INTRODUCTION:
Extensive research appearing recently in
medical and pharmaceutical literature concerns the serious medical problem of
chronic kidney disease (CKD)1,2,3,4,5,6. On the other hand,
mathematics and statistics are well-known for a multitude of applications to
other sciences7, including the medical and pharmaceutical fields8,9,10.
In view of that, the present study focuses on statistical analysis issues of
collecting data, also called pooling of data, which concern the age of CKD
patients, who reached an end-stage renal-disease (ESRD). This stage of the
disease is known to be related to the socio-economic status of patients11
and it causes an economic burden to health systems, taking into account that
usual medical care expenses are often increased because many patients also
suffer from various comorbid conditions12. It has been reported that
“CKD is a worldwide public health problem, a social calamity and an economic
catastrophe”13.
The age of patients, which is the variate under study
in the present article, has been associated with socio-economic factors
underlying this type of disease, and its importance has been stressed on
several occasions in existing pharmacoeconomic literature, as we now explain.
Firstly, a comparison between “developed” and “developing” countries has
clearly shown lower patient ages to be associated with “developing” countries14,15,
thus boosting health costs for the latter countries. Typical examples of younger
patient ages coming from Sub-Saharan African countries include results from
Nigeria15, where 86.5% of ESRD patients were found to be under 60
years, and from Rwanda14, where the percentage of Rwandan patients
under 60 years was 69.7%; furthermore, it was reported that CKD affects young
adults between 20 and 50 years in countries of Sub-Saharan Africa16.
Other examples are Indian data, where the percentage of ESRD patients under the
age of 60 years was 75%13; moreover, it was reported that “in India
the majority of the CKD patients belongs to the younger population when
compared with other countries”17. Secondly, comorbid conditions of
ESRD patients, previously reported to increase health costs, were found to be
also associated with age12.
The aim of this article is to use pooling of data as a
method for obtaining more accurate estimates for main parameters under
evaluation, like the mean and variance of patients age, and also for increasing
the power of statistical tests in a comparison of patients ages between
countries. Pooling of data and statistical issues concerning means and
variances have already been discussed in a more general context in very early
literature for means18, and for variances19. Concerning
the specific area of interest this article is dealing with, pooling of data is
mainly used in renal registries for separate countries or for groups of
countries20. A typical example is the Australia and New Zealand
Dialysis and Transplant Registry15. The problem is that, in many
instances, registries are incomplete or even inexistent21, and thus
it would be nice to make use of the proposed method in order to collect as much
data as possible for statistical analyses. It is also worth mentioning that the
idea of pooling of data for ESRD patients from Sub-Saharan countries was also
reported in previous literature22, but, as mentioned by the authors,
statistical analyses were not done (see top of p. e410). In the following
sections, the proposed method is described in detail and its advantages are
underlined by the means of an application exercise.
MATERIALS AND METHODS:
Age is a continuous variable, and thus it is generally
expressed by its mean and standard deviation (SD). From existing literature, we
noticed that although for some cases the sample sizes of patients are rather
small, inferential results concerning this variable are obtained using the
Student’s t- test23. However, results obtained from this
test might not be reliable for small sample sizes, and non-parametric
statistics should be used instead24. On the contrary, it has been
reported in the latter reference that for large sample sizes the use of
Student’s t- test is highly recommended. Advantages of large sample
sizes are well-known in usual inferential statistical theory, most notably that
they provide more accurate mean estimates and, moreover, when statistical
hypotheses between two population means are tested, the probability of type II
error gets small thus leading to an increased power of Student’s t- test.
Pooling of data:
We concatenated data samples mostly collected from
different centres in some of the countries under examination in order to obtain
larger samples than the initial ones. Besides the aforementioned advantages of
increasing sample sizes, pooling data from different centers is also expected
to limit, to some extent, any effects of the problems of “referral bias” and
“population migration”25, p. c-198, under section “chronic disease
in India”; in other words, problems of selection bias should also be taken into
consideration. This matter is seminal for India, in view of the geographical
dimension and the population size of this country. In India, there is a CKD
registry, that addresses the aforementioned selection bias problems and which
includes a very large number of 52273 adult patients21; this
registry reports a mean equal to 50.1 years and a standard deviation equal to
14.6 years concerning the age of patients. Unfortunately, as mentioned in this
reference (page 4), there is no corresponding ESRD registry in India.
Therefore, pooling of data could provide a useful tool for statistical analyses
of patient ages in this country.
In the present work, examples of pooling of data
concern concatenated samples of adult patients from five different Indian
hospitals13,26,27,28,29, and two concatenated samples of patient
ages from a Malaysian hospital (phases 1 and 2)30. The only problem
is that, for the Malaysian example, patients from phases 1 and 2 are selected
from the same centre; however, since our total sample size is quite large, and
since we noticed that the mean and the SD of patient ages arising from another
study that combines data from five different centres in Malaysia31,
with mean age = 53.7 and SD=14.2, are close to corresponding means and SDs from
phases 1 and 2, presented in Table 1, we reckon that selection bias should not
be a problem in this instance. Remark that we included into our analysis all
patients reported in these phases since only a small proportion belongs to CKD
stages lower than the ESRD stage, and so the results on means and SDs are not
expected to be much affected. For our statistical analysis we shall call
“pooled data” all data obtained from the aforementioned five Indian hospitals
as a single sample and all data obtained from phases 1 and 2 as another single
sample. For comparison reasons, two more countries were also included, namely,
Kenya32 and Greece33. The latter country is the only
“developed” country under study, and so it is expected to have a higher mean
age than the other three countries. The reason for including this particular
country into our analysis is that, apart from using a quite large sample, the
authors also report that “selection of facilities was based on a nationwide
distribution and on the proportion of public and private facilities” (see
subsection “sample and data collection”, towards the bottom of page 86), and
so, their method should account for selection bias. On the other hand, Kenya32
was selected since Sub-Saharan countries have in general very low mean patient
ages14,15. As reported in the literature20, African
countries have very few data on ESRD; for that purpose, the Kenyan data32
included into our analysis arise only from a single centre (the Kenyatta
National Hospital), but in view of the fact that this is the largest referral
hospital of Kenya, we believe that selection bias should be limited.
Statistical methods
homogeneity of data:
Homogeneity of data included into the samples must
not be affected by concatenating them into a pooled sample, which will be used
for further statistical analyses. Concerning our example, this should mean that
pooled patient ages, obtained from a specific country, should, in theory, show
no variability. This will be assessed as follows. Let us consider that
homogeneous populations are a special case of mixture models, with μ1=
μ2 = --- = μg = μ
and σ12 =
σ22 = --- = σg2 =
σ*2 (see Appendix). By the results of the
Appendix, a common population variance σ2 can be estimated by , where denotes the i–th sample, S12 the variance of the i–th sample,
and g the total number of samples. This is the usual common variance
estimator (or “pooled variance”), which is typically denoted as Sg2 = σ*2
(see Appendix). Noticing that SDs concerning the 5 Indian samples are close and
those concerning the 2 Malaysian samples are approximately equal (see Table 1),
we make the assumptions of a common population variance for India and a common
population variance for Malaysia. On the other hand, the mean of the pooled
data (or “pooled mean”), is an estimator for the population mean,
where is the size of sample , and the mean of
the i –th sample (see Appendix).
We need to underline at this point that, by results
of the Appendix, is not approximately equal to the estimate
of the mixture variance unless the
underlying sample groups have approximately equal means, that is, . In this regard, variability in the
pooled data is expressed by the magnitude of expression , used in order to compute the part of variance (see Appendix). A magnitude close to 0
would indicate an approximate homogeneity in the pooled data. Concerning our
example, this is assessed as follows. Since, by results of Table 1, the 5
Indian sample means are close and the 2 Malaysian ones are very similar,
expression (with 2 for Malaysia and 5 for India), is
expected to be close to 0, for both countries. This can be easily verified
since, by using again results of Table 1, is equal to 0.5 for India and 0.0169 for
Malaysia, thus suggesting an approximate homogeneity in the pooled data sets
relative to these countries.
Finally, note that the mixture model approach, used in
this text for assessing homogeneity, provides an alternative to well-known
significance tests like Student’s for comparison between two group means and
one-way ANOVA for comparison between more than two group means. We reckon that
this approach is more adequate than these significance tests for cases where
the null hypothesis of means equality is not rejected, since the latter does
not prove that the null hypothesis is true.
Statistical methods 
the central limit theorem:
The usual central limit theorem (CLT)34 ,
page 112, is expected to apply when variables included into the sample have
equal means and variances, and sample sizes are reasonably large -usually ≥3035.
Since Student’s t-CI35 can be approximated by a z-CI
according to discussion below algebraic expression (5) 34,
page 196, it results that the need for either assuming or testing for a normal
distribution for the samples can be removed.
Concerning our application exercise, assumptions
underlying the CLT hold strong for both the cases of India and Malaysia, since,
by results from the previous subsection, concatenated sample data for these
countries can be considered as homogeneous and all sample sizes involved in the
study are at least equal to 30 (see Table 1); then, by this theorem, means (with ) of samples from a specific country are
all normally distributed and therefore the pooled mean of the country is also
normally distributed. This result is useful for obtaining the CIs for the means
presented in the next section. Furthermore, as a consequence of the CLT,
inference from significance tests using Student’s distribution, can be obtained via a
standard normal distribution36; in the sequel, this result will be
applied to our data for inferential purposes using two-sample Student’s t-
tests.
RESULTS AND DISCUSSION:
Table 1 shows means and variances, obtained in earlier
literature, as well as our pooled means and pooled variances obtained using the
above algebraic expressions for and . The CIs appearing in this table account
for the accuracy of the estimates since a more accurate estimate is associated
with a narrower CI. It is easy to remark that CIs, obtained for cases where
pooled data samples are used, are clearly narrower than those obtained from
single samples.
We now proceed to implementing significance tests for
mean differences between countries considering a common variance for ages,
since we remark that age SDs for all countries used in this study are close
(see Table 1 and Discussion section hereunder). In view of that, we use pooled
SDs for the – tests, which have the form (this is expression appearing under (12)
page 21834, with , and stands for the number of countries under
comparison). Remark that can themselves refer either to an
estimated sample variance (in case of non-pooled data) or to an estimated
pooled variance (in case of pooled data). In order to implement these tests, we
use , as appearing in (12) page 21834,
replacing the common SD by ; note that and , involved in , can be either sample means or pooled
means. Using the discussion mentioned in the previous section36,
computation of -values appearing in Table 2 relies on a standard normal
distribution. For reasons of comparison with most studies in existing
literature15,31, two-sided tests for mean differences are used, and
a -value less than 0.05 is considered to be statistically significant. Results
are presented in Table 2.
Table 1: Accuracy of means of patients’ age
|
India 13
Sample 1
|
n1=201,
Mean=50.27,
SD=13.781
|
95% CI :
[48.326, 52.214]
|
|
India 26
Sample 2
|
n2=30,
Mean=49.72,
SD=13.2
|
95% CI :
[44.9, 54.54]
|
|
India 27
Sample 3
|
n3=39, Mean=51, SD=12.7
|
95% CI :
[46.933, 55.067]
|
|
India 28
Sample 4
|
n 4=111,
Mean=51.4,
SD=14.29
|
95% CI :
[48.687, 54.113]
|
|
India 29
Sample 5
|
n5=108,
Mean=52, SD=11.86
|
95% CI :
[49.72, 54.282]
|
|
India
(pooled data)
|
np=489,
Mean=50.935,
SD=13.382
|
95% CI :
[49.724, 52.145]
|
|
Malaysia - phase
130
Sample 1
|
n1=300,
Mean=55.56,
SD=14.15
|
95%CI :
[53.926, 57.194]
|
|
Malaysia - phase
230
Sample 2
|
n2=300,
Mean=55.3,
SD=14.37
|
95% CI :
[53.641, 56.959]
|
|
Malaysia (pooled
data)
|
np =600,
Mean=55.43,
SD=14.26
|
95% CI :
[54.848, 56.012]
|
Table 2: Effect of pooling of data for comparing
patients’ age differences between countries
|
Between countries
age difference
|
Difference
between sample means for age
|
p-value
|
|
India 26
– Kenya 32
India (pooled) –
Kenya 32
|
5.72
6.935
|
0.0476
< 0.001
|
|
Malaysia - phase
2 30 – India26
Malaysia (pooled)
–
India (pooled)
|
5.58
4.495
|
0.0414
< 0.001
|
|
Greece 33
– Malaysia30 -phase 1
Greece 33
– Malaysia (pooled)
|
2.54
2.67
|
0.0134
0.0012
|
The 95% CIs, presented in Table 1, and the comments
appearing in the previous section, show that the pooled mean concerning
patients’ ages in a particular country is more accurate than each of the
corresponding sample means of the country.
Results of Table 2 show that testing for mean
differences in age between India (pooled data) and Malaysia (pooled data)
provides a very significant 
-value, whereas if the same test was
performed using data for India26, and from the phase 2 for Malaysia30,
the p-value, while still remaining significant, would be however close
to the nominal level. Equivalently, a comparison between India (pooled data)
and Kenya32 (adult patients have mean age= 44, SD=13.98 and sample
size=96) provides a very significant mean difference for ages, whereas if this
test was repeated using only Indian data26, the p-value,
although still remaining significant, would be even closer to the nominal value
than for the previous case. Finally, a comparison between means from Malaysia
(pooled data) and Greece33 on one hand, and Malaysia (phase 1) and
Greece33 on the other hand (where results for adult patients in
Greece are mean age= 58.1, SD= 14.9, sample size of patients on hemodialysis
=642), led to a smaller -value for the former test (see Table 2). Note that, in
case a 1% significance level (instead of 5%) were considered for testing mean
differences for cases where there was no pooling of data, namely, India 26
– Kenya32, Malaysia (phase 2)30 – India26, and
Greece 33 – Malaysia (phase 1)30, the corresponding -values would have been no more
significant, whereas they still remain significant for cases where pooling of
data is used (see Table 2). In view of these results it is interesting to
remark that, although sample means concerning pooled samples and those
concerning cases where no pooling is used are close and even approximately
equal in some instances, corresponding p -values can be quite different (see Table 2). These
findings are in concordance with usual theory since they indicate that
increasing the total sample size leads to low p -values, thus increasing
in turn the power of the test.
CONCLUSION:
Our findings argue in favour of using the method of
pooling of data in order to obtain more accurate estimates for the mean age of
ESRD patients and also for increasing the power of the statistical tests used
for comparing patient ages. This method is simple to implement in practical
applications and has the advantage of not requiring a normal distribution for
the samples. Furthermore, it can prove quite useful for cases where either
there is no registry or there is an incomplete registry for ESRD patients. In
an application exercise, a comparison of patients’ ages by country showed that
Greece had the highest patients’ mean age (as expected, since it officially
belongs to “developed” countries) and Kenya had the lowest one (as expected
from previous discussion). Finally, Malaysia’s mean age was found significantly
higher than that of India. Although these countries are both “developing”, a
much higher GNI per capita as well as a human development index (HDI)
pertaining to Malaysia might be indicators that are in support of this result
(India: GNI=2380 USD and HDI=0.633, Malaysia: GNI=11780 USD and HDI=0.803–see
UNDP link for HDI:
“hdr.undp.org/data-center/human-development-index#/indicies/HDI”).
CONFLICTS OF INTEREST:
The authors have no conflicts of interest regarding
this investigation.
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APPENDIX:
The mean and variance of the pooled data sample are
obtained by using theory concerning mixture models, as follows. Let us, more
generally, assume that data are distributed as a finite mixture with
(say) components and probability density function
(;,), where are the component weights satisfying and , (x;, is the -th component density, with , and and are the corresponding mean and variance of
the -th component. The mean of this mixture
model has the form , and its variance can be written as , with . The mixture mean is estimated by the well-known (unbiased)
pooled sample mean , where are the component sample means, is the proportion of sample data in the –th component, and is the sample size of the –th component. In case the component
variances are all equal, that is, , for , then , and thus, , and can be estimated by the well-known
(unbiased) pooled sample variance, used to estimate a common component
variance, , where are the component sample variances, and are as defined above, when large sample
sizes are involved (note that this expression for is just a generalization of the algebraic
expression under equation (12) , page 218 34, which has the form ). Furthermore, if the component means are
all equal, i.e. , then it results that , and so, , thus implying that the variance of the
mixture is equal to the (common) variance of the components.
The Examples of India and Malaysia