Hypothesis Testing.pptx ( T- test, F- test, U- test , Anova)

Hypothesis testing is the statistical analysis which includes t- test , Z - test, U-test, F-test, Chi square test, Anova .

1. HYPOTHESIS TESTING & PARAMETRIC ANALYSIS MODULE 3

2. What is hypothesis? According to Prof. Morris Hamburg “A hypothesis in statistics is simply a quantitative statement about a population” A statistical hypothesis is an assumption about a population parameter. It is supposition made as a basis for reasoning This assumption may or may not be true. Hypothesis testing refers to the formal procedures used by statisticians to accept or reject statistical hypotheses. To determine whether a statistical hypothesis is true by examining a random sample from the population. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected.

3. Type of Hypothesis Null hypothesis . The null hypothesis, denoted by H o , is usually the hypothesis that sample observations result purely from chance. Alternative hypothesis . The alternative hypothesis, denoted by H 1  or H a , is the hypothesis that sample observations are influenced by some non-random cause. For example, suppose we wanted to determine whether a coin was fair and balanced. A null hypothesis might be that half the flips would result in Heads and half, in Tails. The alternative hypothesis might be that the number of Heads and Tails would be very different. Symbolically, these hypotheses would be expressed as H o : P = 0.5 H a : P ≠ 0.5

4. Hypothesis Tests State the hypotheses Set up a suitable level of significance Setting a test criterion Doing computations Interpret results/ Decision making

5. Decision Errors When a statistical hypothesis is tested there are four possibilities: The hypothesis is true but our test rejects it (Type I error) The hypothesis is false but our test accepts it (Type II error) The hypothesis is true and our test accepts it (Correct decision) The hypothesis is false and our test rejects it

6. Errors in hypothesis testing Type I error . A Type I error occurs when the researcher rejects a null hypothesis when it is true . The probability of committing a Type I error is called the  significance level . This probability is also called  alpha , and is often denoted by α. Type II error . A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called  Beta , and is often denoted by β. The probability of  not  committing a Type II error is called the  Power  of the test.

7. Decision Rules P-value. The strength of evidence in support of a null hypothesis is measured by the  P-value . Suppose the test statistic is equal to  S . The P-value is the probability of observing a test statistic as extreme as  S , assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

8. Decision Rules Region of acceptance. The region of acceptance is a range of values. If the test statistic falls within the region of acceptance, the null hypothesis is not rejected. The region of acceptance is defined so that the chance of making a Type I error is equal to the significance level. The set of values outside the region of acceptance is called the region of rejection. If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance

9. One-Tailed A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution, is called a  one-tailed test . For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.

10. Two Tailed Test A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a  two-tailed test . For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10.

11. Standard Error The standard deviation of the sampling distribution is called standard error The standard error is a statistical term that measures the accuracy with which a sample distribution represents a population by using standard deviation. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error of the mean. The standard error is also inversely proportional to the sample size; the larger the sample size, the smaller the standard error because the statistic will approach the actual value.

12. Parametric Test and Non Parametric Test Parameters: Statistical measurements such as Mean, Variance etc. of the population are called parameters. Parametric tests are those that make assumptions about the parameters of the population distribution from which the sample is drawn. This is often the assumption that the population data are normally distributed. Non-parametric tests are “distribution-free” and, as such, can be used for non-Normal variables. They can thus be applied even if parametric conditions of validity are not met.

13. t- Test A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related in certain features. It is mostly used when the data sets, like the data set recorded as the outcome from flipping a coin 100 times, would follow a normal distribution and may have unknown variances. A t-test is used as a hypothesis testing tool, which allows testing of an assumption applicable to a population.  A t-test looks at the t-statistic, the t-distribution values, and the degrees of freedom to determine the statistical significance.

14. Z-Test A z-test is a statistical test to determine whether two population means are different when the variances are known and the sample size is large. It can be used to test hypotheses in which the z-test follows a normal distribution. A z-statistic, or z-score, is a number representing the result from the z-test. Z-tests are closely related to t-tests, but t-tests are best performed when an experiment has a small sample size. Also, t-tests assume the standard deviation is unknown, while z-tests assume it is known.

15. U-Test The Mann-Whitney U test is the nonparametric equivalent of the two sample t-test. While the t-test makes an assumption about the distribution of a population (i.e. that the sample came from a t-distributed population), the Mann Whitney U Test makes no such assumption. The test compares two populations. The null hypothesis for the test is that the probability is 50% that a randomly drawn member of the first population will exceed a member of the second population.  An alternate null hypothesis is that the two samples come from the same population (i.e. that they both have the same median).

16. Kruskal Wallis - H Test Non parametric alternative to the One Way ANOVA.  The H test is used when the assumptions for ANOVA aren’t met. Sometimes called the one-way ANOVA on ranks, as the ranks of the data values are used in the test rather than the actual data points. The test determines whether the medians of two or more groups are different. Calculate a test statistic and compare it to a distribution cut-off point. The test statistic used is H statistic.  The hypotheses for the test are: H : population medians are equal. H 1 : population medians are not equal. The Kruskal Wallis test tells if there is a significant difference between groups. 

17. Bivariate Analysis Bivariate analysis is stated to be an analysis of any concurrent relation between two variables or attributes. It is one of the simplest forms of statistical analysis, used to find out if there is a relationship between two sets of values. It usually involves the  variables  X and Y. This study explores the relationship of two variables as well as the depth of this relationship to figure out if there are any discrepancies between two variables and any causes of this difference. Some of the examples are percentage table, scatter plot, etc. Univariate analysis  is the analysis of one (“ uni ”) variable. Bivariate analysis is the analysis of exactly two variables. Multivariate analysis  is the analysis of more than two variables.

18. Types of Bivariate Analysis Numerical and Numerical  – In this type, both the variables of bivariate data, independent and dependent, are having numerical values. Categorical and Categorical  – When both the variables are categorical. Numerical and Categorical  – When one variable is numerical and one is categorical.

19. Multivariate Analysis Multivariate means involving multiple dependent variables resulting in one outcome. This explains that the majority of the problems in the real world are Multivariate. For example, we cannot predict the weather of any year based on the season. There are multiple factors like pollution, humidity, precipitation, etc.

20. Advantages and Disadvantages of Multivariate Analysis Advantages The main advantage of multivariate analysis is that since it considers more than one factor of independent variables that influence the variability of dependent variables, the conclusion drawn is more accurate. The conclusions are more realistic and nearer to the real-life situation. Disadvantages The main disadvantage of MVA includes that it requires rather complex computations to arrive at a satisfactory conclusion. Many observations for a large number of variables need to be collected and tabulated; it is a rather time-consuming process.

21. Classification Chart of Multivariate Techniques

22. ANOVA Analysis of variance (ANOVA) is an analysis tool used in statistics that splits an observed aggregate variability found inside a data set into two parts: systematic factors and random factors. The systematic factors have a statistical influence on the given data set, while the random factors do not. Analysts use the ANOVA test to determine the influence that independent variables have on the dependent variable in a regression study. ​

23. One way- Two way Anova One-way  or  two-way  refers to the number of independent variables (IVs) in your Analysis of Variance test. One-way has one independent variable (with 2 levels). For example:  brand of cereal , Two-way has two independent variables (it can have multiple levels). For example:  brand of cereal, calories

24. The Formula for ANOVA is: F=MST​/MSE where: F=ANOVA coefficient MST=Mean sum of squares due to treatment MSE=Mean sum of squares due to error

25. One-way Anova One-way or two-way refers to the number of independent variables in your analysis of variance test. A one-way ANOVA evaluates the impact of a sole factor on a sole response variable. It determines whether all the samples are the same. The one-way ANOVA is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups.

26. Two way Anova A two-way ANOVA is an extension of the one-way ANOVA. With a one-way, you have one independent variable affecting a dependent variable. With a two-way ANOVA, there are two independents. For example, a two-way ANOVA allows a company to compare worker productivity based on two independent variables, such as salary and skill set. It is utilized to observe the interaction between the two factors and tests the effect of two factors at the same time.