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## Effective Use of Statistics in Research – Methods and Tools for Data Analysis

## Role of Statistics in Biological Research

## 1. Establishing a Sample Size

## 2. Testing of Hypothesis

## 3. Data Interpretation Through Analysis

## Types of Statistical Research Methods That Aid in Data Analysis

## 1. Descriptive Analysis

## 2. Inferential Analysis

## 3. Predictive Analysis

## 4. Prescriptive Analysis

## 5. Exploratory Data Analysis

## 6. Causal Analysis

## 7. Mechanistic Analysis

## Important Statistical Tools In Research

## 1. Statistical Package for Social Science (SPSS)

## 2. R Foundation for Statistical Computing

## 3. MATLAB (The Mathworks)

## 4. Microsoft Excel

## 5. Statistical Analysis Software (SAS)

## 6. GraphPad Prism

## Use of Statistical Tools In Research and Data Analysis

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## Basic statistical tools in research and data analysis

## S Bala Bhaskar

## INTRODUCTION

## Quantitative variables

## STATISTICS: DESCRIPTIVE AND INFERENTIAL STATISTICS

Example of descriptive and inferential statistics

## Descriptive statistics

## Measures of central tendency

Example of mean, variance, standard deviation

## Normal distribution or Gaussian distribution

## Skewed distribution

Curves showing negatively skewed and positively skewed distribution

## Inferential statistics

Illustration for null hypothesis

## PARAMETRIC AND NON-PARAMETRIC TESTS

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.

## Parametric tests

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

The formula for paired t -test is:

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

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

## Non-parametric tests

Analogue of parametric and non-parametric tests

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

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

## Tests to analyse the categorical data

## SOFTWARES AVAILABLE FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

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.

## Financial support and sponsorship

There are no conflicts of interest.

6.1 Introduction 6.2 Definitions 6.3 Basic Statistics 6.4 Statistical tests

6.2.1 Error 6.2.2 Accuracy 6.2.3 Precision 6.2.4 Bias

1. Random or unpredictable deviations between replicates, quantified with the "standard deviation". 2. Systematic or predictable regular deviation from the "true" value, quantified as "mean difference" (i.e. the difference between the true value and the mean of replicate determinations). 3. Constant, unrelated to the concentration of the substance analyzed (the analyte). 4. Proportional, i.e. related to the concentration of the analyte. * The "true" value of an attribute is by nature indeterminate and often has only a very relative meaning. Particularly in soil science for several attributes there is no such thing as the true value as any value obtained is method-dependent (e.g. cation exchange capacity). Obviously, this does not mean that no adequate analysis serving a purpose is possible. It does, however, emphasize the need for the establishment of standard reference methods and the importance of external QC (see Chapter 9).

The difference between the (mean) test result obtained from a number of laboratories using the same method and an accepted reference value. The method bias may depend on the analyte level.

The difference between the (mean) test result from a particular laboratory and the accepted reference value.

The difference between the mean of replicate test results of a sample and the ("true") value of the target population from which the sample was taken. In practice, for a laboratory this refers mainly to sample preparation, subsampling and weighing techniques. Whether a sample is representative for the population in the field is an extremely important aspect but usually falls outside the responsibility of the laboratory (in some cases laboratories have their own field sampling personnel).

6.3.1 Mean 6.3.2 Standard deviation 6.3.3 Relative standard deviation. Coefficient of variation 6.3.4 Confidence limits of a measurement 6.3.5 Propagation of errors

Note. When needed (e.g. for the F -test, see Eq. 6.11) the variance can, of course, be calculated by squaring the standard deviation:

m = "true" value (mean of large set of replicates) ¯x = mean of subsamples t = a statistical value which depends on the number of data and the required confidence (usually 95%). s = standard deviation of mean of subsamples n = number of subsamples

m = "true" value x = single measurement t = applicable t tab (Appendix 1) s = standard deviation of set of previous measurements.

Note: This "method-s" or s of a control sample is not a constant and may vary for different test materials, analyte levels, and with analytical conditions.

¯x = mean of duplicates s = known standard deviation of large set

6.3.5.1. Propagation of random errors 6.3.5.2 Propagation of systematic errors

a = ml HCl required for titration sample b = ml HCl required for titration blank s = air-dry sample weight in gram M = molarity of HCl 1.4 = 14×10 -3 ×100% (14 = atomic weight of N) mcf = moisture correction factor

distillation: 0.8%, titration: 0.5%, molarity: 0.2%, sample weight: 0.2%, mcf: 0.2%.

Note. Sample heterogeneity is also represented in the moisture correction factor. However, the influence of this factor on the final result is usually very small.

6.4.1 Two-sided vs. one-sided test 6.4.2 F-test for precision 6.4.3 t-Tests for bias 6.4.4 Linear correlation and regression 6.4.5 Analysis of variance (ANOVA)

- performance of two instruments, - performance of two methods, - performance of a procedure in different periods, - performance of two analysts or laboratories, - results obtained for a reference or control sample with the "true", "target" or "assigned" value of this sample.

1. are A and B different? (two-sided test) 2. is A higher (or lower) than B? (one-sided test).

df 1 = n 1 -1 df 2 = n 2 -1

6.4.3.1. Student's t-test 6.4.3.2 Cochran's t-test 6.4.3.3 t-Test for large data sets (n ³ 30) 6.4.3.4 Paired t-test

1. Student's t-test for comparison of two independent sets of data with very similar standard deviations; 2. the Cochran variant of the t -test when the standard deviations of the independent sets differ significantly; 3. the paired t- test for comparison of strongly dependent sets of data.

¯x = mean of test results of a sample m = "true" or reference value s = standard deviation of test results n = number of test results of the sample.

¯x 1 = mean of data set 1 ¯x 2 = mean of data set 2 s p = "pooled" standard deviation of the sets n 1 = number of data in set 1 n 2 = number of data in set 2.

s 1 = standard deviation of data set 1 s 2 = standard deviation of data set 2 n 1 = number of data in set 1 n 2 = number of data in set 2.

df = n 1 + n 2 - 2

Note. Another illustrative way to perform this test for bias is to calculate if the difference between the means falls within or outside the range where this difference is still not significantly large. In other words, if this difference is less than the least significant difference (lsd). This can be derived from Equation (6.13):

t 1 = t tab at n 1 -1 degrees of freedom t 2 = t tab at n 2 -1 degrees of freedom

= mean of differences within each pair of data s d = standard deviation of the mean of differences n = number of pairs of data

Note. Since such data sets do not have a normal distribution, the "normal" t -test which compares means of sets cannot be used here (the means do not constitute a fair representation of the sets). For the same reason no information about the precision of the two methods can be obtained, nor can the F -test be applied. For information about precision, replicate determinations are needed.

6.4.4.1 Construction of calibration graph 6.4.4.2 Comparing two sets of data using many samples at different analyte levels

1. When the concentration range is so wide that the errors, both random and systematic, are not independent (which is the assumption for the t -tests). This is often the case where concentration ranges of several magnitudes are involved. 2. When pairing is inappropriate for other reasons, notably a long time span between the two analyses (sample aging, change in laboratory conditions, etc.).

Note: Naturally, non-linear higher-order relationships are also possible, but since these are less common in analytical work and more complex to handle mathematically, they will not be discussed here. Nevertheless, to avoid misinterpretation, always inspect the kind of relationship by plotting the data, either on paper or on the computer monitor.

a = intercept of the line with the y-axis b = slope (tangent)

x i = data X ¯x = mean of data X y i = data Y ¯y = mean of data Y

r = 1 perfect positive linear correlation r = 0 no linear correlation (maybe other correlation) r = -1 perfect negative linear correlation

Note. A treatise of the error or uncertainty in the regression line is given.

- The most precise data set is plotted on the x-axis - At least 6, but preferably more than 10 different samples are analyzed - The samples should rather uniformly cover the analyte level range of interest.

Note. In the present example, the scattering of the points around the regression line does not seem to change much over the whole range. This indicates that the precision of laboratory Y does not change very much over the range with respect to laboratory X. This is not always the case. In such cases, weighted regression (not discussed here) is more appropriate than the unweighted regression as used here. Validation of a method (see Section 7.5) may reveal that precision can change significantly with the level of analyte (and with other factors such as sample matrix).

= "fitted" y-value for each x i , (read from graph or calculated with Eq. 6.22). Thus, is the (vertical) deviation of the found y-values from the line. n = number of calibration points. Note: Only the y-deviations of the points from the line are considered. It is assumed that deviations in the x-direction are negligible. This is, of course, only the case if the standards are very accurately prepared.

a = 0.037 ± 2.78 × 0.0132 = 0.037 ± 0.037 and b = 0.626 ± 2.78 × 0.0219 = 0.626 ± 0.061

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## The Beginner's Guide to Statistical Analysis | 5 Steps & Examples

## Table of contents

## Writing statistical hypotheses

- Null hypothesis: A 5-minute meditation exercise will have no effect on math test scores in teenagers.
- Alternative hypothesis: A 5-minute meditation exercise will improve math test scores in teenagers.
- Null hypothesis: Parental income and GPA have no relationship with each other in college students.
- Alternative hypothesis: Parental income and GPA are positively correlated in college students.

## Planning your research design

- In an experimental design , you can assess a cause-and-effect relationship (e.g., the effect of meditation on test scores) using statistical tests of comparison or regression.
- In a correlational design , you can explore relationships between variables (e.g., parental income and GPA) without any assumption of causality using correlation coefficients and significance tests.
- In a descriptive design , you can study the characteristics of a population or phenomenon (e.g., the prevalence of anxiety in U.S. college students) using statistical tests to draw inferences from sample data.

- In a between-subjects design , you compare the group-level outcomes of participants who have been exposed to different treatments (e.g., those who performed a meditation exercise vs those who didn’t).
- In a within-subjects design , you compare repeated measures from participants who have participated in all treatments of a study (e.g., scores from before and after performing a meditation exercise).
- In a mixed (factorial) design , one variable is altered between subjects and another is altered within subjects (e.g., pretest and posttest scores from participants who either did or didn’t do a meditation exercise).
- Experimental
- Correlational

## Measuring variables

- Categorical data represents groupings. These may be nominal (e.g., gender) or ordinal (e.g. level of language ability).
- Quantitative data represents amounts. These may be on an interval scale (e.g. test score) or a ratio scale (e.g. age).

## Sampling for statistical analysis

There are two main approaches to selecting a sample.

- Probability sampling: every member of the population has a chance of being selected for the study through random selection.
- Non-probability sampling: some members of the population are more likely than others to be selected for the study because of criteria such as convenience or voluntary self-selection.

If you want to use parametric tests for non-probability samples, you have to make the case that:

- your sample is representative of the population you’re generalizing your findings to.
- your sample lacks systematic bias.

## Create an appropriate sampling procedure

Based on the resources available for your research, decide on how you’ll recruit participants.

- Will you have resources to advertise your study widely, including outside of your university setting?
- Will you have the means to recruit a diverse sample that represents a broad population?
- Do you have time to contact and follow up with members of hard-to-reach groups?

## Calculate sufficient sample size

To use these calculators, you have to understand and input these key components:

- Significance level (alpha): the risk of rejecting a true null hypothesis that you are willing to take, usually set at 5%.
- Statistical power : the probability of your study detecting an effect of a certain size if there is one, usually 80% or higher.
- Expected effect size : a standardized indication of how large the expected result of your study will be, usually based on other similar studies.
- Population standard deviation: an estimate of the population parameter based on a previous study or a pilot study of your own.

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## Inspect your data

There are various ways to inspect your data, including the following:

- Organizing data from each variable in frequency distribution tables .
- Displaying data from a key variable in a bar chart to view the distribution of responses.
- Visualizing the relationship between two variables using a scatter plot .

## Calculate measures of central tendency

- Mode : the most popular response or value in the data set.
- Median : the value in the exact middle of the data set when ordered from low to high.
- Mean : the sum of all values divided by the number of values.

## Calculate measures of variability

- Range : the highest value minus the lowest value of the data set.
- Interquartile range : the range of the middle half of the data set.
- Standard deviation : the average distance between each value in your data set and the mean.
- Variance : the square of the standard deviation.

Researchers often use two main methods (simultaneously) to make inferences in statistics.

- Estimation: calculating population parameters based on sample statistics.
- Hypothesis testing: a formal process for testing research predictions about the population using samples.

You can make two types of estimates of population parameters from sample statistics:

- A point estimate : a value that represents your best guess of the exact parameter.
- An interval estimate : a range of values that represent your best guess of where the parameter lies.

## Hypothesis testing

- A test statistic tells you how much your data differs from the null hypothesis of the test.
- A p value tells you the likelihood of obtaining your results if the null hypothesis is actually true in the population.

Statistical tests come in three main varieties:

- Comparison tests assess group differences in outcomes.
- Regression tests assess cause-and-effect relationships between variables.
- Correlation tests assess relationships between variables without assuming causation.

## Parametric tests

- A simple linear regression includes one predictor variable and one outcome variable.
- A multiple linear regression includes two or more predictor variables and one outcome variable.

- A t test is for exactly 1 or 2 groups when the sample is small (30 or less).
- A z test is for exactly 1 or 2 groups when the sample is large.
- An ANOVA is for 3 or more groups.

The z and t tests have subtypes based on the number and types of samples and the hypotheses:

- If you have only one sample that you want to compare to a population mean, use a one-sample test .
- If you have paired measurements (within-subjects design), use a dependent (paired) samples test .
- If you have completely separate measurements from two unmatched groups (between-subjects design), use an independent (unpaired) samples test .
- If you expect a difference between groups in a specific direction, use a one-tailed test .
- If you don’t have any expectations for the direction of a difference between groups, use a two-tailed test .

The final step of statistical analysis is interpreting your results.

## Statistical significance

## Effect size

## Decision errors

## Frequentist versus Bayesian statistics

## Is this article helpful?

- Descriptive Statistics | Definitions, Types, Examples
- Inferential Statistics | An Easy Introduction & Examples
- Choosing the Right Statistical Test | Types & Examples

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## Standard statistical tools in research and data analysis

Figure 1. Classification of variables [1]

SOFTWARES FOR STATISTICS, SAMPLE SIZE CALCULATION AND POWER ANALYSIS

- StatPages.net – contains connections to a variety of online power calculators.
- G-Power — a downloadable power analysis software that works on DOS.
- ANOVA power analysis creates an interactive webpage that estimates the power or sample size required to achieve a specified power for one effect in a factorial ANOVA design.
- Sample Power is software created by SPSS. It generates a comprehensive report on the computer screen that may be copied and pasted into another document.

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