## T-Test and Its Nonparametric Mann-Whitney U-Test Statistics

T-test is one of analytical tools utilized to ascertain if the population mean is markedly different to hypothesized value or that of another population. Therefore, t-test measures t-distribution to determine whether two sample means are significantly different (Rosner, 2000). The t-test is employed when dependent variables are continuous ratios or interval scale variables, such as the total pain experienced. In addition, for the t-test to be applicable, independent variables should be two-level categorical ones (Vicky, 2009). Moreover, one can use the t-test even when the sizes of samples are considerably small, provided that the variables are distributed within each population groups. Additionally, the t-test can be used for smaller sample sizes provided that deviation of the counts within the populations is equal (Rosner, 2000). Additionally, statistical procedure of the t-test determines a p-value that shows similarity of results being obtained by chance with a distribution with n-1 degrees of freedom (Rosner, 2000).

Assumptions of the t-test imply that data must come from a continuous variable, the data must be distributed and the measurements independent of each other (Rosner, 2000). The t-test statistics is applied to compare a sample mean to a population mean, in which the test determines whether a sample comes from a population whose mean is known. This statistics is also referred to as one-sample t-test that measures the t value. T-test is also applied to compare means of two samples whether they are significantly different. This type of t-test is called two sample t-tests, which measures the f-distribution (Vicky, 2009). For example, in nursing, two-sample t-test will compare if the total pain in patients related to the treatment differs from the mean pain for the patients allocated to the placebo. The paired t-test compares two quantitative scores taken from the same sample to test whether there was a significant difference between the modes of treatment (Rosner, 2000).

Another use of the t-test or t-distribution is to set confidence limits for the mean. When the means of the samples from a population are normally distributed, 95% of these means will lie within 1.96 standard errors from the true parametric mean (Rosner, 2000). It can then be deduced with confidence that the probability that the true parametric mean lies within 1.96 standard errors is 95%. If the average of a sample lies in excess of 1.96 standard errors from the population mean, it is deduced with 95% certainty that the mean does not belong to the community. Additionally, on-tailed t-test measures a particular difference between two means, that is, whether one sample mean is greater or less than the other. Conversely, two-tailed t-test only measures whether there is a significant difference between the means of the two samples (Vicky, 2009).

Non-parametric statistics is not based upon parameterized groups of probability distributions. Non-parametric statistics includes both inferential and descriptive statistics. In addition, non-parametric statistics does not make any assumptions distribution probabilities for the assessed variables by contrast with parametric statistics (Easton & McColl, 2014). Non-parametric statistics is applied to study the samples that are ranked in an order. Non-parametric statistics is necessary when the data being assessed has a ranking, though no precise arithmetical interpretations (Easton & McColl, 2014). When particular assumptions regarding the underlying study population are questionable, non-parametric tests are used instead of their corresponding parametric tests (Easton & McColl, 2014).

In this regard, Mann-Whitney U-test is the most robust non-parametric statistic mean used to compare two populations (Easton & McColl, 2014). The test determines whether the null hypothesis that the two populations have similar distribution works in opposition to alternative assumption that the two distributions only differ with reference to the median. The Mann-Whitney U-test does not need supposition that the variables are regularly distributed between the two tested samples (Easton & McColl, 2014). In most instances, the Mann-Whitney U-test is used instead of the two-sample t-test parametric counterparts when the researcher questions assumption of normality. This type of statistics is also applicable when the sample data is observed in ranks, instead of direct measurements (Easton & McColl, 2014).

## Description of the Study in the Article by Jean-Louis et al.

The study by Jean-Louis et al. (2013), is a double-blind, randomized, multicenter pre-hospital trial to test efficacy of nitrous oxide for early analgesia treatment in emergency settings. The study aimed at demonstrating effectiveness of pre-mixed nitrous oxide plus oxygen in treating patients suffering from out-of-hospital modest traumatic acute pain (Jean-Louis et al. 2013). The study was carried out in Midi-Pyrenees district. Each patient was identified and eligibility criteria verified. The study examined patients aged 18 years and above with moderately acute pain caused by trauma on a numeric scale of rating of 6, 5, or 4 out of 10 scores. Sequence of randomization was determined using a random numbers table, through a restricted scheme of randomization equilibrated by 12 blocks to ensure nearly equal numbers in every group.

The study included testing of 648 outpatients and random prescription of medication to 60 subjects (Jean-Louis et al. 2013). The study did not record any statistically significant variations within the study categories regarding age (Jean-Louis et al. 2013). Additionally, the research did not observe any meaningful variations in the initial pain rates, IQR of 5 to 6, media 6, or other features (Jean-Louis et al. 2013). Fifteen minutes following the inhalation of the medication, 67% of the subjects under the nitrous oxide therapy showed a 3 or lower NRS score. 27% of the patients in the MA category had a 3 or below NRS rate (delta =40%, 95% CI=17% to 63%, p<0.001). There were significantly lower median pain rates in the category that received nitrous oxide at T15: 2 (IQR of one to four) against five (IQR of three to six, at P < 0.0001. This statistical difference was meaningful after five minutes of administering the therapy (Jean-Louis et al. 2013).

Examination at other evaluation times showed advanced ease of pain with administration of nitrous oxide at five and ten minutes. There was no difference five minutes following the T15 when each patient received the nitrous oxide (Jean-Louis et al. 2013). Inhalation of oxygen and nitrogen seemed clinically superior to the placebo for treating traumatic acute pain in a pre-hospital setting moderately (Jean-Louis et al. 2013). Additionally, the researchers found lower incidences of adverse drug effects without any severe complications. The results of analgesic treatment satisfied the caregivers as well as the patients in both test groups. The study found a variation in pain relief at an interval of five minutes and at all ensuing time points (Jean-Louis et al. 2013). Lastly, the study concluded that nurses can obtain early analgesia for the pre-hospital settings by using 50% oxygen and 50% nitrous oxide (Jean-Louis et al. 2013).

## Use of Statistics in the Study

The study utilized descriptive statistics to analyze the data obtained and reported it as means of samples with the standard deviations. The statistics were also reported as medians with IQRs (inter-quartile ranges) and as proportions with the exact binomial distribution of 95% confidence intervals, CIs (Jean-Louis et al. 2013). Fisher’s exact test and the chi-square tests were used to compare the proportions when appropriate. The t-test, two-tailed t-test was used to compare the means of the samples for data that was normally distributed (Jean-Louis et al. 2013). The non-parametric two-sample Mann-Whitney U-test was used to rank the sum and test the data that did not fit the parametric testing assumptions (Jean-Louis et al. 2013).

All of the participants who went on random assignment of the treatment were analyzed in line with the group treatment in a fashion of intention-to-treat (Jean-Louis et al. 2013). The researchers utilized the last available evaluation for missing values, assuming that there were no further improvements following the dropout. Conversely, all patients with randomized treatments were also included in the analysis of safety. The study also used preliminary observational data about 53 patients taken from one medical facility, to calculate the sample size (Jean-Louis et al. 2013). Statistical analysis showed that sixty percent of the patients with nitrous oxide and oxygen treatment-experienced relief of the pain in fifteen minutes (Jean-Louis et al. 2013). Twenty percent of the patients who received MA experienced pain relief after fifteen minutes. The study approach was designed in a superiority manner and a sample of 52 patients was calculated using a two-tailed test with a power of 90% and a 0.05 type one error. In addition, a p value of less than 0.05 was considered significant (Jean-Louis et al. 2013).

## Appropriateness of the Statistics Used

The use of two-tailed t-test is appropriate in this study since the hypothesis of the study aimed at finding out whether the treatment was effective, that is, the efficacy of nitrous analgesic in treatment of acute traumatic pain (Giuliano & Polanowicz, 2008). The researchers opted to use a two-tailed t-test in order to measure difference between the pain score following the treatment and the pain score following administration of placebo. The Mann-Whitney U-test was also appropriately used for the data sets that did not support the assumptions of its parametric counterpart test (Giuliano & Polanowicz, 2008). The assumption of normality was questionable for some data and the researchers opted to compare medians of the samples (Giuliano & Polanowicz, 2008). Therefore, two-tailed t-test is appropriate when comparing two data sets, though the t-test cannot be used because the assumptions have not been met; in this case, the Mann-Whitney U-test can be used instead.

## Assumptions of the Test

The three assumptions for the t-test were met in this study. The data was derived from a continuous variable. The second assumption that was satisfied is that the data was distributed normally (Giuliano & Polanowicz, 2008). Third, the measurements were independent for the data to be analyzed using the t-test. For the data analyzed using the nonparametric Mann-Whitney U-test, the assumptions of the t-test were not met (Giuliano & Polanowicz, 2008). The assumption of the normal distribution of data was not met as such data was not normally distributed (Giuliano & Polanowicz, 2008).

## The Level of Measurements

Nurse recording included the characteristics of age, height, sex, the type of traumatic injury, weight, blood pressure, baseline pain score, respiratory rate, pulse rate and oxygen saturation. The above measurements were recorded at the enrollment. Assessment of pain was obtained by asking the patients to rate their pain using an integer range of zero to ten, with zero meaning pain-free and ten referring to acute pain. Physiologic parameters recorded in the study include pulse rate, pulse oximetry, noninvasive blood pressure monitoring, respiratory rate, sedation level using a sedation scale as well as adverse events (Jean-Louis et al. 2013). Evaluation of safety and assessment of pain were carried out at the baseline, every five minutes for thirty minutes, and then each fifteen minutes until arrival at the hospital. The independent variables were the treatment, nitrous oxide plus oxygen and the placebo (Jean-Louis et al. 2013). The dependent variable was acute traumatic pain (Jean-Louis et al. 2013).

## Appropriateness of Measurements

The level of measures utilized in the study is appropriate as it helped the researchers to interpret the data obtained from the variables (Giuliano & Polanowicz, 2008). Additionally, this level of measurements was useful for taking decisions about statistical analysis suitable for the values assigned. The measurements were interval level and ordinal level, justifying the t-test and Mann-Whitney U-test statistics, respectively. The ways in which the variables in this study were defined and categorized, in turn, determined appropriateness of the statistics used (Giuliano & Polanowicz, 2008).

## Data Display

Data in this study was presented using tables, graphs and flow charts. A flow diagram was used to show the type and amount of adverse events the patients reported between T30 and randomization (Jean-Louis et al. 2013). In addition, a line graph was used to show the pain scores with indications when each patient received the oxygen and nitrous oxide. The study also used a bar graph to give a numeric rating scale of patients feeling analgesia (Jean-Louis et al. 2013). Foremost, tables showed sedation level, demographic and baseline clinical characteristics, as well as the patients’ clinical characteristics at T15 and adverse events (Jean-Louis et al. 2013).

### Appropriateness of Data Display

The graphs are used to display distribution of the data; that is, to indicate how the cases were distributed across the variable values (Jean-Louis et al. 2013). The form of distribution is described using the measures of central tendency. The bar chart was used to display distribution of the variables measured in discrete categories (Jean-Louis et al. 2013). Additionally, frequency polygon or line graph was used to represent percentages of the cases in each value (Jean-Louis et al. 2013). Finally, tables were also used correctly, to show the exact quantities of the data and display information.