geometric mean.

Although imputation seems natural and intuitive, it has significant implications for statistical inference and, thus, on the reported results of research. [3,4] In this report we discuss the pitfalls of using this common method of imputation in research and in clinical practice.

2. Geometric mean of a non-negative random variable

The so-called 'geometric mean' used in biomedical and psychosocial research is actually the sample geometric mean, that is, the geometric mean of a sample of observations from an underlying distribution.

For example, let [a.sub.i], i=1, ..., n, be a sequence of non-negative numbers; then the geometric mean of the sequence is [nth root of [a.sub.i][a.sub.2] ... [a.sub.n]]. This is what is commonly referred to as the 'geometric mean', but it is important to keep in mind that the sample and population means are completely different concepts. The former is computable based on the sample, while the latter is an unknown quantity, or a parameter in the nomenclature of statistics. The geometric mean described above is actually a sample geometric mean because it is a computable quantity. Unlike the arithmetic, the population geometric mean had never been clearly defined in the literature until the recent work of Feng and colleagues [2,3] who presented a formal definition of this elusive quantity. The population geometric mean of a non-negative variable X defined if either X has non-zero probability at zero (that is, X may equal 0) or X is positive with [absolute value of log X] having a finite mean value, that is, E[absolute value of log X]<[infinity]. Subsequently the definition was further broadened to only require that E[absolute value of log X] exists (which includes E[absolute value of log X]<[infinity] as a special case) and the properties of the population geometric mean were elaborated. [4] This work lays a conceptual foundation to interpret the sample geometric mean (as an estimate of the underlying population geometric mean) and clarifies some ambiguities in using the geometric mean in biomedical research. A brief summary of this work is shown in Box 1.
BOX 1. Relationship of population and sample geometric means

* Suppose F(x) is the probability distribution function of a
non-negative random variable X. If F(0) > 0, then the geometric
mean of X, denoted by [GM.sub.x], is defined as 0. If X is positive
and the expected value of [absolute value of log X] exists, then we
define [GM.sub.x] as exp(E log X). If F(0)[not equal to]0 and the
expected value of [absolute value of log X] does not exist, then it
is not possible to define the geometric mean of X. Hence, we cannot
define the geometric mean for all non-negative random variables,
just like the mean cannot be defined for all random variables.
However, if [GM.sub.x] is well defined, it is the population
geometric mean, which can be used to interpret the sample geometric
mean of X.

* Suppose the geometric mean of X exists, and, [X.sub.i], i=1, ...,n
is a random sample from the distribution of X. Let
[[bar.GM].sub.n] = [nth root of [X.sub.1 ... [X.sub.n] be the
sample geometric mean. Then the sample geometric mean is strongly
consistent [3,4] and, thus, is a consistent estimate of the
population geometric mean.

The geometric mean has a very unusual property. We know that for a positive random variable, arithmetic mean, if it exists, is always positive. However, for some positive random variables, geometric means can be zero. This fact is counterintuitive as the sample geometric mean obtained from a positive random variable is always positive. This unusual property can have significant implications for inference about the population geometric mean when the data in the sample is left-censored due to a detection limit.

Another issue is the relationship between the geometric mean and the arithmetic mean for a positive random variable. In biomedical research, data is often right-skewed with most values close to the lower limit. A popular approach for dealing with this is to log-transform the data, analyze the transformed data, and then transform the result back to the original scale. For a non-negative random variable X, in general there is no connection between the geometric mean [GM.sub.x] and the arithmetic mean E(X), even if they both exist. For example, for two log-normally distributed random variables, if they have the same log-mean values but different log-variances, then their geometric means are equal but their arithmetic means are not equal. This means that we cannot test the hypothesis that they have the same (arithmetic) mean values by testing that the log-transformed data have the same (arithmetic) mean values. This fact is not well appreciated in biomedical research. [2]

3. Geometric mean with detection limit

In this section, we discuss the effect of the naive imputation method on the geometric mean in the presence of a detection limit. Let X be a positive random variable and [delta] be the lower detection limit; X is unobservable (missing) if X < [delta]. A common approach in biomedical research is to define a modified version of X by:


where [eta] is some positive constant. Usually, [eta]= [delta]/2, or a small positive constant less than [delta]. [1,5-15] After the imputation, inference about the population geometric mean of the original data proceeds by treating the imputed data as if they were observed.

To discuss potential effects of this naive imputation on inference about the population geometric mean, we assume that X is a positive random variable, which is the case in most real-study applications.

a) Case 1: [GM.sub.x]>0. The geometric mean of [X.sup.*] can be greater than, less than, or equal to [GM.sub.x] depending on the distribution of X below the detection limit. If the detection limit [delta] is small enough, then with relatively large sample sizes inferences based on the imputed data (such as confidence intervals, the two-sample t-test, and the paired t-test) yield valid results for the original data. However, if [delta] is large, imputation may yield invalid results.

b) Case 2: [GM.sub.x]=0. With the imputation, the geometric mean of [X.sup.*] depends on how the imputed value [eta] is selected and is always greater than [eta]. This means that the estimated geometric mean based on the imputed data may be very far away from the theoretical geometric mean of zero. Another effect is that the imputation brings some arbitrariness into the statistical inference.

Thus whether [GM.sub.x]>0 or [GM.sub.x]=0, imputation has significant implications for inference about the population geometric mean. If [GM.sub.x]>0, inference using common statistical methods is reasonably robust if the detection limit is small; but if [GM.sub.x]=0, any analysis of the geometric mean based on the imputed data is invalid and the result is uninterpretable. Unfortunately, the detection limit makes it impossible to determine whether [GM.sub.x]>0 or [GM.sub.x]=0.

4. Simulation results

As described above, when the sample geometric mean of a positive random variable is 0, the geometric mean of the modified observation (which imputes values below the detection limit [delta]) may be very different from 0 and, thus, inferences based on the modified sample may be misleading.

Suppose Y has a standard log-normal distribution with its probability distribution function [PHI], and U is independent of Y and uniformly distributed on (0,1). Let [C.sub.0] be a positive constant. The random variable X is defined as:


The distribution function of X is:


In the simulation study, [C.sub.0] is set at 0.277602. The data [X.sub.1], ...., [X.sub.n] is generated from the distribution of X defined in equation (1) above.

4.1 Properties of geometric mean

Figure 1 shows the cumulative distribution function of X and Y when [C.sub.0]=0.1. Since [PHI](0.1)=0.01, it is nearly impossible to distinguish between the two distribution function curves in the figure. However, their geometric means are very different. It is easy to prove that GM^=1 and [GM.sub.x]=0, no matter how small [C.sub.0] is (see Example 2 in Feng and colleagues [4] for a proof).


Since X is positive, the sample geometric mean is always positive. Although the sample geometric mean is a consistent estimator of [GM.sub.x] (which is 0 in this case), it may be quite a large number. Table 1 is a sample of n=100 observations from the distribution of Z=100X, where X is defined in equation (1). The sample geometric mean is [[bar.GM].sub.n]=85.72. However, the population geometric mean is actually [GM.sub.n]=0. It is very difficult to imagine that the data in Table 1 is from a distribution with a geometric mean of 0. This strange property of the geometric mean makes it difficult to test whether or not a sequence of positive numbers is a sample from a distribution with a geometric mean equal to 0.

In the simulation study, we set the detection limit at [delta]=0.277602 such that there was a 10% probability that [X.sub.i] is below the detection limit, that is, Pr{X<0.277602}=0.1. No data is observed below this detection limit, so if the value of [delta]/2 is imputed for all cases in which X,falls below [delta]=0.277602, then the modified observations are


Let [[bar.X].sub.n] [1/n] [n.summation over (i=1)] [x.sub.i] and [[bar.X].sup.*.sub.n] = [1/n][n.summation over (i=1)][X.sup.*.sub.i] be the sample means, and let [[bar.GM].sub.n] = [nth root of [X.sub.1]...[X.sub.n]] and [[bar.GM].sub.n.sup.*] = [nth root of ([X.sub.1]...[X.sub.n])] be the sample geometric means of ([X.sub.1], ..., [X.sub.n] and ([X.sub.1.sup.*, ..., [X.sub.n]) respectively.

Table 2 shows the means and standard deviations of [[bar.X].sub.n], [[bar.X].sub.n], [[bar.GM].sub.n], and [[bar.GM].sub.n] for samples of different sizes after 100,000 Monte Carlo replications. In each replicate a random sample [X.sub.1], ..., [X.sub.n] is generated, and [[bar.X].sub.n], [[bar.X].sub.n.sup.*], [[bar.GM].sub.n] and [[bar.X].sub.n.sup.*] are calculated. For each n, the mean and standard deviation of [[bar.GM].sub.n] is the sample mean and sample standard deviation based on 100,000 Monte Carlo replicates.


The same interpretation applies to the other columns in the table. There are two main findings shown in the table:

a) Since the detection limit is relatively small, the difference between the means [[bar.X].sub.n] and [[bar.X].sub.n.sup.*] is very small. They are very close to E([X.sub.1] and E([X.sub.x.sup.*]) even for small sample sizes.

b) The geometric means behave very differently. [[bar.GM].sub.n] converges to 0, while [[bar.GM].sub.n] converges to a constant far away from 0. The sample geometric means [[bar.GM].sub.n] and [[bar.GM].sub.n] also change substantially as the sample size increases.

The panels in Figure 2 show the histograms of [[bar.GM].sub.n] (the 'A' series) and [[bar.GM].sub.n] (the 'B' series) after 100,000 Monte Carlo replications. Although the distribution of [[bar.GM].sub.n] is skewed for relatively small sample sizes (n=10), the skewness almost disappears for relatively large sample sizes. However, the distribution of [[bar.GM].sub.n] is skewed for all sample sizes, particularly for large sample sizes. Since [GM.sub.x]=0, most of the sample geometric means clustered around 0 when n=10,000.


4.2 Hypothesis testing using geometric means

Let [X.sub.11], ..., [X.sub.1,n1] and [X.sub.21], ..., [X.sub.2,n2] be be two independent samples. Suppose we want to test the hypothesis:

[H.sub.0] : [GM.sub.X11]=[GM.sub.X21].

Due to detection limit, only the modified data can be used. The test statistic used in biomedical research is of the form


where [S.sub.k.sup.*2] is the sample variance of log[X.sub.k1.sup.*],...log[X.sub.k,nk.sup.*] (k=1,2).

In the simulation studies, [X.sub.11] has the same distribution as defined in equation (2) with [C.sub.0]=0.277602 and [X.sub.21] is defined as 2[X.sub.11]. Figure 3 shows the histograms of p-values of the test statistic [T.sup.*] for different sample sizes. In our example both samples have the same geometric means, so the null hypothesis is true and the distribution of the p-values of the statistic test [T.sup.*] should be close to the uniform distribution, at least for large sample sizes. However, the histograms shown in Figure 3 clearly indicate otherwise. Thus results of testing the null hypothesis when using the modified data are difficult to interpret and can be quite misleading.

5. Discussion

In this paper we consider the effect of the most common method of data imputation used in biomedical research for results that are below a detection limit. Despite its popularity, this method of using imputed values to compute a sample geometric mean which is used to estimate the population geometric mean (needed in many common statistical analyses) has not been adequately reviewed in the statistical literature. We use simulation studies to show that that the sample geometric mean is a very unstable statistic, so even small modifications introduced by this common imputation method can have a major effect on the estimated true (population) geometric mean and, thus, on statistical inference. [4] The sample geometric mean based on data that includes imputed values can be quite different from the true geometric mean, so the conclusions of hypothesis testing based on the use of modified data can be misleading.

All these problems stem from a very special property of the geometric mean: a positive random variable may have a geometric mean of 0. However, given a random sample from the distribution of a positive random variable, there is no method to determine whether or not the population geometric mean is 0, a problem that is compounded by the detection limit issue that requires the use of imputed values when computing the sample geometric mean. Any computed estimate of the geometric mean under the detection limit is uninterpretable.

Another issue with detection limit is measurement error. In this paper we assume that there is no measurement error from the device or instrument. The effect of potential measurement error on the detection limit requires further investigation.


This study was supported by a pilot grant (PI: Feng) from the Clinical and Translational Sciences Institute at the University of Rochester Medical Center.

Conflict of interest statement

The authors report no conflict of interest related to this manuscript.


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(received, 2015-10-20; accepted, 2015-10-23)

Hongyue Wang obtained her Bachelors of Science in Scientific English from the University of Science and Technology of China in 1995, and a PhD in Statistics from the University of Rochester in 2007. She is a Research Assistant Professor in the Department of Biostatistics and Computational Biology at the University of Rochester Medical Center. Her research interests include longitudinal data analysis, missing data, survival data analysis, and design and analysis of clinical trials. She has extensive collaboration with investigators in infectious diseases, nephrology, neonatology, cardiology, neurodevelopmental and behavioral science, radiation oncology, pediatric surgery, and dentistry. She has published 65 papers about statistical methodology and other topics in peer-reviewed journals.

Hongyue WANG (1) *, Guanqing CHEN (1), Xiang LU (1), Hui ZHANG (2), Changyong FENG (1), (3)

(1) Departments of Biostatistics and Computational Biology, University of Rochester, Rochester, NY

(2) Department of Biostatistics, St. Jude Children's Research Hospital, Memphis, TN

(3) Department of Anesthesiology, University of Rochester, Rochester, NY

* correspondence:

A full-text Chinese translation of this article will be available at http://dx.doi.Org/10.11919/j.issn.1002-0829.215121 on February 26, 2016.
Table 1: A random sample from a distribution with geometric mean 0
(sample size n=100)

166.95   70.67    75.68    2.61     264.39
211.11   127.63   71.91    362.70   73.12
155.42   16.80    109.80   18.34    190.47
152.19   84.66    172.02   45.94    437.89
151.47   513.60   34.72    69.70    492.94
2.67     41.08    205.55   73.19    713.21
108.44   747.98   15.69    59.55    122.46
44.53    191.05   74.81    143.88   194.59
55.08    154.57   53.82    66.33    53.58
118.96   48.13    39.11    690.46   170.17

55.30    129.93   87.55    172.43   59.95
293.65   292.67   369.40   139.59   304.00
29.37    53.43    62.93    137.79   44.72
110.13   53.51    152.44   75.92    60.48
42.03    4.48     82.01    445.03   35.22
182.35   43.62    67.32    37.21    65.01
475.55   0.95     261.28   96.82    168.29
26.63    90.69    141.91   25.92    251.09
17.57    115.23   6.69     49.44    303.29
217.58   62.74    79.84    26.43    106.79

Table 2: Means and standard deviations of sample means and sample
geometric means

            [[X.sub].sub.n]       [[X.sub].sup.*.sub.n]
size        mean         sd          mean         sd

10         1.6459      0.6936       1.6485      0.6930
50         1.6406      0.3068       1.6433      0.3065
100        1.6415      0.2167       1.6442      0.2165
500        1.6418      0.0969       1.6445      0.0968
1,000      1.6413      0.0688       1.6440      0.0688
5,000      1.6411      0.0306       1.6438      0.0306
10,000     1.6413      0.0217       1.6440      0.0216

            [[bar.GM].sub.n]      [[bar.GM].sup.*.sub.n]
size        mean         sd          mean         sd

10         0.9169      0.4460       1.0326      0.3445
50         0.7573      0.3079       0.9887      0.1450
100        0.7027      0.2797       0.9834      0.1017
500        0.5944      0.2389       0.9795      0.0453
1,000      0.5399      0.2464       0.9789      0.0321
5,000      0.3083      0.3056       0.9783      0.0143
10,000     0.1571      0.2651       0.9783      0.0101
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Title Annotation:Biostatistics in psychiatry (29)
Author:Wang, Hongyue; Chen, Guanqing; Lu, Xiang; Zhang, Hui; Feng, Changyong
Publication:Shanghai Archives of Psychiatry
Date:Oct 1, 2015
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