ANOVA and F test , Description and Understanding

By the end of this teaching learning session, co-learners shall be able to :

Explain the concept of Analysis of Variance (ANOVA)

Differentiate between One-way, Two-way, and Repeated Measures ANOVA

Apply ANOVA to experimental/clinical data

Interpret results by comparing Fstat or Fcalculated with Fcritical using the F-table

1. Anova and F-Test Presenter - Dr. Snehashis Singha (JR-2) Peer support - Dr. Vivek Ranjan (JR-2) Department of Pharmacology & Therapeutics King George’s Medical University Lucknow, Uttar Pradesh, India - 226003 Email: [email protected]

2. Specific learning objectives By the end of this teaching learning session, co-learners shall be able to : Explain the concept of Analysis of Variance (ANOVA) Differentiate between One-way, Two-way, and Repeated Measures ANOVA Apply ANOVA to experimental/clinical data Interpret results by comparing F stat or F calculated with F critical using the F-table 11-09-2025 Dr Snehashis Singha 2

3. Contents Introduction Concept of Analysis of Variance (ANOVA) Types of ANOVA Apply ANOVA to clinical data Application of F-test in hypothesis testing Assumptions and limitations of ANOVA Summary 11-09-2025 Dr Snehashis Singha 3

4. ANOVA Analysis of Variance FBS Fasting Blood Sugar SBP Systolic Blood Pressure df Degrees of Freedom SSB Sum of Squares Between groups SSE Sum of Squares Within groups/Error H₀ Null Hypothesis H₁ Alternative Hypothesis μ ( mu) Population Mean ANCOVA Analysis of Covariance MANCOVA Multivariate Analysis of Covariance 11-09-2025 Dr Snehashis Singha 4 Abbreviations

5. Why not t- test ? Why ANOVA is needed ? 11-09-2025 Dr Snehashis Singha 5 Probability of making Type I error 1-(1- α) κ K = number of tests α = Type I error, significance level K= = Number of groups   No. of Groups Probability of making Type I error 3 14% 5 40% 6 54% 7 66%

6. Introduction O AN VA AN alysis O f VA riance Sir Ronald A. Fisher ANOVA is a statistical test that compares the means of three or more groups using the F-test Developed by Sir Ronald A. Fisher 11-09-2025 Dr Snehashis Singha 6

7. STANDARD NORMAL DISTRIBUTION Between the groups Within the groups A B C

8. V ariables affects Independent variable : Value never depends on another variable Dependent variable : Values are predicted from the independent variable Age, Sex Drug, Dose, Dosage Reduction of Blood sugar, Blood pressure 11-09-2025 Dr Snehashis Singha 8

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10. One-way ANOVA Compares means of 3 or more independent groups Extension of independent t -test Does fasting blood sugar (FBS) differ across treatment groups ? Control Drug A Drug B Fasting Blood Sugar 100 85 78 FBS = Dependent Variable Treatment groups = Independent Variable 11-09-2025 Dr Snehashis Singha 10

11. Compares means across two independent variables Extension of One-way ANOVA Does fasting blood sugar (FBS) differ across treatment groups & gender ? Two-way ANOVA Fasting Blood Sugar (mg/dL) Control Drug A Drug B Male 100 85 78 Female 95 82 75 Gender = Independent Variable Treatment groups = Independent Variable FBS = Dependent Variable

12. Fasting Blood Sugar (mg/dL) Control Drug A Drug B Male 100 85 78 Female 95 82 75 Control Drug A Drug B Fasting Blood Sugar (mg/dl) 100 85 78 One-way ANOVA Two-way ANOVA 11-09-2025 Dr Snehashis Singha 12

13. Repeated measures ANOVA Within-subjects ANOVA / ANOVA for correlated samples Extension of paired t -test for >2 time points Used when the same subjects are measured repeatedly over time Systolic blood pressure (SBP) measured at baseline, 2 weeks, 4 weeks, and 8 weeks after treatment Baseline 2 nd week 4 th week 8 th week Systolic BP of same patient (mmHg) 150 140 130 120

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15. Parameter Time Point(s) Suitable Test Bo dy weight Day 43 only (compare across Groups 3, 4, 5 ) FBS Day 43 vs Day 72 (compare across Groups 3, 4, 5 on Day 43 & Day 72 ) Longitudinal change in Body weight Day 7 vs Day 43 vs Day 72 (Same Group across multiple times) One-way ANOVA Two-way ANOVA Repeated Measures ANOVA Dr Snehashis Singha 11-09-2025 15

16. A B C D E Obs 1 10 8 11 6 9 Obs 2 12 7 12 8 10 Obs 3 11 9 13 7 11 Obs 4 9 10 12 7 8 11-09-2025 Dr Snehashis Singha 16 Null hypothesis (H₀): All five drugs have the same mean SBP reduction after 4 weeks, H₀: μA = μB = μC = μD = μE Alternative hypothesis (H₁): At least one drug’s mean SBP reduction differs from the others, H₁: Not all μ’s are equal A clinical trial is conducted to compare the effect of 5 different antihypertensive drugs (A, B, C, D, E) on systolic blood pressure (SBP) reduction after 4 weeks. Twenty patients are randomly assigned, 4 in each group. The SBP reduction (in mmHg) is as follows:

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18. A clinical trial is conducted to compare the effect of 5 different antihypertensive drugs (A, B, C, D, E) on systolic blood pressure (SBP) reduction after 4 weeks. Twenty patients are randomly assigned, 4 in each group. The SBP reduction (in mmHg) is as follows: 11-09-2025 Dr Snehashis Singha 18 A B C D E Obs 1 10 8 11 6 9 Obs 2 12 7 12 8 10 Obs 3 11 9 13 7 11 Obs 4 9 10 12 7 8 Mean Values of each group 10.5 8.5 12 7 9.5 Mean Values of all groups = 9.5 (Grand Mean) A B C D E Obs 1 10 8 11 6 9 Obs 2 12 7 12 8 10 Obs 3 11 9 13 7 11 Obs 4 9 10 12 7 8 Mean Values of each group Mean Values of all groups

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20. SSB (Sum of Squares Between groups) A B C D E Obs1 10 8 11 6 9 Obs2 12 7 12 8 10 Obs3 11 9 13 7 11 Obs4 9 10 12 7 8 Mean Values of each gp . 10.5 8.5 12 7 9.5 Grand mean 9.5 Drug A : 4 x (10.5−9.5)² = 4 x (1) = 4 SSB = 4 + 4 + 25 + 25 + 0 = 58 Drug B : 4 x (8.5−9.5)² = 4 x (1) = 4 Drug C : 4 x (12−9.5)² = 4 x (2.5²) = 25 Drug D : 4 x (7−9.5)² = 4 x (2.5²) = 25 Drug E : 4(9.5−9.5)² = 20 Dr Snehashis Singha

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22. A B C D E Obs1 10 8 11 6 9 Obs2 12 7 12 8 10 Obs3 11 9 13 7 11 Obs4 9 10 12 7 8 Mean Values of each gp . 10.5 8.5 12 7 9.5 Grand mean 9.5 Drug A = (10-10.5) 2 + (12-10.5) 2 + (11-10.5) 2 + (9-10.5) 2 = 5 Drug B = (−0.5)² + (−1.5)² + 0.5² + 1.5² = 5 Drug C = (−1)² + 0² + 1² + 0² = 2 Drug D = (−1)² + 1² + 0² + 0² = 2 Drug E = (−0.5)² + 0.5² + 1.5² + (−1.5)² = 5 SSE = 5 + 5 + 2 + 2 + 5 = 19 SSE (Sum of Squares Within groups/Error) 22 Dr Snehashis Singha

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24. Df between groups = a − 1 5 − 1 = 4 A B C D E Obs1 10 8 11 6 9 Obs2 12 7 12 8 10 Obs3 11 9 13 7 11 Obs4 9 10 12 7 8 Mean Values of each gp . 10.5 8.5 12 7 9.5 Grand mean 9.5 a = number of groups = 5 n = number of observations per group = 4 N = total number of observations = a × n = 20 Degree of Freedom ( df ) Df within groups = a(n − 1) 5×(4-1) = 15 11-09-2025 Dr Snehashis Singha 24

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26. df (Degrees of Freedom) SS (Sum of Squares) [Variation] MS (Mean Square) [Variance] (F-ratio) Compare F stat vs F critical Between Groups a − 1 SSB MSB = F stat = ( F stat vs F critical ) Within Groups (Error) a(n − 1) SSE MSE = Total SST = SSB + SSE df (Degrees of Freedom) SS (Sum of Squares) [Variation] MS (Mean Square) [Variance] (F-ratio) Compare F stat vs F critical Between Groups a − 1 SSB ( F stat vs F critical ) Within Groups (Error) a(n − 1) SSE Total SST = SSB + SSE 4 15 19 77 19 58 11.45 1.267 14.5 ?

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28. Source : Sarty GE. Appendix: Tables – Introduction to Applied Statistics for Psychology Students. Introduction to Applied Statistics for Psychology Students. University of Saskatchewan; 2022. Available from: https://www.saskoer.ca/introtoappliedstatsforpsych/back-matter/appendix-tables/ F- distribution Table 11-09-2025 28

29. F critical > F stat ----- Fail to Reject H F critical < F stat ----- Reject H Numerator degree of freedom = Degree of freedom between groups = 4 Denominator degree of freedom = Degree of freedom within groups = 15 F critical = F (4,15) 11-09-2025 Dr Snehashis Singha 29 F stat =   F critical =  

30. F critical > F stat ----- Fail to Reject H F critical < F stat ----- Reject H F critical ( 3.06 ) < F stat ( 11.45 ) Reject H We therefore reject the null hypothesis of equal mean SBP reduction across drugs A,B,C,D,E and accept the alternative that at least one drug’s mean SBP reduction differs significantly from the others Null hypothesis (H₀): All five drugs have the same mean SBP reduction after 4 weeks H₀: μA = μB = μC = μD = μE Alternative hypothesis (H₁): At least one drug’s mean SBP reduction differs from the others H₁: Not all μ’s are equal 11-09-2025 Dr Snehashis Singha 31

31. ANOVA Adjusts for Covariates ANOVA Multivariate analysis ANCOVA (Analysis of Covariance) MANCOVA (Multivariate Analysis of Covariance) Independent Variables : Drug A vs Drug B , Dependent Variable 1 : Blood Sugar, Dependent variable 2 : Serum Cholesterol, COVARIATES : Baseline levels, Age, BMI, Independent Variables : Drug A vs Drug B, Dependent Variables : Reduction in Blood pressure after treatment, COVARIATE : Baseline blood pressure, Age, BMI 11-09-2025 Dr Snehashis Singha 32

32. CONFOUNDERS COVARIATES “All confounders are covariates, but not all covariates are confounders” Does smoking cause lung cancer? 11-09-2025 Dr Snehashis Singha 33

33. Assumptions for ANOVA Parametric test, normally distributed population Homogeneity of variance Samples are drawn independently of each other Within each sample, the observations are sampled randomly and independently of each other 11-09-2025 Dr Snehashis Singha 34

34. Limitations of ANOVA Assumes normal distribution of data in each group Requires homogeneity of variances Observations must be independent within and across groups Sensitive to outliers, which can distort results Indicates only overall group differences, not which groups differ specifically 11-09-2025 Dr Snehashis Singha 35

35. 11-09-2025 Dr Snehashis Singha 36 Group1 control Group 3 Group 2 Group 4 Post Hoc Tests

36. Group1 control Post Hoc Test: Dunnett’s 11-09-2025 Dr Snehashis Singha 37 Group 3 Group 2 Group 4

37. Group1 control 11-09-2025 Dr Snehashis Singha 38 Post Hoc Test : Tukey’s Group 3 Group 2 Group 4

38. Summary ANOVA is a statistical method to compare means of three or more groups. Types include One-way, Two-way, and Repeated Measures ANOVA The procedure involves calculating SSB, SSE, degrees of freedom, mean squares Results are interpreted by comparing calculated F with critical F from the F table If calculated F exceeds critical F, the null hypothesis is rejected, indicating a significant difference ANOVA assumes normal distribution, homogeneity of variance, and independence of observations 11-09-2025 Dr Snehashis Singha 39

39. Further reading Family-wise Type I error percentage calculation (FWER) Post hoc tests in ANOVA - Tukey’s, Bonferroni, Dunnett’s methods Assumption testing - normality (Shapiro-Wilk) Use of statistical software (SPSS, R, GraphPad) for ANOVA analysis 11-09-2025 Dr Snehashis Singha 40

40. Specific learning objectives achieved By the end of this teaching learning session, I hope co-learners are now able to : Explain the concept of Analysis of Variance (ANOVA) Differentiate between One-way, Two-way, and Repeated Measures ANOVA Apply ANOVA to experimental/clinical data Interpret results by comparing F stat or F calculated with F critical using the F-table 11-09-2025 Dr Snehashis Singha 41

41. References Rosner B. Fundamentals of biostatistics. Belmont: Duxbury Press; 2006. pp. 557–581 Thompson HW, Mera R, Prasad C. The Analysis of Variance (ANOVA). Nutr Neurosci . 1999;2(1):43-55 Sarkar S, Srivastava V, Mohanty M. Postgraduate Pharmacology. 2nd ed. New Delhi: Paras Medical Publisher; 2025 Banerjee B. Mahajan's methods in biostatistics for medical students and research workers. 9th ed. New Delhi: Jaypee Brothers Medical Publishers; 2018 11-09-2025 Dr Snehashis Singha 42

42. Audience questions ? 11-09-2025 Dr Snehashis Singha 43

43. Questions What is ANOVA and what does it test? How does ANOVA differ from a t-test? What are the main types of ANOVA? What is the difference between a confounder and a covariate? 11-09-2025 Dr Snehashis Singha 44

44. Thank You