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# Inferential Statistics

## Preparing Data

❶In the Regression-Discontinuity Design , we need to be especially concerned about curvilinearity and model misspecification. Because the analyses differ for each, they are presented separately.

## Types of Statistical Tests   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:. The variance of a sample is defined by slightly different formula:.

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. The SD of a sample is defined by slightly different formula:. An example for calculation of variation and SD is illustrated in Table 2.

Most of the biological variables usually cluster around a central value, with symmetrical positive and negative deviations about this point. 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.

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.

Alternative hypothesis H 1 and H a denotes that a statement between the variables is expected to be true. 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 ]. However, if null hypotheses H0 is incorrectly rejected, this is known as a Type I error. Numerical data quantitative variables that are normally distributed are analysed with parametric tests.

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. 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.

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:. The group variances can be compared using the F -test. If F differs significantly from 1. 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. 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. 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: 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.

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. 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. Therefore, it is useful when it is difficult to measure the values. 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. 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: 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.

The Kruskal—Wallis test is a non-parametric test to analyse the variance. The data values are ranked in an increasing order, and the rank sums calculated followed by calculation of the test statistic. 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.

The Friedman test is a non-parametric test for testing the difference between several related samples. When it comes to statistic analysis, there are two classifications: In a nutshell, descriptive statistics intend to describe a big hunk of data with summary charts and tables, but do not attempt to draw conclusions about the population from which the sample was taken. You are simply summarizing the data you have with pretty charts and graphs—kind of like telling someone the key points of a book executive summary as opposed to just handing them a thick book raw data.

Conversely, with inferential statistics, you are testing a hypothesis and drawing conclusions about a population, based on your sample. To understand the simple difference between descriptive and inferential statistics, all you need to remember is that descriptive statistics summarize your current dataset and inferential statistics aim to draw conclusions about an additional population outside of your dataset.

This would sure be easier for someone to interpret than a big spreadsheet. There are hundreds of ways to visualize data, including data tables, pie charts, line charts, etc. Note that the analysis is limited to your data and that you are not extrapolating any conclusions about a full population. Descriptive statistic reports generally include summary data tables kind of like the age table above , graphics like the charts above , and text to explain what the charts and tables are showing.

Objective randomization allows properly inductive procedures. The statistical analysis of a randomized experiment may be based on the randomization scheme stated in the experimental protocol and does not need a subjective model. However, at any time, some hypotheses cannot be tested using objective statistical models, which accurately describe randomized experiments or random samples.

In some cases, such randomized studies are uneconomical or unethical. It is standard practice to refer to a statistical model, often a linear model, when analyzing data from randomized experiments. However, the randomization scheme guides the choice of a statistical model. It is not possible to choose an appropriate model without knowing the randomization scheme. Different schools of statistical inference have become established.

These schools—or "paradigms"—are not mutually exclusive, and methods that work well under one paradigm often have attractive interpretations under other paradigms. The classical or frequentist paradigm, the Bayesian paradigm, and the AIC -based paradigm are summarized below. The likelihood-based paradigm is essentially a sub-paradigm of the AIC-based paradigm. This paradigm calibrates the plausibility of propositions by considering notional repeated sampling of a population distribution to produce datasets similar to the one at hand.

By considering the dataset's characteristics under repeated sampling, the frequentist properties of a statistical proposition can be quantified—although in practice this quantification may be challenging. One interpretation of frequentist inference or classical inference is that it is applicable only in terms of frequency probability ; that is, in terms of repeated sampling from a population.

However, the approach of Neyman  develops these procedures in terms of pre-experiment probabilities. That is, before undertaking an experiment, one decides on a rule for coming to a conclusion such that the probability of being correct is controlled in a suitable way: In contrast, Bayesian inference works in terms of conditional probabilities i. The frequentist procedures of significance testing and confidence intervals can be constructed without regard to utility functions.

However, some elements of frequentist statistics, such as statistical decision theory , do incorporate utility functions. Loss functions need not be explicitly stated for statistical theorists to prove that a statistical procedure has an optimality property. The Bayesian calculus describes degrees of belief using the 'language' of probability; beliefs are positive, integrate to one, and obey probability axioms. Bayesian inference uses the available posterior beliefs as the basis for making statistical propositions.

There are several different justifications for using the Bayesian approach. Many informal Bayesian inferences are based on "intuitively reasonable" summaries of the posterior. For example, the posterior mean, median and mode, highest posterior density intervals, and Bayes Factors can all be motivated in this way.

While a user's utility function need not be stated for this sort of inference, these summaries do all depend to some extent on stated prior beliefs, and are generally viewed as subjective conclusions. Methods of prior construction which do not require external input have been proposed but not yet fully developed. Formally, Bayesian inference is calibrated with reference to an explicitly stated utility, or loss function; the 'Bayes rule' is the one which maximizes expected utility, averaged over the posterior uncertainty.

Formal Bayesian inference therefore automatically provides optimal decisions in a decision theoretic sense. Given assumptions, data and utility, Bayesian inference can be made for essentially any problem, although not every statistical inference need have a Bayesian interpretation. Analyses which are not formally Bayesian can be logically incoherent ; a feature of Bayesian procedures which use proper priors i. Some advocates of Bayesian inference assert that inference must take place in this decision-theoretic framework, and that Bayesian inference should not conclude with the evaluation and summarization of posterior beliefs.

The Akaike information criterion AIC is an estimator of the relative quality of statistical models for a given set of data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. Thus, AIC provides a means for model selection. AIC is founded on information theory: In doing so, it deals with the trade-off between the goodness of fit of the model and the simplicity of the model.

The minimum description length MDL principle has been developed from ideas in information theory  and the theory of Kolmogorov complexity. However, if a "data generating mechanism" does exist in reality, then according to Shannon 's source coding theorem it provides the MDL description of the data, on average and asymptotically. ## Main Topics

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Inferential Analysis: From Sample to Population Introduction Inferential analysis is used to generalize the results obtained from a random (probability) sample back to .

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With inferential statistics, you are trying to reach conclusions that extend beyond the immediate data alone. Analysis of Covariance (ANCOVA), regression analysis, and many of the multivariate methods like factor analysis, multidimensional scaling, cluster analysis, discriminant function analysis, and so on. An understanding of that.

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Module 5: Data Preparation and Analysis Preparing Data. After data collection, the researcher must prepare the data to be analyzed. Organizing the data correctly . Research Methodology Sample Paper on Inferential Statistics Inferential Statistics Inferential statistics is a procedure used by researchers to draw conclusions based on data that is beyond simple description (Clayton, ).

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This article explains the difference between descriptive and inferential statistic methods. In short, descriptive statistics are limited to your dataset, while inferential statistics attempt to draw conclusions about a population. When it comes to statistic analysis, there are two classifications: That’s enough on market research. terminology of data analysis, and be prepared to learn about using JMP for data analysis. Introduction: A Common Language for Researchers Research in the social sciences is a diverse topic.