Basics of Pharmaceutical Statistics Fundamentals of Hypothesis Testing PREPARED BY: - Dr. PORAS PATEL Mob : 091-7874572005 Email Id: poras1980@gmail.com

What is biostatistics Statistics is the science and art of collecting, summarizing, and analyzing data that are subject to random variation. Biostatistics is the application of statistics and mathematical methods to the design and analysis of health, biomedical, and biological studies. Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Different Tests of Significance One-Sample (t-test or z-test) Compares one sample mean versus a population mean Two-Sample (t-test) Compares one sample mean versus another sample mean Independent t-tests (equal samples) Dependent t-tests (dependent/paired samples) Analysis of Variance (ANOVA) Comparing several samples mean. It can be classified as One Way & Two Way and others like Ancova. Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

How to properly use Biostatistics Develop an underlying question of interest Generate a hypothesis Design a study (Protocol) Collect Data Analyze Data Descriptive statistics Statistical Inference Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Relationship between population and sample (Simple random sampling) Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Sampling Techniques Population Simple Random Sample Stratified Random Systematic Sampling Cluster Sampling Convenience Sampling Bias free sample Bias free sample Biased sample Bias free sample Biased sample Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Impact of Misuse of statistics About 25% of biological research is flawed because of incorrect conclusions drawn from confounded experimental designs and misuse of statistical methods Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

What is a Hypothesis? A hypothesis is a claim (assumption) about the population parameter Difference between the value of sample statistic and the corresponding hypothesized parameter value is called hypothesis testing. Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Hypothesis Testing Process Assume the population mean age is 50. Identify the Population ( ) Take a Sample No, not likely! REJECT Null Hypothesis Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

... Therefore, we reject the null hypothesis that m = 50. Reason for Rejecting H0 Sampling Distribution of It is unlikely that we would get a sample mean of this value ... ... Therefore, we reject the null hypothesis that m = 50. ... if in fact this were the population mean. m = 50 20 If H0 is true Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Components of Biostatistics Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Normal Distribution Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Normal distribution BELL-SHAPED symmetrical about the mean (No skewness) total area under curve = 1 approximately 68% of distribution is within one standard deviation of the mean approximately 95% of distribution is within two standard deviations of the mean approximately 99.7% of distribution is within 3 standard deviations of the mean Mean = Median = Mode Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Empirical Rule 68% About 68% of the area lies within 1 standard deviation of the mean About 95% of the area lies within 2 standard deviations This rule has been discussed earlier. Emphasize that there is still 0.3% of the distribution falling outside the 3 standard deviation limits. About 99.7% of the area lies within 3 standard deviations of the mean Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Copyright © Dr. Poras Patel Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Level of Significance, Is designated by , (level of significance) Typical values are .01, .05, .10 Is selected by the researcher at the beginning Provides the critical value(s) of the test Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The z-Test for Comparing Population Means Critical values for standard normal distribution Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Level of Significance and the Rejection Region I claim that mean CVD in the INDIA is atleast 3 out of 100 people! a H0: m ³ 3 H1: m < 3 Critical Value(s) Rejection Regions a H0: m £ 3 H1: m > 3 a/2 H0: m = 3 H1: m ¹ 3 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Hypothesis Testing State the research question. State the statistical hypothesis. Set decision rule. Calculate the test statistic. Decide if result is significant. Interpret result as it relates to your research question. Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Rejection & Non-Rejection Regions = Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The Null Hypothesis, H0 States the assumption (numerical) to be tested e.g.: The average number of CVD in INDIA is at least three ( ) Is always about a population parameter ( ), not about a sample statistic ( ) Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The Null Hypothesis, H0 (continued) Begins with the assumption that the null hypothesis is true Similar to the notion of innocent until proven guilty Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The Alternative Hypothesis, H1 Is the opposite of the null hypothesis e.g.: The average number of CVD in INDIA is less than 3 ( ) Never contains the “=” sign May or may not be accepted Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

General Steps in Hypothesis Testing e.g.: Test the assumption that the true mean number of of CVD in INDIA is at least three ( Known) State the H0 State the H1 Choose Choose n Choose Test Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

General Steps in Hypothesis Testing (continued) Set up critical value(s) 7. Collect data 8. Compute test statistic and p-value 9. Make statistical decision 10. Express conclusion 100 persons surveyed Computed test stat =-2, p-value = .0228 Reject null hypothesis The true mean number of CVD is less than 3 in human population. Reject H0 a Z -1.645 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The z-Test for Comparing Population Means Critical values for standard normal distribution Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

p-Value Approach to Testing Convert Sample Statistic (e.g. ) to Test Statistic (e.g. Z, t or F –statistic) Obtain the p-value from a table or computer Compare the p-value with If p-value , do not reject H0 If p-value , reject H0 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Comparison of Critical-Value & P-Value Approaches Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Result Probabilities H0: Innocent Jury Trial Hypothesis Test The Truth Verdict Innocent Guilty Decision H True H False Do Not Type II Innocent Correct Error Reject 1 - a Error ( b ) H Type I Reject Power Guilty Error Correct Error H (1 - b ) ( a ) Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Type I & II Errors Have an Inverse Relationship If you reduce the probability of one error, the other one increases so that everything else is unchanged. b a Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Critical Values Approach to Testing Convert sample statistic (e.g.: ) to test statistic (e.g.: Z, t or F –statistic) Obtain critical value(s) for a specified from a table or computer If the test statistic falls in the critical region, reject H0 Otherwise do not reject H0 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

One-tail Z Test for Mean ( Known) Assumptions Population is normally distributed If not normal, requires large samples Null hypothesis has or sign only Z test statistic Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Rejection Region Reject H0 Reject H0 Z Z H0: m ³ m0 H1: m < m0 Z Small values of Z don’t contradict H0 Don’t Reject H0 ! Z Must Be Significantly Below 0 to reject H0 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example: One Tail Test H0: m £ 368 H1: m > 368 Q. Does an average box of cereal contain more than 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified s to be 15 grams. Test at the a = 0.05 level. 368 gm. H0: m £ 368 H1: m > 368 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Finding Critical Value: One Tail Standardized Cumulative Normal Distribution Table (Portion) What is Z given a = 0.05? .05 Z .04 .06 1.6 .9495 .9505 .9515 .95 a = .05 1.7 .9591 .9599 .9608 1.8 .9671 .9678 .9686 1.645 Z Critical Value = 1.645 1.9 .9738 .9744 .9750 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example Solution: One Tail Test H0: m £ 368 H1: m > 368 Test Statistic: Decision: Conclusion: a = 0.5 n = 25 Critical Value: 1.645 Reject Do Not Reject at a = .05 .05 No evidence that true mean is more than 368 1.645 Z 1.50 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

p -Value Solution p-Value is P(Z ³ 1.50) = 0.0668 Z 1.50 Use the alternative hypothesis to find the direction of the rejection region. P-Value =.0668 1.0000 - .9332 .0668 Z 1.50 From Z Table: Lookup 1.50 to Obtain .9332 Z Value of Sample Statistic Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

(p-Value = 0.0668) ³ (a = 0.05) Do Not Reject. p -Value Solution (continued) (p-Value = 0.0668) ³ (a = 0.05) Do Not Reject. p Value = 0.0668 Reject a = 0.05 Z 1.645 1.50 Test Statistic 1.50 is in the Do Not Reject Region Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example: Two-Tail Test Q. Does an average box of cereal contain 368 grams of cereal? A random sample of 25 boxes showed = 372.5. The company has specified s to be 15 grams. Test at the a = 0.05 level. 368 gm. H0: m = 368 H1: m ¹ 368 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example Solution: Two-Tail Test H0: m = 368 H1: m ¹ 368 Test Statistic: Decision: Conclusion: a = 0.05 n = 25 Critical Value: ±1.96 Reject Do Not Reject at a = .05 .025 .025 No Evidence that True Mean is Not 368 -1.96 1.96 Z 1.50 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

(p Value = 0.1336) ³ (a = 0.05) Do Not Reject. p-Value Solution (p Value = 0.1336) ³ (a = 0.05) Do Not Reject. p Value = 2 x 0.0668 Reject Reject a = 0.05 Z 1.50 1.96 Test Statistic 1.50 is in the Do Not Reject Region Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Connection to Confidence Intervals Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

What is a t Test? Commonly Used Definition: Comparing two means to see if they are significantly different from each other Technical Definition: Any statistical test that uses the t family of distributions t Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Independent Samples t Test Use this test when you want to compare the means of two independent samples on a given variable “Independent” means that the members of one sample do not include, and are not matched with, members of the other sample Example: Compare the average height of 50 randomly selected men to that of 50 randomly selected women Independent Mean #1 Independent Mean #2 Compare using t test Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Dependent Samples t Test Used to compare the means of a single sample or of two matched or paired samples Example: If a group of students took a math test in March and that same group of students took the same math test two months later in May, we could compare their average scores on the two test dates using a dependent samples t test Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Comparing the Two t Tests Independent Samples Tests the equality of the means from two independent groups (diagram below) Relies on the t distribution to produce the probabilities used to test statistical significance Dependent Samples Tests the equality of the means between related groups or of two variables within the same group (diagram below) Relies on the t distribution to produce the probabilities used to test statistical significance Treatment group Person #1 Control group Person #2 Before treatment Person #1 After treatment Person #1 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Types One sample compare with population Unpaired compare with control same subjects: pre-post Z-test large samples >30 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Compare Means (or medians)Example: Compare blood presures of two or more groups, or compare BP of one group with a theoretical value. 1 Group: One Sample t test Wilcoxon rank sum test 2 Groups: Unpaired t test Paired t test Mann-Whitney t test Welch’s corrected t test Wilcoxon matched pairs test Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Repeated measures ANOVA Kruskal-Wallis test Friedman test 3-26 Groups: One-way ANOVA Repeated measures ANOVA Kruskal-Wallis test Friedman test (All with post tests) Raw data Average data Mean, SD, & NAverage data Mean, SEM, & N Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Statistical Analysis Is there a difference? control group mean treatment group mean Is there a difference? Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

What does difference mean? The mean difference is the same for all three cases medium variability high variability low variability Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

What does difference mean? medium variability high variability Which one shows the greatest difference? low variability Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

t Test: Unknown Assumption Population is normally distributed If not normal, requires a large sample T test statistic with n-1 degrees of freedom Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example: One-Tail t Test Does an average box of cereal contain more than 368 grams of cereal? A random sample of 36 boxes showed X = 372.5, and s = 15. Test at the a = 0.01 level. 368 gm. H0: m £ 368 H1: m > 368 s is not given Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Example Solution: One-Tail H0: m £ 368 H1: m > 368 Test Statistic: Decision: Conclusion: a = 0.01 n = 36, df = 35 Critical Value: 2.4377 Reject Do Not Reject at a = .01 .01 No evidence that true mean is more than 368 2.4377 t35 1.80 Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The t Table Since it takes into account the changing shape of the distribution as n increases, there is a separate curve for each sample size (or degrees of freedom). However, there is not enough space in the table to put all of the different probabilities corresponding to each possible t score. The t table lists commonly used critical regions (at popular alpha levels). Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Z-distribution versus t-distribution Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

The z-Test for Comparing Population Means Critical values for standard normal distribution Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

Summary We can use the z distribution for testing hypotheses involving one or two independent samples To use z, the samples are independent and normally distributed The sample size must be greater than 30 Population parameters must be known Copyright © Dr. Poras Patel. Copy Without Permission will be Liable for Legal Action.

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