Parametric tests

Department of Community Medicine, BMC, Sagar

1. PARAMETRIC TESTS Guided by- Presented by- Dr. Shraddha Mishra Dr. Shefali Jain (Associate Professor) (P.G. 2 nd year)

2. Let’s take an example to understand the concept of Hypothesis Testing. A person is on trial for a criminal offense and the judge needs to provide a verdict on his case. Now, there are four possible combinations in such a case: First Case: The person is innocent and the judge identifies the person as innocent Second Case: The person is innocent and the judge identifies the person as guilty Third Case: The person is guilty and the judge identifies the person as innocent Fourth Case: The person is guilty and the judge identifies the person as guilty

3. As you can clearly see, there can be two types of error in the judgment – Type 1 error, when the verdict is against the person while he was innocent and Type 2 error, when the verdict is in favor of Person while he was guilty According to the Presumption of Innocence, the person is considered innocent until proven guilty. That means the judge must find the evidence which convinces him “beyond a reasonable doubt”. This phenomenon of  “Beyond a reasonable doubt”  can be understood as  Probability (Judge Decided Guilty | Person is Innocent) should be small

4. The basic concepts of Hypothesis Testing are actually quite analogous to this situation. We consider  the Null Hypothesis  to be true until we find strong evidence against it. Then. we accept the  Alternate Hypothesis . We also determine the  Significance Level (⍺ ) which can be understood as the Probability of (Judge Decided Guilty | Person is Innocent) in the previous example. Thus, if ⍺ is smaller, it will require more evidence to reject the Null Hypothesis.

5. Directional Hypothesis In the Directional Hypothesis, the null hypothesis is rejected if the test score is too large (for right-tailed and too small for left tailed). Thus, the rejection region for such a test consists of one part, which is right from the center

6. Non-Directional Hypothesis In a Non-Directional Hypothesis test, the Null Hypothesis is rejected if the test score is either too small or too large. Thus, the rejection region for such a test consists of two parts: one on the left and one on the right

7. p-value has the benefit that we  only need one value  to make a decision about the hypothesis. We don’t need to compute two different values like critical value and test scores. Another benefit of using p-value is that we can test at  any desired level of significance  by comparing this directly with the significance level

8. Critical Value is the cut off value between Acceptance Zone and Rejection Zone. We compare our test score to the critical value and if the test score is greater than the critical value, that means our test score lies in the Rejection Zone and we reject the Null Hypothesis. On the opposite side, if the test score is less than the Critical Value, that means the test score lies in the Acceptance Zone and we fail to reject the null Hypothesis

9. Statistical Test These are intended to decide whether a hypothesis about distribution of one or more populations should be rejected or accepted. These may be: Parametric Test Non Parametric Test

10. These tests the statistical significance of the:- 1) Difference in sample and population means. 2) Difference in two sample means 3) Several population means 4) Difference in proportions between sample and population 5) Difference in proportions between two independent populations 6) Significance of association between two variables 

11. System for statistical Analysis State the Research Hypothesis State the Level of Significance Calculate the test statistic Compare the calculated test statistic with the tabulated values Decision Statement of Result

12. when to use parametric test? To use parametric test the following conditions have to be satisfied: Data must be either in Interval scale or ratio scale. Subjects should be randomly selected. Data should be normally distributed

13. Determination of parametric test Interval scale: interval between observation in terms of fixed unit of measurement. Eg . Measures of temperature. Ratio scale: The scale has a fundamental zero point. Eg . Age, income. (IN CASE OF NOMINAL AND ORDINAL SCALE NON- PARAMETRIC TEST IS USED)

14. Types of parametric tests Large sample tests. - Z test Small sample test - t-test Independent /unpaired – t test Paired t-test ANOVA (analysis of variance) one way analysis of variance Two way analysis of variance

15. STUDENT’S T-TEST Developed by Prof W.S Gossett in 1908 , who published statistical papers under the pen name of ‘Student’. Thus the test is known as Student’s ‘t’ test. Indications for the test:- 1. When samples are small 2. Population variance are not known.

16. Uses 1. Two means of small independent samples 2. Sample mean and population mean 3. Two proportions of small independent samples 

17. Assumptions made in the use of ‘t’ test 1. Samples are randomly selected 2. Data utilised is Quantitative 3. Variable follow normal distribution 4. Sample variances are mostly same in both the groups under the study 5. Samples are small, mostly lower than 30

18. ONE SAMPLE T-TEST When compare the mean of a single group of observations with a specified value In one sample t-test, we know the population mean. We draw a random sample from the population and then compare the sample mean with the population mean and make a statistical decision as to whether or not the sample mean is different from the population

19. Now we compare calculated value with table value at certain level of significance (generally 5% or 1%) If absolute value of ‘t’ obtained is greater than table value then reject the null hypothesis and if it is less than table value, the null hypothesis may be accepted

20. Two Sample ‘t’ test A. Unpaired Two sample ‘t’- test Unpaired t- test is used when we wish to compare two means Used when the two independent random samples come from the normal populations having unknown or same variance We test the null hypothesis, that the two population means are same i.e µ1= µ2 against an appropriate one sided or two sided alternative hypothesis

21. Assumptions: The samples are random & independent of each other The distribution of dependent variable is normal. The variances are equal in both the groups

22. PAIRED TWO-SAMPLES T-TEST Used when we have paired data of observations from one sample only, when each individual gives a pair of observations. Same individuals are studied more than once in different circumstances- measurements made on the same people before and after interventions

23.  Assumptions: The outcome variable should be continuous The difference between pre-post measurements should be normally distributed  Instead of using a series of individual comparisons we examine the differences among the groups through an analysis that considers the variation among all groups at once. i.e. ANALYSIS OF VARIANCE

24. Analysis of Variance(ANOVA) Given by Sir Ronald Fisher The principle aim of statistical models is to explain the variation in measurements. Analysis of variance (ANOVA) is a statistical technique that is used to check if the means of two or more groups are significantly different from each other. ANOVA checks the impact of one or more factors by comparing the means of different samples .

25. Another measure to compare the samples is called a t-test. When we have only two samples, t-test and ANOVA give the same results. However, using a t-test would not be reliable in cases where there are more than 2 samples. If we conduct multiple t-tests for comparing more than two samples, it will have a compounded effect on the error rate of the result. Assumptions for ANOVA Sample population can be easily approximated to normal distribution. All populations have same Standard Deviation. Individuals in population are selected randomly. Independent samples

26. The Null hypothesis in ANOVA is valid when all the sample means are equal, or they don’t have any significant difference. Thus, they can be considered as a part of a larger set of the population. On the other hand, the alternate hypothesis is valid when at least one of the sample means is different from the rest of the sample means. 

27. ANOVA compares variance by means of a simple ratio, called F-Ratio F=Variance between groups/Variance within groups The resulting F statistics is then compared with critical value of F (critic), obtained from F tables in much the same way as was done with ‘t’ If the calculated value exceeds the critical value for the appropriate level of α, the null hypothesis will be rejected.

28. Variance between groups

29. F- statistics F=Variance between groups/Variance within groups

30. A F test is therefore a test of the Ratio of Variances F Tests can also be used on their own, independently of the ANOVA technique, to test hypothesis about variances. In ANOVA, the F test is used to establish whether a statistically significant difference exists in the data being tested.

31. One Way ANOVA If the various experimental groups differ in terms of only one factor at a time- a one way ANOVA is used e.g. A study to assess the effectiveness of four different antibiotics on S Sanguis E.g. You have a group of individuals randomly split into smaller groups and completing different tasks. For example, you might be studying the effects of tea on weight loss and form three groups: green tea, black tea, and no tea.

32. Two Way ANOVA If the various groups differ in terms of two or more factors at a time, then a Two Way ANOVA is performed e.g. A study to assess the effectiveness of four different antibiotics on S Sanguis in three different age groups E.g. you might want to find out if there is an interaction between income and gender for anxiety level at job interviews. The anxiety level is the outcome, or the variable that can be measured. Gender and Income are the two  categorical variables .

33. Pearson’s Correlation Coefficient Correlation is a technique for investigating the relationship between two quantitative, continuous variables Pearson’s Correlation Coefficient(r) is a measure of the strength of the association between the two variables.

34. Assumptions Made in Calculation of ‘r’ 1. Subjects selected for study with pair of values of X & Y are chosen with random sampling procedure. 2. Both X & Y variables are continuous 3. Both variables X & Y are assumed to follow normal distribution

35.  Steps The first step in studying the relationship between two continuous variables is to draw a scatter plot of the variables to check for linearity. The correlation coefficient should not be calculated of the relationship is not linear For correlation only purposes, it does not matter on which axis the variables are plotted

36. However, conventionally, the independent variable is plotted on X axis and dependent variable on Y-axis The nearer the scatter of points is to a straight line, the higher the strength of association between the variables. 

37. Types of Correlation Perfect Positive Correlation r=+1 Partial Positive Correlation 0<r<+1 Perfect negative correlation r=-1 Partial negative correlation 0>r>-1 No Correlation

38. Z Test z  tests are a statistical way of testing a hypothesis when either: We know the population variance, or We do not know the population variance but our sample size is large n ≥ 30 If we have a sample size of less than 30 and do not know the population variance, then we must use a t-test

39. Assumptions to apply Z test The sample must be randomly selected Data must be quantitative Samples should be larger than 30 Data should follow normal distribution Sample variances should be almost the same in both the groups of study 

40. If the SD of the populations is known, a Z test can be applied even if the sample is smaller than 30

41. Indications for Z Test To compare sample mean with population mean To compare two sample means To compare sample proportion with population proportion To compare two sample proportions

42. Steps 1. Define the problem 2. State the null hypothesis (H0) & alternate hypothesis (H1) 3. Find Z value Z= Observed mean-Mean Standard Error 4. Fix the level of significance 5. Compare calculated Z value with the value in Z table at corresponding degree significance level. If the observed Z value is greater than theoritical Z value, Z is significant, reject null hypothesis and accept alternate hypothesis

43. One tailed and Two tailed Z tests • Z values on each side of mean are calculated as +Z or as -Z. A result larger than difference between sample mean will give +Z and result smaller than the difference between mean will give -Z

44. Two sample Z test

45. E.g. for two tailed: In a test of significance, when one wants to determine whether the mean IQ of malnourished children is different from that of well nourished and does not specify higher or lower, the P value of an experiment group includes both sides of extreme results at both ends of scale, and the test is called two tailed test. E.g. for single tailed: In a test of significance when one wants to know specifically whether a result is larger or smaller than what occur by chance, the significant level or P value will apply to relative end only e.g. if we want to know if the malnourished have lesser mean IQ than the well nourished, the result will lie at one end ( tail )of the distribution, and the test is called single tailed test

46. Conclusion Tests of significance play an important role in conveying the results of any research & thus the choice of an appropriate statistical test is very important as it decides the fate of outcome of the study. Hence the emphasis placed on tests of significance in clinical research must be tempered with an understanding that they are tools for analyzing data & should never be used as a substitute for knowledgeable interpretation of outcomes.

47. THANK YOU