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Data Science: the impact of statistics

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• Published: 16 February 2018
• Volume 6 , pages 189–194, ( 2018 )

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• Claus Weihs 1 &
• Katja Ickstadt 2  

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In this paper, we substantiate our premise that statistics is one of the most important disciplines to provide tools and methods to find structure in and to give deeper insight into data, and the most important discipline to analyze and quantify uncertainty. We give an overview over different proposed structures of Data Science and address the impact of statistics on such steps as data acquisition and enrichment, data exploration, data analysis and modeling, validation and representation and reporting. Also, we indicate fallacies when neglecting statistical reasoning.

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Data Analysis

Data science vs. statistics: two cultures?

Data Science: An Introduction

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1 Introduction and premise

Data Science as a scientific discipline is influenced by informatics, computer science, mathematics, operations research, and statistics as well as the applied sciences.

In 1996, for the first time, the term Data Science was included in the title of a statistical conference (International Federation of Classification Societies (IFCS) “Data Science, classification, and related methods”) [ 37 ]. Even though the term was founded by statisticians, in the public image of Data Science, the importance of computer science and business applications is often much more stressed, in particular in the era of Big Data.

Already in the 1970s, the ideas of John Tukey [ 43 ] changed the viewpoint of statistics from a purely mathematical setting , e.g., statistical testing, to deriving hypotheses from data ( exploratory setting ), i.e., trying to understand the data before hypothesizing.

Another root of Data Science is Knowledge Discovery in Databases (KDD) [ 36 ] with its sub-topic Data Mining . KDD already brings together many different approaches to knowledge discovery, including inductive learning, (Bayesian) statistics, query optimization, expert systems, information theory, and fuzzy sets. Thus, KDD is a big building block for fostering interaction between different fields for the overall goal of identifying knowledge in data.

Nowadays, these ideas are combined in the notion of Data Science, leading to different definitions. One of the most comprehensive definitions of Data Science was recently given by Cao as the formula [ 12 ]:

data science = (statistics + informatics + computing + communication + sociology + management) | (data + environment + thinking) .

In this formula, sociology stands for the social aspects and | (data + environment + thinking) means that all the mentioned sciences act on the basis of data, the environment and the so-called data-to-knowledge-to-wisdom thinking.

A recent, comprehensive overview of Data Science provided by Donoho in 2015 [ 16 ] focuses on the evolution of Data Science from statistics. Indeed, as early as 1997, there was an even more radical view suggesting to rename statistics to Data Science [ 50 ]. And in 2015, a number of ASA leaders [ 17 ] released a statement about the role of statistics in Data Science, saying that “statistics and machine learning play a central role in data science.”

In our view, statistical methods are crucial in most fundamental steps of Data Science. Hence, the premise of our contribution is:

Statistics is one of the most important disciplines to provide tools and methods to find structure in and to give deeper insight into data, and the most important discipline to analyze and quantify uncertainty.

This paper aims at addressing the major impact of statistics on the most important steps in Data Science.

2 Steps in data science

One of forerunners of Data Science from a structural perspective is the famous CRISP-DM (Cross Industry Standard Process for Data Mining) which is organized in six main steps: Business Understanding, Data Understanding, Data Preparation, Modeling, Evaluation, and Deployment [ 10 ], see Table  1 , left column. Ideas like CRISP-DM are now fundamental for applied statistics.

In our view, the main steps in Data Science have been inspired by CRISP-DM and have evolved, leading to, e.g., our definition of Data Science as a sequence of the following steps: Data Acquisition and Enrichment, Data Storage and Access , Data Exploration, Data Analysis and Modeling, Optimization of Algorithms , Model Validation and Selection, Representation and Reporting of Results, and Business Deployment of Results . Note that topics in small capitals indicate steps where statistics is less involved, cp. Table  1 , right column.

Usually, these steps are not just conducted once but are iterated in a cyclic loop. In addition, it is common to alternate between two or more steps. This holds especially for the steps Data Acquisition and Enrichment , Data Exploration , and Statistical Data Analysis , as well as for Statistical Data Analysis and Modeling and Model Validation and Selection .

Table  1 compares different definitions of steps in Data Science. The relationship of terms is indicated by horizontal blocks. The missing step Data Acquisition and Enrichment in CRISP-DM indicates that that scheme deals with observational data only. Moreover, in our proposal, the steps Data Storage and Access and Optimization of Algorithms are added to CRISP-DM, where statistics is less involved.

The list of steps for Data Science may even be enlarged, see, e.g., Cao in [ 12 ], Figure 6, cp. also Table  1 , middle column, for the following recent list: Domain-specific Data Applications and Problems, Data Storage and Management, Data Quality Enhancement, Data Modeling and Representation, Deep Analytics, Learning and Discovery, Simulation and Experiment Design, High-performance Processing and Analytics, Networking, Communication, Data-to-Decision and Actions.

In principle, Cao’s and our proposal cover the same main steps. However, in parts, Cao’s formulation is more detailed; e.g., our step Data Analysis and Modeling corresponds to Data Modeling and Representation, Deep Analytics, Learning and Discovery . Also, the vocabularies differ slightly, depending on whether the respective background is computer science or statistics. In that respect note that Experiment Design in Cao’s definition means the design of the simulation experiments.

In what follows, we will highlight the role of statistics discussing all the steps, where it is heavily involved, in Sects.  2.1 – 2.6 . These coincide with all steps in our proposal in Table  1 except steps in small capitals. The corresponding entries Data Storage and Access and Optimization of Algorithms are mainly covered by informatics and computer science , whereas Business Deployment of Results is covered by Business Management .

2.1 Data acquisition and enrichment

Design of experiments (DOE) is essential for a systematic generation of data when the effect of noisy factors has to be identified. Controlled experiments are fundamental for robust process engineering to produce reliable products despite variation in the process variables. On the one hand, even controllable factors contain a certain amount of uncontrollable variation that affects the response. On the other hand, some factors, like environmental factors, cannot be controlled at all. Nevertheless, at least the effect of such noisy influencing factors should be controlled by, e.g., DOE.

DOE can be utilized, e.g.,

to systematically generate new data ( data acquisition ) [ 33 ],

for systematically reducing data bases [ 41 ], and

for tuning (i.e., optimizing) parameters of algorithms [ 1 ], i.e., for improving the data analysis methods (see Sect.  2.3 ) themselves.

Simulations [ 7 ] may also be used to generate new data. A tool for the enrichment of data bases to fill data gaps is the imputation of missing data [ 31 ].

Such statistical methods for data generation and enrichment need to be part of the backbone of Data Science. The exclusive use of observational data without any noise control distinctly diminishes the quality of data analysis results and may even lead to wrong result interpretation. The hope for “The End of Theory: The Data Deluge Makes the Scientific Method Obsolete” [ 4 ] appears to be wrong due to noise in the data.

Thus, experimental design is crucial for the reliability, validity, and replicability of our results.

2.2 Data exploration

Exploratory statistics is essential for data preprocessing to learn about the contents of a data base. Exploration and visualization of observed data was, in a way, initiated by John Tukey [ 43 ]. Since that time, the most laborious part of data analysis, namely data understanding and transformation, became an important part in statistical science.

Data exploration or data mining is fundamental for the proper usage of analytical methods in Data Science. The most important contribution of statistics is the notion of distribution . It allows us to represent variability in the data as well as (a-priori) knowledge of parameters, the concept underlying Bayesian statistics. Distributions also enable us to choose adequate subsequent analytic models and methods.

2.3 Statistical data analysis

Finding structure in data and making predictions are the most important steps in Data Science. Here, in particular, statistical methods are essential since they are able to handle many different analytical tasks. Important examples of statistical data analysis methods are the following.

Hypothesis testing is one of the pillars of statistical analysis. Questions arising in data driven problems can often be translated to hypotheses. Also, hypotheses are the natural links between underlying theory and statistics. Since statistical hypotheses are related to statistical tests, questions and theory can be tested for the available data. Multiple usage of the same data in different tests often leads to the necessity to correct significance levels. In applied statistics, correct multiple testing is one of the most important problems, e.g., in pharmaceutical studies [ 15 ]. Ignoring such techniques would lead to many more significant results than justified.

Classification methods are basic for finding and predicting subpopulations from data. In the so-called unsupervised case, such subpopulations are to be found from a data set without a-priori knowledge of any cases of such subpopulations. This is often called clustering.

In the so-called supervised case, classification rules should be found from a labeled data set for the prediction of unknown labels when only influential factors are available.

Nowadays, there is a plethora of methods for the unsupervised [ 22 ] as well for the supervised case [ 2 ].

In the age of Big Data, a new look at the classical methods appears to be necessary, though, since most of the time the calculation effort of complex analysis methods grows stronger than linear with the number of observations n or the number of features p . In the case of Big Data, i.e., if n or p is large, this leads to too high calculation times and to numerical problems. This results both, in the comeback of simpler optimization algorithms with low time-complexity [ 9 ] and in re-examining the traditional methods in statistics and machine learning for Big Data [ 46 ].

Regression methods are the main tool to find global and local relationships between features when the target variable is measured. Depending on the distributional assumption for the underlying data, different approaches may be applied. Under the normality assumption, linear regression is the most common method, while generalized linear regression is usually employed for other distributions from the exponential family [ 18 ]. More advanced methods comprise functional regression for functional data [ 38 ], quantile regression [ 25 ], and regression based on loss functions other than squared error loss like, e.g., Lasso regression [ 11 , 21 ]. In the context of Big Data, the challenges are similar to those for classification methods given large numbers of observations n (e.g., in data streams) and / or large numbers of features p . For the reduction of n , data reduction techniques like compressed sensing, random projection methods [ 20 ] or sampling-based procedures [ 28 ] enable faster computations. For decreasing the number p to the most influential features, variable selection or shrinkage approaches like the Lasso [ 21 ] can be employed, keeping the interpretability of the features. (Sparse) principal component analysis [ 21 ] may also be used.

Time series analysis aims at understanding and predicting temporal structure [ 42 ]. Time series are very common in studies of observational data, and prediction is the most important challenge for such data. Typical application areas are the behavioral sciences and economics as well as the natural sciences and engineering. As an example, let us have a look at signal analysis, e.g., speech or music data analysis. Here, statistical methods comprise the analysis of models in the time and frequency domains. The main aim is the prediction of future values of the time series itself or of its properties. For example, the vibrato of an audio time series might be modeled in order to realistically predict the tone in the future [ 24 ] and the fundamental frequency of a musical tone might be predicted by rules learned from elapsed time periods [ 29 ].

In econometrics, multiple time series and their co-integration are often analyzed [ 27 ]. In technical applications, process control is a common aim of time series analysis [ 34 ].

2.4 Statistical modeling

Complex interactions between factors can be modeled by graphs or networks . Here, an interaction between two factors is modeled by a connection in the graph or network [ 26 , 35 ]. The graphs can be undirected as, e.g., in Gaussian graphical models, or directed as, e.g., in Bayesian networks. The main goal in network analysis is deriving the network structure. Sometimes, it is necessary to separate (unmix) subpopulation specific network topologies [ 49 ].

Stochastic differential and difference equations can represent models from the natural and engineering sciences [ 3 , 39 ]. The finding of approximate statistical models solving such equations can lead to valuable insights for, e.g., the statistical control of such processes, e.g., in mechanical engineering [ 48 ]. Such methods can build a bridge between the applied sciences and Data Science.

Local models and globalization Typically, statistical models are only valid in sub-regions of the domain of the involved variables. Then, local models can be used [ 8 ]. The analysis of structural breaks can be basic to identify the regions for local modeling in time series [ 5 ]. Also, the analysis of concept drifts can be used to investigate model changes over time [ 30 ].

In time series, there are often hierarchies of more and more global structures. For example, in music, a basic local structure is given by the notes and more and more global ones by bars, motifs, phrases, parts etc. In order to find global properties of a time series, properties of the local models can be combined to more global characteristics [ 47 ].

Mixture models can also be used for the generalization of local to global models [ 19 , 23 ]. Model combination is essential for the characterization of real relationships since standard mathematical models are often much too simple to be valid for heterogeneous data or bigger regions of interest.

2.5 Model validation and model selection

In cases where more than one model is proposed for, e.g., prediction, statistical tests for comparing models are helpful to structure the models, e.g., concerning their predictive power [ 45 ].

Predictive power is typically assessed by means of so-called resampling methods where the distribution of power characteristics is studied by artificially varying the subpopulation used to learn the model. Characteristics of such distributions can be used for model selection [ 7 ].

Perturbation experiments offer another possibility to evaluate the performance of models. In this way, the stability of the different models against noise is assessed [ 32 , 44 ].

Meta-analysis as well as model averaging are methods to evaluate combined models [ 13 , 14 ].

Model selection became more and more important in the last years since the number of classification and regression models proposed in the literature increased with higher and higher speed.

2.6 Representation and reporting

Visualization to interpret found structures and storing of models in an easy-to-update form are very important tasks in statistical analyses to communicate the results and safeguard data analysis deployment. Deployment is decisive for obtaining interpretable results in Data Science. It is the last step in CRISP-DM [ 10 ] and underlying the data-to-decision and action step in Cao [ 12 ].

Besides visualization and adequate model storing, for statistics, the main task is reporting of uncertainties and review [ 6 ].

3 Fallacies

The statistical methods described in Sect.  2 are fundamental for finding structure in data and for obtaining deeper insight into data, and thus, for a successful data analysis. Ignoring modern statistical thinking or using simplistic data analytics/statistical methods may lead to avoidable fallacies. This holds, in particular, for the analysis of big and/or complex data.

As mentioned at the end of Sect.  2.2 , the notion of distribution is the key contribution of statistics. Not taking into account distributions in data exploration and in modeling restricts us to report values and parameter estimates without their corresponding variability. Only the notion of distributions enables us to predict with corresponding error bands.

Moreover, distributions are the key to model-based data analytics. For example, unsupervised learning can be employed to find clusters in data. If additional structure like dependency on space or time is present, it is often important to infer parameters like cluster radii and their spatio-temporal evolution. Such model-based analysis heavily depends on the notion of distributions (see [ 40 ] for an application to protein clusters).

If more than one parameter is of interest, it is advisable to compare univariate hypothesis testing approaches to multiple procedures, e.g., in multiple regression, and choose the most adequate model by variable selection. Restricting oneself to univariate testing, would ignore relationships between variables.

Deeper insight into data might require more complex models, like, e.g., mixture models for detecting heterogeneous groups in data. When ignoring the mixture, the result often represents a meaningless average, and learning the subgroups by unmixing the components might be needed. In a Bayesian framework, this is enabled by, e.g., latent allocation variables in a Dirichlet mixture model. For an application of decomposing a mixture of different networks in a heterogeneous cell population in molecular biology see [ 49 ].

A mixture model might represent mixtures of components of very unequal sizes, with small components (outliers) being of particular importance. In the context of Big Data, naïve sampling procedures are often employed for model estimation. However, these have the risk of missing small mixture components. Hence, model validation or sampling according to a more suitable distribution as well as resampling methods for predictive power are important.

4 Conclusion

Following the above assessment of the capabilities and impacts of statistics our conclusion is:

The role of statistics in Data Science is under-estimated as, e.g., compared to computer science. This yields, in particular, for the areas of data acquisition and enrichment as well as for advanced modeling needed for prediction.

Stimulated by this conclusion, statisticians are well-advised to more offensively play their role in this modern and well accepted field of Data Science.

Only complementing and/or combining mathematical methods and computational algorithms with statistical reasoning, particularly for Big Data, will lead to scientific results based on suitable approaches. Ultimately, only a balanced interplay of all sciences involved will lead to successful solutions in Data Science.

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Acknowledgements

The authors would like to thank the editor, the guest editors and all reviewers for valuable comments on an earlier version of the manuscript. They also thank Leo Geppert for fruitful discussions.

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Weihs, C., Ickstadt, K. Data Science: the impact of statistics. Int J Data Sci Anal 6 , 189–194 (2018). https://doi.org/10.1007/s41060-018-0102-5

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• v.60(9); 2016 Sep

Basic statistical tools in research and data analysis

Zulfiqar ali.

Department of Anaesthesiology, Division of Neuroanaesthesiology, Sheri Kashmir Institute of Medical Sciences, Soura, Srinagar, Jammu and Kashmir, India

S Bala Bhaskar

1 Department of Anaesthesiology and Critical Care, Vijayanagar Institute of Medical Sciences, Bellary, Karnataka, India

Statistical methods involved in carrying out a study include planning, designing, collecting data, analysing, drawing meaningful interpretation and reporting of the research findings. The statistical analysis gives meaning to the meaningless numbers, thereby breathing life into a lifeless data. The results and inferences are precise only if proper statistical tests are used. This article will try to acquaint the reader with the basic research tools that are utilised while conducting various studies. The article covers a brief outline of the variables, an understanding of quantitative and qualitative variables and the measures of central tendency. An idea of the sample size estimation, power analysis and the statistical errors is given. Finally, there is a summary of parametric and non-parametric tests used for data analysis.

INTRODUCTION

Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population.[ 1 ] This requires a proper design of the study, an appropriate selection of the study sample and choice of a suitable statistical test. An adequate knowledge of statistics is necessary for proper designing of an epidemiological study or a clinical trial. Improper statistical methods may result in erroneous conclusions which may lead to unethical practice.[ 2 ]

Variable is a characteristic that varies from one individual member of population to another individual.[ 3 ] Variables such as height and weight are measured by some type of scale, convey quantitative information and are called as quantitative variables. Sex and eye colour give qualitative information and are called as qualitative variables[ 3 ] [ Figure 1 ].

Classification of variables

Quantitative variables

Quantitative or numerical data are subdivided into discrete and continuous measurements. Discrete numerical data are recorded as a whole number such as 0, 1, 2, 3,… (integer), whereas continuous data can assume any value. Observations that can be counted constitute the discrete data and observations that can be measured constitute the continuous data. Examples of discrete data are number of episodes of respiratory arrests or the number of re-intubations in an intensive care unit. Similarly, examples of continuous data are the serial serum glucose levels, partial pressure of oxygen in arterial blood and the oesophageal temperature.

A hierarchical scale of increasing precision can be used for observing and recording the data which is based on categorical, ordinal, interval and ratio scales [ Figure 1 ].

Categorical or nominal variables are unordered. The data are merely classified into categories and cannot be arranged in any particular order. If only two categories exist (as in gender male and female), it is called as a dichotomous (or binary) data. The various causes of re-intubation in an intensive care unit due to upper airway obstruction, impaired clearance of secretions, hypoxemia, hypercapnia, pulmonary oedema and neurological impairment are examples of categorical variables.

Ordinal variables have a clear ordering between the variables. However, the ordered data may not have equal intervals. Examples are the American Society of Anesthesiologists status or Richmond agitation-sedation scale.

Interval variables are similar to an ordinal variable, except that the intervals between the values of the interval variable are equally spaced. A good example of an interval scale is the Fahrenheit degree scale used to measure temperature. With the Fahrenheit scale, the difference between 70° and 75° is equal to the difference between 80° and 85°: The units of measurement are equal throughout the full range of the scale.

Ratio scales are similar to interval scales, in that equal differences between scale values have equal quantitative meaning. However, ratio scales also have a true zero point, which gives them an additional property. For example, the system of centimetres is an example of a ratio scale. There is a true zero point and the value of 0 cm means a complete absence of length. The thyromental distance of 6 cm in an adult may be twice that of a child in whom it may be 3 cm.

STATISTICS: DESCRIPTIVE AND INFERENTIAL STATISTICS

Descriptive statistics[ 4 ] try to describe the relationship between variables in a sample or population. Descriptive statistics provide a summary of data in the form of mean, median and mode. Inferential statistics[ 4 ] use a random sample of data taken from a population to describe and make inferences about the whole population. It is valuable when it is not possible to examine each member of an entire population. The examples if descriptive and inferential statistics are illustrated in Table 1 .

Example of descriptive and inferential statistics

Descriptive statistics

The extent to which the observations cluster around a central location is described by the central tendency and the spread towards the extremes is described by the degree of dispersion.

Measures of central tendency

The measures of central tendency are mean, median and mode.[ 6 ] Mean (or the arithmetic average) is the sum of all the scores divided by the number of scores. Mean may be influenced profoundly by the extreme variables. For example, the average stay of organophosphorus poisoning patients in ICU may be influenced by a single patient who stays in ICU for around 5 months because of septicaemia. The extreme values are called outliers. The formula for the mean is

where x = each observation and n = number of observations. Median[ 6 ] is defined as the middle of a distribution in a ranked data (with half of the variables in the sample above and half below the median value) while mode is the most frequently occurring variable in a distribution. Range defines the spread, or variability, of a sample.[ 7 ] It is described by the minimum and maximum values of the variables. If we rank the data and after ranking, group the observations into percentiles, we can get better information of the pattern of spread of the variables. In percentiles, we rank the observations into 100 equal parts. We can then describe 25%, 50%, 75% or any other percentile amount. The median is the 50 th percentile. The interquartile range will be the observations in the middle 50% of the observations about the median (25 th -75 th percentile). Variance[ 7 ] is a measure of how spread out is the distribution. It gives an indication of how close an individual observation clusters about the mean value. The variance of a population is defined by the following formula:

where σ 2 is the population variance, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The variance of a sample is defined by slightly different formula:

where s 2 is the sample variance, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. The formula for the variance of a population has the value ‘ n ’ as the denominator. The expression ‘ n −1’ is known as the degrees of freedom and is one less than the number of parameters. Each observation is free to vary, except the last one which must be a defined value. The variance is measured in squared units. To make the interpretation of the data simple and to retain the basic unit of observation, the square root of variance is used. The square root of the variance is the standard deviation (SD).[ 8 ] The SD of a population is defined by the following formula:

where σ is the population SD, X is the population mean, X i is the i th element from the population and N is the number of elements in the population. The SD of a sample is defined by slightly different formula:

where s is the sample SD, x is the sample mean, x i is the i th element from the sample and n is the number of elements in the sample. An example for calculation of variation and SD is illustrated in Table 2 .

Example of mean, variance, standard deviation

Normal distribution or Gaussian distribution

Most of the biological variables usually cluster around a central value, with symmetrical positive and negative deviations about this point.[ 1 ] The standard normal distribution curve is a symmetrical bell-shaped. In a normal distribution curve, about 68% of the scores are within 1 SD of the mean. Around 95% of the scores are within 2 SDs of the mean and 99% within 3 SDs of the mean [ Figure 2 ].

Normal distribution curve

Skewed distribution

It is a distribution with an asymmetry of the variables about its mean. In a negatively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the right of Figure 1 . In a positively skewed distribution [ Figure 3 ], the mass of the distribution is concentrated on the left of the figure leading to a longer right tail.

Curves showing negatively skewed and positively skewed distribution

Inferential statistics

In inferential statistics, data are analysed from a sample to make inferences in the larger collection of the population. The purpose is to answer or test the hypotheses. A hypothesis (plural hypotheses) is a proposed explanation for a phenomenon. Hypothesis tests are thus procedures for making rational decisions about the reality of observed effects.

Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty).

In inferential statistics, the term ‘null hypothesis’ ( H 0 ‘ H-naught ,’ ‘ H-null ’) denotes that there is no relationship (difference) between the population variables in question.[ 9 ]

Alternative hypothesis ( H 1 and H a ) denotes that a statement between the variables is expected to be true.[ 9 ]

The P value (or the calculated probability) is the probability of the event occurring by chance if the null hypothesis is true. The P value is a numerical between 0 and 1 and is interpreted by researchers in deciding whether to reject or retain the null hypothesis [ Table 3 ].

P values with interpretation

If P value is less than the arbitrarily chosen value (known as α or the significance level), the null hypothesis (H0) is rejected [ Table 4 ]. However, if null hypotheses (H0) is incorrectly rejected, this is known as a Type I error.[ 11 ] Further details regarding alpha error, beta error and sample size calculation and factors influencing them are dealt with in another section of this issue by Das S et al .[ 12 ]

Illustration for null hypothesis

PARAMETRIC AND NON-PARAMETRIC TESTS

Numerical data (quantitative variables) that are normally distributed are analysed with parametric tests.[ 13 ]

Two most basic prerequisites for parametric statistical analysis are:

• The assumption of normality which specifies that the means of the sample group are normally distributed
• The assumption of equal variance which specifies that the variances of the samples and of their corresponding population are equal.

However, if the distribution of the sample is skewed towards one side or the distribution is unknown due to the small sample size, non-parametric[ 14 ] statistical techniques are used. Non-parametric tests are used to analyse ordinal and categorical data.

Parametric tests

The parametric tests assume that the data are on a quantitative (numerical) scale, with a normal distribution of the underlying population. The samples have the same variance (homogeneity of variances). The samples are randomly drawn from the population, and the observations within a group are independent of each other. The commonly used parametric tests are the Student's t -test, analysis of variance (ANOVA) and repeated measures ANOVA.

Student's t -test

Student's t -test is used to test the null hypothesis that there is no difference between the means of the two groups. It is used in three circumstances:

where X = sample mean, u = population mean and SE = standard error of mean

where X 1 − X 2 is the difference between the means of the two groups and SE denotes the standard error of the difference.

• To test if the population means estimated by two dependent samples differ significantly (the paired t -test). A usual setting for paired t -test is when measurements are made on the same subjects before and after a treatment.

The formula for paired t -test is:

where d is the mean difference and SE denotes the standard error of this difference.

The group variances can be compared using the F -test. The F -test is the ratio of variances (var l/var 2). If F differs significantly from 1.0, then it is concluded that the group variances differ significantly.

Analysis of variance

The Student's t -test cannot be used for comparison of three or more groups. The purpose of ANOVA is to test if there is any significant difference between the means of two or more groups.

In ANOVA, we study two variances – (a) between-group variability and (b) within-group variability. The within-group variability (error variance) is the variation that cannot be accounted for in the study design. It is based on random differences present in our samples.

However, the between-group (or effect variance) is the result of our treatment. These two estimates of variances are compared using the F-test.

A simplified formula for the F statistic is:

where MS b is the mean squares between the groups and MS w is the mean squares within groups.

Repeated measures analysis of variance

As with ANOVA, repeated measures ANOVA analyses the equality of means of three or more groups. However, a repeated measure ANOVA is used when all variables of a sample are measured under different conditions or at different points in time.

As the variables are measured from a sample at different points of time, the measurement of the dependent variable is repeated. Using a standard ANOVA in this case is not appropriate because it fails to model the correlation between the repeated measures: The data violate the ANOVA assumption of independence. Hence, in the measurement of repeated dependent variables, repeated measures ANOVA should be used.

Non-parametric tests

When the assumptions of normality are not met, and the sample means are not normally, distributed parametric tests can lead to erroneous results. Non-parametric tests (distribution-free test) are used in such situation as they do not require the normality assumption.[ 15 ] Non-parametric tests may fail to detect a significant difference when compared with a parametric test. That is, they usually have less power.

As is done for the parametric tests, the test statistic is compared with known values for the sampling distribution of that statistic and the null hypothesis is accepted or rejected. The types of non-parametric analysis techniques and the corresponding parametric analysis techniques are delineated in Table 5 .

Analogue of parametric and non-parametric tests

Median test for one sample: The sign test and Wilcoxon's signed rank test

The sign test and Wilcoxon's signed rank test are used for median tests of one sample. These tests examine whether one instance of sample data is greater or smaller than the median reference value.

This test examines the hypothesis about the median θ0 of a population. It tests the null hypothesis H0 = θ0. When the observed value (Xi) is greater than the reference value (θ0), it is marked as+. If the observed value is smaller than the reference value, it is marked as − sign. If the observed value is equal to the reference value (θ0), it is eliminated from the sample.

If the null hypothesis is true, there will be an equal number of + signs and − signs.

The sign test ignores the actual values of the data and only uses + or − signs. Therefore, it is useful when it is difficult to measure the values.

Wilcoxon's signed rank test

There is a major limitation of sign test as we lose the quantitative information of the given data and merely use the + or – signs. Wilcoxon's signed rank test not only examines the observed values in comparison with θ0 but also takes into consideration the relative sizes, adding more statistical power to the test. As in the sign test, if there is an observed value that is equal to the reference value θ0, this observed value is eliminated from the sample.

Wilcoxon's rank sum test ranks all data points in order, calculates the rank sum of each sample and compares the difference in the rank sums.

Mann-Whitney test

It is used to test the null hypothesis that two samples have the same median or, alternatively, whether observations in one sample tend to be larger than observations in the other.

Mann–Whitney test compares all data (xi) belonging to the X group and all data (yi) belonging to the Y group and calculates the probability of xi being greater than yi: P (xi > yi). The null hypothesis states that P (xi > yi) = P (xi < yi) =1/2 while the alternative hypothesis states that P (xi > yi) ≠1/2.

Kolmogorov-Smirnov test

The two-sample Kolmogorov-Smirnov (KS) test was designed as a generic method to test whether two random samples are drawn from the same distribution. The null hypothesis of the KS test is that both distributions are identical. The statistic of the KS test is a distance between the two empirical distributions, computed as the maximum absolute difference between their cumulative curves.

Kruskal-Wallis test

The Kruskal–Wallis test is a non-parametric test to analyse the variance.[ 14 ] It analyses if there is any difference in the median values of three or more independent samples. The data values are ranked in an increasing order, and the rank sums calculated followed by calculation of the test statistic.

Jonckheere test

In contrast to Kruskal–Wallis test, in Jonckheere test, there is an a priori ordering that gives it a more statistical power than the Kruskal–Wallis test.[ 14 ]

Friedman test

The Friedman test is a non-parametric test for testing the difference between several related samples. The Friedman test is an alternative for repeated measures ANOVAs which is used when the same parameter has been measured under different conditions on the same subjects.[ 13 ]

Tests to analyse the categorical data

Chi-square test, Fischer's exact test and McNemar's test are used to analyse the categorical or nominal variables. The Chi-square test compares the frequencies and tests whether the observed data differ significantly from that of the expected data if there were no differences between groups (i.e., the null hypothesis). It is calculated by the sum of the squared difference between observed ( O ) and the expected ( E ) data (or the deviation, d ) divided by the expected data by the following formula:

A Yates correction factor is used when the sample size is small. Fischer's exact test is used to determine if there are non-random associations between two categorical variables. It does not assume random sampling, and instead of referring a calculated statistic to a sampling distribution, it calculates an exact probability. McNemar's test is used for paired nominal data. It is applied to 2 × 2 table with paired-dependent samples. It is used to determine whether the row and column frequencies are equal (that is, whether there is ‘marginal homogeneity’). The null hypothesis is that the paired proportions are equal. The Mantel-Haenszel Chi-square test is a multivariate test as it analyses multiple grouping variables. It stratifies according to the nominated confounding variables and identifies any that affects the primary outcome variable. If the outcome variable is dichotomous, then logistic regression is used.

SOFTWARES AVAILABLE FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

Numerous statistical software systems are available currently. The commonly used software systems are Statistical Package for the Social Sciences (SPSS – manufactured by IBM corporation), Statistical Analysis System ((SAS – developed by SAS Institute North Carolina, United States of America), R (designed by Ross Ihaka and Robert Gentleman from R core team), Minitab (developed by Minitab Inc), Stata (developed by StataCorp) and the MS Excel (developed by Microsoft).

There are a number of web resources which are related to statistical power analyses. A few are:

• StatPages.net – provides links to a number of online power calculators
• G-Power – provides a downloadable power analysis program that runs under DOS
• Power analysis for ANOVA designs an interactive site that calculates power or sample size needed to attain a given power for one effect in a factorial ANOVA design
• SPSS makes a program called SamplePower. It gives an output of a complete report on the computer screen which can be cut and paste into another document.

It is important that a researcher knows the concepts of the basic statistical methods used for conduct of a research study. This will help to conduct an appropriately well-designed study leading to valid and reliable results. Inappropriate use of statistical techniques may lead to faulty conclusions, inducing errors and undermining the significance of the article. Bad statistics may lead to bad research, and bad research may lead to unethical practice. Hence, an adequate knowledge of statistics and the appropriate use of statistical tests are important. An appropriate knowledge about the basic statistical methods will go a long way in improving the research designs and producing quality medical research which can be utilised for formulating the evidence-based guidelines.

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