Alnory A^{1*}, Sondous A^{2} and Abdalajabar A^{2}
^{1}Department of Biostatistics and Epidemiology, University of Gezira, Gezira, Sudan
^{2}Department of Applied Statistics, University of Gezira, Gezira, Sudan
Received date: November 14, 2017; Accepted date: November 25, 2017; Published date: December 10, 2017
Citation: Alnory A, Sondous A,Abdalajabar A(2017) Statistical Comparison to Determine First Line Regimen of Two AntiMalarial for Uncomplicated Plasmodium falciparum Malaria among Children under 5 Years.J Biomed Sci Appl Vol. 1 No. 2:9.
Comparison of regimen efficacy in clinical medicine has been utilized to improve the effect of drugs against disease The pairs alternative ttest method approach is used in this experimental study to derive efficacy estimates of two antimalarial regimens. This study aimed at comparing mostly used drugs against malaria in Africa in the present time, namely artesunatemefloquine and artemetherlumefantrine to determine which of them is more effective treatment of the disease among children under 5 years of age and thus may be considered as a first line treatment, especially when laboratory tests are not available. The results achieved in this study were based on hypothetical experimental data. The data which represented 12 pairs of match’s children were analyzed using t tests with the help of SSS program. The study concluded that artesunatemefloquine is more effective as a first line treatment than artemetherlumefantrine in children less than five years age of both sexes.
Malaria; Children; Sudan; Regimen
The main objective of this study is to enhance methodology in epidemiological research and illustrate a procedure to test therapeutic responses post antimalarial treatment of two regimens. The purpose of this kind of study is to illustrate to young researcher in the heath field who are not familiar with statistical experimental design how to measure and compare antimalarial clinical efficacy. Many studies have been made to test the superiority of a traditional drug against another drug [1]. This has become a routine for pharmacists to test their drugs. Malaria in Africa has, especially, become a problematic endemic disease resisting various kinds of drugs and such statistical tests are continuously needed in any attempt to control the disease. For this reason the study aimed at comparing two mostly used drugs against malaria in the present time, namely artesunatemefloquine and artemetherlumefantrine to determine which of them is more effective treatment of the disease among children under 5 years of age. Rapid treatment of malaria is necessary to avoid lifethreatening complications, especially in the absence of laboratory tests, so comparing the two most important treatments currently in use in most African countries in the malaria zone will enable us to identify the best treatment as a first line to treat malaria in such cases. The study is basically concerned with explaining the methodology rather than obtaining robust results.
The t distribution
Definition: If Y is a standard normal random variable and Z is a Gamma random variable with parameters n and h=1, independent of Y. Then X a random variable has a standard Student's t distribution with n degrees of freedom if it can be written as a ratio
Equivalently, we can write:
Where χ^{2}  Chisquare is a random variable with n degrees of freedom (dividing by n a Chisquare random variable with n degrees of freedom, one obtains a Gamma random variable with parameters α=1/2 and β=2 [2].
Definitions the distribution related to the t distribution
Definition of normal distribution: The normal distribution, also known as the Gaussian distribution, is a probability distribution most often used to describe the behavior of a variable clustered around a mean. When graphed, it takes the shape of a bell curve where the peak of the bell is the mean μ, and the width is determined by the standard deviation σ. The equation
Calculate a normal distribution is, where the standard normal distribution occurs when mean=0 and variance=1. By virtue of the central limit theorem (which states that the distribution of the mean of a large data set tends to a normal distribution), the normal distribution is very useful in statistical analysis of populations and is often encountered in natural sciences [3].
Characteristic function
• The expected value of a normal random variable X is E(x)=μ.
• The variance of a normal random variable X is Var(x)=σ^{2}.
• The moment generating function of a normal random variable X is defined for any
If μ=0 and σ^{2}=1 then the distribution called standard normal distribution [3].
Derivation of tdistribution: For random samples from normal distribution with mean μ and standard deviation σ, σ^{2}/n has a standard normal distribution with mean μ and variance σ^{2}/n.
There for the statistic:
Has a standard normal distribution with mean 0 and variance 1.
The main problem here is that the standard deviation is unknown and it is necessary to use the sample standard deviation but this needs the following amendment. To find the corresponding distribution we start with the joint probability density function for the variables (s, t).
where x has a standard normal distribution with parameters 0,1 and y has a gamma distribution with α=v/2 and β=2.
For  ∞ < x < ∞ and y > 0, and f(x, y) = 0 elsewhere, hence. If t is related to x with following relationship
Hence
And
Hence
and we get
Substituting in (1)
(2)
Now substitute
So that
Substituting in (2), we get
And this is the probability density function of t [4].
Independent ttest assumptions
• The two samples are randomly and independently selected from the two populations.
• The two populations are normally distributed.
• The two sample variances are equal.
Paired ttest assumptions
• Since the observations are paired in this case we test the differences.
• The population of differences has a normal distribution.
• The differences are independent.
• The differences have equal mean and variance
The tTest for pairs (Equal samples): The t test for pairs is used when the sample size is too small and the variance of the distribution of pairs is not known. Such usage of small sampling theory in clinical medicine is governed by a number of axioms that have to be satisfied when selecting the sample. These axioms are:
1 The selected sample pairs have to be in the same age range.
2 The selected sample pairs have to be of the same sex.
3 The selected sample pairs have to be of the same prognosis.
4 Occupation period of the disease should approximately be equal for the selected pairs patients.
The satisfaction of the axioms above are more important that the sample size [5] (Table 1).
Pairs of  Treatment  

Patients  Treatment A A 
Treatment B B 
Difference AB 

1  X_{1}  Y_{1}  D1  
2  X_{2}  Y_{2}  D_{2}  
3  X_{3}  Y_{3}  D_{3}  
4  X_{4}  Y_{4}  D_{4}  
5  X_{5}  Y_{5}  D_{5}  
6  X_{6}  Y_{6}  D_{6}  
7  X_{7}  Y_{7}  D_{7}  
8  X_{8}  Y_{8}  D_{8}  
9  X_{9}  Y_{9}  D_{9}  
10  X_{10}  Y_{10}  D_{10}  
11  X_{11}  Y_{11}  D_{11}  
12  X_{12}  Y_{12}  D_{12}  
Total  
Mean 
Table 1: Cell readings of 12 Alternate pairs given two different treatments.
The table above provides an example of applying the t test. We have 12 matched pairs which were subjected to two different treatments A and B and when the number of pairs is equal (12 in this case) the model works with the differences between the two treatments for testing the null hypothesis that:
To apply the test first calculates the average difference and the standard error of the mean SD:
Difference To obtain this the common variance is calculated as
This is then divided by n1 to give SD2, then we get the standard error of the mean by taking the square root of SD2 divided by the sample size i.e., and finally we divide by the standard error of the mean to get the t statistics as .
Data:
To collect data on comparative treatments effect is always a complicated task. This especially true when comparing the effect of two antimalarial drugs. The difficulty arises for a number of reasons.
1. If the plasmodium parasite is scanty, it does not appear in the blood film after taking any of the antimalarial drugs although it is present in the patient blood. This is a characteristic of people who are repeatedly infected by malaria. For this reason the study should include in the experiment patients who develop the disease for the first time with severe but uncomplicated symptoms.
2. While the treatments are different in two groups, researchers should try to keep as many of the conditions the same as possible. The alternative pairs who will be given the two drugs alternatively should be of the same age group, same sex and having the same prognosis.
3. The inclusion criteria of severe malaria means that only patients with an account of malaria parasite in 100 fields of an oil emersion lens should have a mean number of parasites of 30 and over to satisfy normality assumption. Under such criteria the size of the comparison sample cannot be large because the researcher has to wait for a patient with all the above characteristics to report to health facility. This means that samples are longitudinal in nature.
4. Because samples sizes will be very small, the researcher will have to use small sampling theory tests such as tstudent, chisquare or F test. Such framework can only be applied in clinical medicine where experiments are highly controlled. This makes things even more complicated because the patients should be inpatients. Moreover, the researcher should have agreement of collaboration of a doctor and a pathologist working in the facility, in addition to strict ethical considerations and of course financial support.
5. A hypothetical trial experimental trial is presented here to illustrate the application of the procedure using SPSS program. The bench mark parasite counts per 100 fields for hypothetical cohorts of male and female children 04 years are based on educated guessing. Treatments to be compared are artesunatemefloquine and artemetherlumefantrine. The hypothetical cohorts are presented in Tables 2 and 3 successively. Each table shows Parasite count before treatment and Parasite count 24 hours after treatment (Tables 4 and 5).
Parasite count before treatment  Parasite count 24 hours after treatment  

Patient pairs  Patient 1  Patient 2  artesunatemefloquine  artemetherlumefantrine 
1  109  106  33  38 
2  101  103  36  20 
3  111  109  45  25 
4  107  105  50  33 
5  102  101  44  19 
6  107  106  51  32 
7  105  104  34  27 
8  115  108  50  22 
9  110  116  35  43 
10  107  113  35  31 
11  108  102  46  29 
12  107  112  34  32 
Table 2: Cell readings of 12 Alternate pairs of children (04) given two different antimalarial treatments. Source: hypothetical data (Males A).
Patient pairs  Parasite count beforetreatment  Parasite count 24 hours after treatment  

Patient 1  Patient 2  artesunatemefloquine  artemetherlumefantrine  
1  108  104  30  22 
2  105  105  37  22 
3  112  101  25  15 
4  117  105  40  30 
5  112  105  45  29 
6  101  106  52  22 
7  103  104  33  26 
8  105  104  50  20 
9  116  115  55  41 
10  105  112  35  30 
11  102  100  36  24 
12  104  100  34  22 
Table 3 Cell readings of 12 Alternate pairs of children (04) given two different antimalarial treatments. Source: hypothetical data (Females B).

N  Mean  Std. Deviation  Std. Error Mean 

Patient 1 1males (04) before treatment  7  108.4286  3.25869  1.23167 
5  106.0000  4.35890  1.94936  
Patient t 1 females(04) before treatment  7  105.1429  5.01427  1.89521 
5  110.8000  4.54973  2.03470 
Table 4: Descriptive statistics. Source: SPSS output based on hypothetical samples of males and females under 5.
ttest for Equality of Means  

T  df  Sig. (2tailed)  Mean Difference  Std. Error Difference  95% Confidence Interval of the Difference  
Lower  Upper  
Experiment 1males (04) before treatment  Equal variances assumed  1.110  10  .293  2.42857  2.18865  2.44805  7.30520 
Equal variances not assumed  1.053  7.079  .327  2.42857  2.30586  3.01161  7.86875  
Experiment 2 females(04) before treatment  Equal variances assumed  1.999  10  .074  5.65714  2.83039  11.96364  .64936 
Equal variances not assumed  2.034  9.290  .071  5.65714  2.78062  11.91755  .60326 
Table 5: ttest of equality of means. Source: SPSS output based on hypothetical samples of males and females under 5.
The results shown here are, of course, illustrative: All results are based on output from SPSS package
To remind the reader, our experiment involves two outcomes. Malaria parasite count in 100 fields of an emersion lens for twelve pairs of male children represents experiment 1 denoted in the analysis as “patient 1”. Similar experiment for females is denoted as “patient 2”.
Test of independence between males and females samples
Tests the significance of the difference between the two samples means. The procedure provides descriptive statistics for each test variables together with a test of variance equality and a 95% confidence interval for the difference. The two variables involved in our study are the parasite densities of males and females before taking any treatment in the first experiment (patient 1). For simplicity artesunatemefloquine is referred to as “Quartum” and artemetherlumefantrine as “Artemether”.
Usually, the groups in a twosample t test are fixed by design, and the grouping variable has one value for each group. However, there are times when assignment to one of two groups can be made on the basis of an existing scale variable [6]. With the IndependentSamples t Test procedure, all we need to provide is a cut point. The SPSS program divides the sample in two at the cut point and performs the t test. The virtue of this method is that the cut point can easily be changed without the need to recreate the grouping variable by hand every time. The procedure first produces (Table 4).
The descriptive table displays the sample size cut of points, mean, standard deviation, and standard error for both groups. On average parasite count for males (3.5 units of the mean) is less than the comparison group (females), and they vary a little less around their standard deviations (Table 4).
The procedure produces two tests of the difference between the two groups (Table 3). One test assumes that the variances of the two groups are equal and the other assumes that they are not. The procedure produces ttest for both. The average significance value of the statistic is 0.191. Because this value is greater than 0.10, we can assume that the two groups have equal variances and ignore the second test (Table 5).
The 95% Confidence Interval of the Difference provides an estimate of the boundaries between which the true mean difference lies in 95% of all possible random samples of 12 pairs. Since the significance values of the test are greater than 0.05, we can safely conclude that the average of 0.191 malaria parasites per 100 fields is not due to chance alone and the assumption that the two sample means are equal cannot be accepted. Hence the two samples are independent.
The same test was done for experiment two (patient 2) and arrived to the same conclusion i.e., the two samples are independent.
Pairedsamples t test
The PairedSamples T Test procedure compares the means of two variables for single pairs. The procedure computes the differences between values of the two variables for each case and tests whether the average differs from 0. In this study on comparative malaria treatment, number of parasites per 100 fields of an oil emersion lens are counted at the beginning of the study, given treatment 1, (a) and measured again. The same procedure was repeated for treatment 2 (Quartum). A third comparison is done for both treatment after taking the drug and test that the difference does differ from 0. Thus, each subject has two counts, often called before and after measures and after treatment for both. An alternative design for which this test is used is a matchedpair (see chapter one for pairs matching criteria) in which each record in the data file contains the response for the patient and also for his or her matched pair. It is the second procedure that is used here.
The procedure provides statistics: for each variable: mean, sample size, standard deviation, and standard error of the mean. For each pair of variables: correlation, average difference in means, t test, and 95% confidence interval for mean difference (one can specify the confidence level).
The descriptive table displays the mean, sample size, standard deviation, and standard error for both groups. Across all 12 pairs, parasite load dropped substantially, a drop of 61% after 24 hours from taking Artemether. Now we have to confirm that this large drop is not due to chance. But before that we have to establish that the variables under the study are approximately normal (Table 6).
Mean  N  Std. Deviation  Std. Error Mean  

Pair 1  Parasite before Artemether administration for male  107.42  12  3.777  1.090 
Parasite after Artemether administration for male  41.92  12  8.743  2.524 
Table 6: Paired samples statistics for male. Source: SPSS output based on hypothetical samples of males and females under 5.
Tests of normality
The tests of normality overlay a normal curve on actual data, to assess the fit. A significant test means the fit is poor. For Patient 1 (parasite load before treatment  males) the test is not significant; they fit the normal curve well. The conclusion applies for patient 2. (Parasite load before treatment  females). These are the original pairs which are normally distributed so are any other distributions based on them. We test whether the proportion of parasites in the blood follows normal distribution. The test results are shown in Table 7. Both have no significance according KolmogorovSmirnov and ShapiroWilk tests suggesting that both are approximately normal (Table 8). The fit displays approximately normal curves for both variables (Figures 1A and 1B).
Factor  KolmogorovSmirnov  ShapiroWilk  


Statistic  df  Sig.  Statistic  df  Sig.  
Data  Patient 1  .206  12  .170  .955  12  .707 
Patient 2  .175  12  .200*  .948  12  .613 
Table 7: Tests of normality. Source: SPSS output based on hypothetical samples of males and females under 5.
Analyses for males
In the descriptive analysis above we have shown that administration of Artemether to male children 04 reduces the parasite load by 61% after 24 hours of administration. Now we test the significance of this drop. The test begins by showing correlation for Artemether before and after. At 0.45 the correlation between the before treatment and after treatment levels is not statistically significant. Levels were lower overall, but the change was inconsistent across subjects (Table 8).
N  Correlation  Sig.  

Pair 1  Parasite before Artimether administration for male & Parasite after Artimether administration for males  12  .450  .142 
Table 8 Paired samples correlations for males. Source: SPSS output based on hypothetical samples of males and females under 5.
In Table 7, the Mean column in the pairedsamples t test table displays the average difference in parasite count for Artemether experiment between before and after treatment (Table 9).
Paired Differences  T  df  Sig. (2tailed)  

Mean  Std. Deviation  Std. Error Mean  95% Confidence Interval of the Difference  
Lower  Upper  
Pair 1  Parasite before Artemether administration for male  Parasite after Artemether administration for male  65.500  7.810  2.255  60.538  70.462  29.051  11  .000 
Table 9 Paired samples test for males. Source: SPSS output based on hypothetical samples of males and females under 5.
The Std. Deviation column displays the standard deviation of the average difference score. The Std. Error Mean column provides
an index of the variability one can expect in repeated random samples of 12 patients as in this study. The 95% Confidence Interval of the Difference provides an estimate of the boundaries between which the true mean difference lies in 95% of our sample of 12. The t statistic is obtained by dividing the mean difference by its standard error.
The Sig. (2tailed) column displays the probability of obtaining a t statistic whose absolute value is equal to or greater than the obtained t statistic. Since the significance value for drop in parasite is less than 0.05, we can conclude that the average drop of 62% of parasite per patient is not due to chance variation, and can be attributed to the Artemether. The conclusion from this is that Artemether is a very effective drug for malaria treatment among male’s children 04.
The same tests were performed for female’s children which indicated that Across all 12 pairs, parasite load dropped substantially, a drop of 73% after 48 hours from taking Artemether with higher correlation (0.599) and a significant difference from 0.
Analyses for males
The descriptive Table 10, displays the mean, sample size, standard deviation, and standard error for both groups. Across all 12 pairs, parasite load dropped substantially, a drop of 78% after 24 hours from taking Quartum. Now we have to confirm that this large drop is not due to chance (Table 10).
The descriptive Table 10, displays the mean, sample size, standard deviation, and standard error for both groups. Across all 12 pairs, parasite load dropped substantially, a drop of 78% after 24 hours from taking Quartum. Now we have to confirm that this large drop is not due to chance (Table 10).
Mean  N  Std. Deviation  Std. Error Mean  

Pair 1  Parasite before Quartum administration for males  107.0833  12  4.66044  1.34535 
Parasite After Quartum administration for males  29.2500  12  7.16208  2.06752 
Table 10 Paired samples statistics for males. Source: SPSS output based on hypothetical samples of males and females under 5.
In Table 11, the Mean column in the pairedsamples t test table displays the average difference in parasite count for Quartum experiment between before and after treatment (Table 12).

N  Correlation  Sig.  

Pair 1  Parasite before Quartum administration for males & Parasite After Quartum administration for males  12  .609  .010 
Table 11 Paired samples correlations for males. Source: SPSS output based on hypothetical samples of males and females under 5.
Paired Differences  t  df  Sig. (2tailed)  

Mean  Std. Deviation  Std. Error Mean  95% Confidence Interval of the Difference  
Lower  Upper  
Pair 1  Parasite before Quartum administration for males  Parasite After Quartum administration for males  77.83333  5.74984  1.65983  74.18006  81.48660  46.892  11  .000 
Table 12: Paired samples test for males. Source: SPSS output based on hypothetical samples of males and females under 5.
The Std. Deviation column displays the standard deviation of the average difference score. The Std. Error Mean column provides an index of the variability one can expect in repeated random samples of 12 patients as in this study. The 95% Confidence Interval of the Difference provides an estimate of the boundaries between which the true mean difference lies in 95% of our sample of 12. The t statistic is obtained by dividing the mean difference by its standard error.
The Sig. (2tailed) column displays the probability of obtaining a t statistic whose absolute value is equal to or greater than the obtained t statistic. Since the significance value for drop in parasite is less than 0.05, we can conclude that the average drop of 78% of parasite per patient is not due to chance variation, and can be attributed to the Quartum.
The conclusion from this is that Quartum is a very effective drug for malaria treatment among male children 04. The same tests were performed for female’s children which indicated that Across all 12 pairs, parasite load dropped substantially, a drop of 80% after 24 hours from taking Quartum with even higher correlation (0.795) and a significant difference from 0.
Comparison between quartum and artimether
Comparison between quartum and artemether for males: The descriptive Table 13, displays the mean, sample size, standard deviation, and standard error for both groups. For both treatment parasite loads dropped. The percentage difference of the dropped is 71% in favor of Quartum. Now we have to confirm that this large drop is not due to chance (Table 13).
Mean  N  Std. Deviation  Std. Error Mean  

Pair 1  Parasite after Artemether administration for males  41.92  12  8.743  2.524 
Parasite After Quartum administration for males  29.2500  12  7.16208  2.06752 
Table 13: Paired samples statistics for males. Source: SPSS output based on hypothetical samples of males and females under 5.
The test begins by showing correlation for parasite after Quartum and Artemether administration at (0.357) the correlation between the Quartum treatment and Artemether treatment levels is inverse butt statistically insignificant. Levels were lower overall, but the change was inconsistent across subjects (Table 14).
N  Correlation  Sig.  

Pair 1  Parasite after Artemether administration for males & Parasite After Quartum administration for males  12  .0.357  .255 
Table 14: Paired samples correlations for males. Source: SPSS output based on hypothetical samples of males and females under 5.
In Table 15, the Mean column in the pairedsamples t test table displays the average difference in parasite count for Quartum and Artemether experiment after treatment.
Paired Differences  T  df  Sig. (2tailed)  

Mean  Std. Deviation  Std. Error Mean  95% Confidence Interval of the Difference  
Lower  Upper  
Pair 1  Parasite after Artemether administration for males  Parasite After Quartum administration for males  12.66667  13.13104  3.79061  4.32360  21.00973  3.342  11  .007 
Table 15: Paired samples test for males. Source: SPSS output based on hypothetical samples of males and females under 5.
The Std. Deviation column displays the standard deviation of the average difference score. The Std. Error Mean column provides an index of the variability one can expect in repeated random samples of 12 patients as in this study. The 95% Confidence Interval of the Difference provides an estimate of the boundaries between which the true mean difference lies in 95% of our sample of 12. The t statistic is obtained by dividing the mean difference by its standard error.
The Sig. (2tailed) column displays the probability of obtaining a t statistic whose absolute value is equal to or greater than the obtained t statistic. Since the significance value for drop in parasite is less than 0.05, we can conclude that the average drop of (71%) of parasite per patient is not due to chance variation, and can be attributed to the treatment.
The conclusion from this is that Quartum is more effective drug for malaria treatment among male’s children 04 than Artemether. The conclusion from this is that Quartum is a very effective drug for malaria treatment among male children 04. The same tests were performed for female’s children which indicated that across all 12 pairs, parasite load dropped substantially, a drop of 80% after 24 hours from taking Quartum with moderate correlation (0.524) and a significant difference from 0. The conclusion to be drawn from this is that Quartum is more effective drug for malaria treatment among female’s children 04 than Artemether.
At present, probably the most common drugs commonly used, at last in Africa, and has some potency are the socalled artesunatemefloquine and artemetherlumefantrine, but this does not prevent that there is resistance to these drugs from the majority of patients. Using a MonteCarlo like trial, this illustrative study calculated the cure rate of malaria using these regimens found that both are highly effective. The study revealed that artemetherlumefantrine is more effective drug for malaria treatment among both males and females children 04 than Artemether This result has been confirmed for both males and females. If these results were drawn from real controlled data, then one would recommend artesunatemefloquine as first line antimalarial. We recommend that such trial be sponsored and conducted in Africa.
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