Unit-4 Basics of Testing of Hypothesis Long Essays 1. Discuss about the hypothesis testing of parametric data The parametric test is the hypothesis test which provides generalisations for making statements about the mean of the parent population. A t-test based on Student’s t-statistic, which is often used in this regard. The t-statistic rests on the underlying assumption that there is the normal distribution of variable and the mean in known or assumed to be known. The population variance is calculated for the sample. It is assumed that the variables of interest, in the population are measured on an interval scale. A parametric statistical test is one that makes assumptions about the parameters (defining properties) of the population distribution(s) from which one's data are drawn, while a non-parametric test is one that makes no such assumptions. In parametric statistics it is assumed that the data has come from a type of probability distribution and makes inferences about the parameters of the distribution. In general, parametric methods make more assumptions than nonparametric methods. If those extra assumptions are correct, parametric methods can produce more accurate and precise estimates. For this reason they are described as having more statistical power. However, if assumptions made in the parametric analysis are incorrect then these methods can be very misleading. The concept of robustness refers to the likelihood of getting a misleading result, and parametric methods are less robust than non-parametric alternatives. In selection of method there is a trade-off to be made of simplicity and power versus robustness. Which is more appropriate depends on the specifics of the phenomenon being studied. ● A statistical test, in which specific assumptions are made about the population parameter is known as the parametric test. ● In the parametric test, the test statistic is based on distribution. ● In the parametric test, it is assumed that the measurement of variables of interest is done on interval or ratio level. ● In general, the measure of central tendency in the parametric test is mean, ● In the parametric test, there is complete information about the population. ● The applicability of parametric test is for variables only. ● For measuring the degree of association between two quantitative variables, Pearson’s coefficient of correlation is used in the parametric test. 2. Explain the hypothesis testing of non-parametric data. The nonparametric test is defined as the hypothesis test which is not based on underlying assumptions, i.e. it does not require population’s distribution to be denoted by specific parameters. The test is mainly based on differences in medians. Hence, it is alternately known as the distribution-free test. The test assumes that the variables are measured on a nominal or ordinal level. It is used when the independent variables are non-metric. A nonparametric test is a hypothesis test that does not require the population's distribution to be characterized by certain parameters. However, nonparametric tests are not completely free of assumptions about your data. For instance, nonparametric tests require the data to be an independent random sample. Nonparametric statistics refer to a statistical method wherein the data is not required to fit a normal distribution. Nonparametric statistics uses data that is often ordinal, meaning it does not rely on numbers, but rather a ranking or order of sorts. Non-parametric statistics techniques do not rely on data belonging to any particular distribution. Sometimes these are called distribution-free methods, which do not rely on assumptions that the data are drawn from a given probability distribution. Sometimes in a complex system, individual variables are assumed to be parametric but not the connection between variables. Examples here include nonparametric regression and non-parametric hierarchical Bayesian models. ● A statistical test used in the case of non-metric independent variables is called nonparametric test. ● The test statistic is arbitrary in the case of the nonparametric test. ● The variables of interest are measured on nominal or ordinal scale. ● The measure of central tendency in the case of the nonparametric test is median. ● In the nonparametric test, there is no information about the population. ● Nonparametric test applies to both variables and attributes. ● Spearman’s rank correlation is used in the nonparametric test. 3. Explain hypothesis testing in detail. Discuss the clinical versus the statistical significance of hypothesis testing. Hypothesis test as the formal procedures that statisticians use to test whether a hypothesis can be accepted or not. A hypothesis is an assumption about something. Hypothesis testing is about testing to see whether the stated hypothesis is acceptable or not. During our hypothesis testing, we want to gather as much data as we can so that we can prove our hypothesis one way or another. There is a proper four-step method in performing a proper hypothesis test: ● Write the hypothesis ● Create an analysis plan ● Analyze the data ● Interpret the result Hypothesis The first step is that of writing the hypothesis. You actually have two hypotheses to write. One is called the null hypothesis. This is the hypothesis based on chance. Think of this as the hypothesis that states how you would expect things to work without any external factors to change it. The other hypothesis is called the alternative hypothesis. This is the hypothesis that shows a change from the null hypothesis that is caused by something. In hypothesis testing, we just test to see if our data fits our alternative hypothesis or if it fits the null hypothesis. We don't worry about what is causing our data to shift from the null hypothesis if it does. Keep in mind, when writing your null hypothesis and alternative hypothesis, they must be written in such a way so that if the null hypothesis is false, then the alternative hypothesis is true and vice versa. Analysis Plan The second step is to create an analysis plan. This involves deciding how to read your results to know whether your null hypothesis is true or your alternative hypothesis is true. Usually, this involves analyzing just one single test statistic. There are two ways to read your results: P-value method and the region of acceptance method. The P-value is the probability of observing the desired statistic. If this P-value is less than the significance level, then the null hypothesis is not valid. The significance level is the probability of making the mistake of saying that the null hypothesis is not valid when it actually is true. The region of acceptance is a chosen range of values that results in the null hypothesis being stated as valid. Data Analysis The third step is that of analyzing the data. It is the putting step two into action. It is in this step that the data is analyzed and either a P-value is found, or the data's region is found. Interpretation The fourth step involves interpreting the results. It is in this step that the data is compared to the region of acceptance or the significance level. If the P-value is less than the significance level, then the null hypothesis is not valid. If the data is within the region of acceptance, then the null hypothesis is valid. 4. What is hypothesis? What are different types of hypothesis? Explain how you will formulate a hypothesis with a suitable example. A hypothesis test is a statistical test that is used to determine whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. A hypothesis test examines two opposing hypotheses about a population: the null hypothesis and the alternative hypothesis. The null hypothesis (H 0) is a hypothesis which the researcher tries to disprove, reject or nullify. A null hypothesis is a statistical hypothesis that is tested for possible rejection under the assumption that it is true (usually that observations are the result of chance). The alternative hypothesis is the hypothesis used in hypothesis testing that is contrary to the null hypothesis. It is usually taken to be that the observations are the result of a real effect (with some amount of chance variation superposed). Procedure of Testing Hypothesis ● Specify the hypotheses. First, the manager formulates the hypotheses. ● Determine the power and sample size for the test. ● Choose a significance level (also called alpha or α). ● Collect the data. ● Compare the p-value from the test to the significance level. ● Decide whether to reject or fail to reject the null hypothesis. 5. Define correlation and regression. What are the different measures of correlation? Explain which measures are used for computation of correlation. Or Note on correlation and regression & their applications. (Short Essay) Or What is Correlation? Name different types of Correlation. Which are the different measures of correlation? (Short Essay) Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables. Although frequently confused, they are quite different. Correlation measures the association between two variables and quantities the strength of their relationship. Correlation evaluates only the existing data. Regression uses the existing data to define a mathematical equation which can be used to predict the value of one variable based on the value of one or more other variables and can therefore be used to extrapolate between the existing data. The regression equation can therefore be used to predict the outcome of observations not previously seen or tested. Correlation is a bivariate analysis that measures the strengths of association between two variables. In statistics, the value of the correlation coefficient varies between +1 and -1. When the value of the correlation coefficient lies around ± 1, then it is said to be a perfect degree of association between the two variables. As the correlation coefficient value goes towards 0, the relationship between the two variables will be weaker. Usually, in statistics, we measure three types of correlations: Pearson correlation, Kendall rank correlation and Spearman correlation. Carl Pearson’s coefficient of correlation (r) is widely used in statistics to measure the degree of the relationship between linear related variables. Spearman’s Rank correlation coefficient is used to identify and test the strength of a relationship between two sets of data. It is often used as a statistical method to aid with either proving or disproving a hypothesis e.g. the depth of a river does not progressively increase the further from the river bank. The formula used to calculate Spearman’s Rank is shown below. The most commonly used techniques for investigating the relationship between two quantitative variables are correlation and linear regression. Correlation quantifies the strength of the linear relationship between a pair of variables, whereas regression expresses the relationship in the form of an equation. 6. Discuss various steps involved in testing the significance of single mean and difference between two means in small samples using student’s test. 7. Explain the following: null hypothesis, level of significance c) power of test, d) p value. The null hypothesis (H 0) is a hypothesis which the researcher tries to disprove, reject or nullify. A null hypothesis is a statistical hypothesis that is tested for possible rejection under the assumption that it is true (usually that observations are the result of chance). The probability of rejecting the null hypothesis in a statistical test when it is true is called significance level. The null hypothesis is rejected if the p-value is less than a predetermined level, α. α is called the significance level, and is the probability of rejecting the null hypothesis given that it is true (a type I error). It is usually set at or below 5%. The significance level, also denoted as alpha or α, is the probability of rejecting the null hypothesis when it is true. For example, a significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. Statistically significant is the likelihood that a relationship between two or more variables is caused by something other than random chance. Statistical hypothesis testing is used to determine whether the result of a data set is statistically significant. To find the significance level, subtract the number shown from one. For example, a value of ".01" means that there is a 99% (1-.01=.99) chance of it being true. The power of any test of statistical significance is defined as the probability that it will reject a false null hypothesis. Statistical power is inversely related to beta or the probability of making a Type II error. Increase the power of a hypothesis test ● Use a larger sample. ● Improve your process. ● Use a higher significance level (also called alpha or α). ● Choose a larger value for Differences. ● Use a directional hypothesis (also called one-tailed hypothesis). ● A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis. The p-value is defined as the probability of obtaining a result equal to or "more extreme" than what was actually observed, when the null hypothesis is true. In frequentist inference, the p-value is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing. Short Essay 1. Explain α & β errors in hypothesis testing. Type I error (α error) When the null hypothesis is true and you reject it, you make a type I error. The probability of making a type I error is α, which is the level of significance you set for your hypothesis test. An α of 0.05 indicates that you are willing to accept a 5% chance that you are wrong when you reject the null hypothesis. To lower this risk, you must use a lower value for α. However, using a lower value for alpha means that you will be less likely to detect a true difference if one really exists. Type II error (β error) When the null hypothesis is false and you fail to reject it, you make a type II error. The probability of making a type II error is β, which depends on the power of the test. You can decrease your risk of committing a type II error by ensuring your test has enough power. You can do this by ensuring your sample size is large enough to detect a practical difference when one truly exists. The probability of rejecting the null hypothesis when it is false is equal to 1–β. This value is the power of the test. 2. Classify and explain the measures of central tendency. The most common measures of central tendency are the arithmetic mean, the median and the mode. A central tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. The central tendency of a distribution is typically contrasted with its dispersion or variability; dispersion and central tendency are the often characterized properties of distributions. Analysts may judge whether data has a strong or a weak central tendency based on its dispersion. The following may be applied to one-dimensional data. Depending on the circumstances, it may be appropriate to transform the data before calculating a central tendency. Examples are squaring the values or taking logarithms. Whether a transformation is appropriate and what it should be, depend heavily on the data being analyzed. Arithmetic mean (or simply, mean) – the sum of all measurements divided by the number of observations in the data set. Median – the middle value that separates the higher half from the lower half of the data set. The median and the mode are the only measures of central tendency that can be used for ordinal data, in which values are ranked relative to each other but are not measured absolutely. Mode – the most frequent value in the data set. This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments. Geometric mean – the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale. Harmonic mean – the reciprocal of the arithmetic mean of the reciprocals of the data values. This measure too is valid only for data that are measured absolutely on a strictly positive scale. Weighted arithmetic mean – an arithmetic mean that incorporates weighting to certain data elements. 3. Compare and contrast nonparametric and parametric data. Or Distinguish between parametric and nonparametric tests. The fundamental differences between parametric and nonparametric test are: ● A statistical test, in which specific assumptions are made about the population parameter is known as the parametric test. A statistical test used in the case of non-metric independent variables is called nonparametric test. ● In the parametric test, the test statistic is based on distribution. On the other hand, the test statistic is arbitrary in the case of the nonparametric test. ● In the parametric test, it is assumed that the measurement of variables of interest is done on interval or ratio level. As opposed to the nonparametric test, wherein the variable of interest are measured on nominal or ordinal scale. ● In general, the measure of central tendency in the parametric test is mean, while in the case of the nonparametric test is median. ● In the parametric test, there is complete information about the population. Conversely, in the nonparametric test, there is no information about the population. ● The applicability of parametric test is for variables only, whereas nonparametric test applies to both variables and attributes. ● For measuring the degree of association between two quantitative variables, Pearson’s coefficient of correlation is used in the parametric test, while spearman’s rank correlation is used in the nonparametric test. 4. Classify and explain the tests used for hypothesis testing of parametric data. A parametric statistical test is a test whose model specifies certain conditions about the parameters of the population from which the research sample was drawn. The observations must be independent, the observations must be drawn from normally distributed populations, these populations must have the same variances and variables involved must have been measured in at least an interval scale. Commonly used parametric tests are described below: Pearson Correlation Coefficient: The correlation coefficient (r) is a value that two continuous variables from the same subject correlate to each other. An r value of 1.0 means the data are completely positively correlated and 1 variable can be used to compute the other. An r of zero means that the two variables are completely random. An r of -1.0 is completely negatively correlated. The important thing to remember is that this is only an association and does not imply a cause-and-effect relationship. Student t-Test: The Student t-test is probably the most widely used parametric test. It was developed by a statistician working at the Guinness brewery and is called the Student t-test because of proprietary rights. A single sample t-test is used to determine whether the mean of a sample is different from a known average. A two-sample t-test is used to establish whether a difference occurs between the means of two similar data sets. The t-test uses the mean, standard deviation, and number of samples to calculate the test statistic. In a data set with a large number of samples, the critical value for the Student t-test is 1.96 for α of 0.05, obtained from a t-test table. The calculation to determine the t-value is relatively simple, but it can be found easily on-line or in any elementary statistics book. The z-Test: The next test, which is very similar to the Student t-test, is the z-test. However, with the z-test, the variance of the standard population, rather than the standard deviation of the study groups, is used to obtain the z-test statistic. Using the z-chart, like the t-table, we see what percentage of the standard population is outside the mean of the sample population. If, like the t- test, greater than 95% of the standard population is on one side of the mean, the p-value is less than 0.05 and statistical significance is achieved. As some assumption of sample size exists in the calculation of the z-test, it should not be used if sample size is less than 30. If both the n and the standard deviation of both groups are known, a two sample t-test is best. ANOVA: Analysis of variance (ANOVA) is a test that incorporates means and variances to determine the test statistic. The test statistic is then used to determine whether groups of data are the same or different. When hypothesis testing is being performed with ANOVA, the null hypothesis is stated such that all groups are the same. The test statistic for ANOVA is called the F-ratio. As with the t- and z-statistics, the F-statistic is compared with a table to determine whether it is greater than the critical value. In interpreting the F-statistic, the degrees of freedom for both the numerator and the denominator are required. The degrees of freedom in the numerator are the number of groups minus 1, and the degrees of freedom in the denominator are the number of data points minus the number of group. 5. Explain linear regression? How is it applied for pharmaceutical science? Or What is linear regression? How is it applied for pharmaceutical science? Regression uses the existing data to define a mathematical equation which can be used to predict the value of one variable based on the value of one or more other variables and can therefore be used to extrapolate between the existing data. The regression equation can therefore be used to predict the outcome of observations not previously seen or tested. Regression analysis mathematically describes the dependence of the Y variable on the X variable and constructs an equation which can be used to predict any value of Y for any value of X. It is more specific and provides more information than does correlation. Unlike correlation, however, regression is not scale independent and the derived regression equation depends on the units of each variable involved. As with correlation, regression assumes that each of the variables is normally distributed with equal variance. In addition to deriving the regression equation, regression analysis also draws a line of best fit through the data points of the scattergram. These “regression lines” may be linear, in which case the relationship between the variables fits a straight line, or nonlinear, in which case a polynomial equation is used to describe the relationship. Regression (also known as simple regression, linear regression, or least squares regression) fits a straight line equation of the following form to the data: Y = a + bX where Y is the dependent variable, X is the single independent variable, a is the Y-intercept of the regression line, and b is the slope of the line (also known as the regression coefficient). APPLICATION OF LINEAR REGRESSION Construction of a standard curve in drug analysis using linear regression. Known amounts of drug are subjected to an assay procedure, and a plot of percentage recovered (or amount recovered) versus amount added is constructed. Theoretically, the relationship is usually a straight line. A knowledge of the line parameters A and B can be used to predict the amount of drug in an unknown sample based on the assay results. In most practical situations, A and B are unknown. The least squares estimates a and b of these parameters are used to compute drug potency (X) based on the assay response (y) 6. Explain Perason’s correlation and Spearmann’s Correlation. Correlation is a bivariate analysis that measures the strengths of association between two variables. In statistics, the value of the correlation coefficient varies between +1 and -1. When the value of the correlation coefficient lies around ± 1, then it is said to be a perfect degree of association between the two variables. As the correlation coefficient value goes towards 0, the relationship between the two variables will be weaker. Spearman’s Rank correlation coefficient is used to identify and test the strength of a relationship between two sets of data. It is often used as a statistical method to aid with either proving or disproving a hypothesis e.g. the depth of a river does not progressively increase the further from the river bank. The formula used to calculate Spearman’s Rank is shown below. 7. Define (a) α & β errors (b) confidence interval (c) power of the test. a) α- 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, and is often denoted by α. β-error - A Type II error occurs when the researcher accepts 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. b) Confidence interval - Statisticians use a confidence interval to express the degree of uncertainty associated with a sample statistic. A confidence interval is an interval estimate combined with a probability statement. A confidence interval is a range around a measurement that conveys how precise the measurement is. For most chronic disease and injury programs, the measurement in question is a proportion or a rate (the percent of New Yorkers who exercise regularly or the lung cancer incidence rate). Confidence intervals are often seen on the news when the results of polls are released. c) The power of a statistical test gives the likelihood of rejecting the null hypothesis when the null hypothesis is false. Just as the significance level (alpha) of a test gives the probability that the null hypothesis will be ejected when it is actually true (a wrong decision), power quantifies the chance that the null hypothesis will be rejected when it is actually false (a correct decision). Thus, power is the ability of a test to correctly reject the null hypothesis. Or The power of a test is the probability of making the correct decision if the alternative hypothesis is true. That is, the power of a hypothesis test is the probability of rejecting the null hypothesis H0 when the alternative hypothesis HA is the hypothesis that is true 8. What are the underlying assumptions of one way ANOVA? Explain under what circumstances ANOVA is the most preferred type of statistical data analysis? The point of conducting an experiment is to find a significant effect between the stimuli being tested. To do this various statistical tests are used. In a psychology experiment an independent variable and dependant variable are the stimuli being manipulated and the behaviour being measured. Statistical tests are carried out to confirm if the behaviour occurring is more than chance. The t-test compares the means between 2 samples and is simple to conduct, but if there is more than 2 conditions in an experiment a ANOVA is required. The fact the ANOVA can test more than one treatment is a major advantage over other statistical analysis. ANOVA’s use an F-ratio as its significance statistic which is variance because it is impossible to calculate the sample means difference with more than two samples. The ANOVA is an important test because it enables us to see for example how effective two different types of treatment are and how durable they are. Effectively a ANOVA can tell us how well a treatment work, how long it lasts and how budget friendly it will be an example being intensive early behavioural intervention (EIBI) for autistic children which lasts a long time with a lot hour, has amazing results but costs a lot of money. The ANOVA is able to tell us if another therapy can do the same task in shorter amount of time and therefor costing less and making the treatment more accessible. Conducting this test would also help establish concurrent validity for the therapy against EIBI. The F-ratio tells the researcher how big of a difference there is between the conditions and the effect is more than just chance. Testing of the Assumptions ● The population in which samples are drawn should be normally distributed. ● Independence of cases: the sample cases should be independent of each other. ● Homogeneity of variance: Homogeneity means that the variance among the groups should be approximately equal. These assumptions can be tested using statistical software (like Intellectus Statistics!). The assumption of homogeneity of variance can be tested using tests such as Levene’s test or the Brown-Forsythe Test. Normality of the distribution of the scores can be tested using plots, the values of skewness and kurtosis, or using tests such as Shapiro-Wilk or Kolmogorov-Smirnov. The assumption of independence can be determined from the design of the study. It is important to note that ANOVA is not robust to violations to the assumption of independence. This is to say, that even if you violate the assumptions of homogeneity or normality, you can conduct the test and basically trust the findings. However, violations to independence assumption one cannot trust those ANOVA results. In general, with violations of homogeneity the study can probably carry on if you have equal sized groups. With violations of normality, continuing with the ANOVA should be ok if you have a large sample size and equal sized groups. 9. For what type of data is Chi Square test performed? Of What are the assumptions under which Chi-square test can be applied to analyze the data. For what type of data Chi-square test is applied. a) Sample size assumption: The chi-square test can be used to determine differences in proportions using a two-by-two contingency table. It is however important to understand that the chi-square tests yields only an approximate p-value, on which a correction factor is then applied. This only works well when your datasets are large enough. When sample sizes are small, as indicated by more than 20% of the contingency cells having expected values < 5 a Fisher's exact test maybe more appropriate. This test is one of a class of “exact tests”, because the significance of the deviation from a “null hypothesis” can be calculated exactly, rather than relying on an approximation. b) Independence assumption: Secondly, the chi-square test cannot be used on correlated data. When you are looking to test differences in proportions among matched pairs in a before/after scenario, an appropriate choice would be the McNemar's test. In essence, it is a chi-square goodness of fit test on the two discordant cells, with a null hypothesis stating that 50% of the changes (agreements or disagreements) go in each direction. This test requires the same subjects to be included in the before and after measurements i.e. the pairs should be matched one-on-one. When you choose to analyse your data using a chi-square test for independence, you need to make sure that the data you want to analyse "passes" two assumptions. You need to do this because it is only appropriate to use a chi-square test for independence if your data passes these two assumptions. If it does not, you cannot use a chi-square test for independence. These two assumptions are: Assumption #1: Your two variables should be measured at an ordinal or nominal level (i.e., categorical data). You can learn more about ordinal and nominal variables in our article: Types of Variable. Assumption #2: Your two variable should consist of two or more categorical, independent groups. Example independent variables that meet this criterion include gender (2 groups: Males and Females), ethnicity (e.g., 3 groups: Caucasian, African American and Hispanic), physical activity level (e.g., 4 groups: sedentary, low, moderate and high), profession (e.g., 5 groups: surgeon, doctor, nurse, dentist, therapist), and so forth. 10. Write a note on null and alternate hypothesis. The null hypothesis (H 0) is a hypothesis which the researcher tries to disprove, reject or nullify. The 'null' often refers to the common view of something, while the alternative hypothesis is what the researcher really thinks is the cause of a phenomenon. The general procedure for null hypothesis testing is as follows: ● State the null and alternative hypotheses ● Specify α and the sample size ● Select an appropriate statistical test ● Collect data (note that the previous steps should be done prior to collecting data) ● Compute the test statistic based on the sample data ● Determine the p-value associated with the statistic ● Decide whether to reject the null hypothesis by comparing the p-value to α (i.e. reject the null hypothesis if p < α) ● Report your results, including effect sizes (as described in Effect Size) 11. Explain Wilcoxon signed rank test and Mann Whitney U test. The W ilcoxon signed-rank test is a non-parametric statistical hypothesis test used when comparing two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ (i.e. it is a paired difference test). It can be used as an alternative to the paired Student's t-test, t-test for matched pairs, or the t-test for dependent samples when the population cannot be assumed to be normally distributed. The sign test and Wilcoxon signed rank test (Mann—Whitney U-test) are nonparametric tests for the comparison of paired samples. These data result from designs where each treatment is assigned to the same person or object (or at least subjects that are very much alike). If two treatments are to be compared where the observations have been obtained from two independent groups, the nonparametric Wilcoxon rank sum test (also known as the Mann—Whitney U-test) is an alternative to the two independent sample t test. The Wilcoxon rank sum test is applicable if the data are at least ordinal (i.e., the observations can be ordered). This nonparametric procedure tests the equality of the distributions of the two treatments. 12. Write brief note on statistical software of SPSS, SAS, and Epi Info. Statistical software are specialized computer programs for analysis in statistics and econometrics. The software name originally stood for Statistical Package for the Social Sciences (SPSS), reflecting the original market, although the software is now popular in other fields as well, including the health sciences and marketing. SPSS is a widely used program for statistical analysis in social science. It is also used by market researchers, health researchers, survey companies, government, education researchers, marketing organizations, data miners,[3] and others. The original SPSS manual (Nie, Bent & Hull, 1970) has been described as one of "sociology's most influential books" for allowing ordinary researchers to do their own statistical analysis.[4] In addition to statistical analysis, data management (case selection, file reshaping, creating derived data) and data documentation (a metadata dictionary was stored in the datafile) are features of the base software. Statistics included in the base software: ● Descriptive statistics: Cross tabulation, Frequencies, Descriptives, Explore, Descriptive Ratio Statistics ● Bivariate statistics: Means, t-test, ANOVA, Correlation (bivariate, partial, distances), Nonparametric tests ● Prediction for numerical outcomes: Linear regression ● Prediction for identifying groups: Factor analysis, cluster analysis (two-step, K-means, hierarchical), Discriminant SAS (Statistical Analysis System)[1] is a software suite developed by SAS Institute for advanced analytics, multivariate analyses, business intelligence, data management, and predictive analytics. SAS was developed at North Carolina State University from 1966 until 1976, when SAS Institute was incorporated. SAS was further developed in the 1980s and 1990s with the addition of new statistical procedures, additional components and the introduction of JMP. A point-and-click interface was added in version 9 in 2004. A social media analytics product was added in 2010. Epi Info is public domain statistical software for epidemiology developed by Centers for Disease Control and Prevention(CDC) in Atlanta, Georgia (USA). Epi Info has been in existence for over 20 years and is currently available for Microsoft Windows. The program allows for electronic survey creation, data entry, and analysis. Within the analysis module, analytic routines include t-tests, ANOVA, nonparametric statistics, cross tabulations and stratification with estimates of odds ratios, risk ratios, and risk differences, logistic regression (conditional and unconditional), survival analysis (Kaplan Meier and Cox proportional hazard), and analysis of complex survey data. The software is in the public domain, free, and can be downloaded. 13. Classify and explain different types of t-tests and explain them. 14. Explain the uses of Chi Square test giving suitable examples. ● Chi Square test is most commonly used when data are in frequencies such as the number of responses in two or more categories. ● Chi Square test is an important non-parametric test as no rigid assumptions are necessary in regard to the type of population, no need of parameter values and relatively less mathematical details are involved. ● Chi Square test can also be applied to a complex contingency table with several classes and as such is a very useful test in research work. ● Chi Square test is used for testing of hypothesis and is not useful for estimation. ● Chi Square test for nominal data i.e. data with no natural order (Ex. Gender, color, etc) ● The important applications of Chi Square test in medical statistics are ▪ Test of proportions (compare frequencies of diabetics & non-diabetics in group weighing 40-50kg, 50-60kg, 60-70kg & >70kg) ▪ Test of associations (smoking & cancer, treatment & outcome of the treatment, vaccination & immunity) ▪ Test of goodness of fit (determine if actual numbers are similar to the expected / theoretical numbers) ▪ Test of independence of attributes ▪ Test of homogeneity. 15. Describe ANOVA by stating related assumptions. Explain why Student’s t-test cannot be applied where ANOVA has to be applied. To use the ANOVA test we made the following assumptions: ● Each group sample is drawn from a normally distributed population ● All populations have a common variance ● All samples are drawn independently of each other ● Within each sample, the observations are sampled randomly and independently of each other ● Factor effects are additive ● When utilizing a t-test or ANOVA, certain assumptions have to be in place. In other words, a statistical test cannot be arbitrarily used, but a specific set of conditions must be met for the statistical test to be deemed appropriate and meaningful. These conditions are known as model assumptions. ● The model assumptions for t-test or ANOVA include independence, normality, and homogeneity of variances. ● The observations are from a random sample and they are independent from each other ● The observations are assumed to be normally distributed within each group o ANOVA is still appropriate if this assumption is not met but the sample size in each group is large (> 30) ● The variances are approximately equal between groups o If the ratio of the largest SD / smallest SD < 2, this assumption is considered to be met. ● It is not required to have equal sample sizes in all groups. While the t-test is limited to comparing means of two groups, one-way ANOVA can compare more than two groups. Therefore, the t-test is considered a special case of one-way ANOVA. These analyses do not, however, necessarily imply any causality (i.e., a causal relationship between the left-hand and right-hand side variables) 16. What is ANOVA? Explain the method of one way ANOVA. Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences among group means and their associated procedures (such as "variation" among and between groups), developed by statistician and evolutionary biologist Ronald Fisher. The ANOVA table also shows the statistics used to test hypotheses about the population means. Ratio of and. When the null hypothesis of equal means is true, the two mean squares estimate the same quantity (error variance), and should be of approximately equal magnitude. In other words, their ratio should be close to 1. The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of two or more independent (unrelated) groups (although you tend to only see it used when there are a minimum of three, rather than two groups). 17. Explain the need for testing of hypothesis in pharmaceutical research. A statistical hypothesis is an assumption about the value of population parameter. Hypothesis testing is the process of testing the validity of the statistical hypothesis based on a random sample drawn from the population. Hypothesis testing is performed regularly in many industries. In pharmaceutical research, must perform many hypothesis tests on new drug products before they are deemed to be safe and effective by the federal food and drug administration (FDA). In these instances, the drug is hypothesized to be both unsafe and ineffective. Then, if the sample results from the studies performed provide “significant” evidence to the contrary, the FDA will allow the company to market the drug. Hypothesis testing is used by pharmaceutical companies to ascertain whether a drug is effective against a certain disease, by neuroscientists to determine whether neuroplasticity-based therapy helps stroke patients. In most experiments in pharmaceutical research, the variance is unknown. Usually, the only estimate of the variance comes from the experimental data itself. Use of the cumulative standard normal distribution to determine probabilities for the comparison of a mean to a known value is valid only if the variance is known. 18. Explain Chi-square test. A chi-squared test, also referred to as test (or chi-square test), is any statistical hypothesis test in which the sampling distribution of the test statistic is a chi-square distribution when the null hypothesis is true. Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can then be used to reject the null hypothesis that the data are independent. Also considered a chi-square test is a test in which this is asymptotically true, meaning that the sampling distribution (if the null hypothesis is true) can be made to approximate a chi-square distribution as closely as desired by making the sample size large enough. The chi-squared test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. 19. List the pharmaceutical applications of Student’s t-test. Student t-test: Student t-test, in statistics, a method of testing hypotheses about the mean of a small sample drawn from a normally distributed population when the population standard deviation is unknown. We can use this test under the assuming for the sample size is lesser than 30, observations should be independent from each other, one observation is not related or does not affect another observations, data should be followed normally distributed and data should be randomly selected from a population, where each item has an equal chance of being selected. There are two type of Student t-test under one sample and two sample. One sample student t-test is a statistical procedure used to examine the mean difference between the sample and the known value of the population mean. It is used to determine if a mean response changes under different experimental conditions. In other hand, two-sample t-test is used to compare the means of two independent populations, denoted µ1 and µ2 with standard deviation of the populations should be equal. This test has ubiquitous application in the analysis of controlled clinical trials. For example in clinical trials, the comparison of mean decreases in diastolic blood pressure between two groups of patients receiving different antihypertensive agents, or estimating pain relief from a new treatment relative to that of a placebo based on subjective assessment of percent improvement in two parallel groups. Short Notes 1. What is Chi-square test? A chi-squared test, also written as χ2 test, is any statistical hypothesis test wherein the sampling distribution of the test statistic is a chi-squared distribution when the null hypothesis is true. Without other qualification, 'chi-squared test' often is used as short for Pearson's chi-squared test. Chi-squared tests are often constructed from a sum of squared errors, or through the sample variance. Test statistics that follow a chi-squared distribution arise from an assumption of independent normally distributed data, which is valid in many cases due to the central limit theorem. A chi-squared test can be used to attempt rejection of the null hypothesis that the data are independent. 2. Power of study The power of any test of statistical significance is defined as the probability that it will reject a false null hypothesis. Statistical power is the likelihood that a study will detect an effect when there is an effect there to be detected. 3. R values of Correlation In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatter plot. The value of r is always between +1 and –1. To interpret its value, see which of the following values your correlation r is closest to: Exactly –1. 4. Explain: Range, Inter-quartile range & Variance The range is simply the highest value minus the lowest value. But range does not provide a sufficient picture about the dispersion. Range = U – L Interquartile range is an extension of the range that considers quartiles within a data set. Quartiles of a data set are three points that divide the data set into four parts. The three values are first quartile or Q1 which mainly represent the initial 25% of the data set, second quartile (or median) or Q2, which represents the initial 50% of the data set and third quartile or Q3, which represents the initial 75% of the data set. Interquartile range is the difference between Q3 andQ 1. The interquartile range summarizes the spread or variation of values in a data set especially around the median. However, like range it provides incomplete information about the data. Inter quartile range = Q3-Q1 Semi Inter quartile range or quartile deviation = The variance and standard deviation describe how far or close the numbers or observations of a data set lie from the mean (or average). Variance is the measure of the average distance between each of a set of data points and their mean value; equal to the sum of the squares of the deviation from the mean value. Standard deviation though calculated as the square root of the variance is the absolute value calculated to indicate the extent of deviation from the average of the data set. Variance for Ungrouped data for Grouped Data 5. What is ANOVA? Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences among group means and their associated procedures (such as "variation" among and between groups), developed by statistician and evolutionary biologist Ronald Fisher. 6. Student’s t-test A t-test is any statistical hypothesis test in which the t est statistic follows a Student's t-distribution under the null hypothesis. It can be used to determine if two sets of data are significantly different from each other. 7. Applications of Student’s t-test. Among the most frequently used t-tests are: ● A one-sample location test of whether the mean of a population has a value specified in a null hypothesis. ● A two-sample location test of the null hypothesis such that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as "unpaired" or "independent samples" t-tests, as they are typically applied when the statistical unitsunderlying the two samples being compared are non-overlapping.[8] ● A test of the null hypothesis that the difference between two responses measured on the same statistical unit has a mean value of zero. For example, suppose we measure the size of a cancer patient's tumor before and after a treatment. If the treatment is effective, we expect the tumor size for many of the patients to be smaller following the treatment. This is often referred to as the "paired" or "repeated measures" t-test:[8][9] see paired difference test. ● A test of whether the slope of a regression line differs significantly from 0. 8. Standard error of mean. A standard error is the standard deviation of the sampling distribution of a statistic. Standard error is a statistical term that measures the accuracy with which a sample represents a population. In statistics, a sample mean deviates from the actual mean of a population; this deviation is the standard error. 9. Pearson’s Correlation Correlation is a bivariate analysis that measures the strengths of association between two variables. In statistics, the value of the correlation coefficient varies between +1 and -1. When the value of the correlation coefficient lies around ± 1, then it is said to be a perfect degree of association between the two variables. As the correlation coefficient value goes towards 0, the relationship between the two variables will be weaker. 10. P-value A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis, so you reject the null hypothesis. A large p-value (> 0.05) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis. The p-value is defined as the probability of obtaining a result equal to or "more extreme" than what was actually observed, when the null hypothesis is true. In frequentist inference, the p-value is widely used in statistical hypothesis testing, specifically in null hypothesis significance testing. 11. Sign test The Sign test is a non-parametric test that is used to test whether or not two groups are equally sized. The sign test is used when dependent samples are ordered in pairs, where the bivariate random variables are mutually independent. It is based on the direction of the plus and minus sign of the observation, and not on their numerical magnitude. It is also called the binominal sign test, with p = 0.5. The sign test is considered a weaker test, because it tests the pair value below or above the median and it does not measure the pair difference. 12. What is type I and type II errors in hypothesis testing Or What is α & β error α- 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, and is often denoted by α. β-error - A Type II error occurs when the researcher accepts 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. 13. One-tailed and two-tailed tests. The extreme values to only one side of the mean in which case the region of significance will be a region to one-sideone side of the mean in which case the region of significance will be a region to one-side of the distribution. The area of such a region will be equal to the level of significance itself. Such test is called a one tailed test. The test of acceptance or non-acceptance of a hypothesis, we concentrated on the value of z on both sides of the mean. This can be categorically stated that the focus of attention lies in the two tails of the distribution and hence such test is called two tailed test. 14. Mann Whiney U tests Or Wilcoxon rank sum test The sign test and Wilcoxon signed rank test (Mann—Whitney U-test) are nonparametric tests for the comparison of paired samples. These data result from designs where each treatment is assigned to the same person or object (or at least subjects that are very much alike). If two treatments are to be compared where the observations have been obtained from two independent groups, the nonparametric Wilcoxon rank sum test (also known as the Mann—Whitney U-test) is an alternative to the two independent sample t test. The Wilcoxon rank sum test is applicable if the data are at least ordinal (i.e., the observations can be ordered). This nonparametric procedure tests the equality of the distributions of the two treatments. 15. Explain one way ANOVA. The one-way analysis of variance (ANOVA) is used to determine whether there are any statistically significant differences between the means of three or more independent (unrelated) groups. 16. Define Regression. Explain types of regression Or What is linear regression? Regression (also known as simple regression, linear regression, or least squares regression) fits a straight line equation of the following form to the data: Y = a + bX where Y is the dependent variable, X is the single independent variable, a is the Y-intercept of the regression line, and b is the slope of the line (also known as the regression coefficient). Two types of regressions i.e., Regression lines of x on y and Regression lines of y on x. 17. Types of correlation Correlation are of three types: ● Positive Correlation ● Negative Correlation ● No correlation In correlation, when values of one variable increase with the increase in another variable, it is supposed to be a positive correlation. On the other hand, if the values of one variable decrease with the decrease in another variable, then it would be a negative correlation. There might be the case when there is no change in a variable with any change in another variable. In this case, it is defined as no correlation between the two. 18. Paired t-test The purpose of the test is to determine whether there is statistical evidence that the mean difference between paired observations on a particular outcome is significantly different from zero. The Paired Samples t Test is a parametric test. This test is also known as: Dependent t Test. 19. Degree of freedom In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. The number of independent ways by which a dynamic system can move, without violating any constraint imposed on it, is called number of degrees of freedom. In other words, the number of degrees of freedom can be defined as the minimum number of independent coordinates that can specify the position of the system completely. Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter are called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter are equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself 20. Confidence intervals Confidence interval - Statisticians use a confidence interval to express the degree of uncertainty associated with a sample statistic. A confidence interval is an interval estimate combined with a probability statement. A confidence interval is a range around a measurement that conveys how precise the measurement is. For most chronic disease and injury programs, the measurement in question is a proportion or a rate (the percent of New Yorkers who exercise regularly or the lung cancer incidence rate). Confidence intervals are often seen on the news when the results of polls are released. 21. Difference between statistics and parameter. Parameters and statistics are both numbers which are calculated. The difference between these two terms comes from where you get the numbers from. A parameter is any number calculated from a population. A statistic is any number calculated from a sample. 22. Difference between ANOVA & Student’s t-test 23. Differentiate parametric and nonparametric data A parametric statistical test is one that makes assumptions about the parameters (defining properties) of the population distribution(s) from which one's data are drawn, while a non-parametric test is one that makes no such assumptions. Nonparametric tests are also called distribution-free tests because they don't assume that your data follow a specific distribution. You may have heard that you should use nonparametric tests when your data don't meet the assumptions of the parametric test, especially the assumption about normally distributed data.
