Parametric test - t Test, ANOVA, ANCOVA, MANOVA

parametric test - t test, ANOVA, ANCOVA, MANOVA

1. Parametric Test : t 2 test anova ancova manova Princy Francis M I st Yr MSc(N) JMCON

2. DEFINITION Statistics is a branch of science that deals with the collection, organisation, analysis of data and drawing of inferences from the samples to the whole population. Statistical tests are intended to decide whether a hypothesis about distribution of one or more populations or samples should be rejected or accepted.

3. DEFINITION Inferential statistics is the statistics that permit inferences on whether the results observed in a sample are likely to occur in the larger population Parametric test is a class of statistical tests that involve assumptions about the distribution of the variables and estimation of a parameter.

4. Parametric test Parametric test is a statistical test that makes assumptions about the parameters of the population distribution(s) from which one’s data is drawn. most commonly used. the findings are inferred to the parameters of a normally distributed populations. Numerical data (quantitative variables) that are normally distributed are analysed with parametric tests. P arametric tests are done on the basis of mean and standard deviation

5. ASSUMPTIONS It requires 3 assumptions for using : the sample was drawn from a population for which the variance can be calculated. It is expected to be in normal distribution . the levels of measurement should be atleast interval data or ordinal data with an approximately normal distribution. the data can be treated as random samples .

6. Application of parametric test • Used for Quantitative data. • Used for continuous variables. • Used when data are measured on approximate interval or ratio scales of measurement. • Data should follow normal distribution.

7. PARAMETRIC STATISTICAL ANALYSIS Student's t-test Analysis of variance (ANOVA) A nalysis of Covariance (ANCOVA) M ultivariate analysis of variance (MANOVA)

8. Student's  t -test Developed by Prof.W.S.Gossett Student's  t -test is used to test the null hypothesis that there is no difference between the means of the two groups Indication for t test When samples are small (<30) Population variance are not known

9. TYPES One-sample  t -test Independent Two Sample T Test (the unpaired  t -test) The paired  t -test

10. One-sample  t -test To test if a sample mean (as an estimate of a population mean) differs significantly from a given population mean. The mean of one sample is compared with population mean where  = sample mean,  u  = population mean and S = standard deviation, n = sample size  

11. Example A random sample of size 20 from a normal population gives a sample mean of 40, standard deviation of 6. Test the hypothesis is population mean is 44. Check whether there is any difference between mean. H : There is no significant difference between sample mean and population mean H 1 : There is significant difference between sample mean and population mean mean = 40 , , n = 20 and S = 6  

12. t calculated = 2.981 t table value = 2.093 t calculated > t table value ; Reject H 0.

13. Independent Two Sample T Test (the unpaired  t -test) To test if the population means estimated  by two independent samples  differ significantly. Two different samples with same mean at initial point and compare mean at the end

14. t = Where 1 - 2   is the difference between the means of the two groups and S denotes the standard deviation. Example : Compare the height of girls and boys. compare 2 stress reduction interventions  

15. ILLUSTRATION Mean Hb level of 5 male are 10, 11, 12.5, 10.5, 12 and 5 female are 10, 17.5, 14.2,15 and 14.1 . Test whether there is any significant difference between Hb values. H : There is no significant difference between Hb Level H 1 : There is significant difference between Hb level. t =  

16. = 11.2 , 14.16 , = 1.075, = 7.293 t calculated = 2.287, t table = 2.306, t calculated > t table value ; reject H 0.   X 1 X 2 X 1 - X 2 - (X 1 - ) 2 (X 2 - ) 2 10 11 12.5 10.5 12 10 17.5 14.2 15 14.1 -1.2 - 0.2 1.3 -0.7 0.8 -4.16 3.34 0.04 0.84 -0.06 1.44 0.04 1.69 0.49 0.64 17.305 11.156 0.0016 0.706 0.0036 = 56 70.8     4.3 29.172 X 1 X 2 10 11 12.5 10.5 12 10 17.5 14.2 15 14.1 -1.2 - 0.2 1.3 -0.7 0.8 -4.16 3.34 0.04 0.84 -0.06 1.44 0.04 1.69 0.49 0.64 17.305 11.156 0.0016 0.706 0.0036 70.8     4.3 29.172

17. The paired  t -test To test if the population means estimated  by two dependent samples  differ significantly . A usual setting for paired  t -test is when measurements are made on the same subjects before and after a treatment. where   is the mean difference and S d denotes the standard deviation of the difference.  

18. Example Systolic BP of 5 patients before and after a drug therapy is Before 160, 150, 170, 130, 140 After 140, 110, 120, 140, 130 Test whether there is any significant difference between BP level. H : There is no significant difference between BP Level before and after drug H 1 : There is significant difference between BP level before and after drug

19. = 22, S d = 23.875 t calculated = 2.060, t table = 2.567, t calculated < t table value ; Accept H 0.   Before After d d- (d- ) 2 160 150 170 130 140 140 110 120 140 130 20 40 50 -10 10 -2 18 28 -32 -12 4 324 784 1024 144     = 110   2280 Before After d 160 150 170 130 140 140 110 120 140 130 20 40 50 -10 10 -2 18 28 -32 -12 4 324 784 1024 144       2280

20. ANALYSIS OF VARIANCE (ANOVA) R. A. Fischer. The Student's  t -test cannot be used for comparison of three or more groups. The purpose of ANOVA is to test if there is any significant difference between the means of two or more groups. The analysis of variance is the systematic algebraic procedure of decomposing the overall variation in the responses observed in an experiment into variation. Two variances – (a) between-group variability and (b) within-group variability that is variation existing between the samples and variations existing within the sample. The within- group variability (error variance) is the variation that cannot be accounted for in the study design. The between-group (or effect variance) is the result of treatment

21. ASSUMPTIONS OF ANOVA The population in which samples are drawn should be normal The sample observations are independent of each other The samples are selected at random The samples are drawn from population having equal variance The sample size should not differ widely The various effects(treatment and errors) are additive in nature The experimental error are normally and independently distributed with mean Zero

22. A simplified formula for the  F  statistic is where  MST  is the mean squares between the groups and  MSE  is the mean squares within groups

23. TYPES ONE WAY ANOVA T WO WAY ANOVA

24. ONE WAY ANOVA It compares three or more unmatched groups when data are categorized in one way. Total sum of Square(TSS) = treatment sum of square(SST) + error sum of square(SSE) Example. 1. Compare control group with three different doses of aspirin in rats 2. Effect of supplementation of vit C in each subject before, during and after the treatment.

25. One way ANOVA table Source d.f Sum of Square Mean Square F Between treatment Within treatment error t-1   n-t SST   SSE MST = SST/t-1   MSE = SSE/n-t F= MST/MSE F (t-1),(n-t) Total n-1 TSS     Source d.f Sum of Square Mean Square F Between treatment Within treatment error t-1   n-t SST   SSE MST = SST/t-1   MSE = SSE/n-t Total n-1 TSS    

26. TWOWAY ANOVA It is used to determine the effect of two nominal predictor variables on a continuous outcome variable. A two-way ANOVA test analyzes the effect of the independent variables on the expected outcome along with their relationship to the outcome itself. Example : Effect of two antihypertensive drugs in two different doses Comparing the employee productivity based on the working hours and working conditions

27. Two way ANOVA table Source d.f Sum of Square Mean Square F Treatment Blocks Error t-1 r-1 (r-1)(t-1) SST SSB SSE MST = SST/t-1 MSB = SSB/r-1 MSE = SSE/(r-1)(t-1) F t = MST/MSE F (t-1), (t-1) (r-1) F b = MSB/MSE F (r-1), (t-1) (r-1) Total rt-1 TSS     Source d.f Sum of Square Mean Square F Treatment Blocks Error t-1 r-1 (r-1)(t-1) SST SSB SSE MST = SST/t-1 MSB = SSB/r-1 MSE = SSE/(r-1)(t-1) Total rt-1 TSS    

28. Difference between one & two way ANOVA a one-way ANOVA is used to determine if there is a difference in the mean height of stalks of three different types of seeds. Since only 1 factor that could be making the heights different. if three different types of seeds, and then add the possibility that three different types of fertilizer is used The mean height of the stalks could be different for a combination of several reasons. Two factors (type of seed and type of fertilizer), use a two-way ANOVA. 

29. ANALYSIS OF COVARIANCE (ANCOVA ) ACOVA is a technique that combines the feature of analysis of variance and regression . It is used to increase the precision of treatment comparisons. This method is based on the fact that there are some extraneous sources of variation which also contribute to the experimental error but are not controlled. These additional variations are known as the ancillary or concomitant variates .

30. DEFINITION The very logical procedure, which reduces the experimental error by eliminating from it the effects of variations in the concomitant variate and thus increase the precision of the main variate on the concomitant variate is known as Analysis of Covariance (ANCOVA)

31. Example : Effect of 3 diet on gaining weight of animal with different age group and different initial weight will influence animal performance and precision of experiment.

32. Assumptions All assumptions of ANOVA is applicable here too. In addition, it is assumed that, The relationship between X and Y is linear The relationship is same for each treatment The covariates are not affected by treatment The observations are from normal populations

33. USES To increase the precision in a randomized experiment To remove the effects of disturbing variables in observational studies To throw light on the nature of treatment effects To analyse the data when some observations are missing To fit regression in multiple classification

34. WORKING OF ANCOVA ANCOVA works by adjusting the total SS, group SS, and error SS of the independent variable to remove the influence of the covariate Y ij = µ + t i + β ( x ij - ) + e ij Where Y ij is the j th observation on the response variable under i th treatment , µ is mean of x variable, t i is i th treatment effect β linear regression coefficient e ij random error component which is independently and normally distributed with mean zero and variance  

35. Sum of Squares

36. Sum of Products To control for the covariate, the sum of products (SP) for the dependent variable and covariate must also be used. x is the covariate, and y is the dependent variable. i is the individual subject, and j is the group.

37. Error Sum of Products This is the sum of the products of the dependent variable and residual minus the group means of the dependent variable and residual.

38. Adjusting the Sum of Squares Using the SS’s for the covariate and the dependent variable, and the SP’s, adjust the SS’s for the dependent variable

39. Test Statistic F = Mean square of β / adjusted error mean square

40. ADVANTAGES Better power Improved ability to detect and estimate interactions The availability of extensions to deal with measurement error in the covariates.  

41. DISADVANTAGES There will be cost of introducing the blocking factor. It may be difficult to find blocking factors that are highly correlated with the dependent variable. Loss of power may occur if a poorly correlated blocking factor is used.

42. MULTIVARIATE ANALYSIS OF VARIANCE (MANOVA) variation of ANOVA. MANOVA assesses the statistical significance of the effect of one or more independent variables on a set of two or more dependent variables. MANOVA has the ability to examine more than one dependent variable at once or simultaneous effect of independent variables on multiple dependent variables. Control Type I error.

43. EXAMPLE Case study on 2 different text books & student improvement in maths and physics. Dependent variables here are improvement in maths and physics. Hypothesis: both the Dependent Variable are affected by difference in text books.

44. ASSUMPTIONS Multivariate normality : dependent variables should be normally distributed within groups. Linear combinations of dependent variables must be distributed. All subjects of variables must have multivariate normal distribution. Homogeneity of covariance matrices. The inter correlations (co variances) of the multiple dependent variable across the cells of design.

45. ASSUMPTIONS Independence of observations Subject score on dependent variables are not influenced or related to other subject scores. Linearity Linear relationship against all pairs of dependent variables, all pairs of covariates, all dependent variable – covariate pairs in each cell. Therefore if relationship deviates from linearity the power of analysis will be compromised.

46. ADVANTAGES MANOVA enables to test multiple dependent variables. MANOVA can protect against Type I errors

47. DISADVANTAGES MANOVA is many times more complicated than ANOVA, making it a challenge to see which independent variables are affecting dependent variables. One degree of freedom is lost with the addition of each new variable.

48. SUMMARY OF PARAMETRIC TESTS APPLIED FOR DIFFERENT TYPE OF DATA Type of Group Parametric test Comparison of two paired groups Paired t test Comparison of two unpaired groups Unpaired two sample t test Comparison of population and sample drawn from the same population One sample t test Comparison of 3 or more matched groups but varied in 1 factors One Way Anova Comparison of 3 or more matched groups but varied in 2 factor Two Way Anova

49. JOURNAL ABSTRACT A Multivariate Analysis (MANOVA) of Where Adult Learners Are In Higher Education American institutions of higher education were originally established with the purpose of educating the advantaged youth. Due to this increase in adults re-entering the academy, it is appropriate and timely to ask where these students are attending school, what is known about their distribution in the higher education system, and whether they are assembled in one type of institution or evenly distributed among institutions.

50. Therefore, the purpose of this study was to determine where undergraduate adult students are located within the 4-year private, public, and for-profit universities offering undergraduate degrees in the United States. This study utilized descriptive and multivariate analyses of variance (MANOVA) statistical analyses. Descriptive analysis provided the number, means, and standard deviations for college and university enrolments obtained from the Integrated Postsecondary Education Data System (IPEDS) of the National Center for Education Statistics (NCES) to answer two research questions. Two MANOVAs and comparative designs were employed to examine electronic data accessed through IPEDS. Undergraduate students under the age of 25 are enrolling in 4-year public and private universities in the United States at about double the enrolment rate as that of for-profit universities.

51. Using ANOVA to Examine the Relationship between Safety & Security and Human Development This study aimed to examine the relationship between safety and security index and human development. The sample consisted of 53 African countries. The research question is Does a statistical significant relationship exist between safety and security index and human development. In the process of examining the relationship between variables, researchers can use t test or ANOVA to compare the means of two groups on the dependent variable.

52. The main difference between t-test and ANOVA is that t test can only be used to compare two groups while ANOVA can be used to compare two or more groups. In the process of selecting the data analysis technique for this study, considered both ANOVA and t-test. The advantage ANOVA has over t-test is that the post-hoc tests of ANOVA allow to better controlling type 1 error. Therefore, in order to control type 1 error, chose ANOVA as data analysis technique for this study. The results indicated that there is a statistically significant relationship with strong effect size between safety and security index and human development.

53. ASSIGNMENT Write an assignment on parametric statistical application in nursing .

54. REFERENCES Mahajan BK. Methods in Biostatistics. Sixth edition. New Delhi: Jaypee brothers medical publishers. 2003 Rao SSSP. Biostatistics. Third edition. New Delhi: Prentice Hall India Pvt Ltd;2004. Agarwal LB. Basic Biostatistics. Fourth edition. India: Newage international publishers:2006 Ali Z, Bhaskar BS. Basic Statistical Tools in research and data analysis. Indian J Anaesth . 2016 Sep; 60(9): 662–669. Sundaram RK, Dwivedi SN, Sreenivas V. Medical Statistics : Principles and methods. Second edition. New Delhi: Wolter Kluwer publication; 2015.