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  1. 1. ANALYSIS OF VARIANCE (ANOVA) or F-TEST Harold P. Mediodia Antipolo City Dec 2008
  2. 2. • used in determining the significant differences among the means of more than two groups • variation both between and within each of the group is analyzed to establish the F value One-way Analysis of Variance (One-Way ANOVA)
  3. 3. Example: A study was conducted to determine the effect of different stocking densities on the gel strength of agar gel produced by Gracilariopsis bailinae. They used three different stocking densities: 500, 1000, 2000 g/m². They extracted agar gel and measured the gel strength in g/cm².
  4. 4. The One-Way ANOVA is useful in this case for the following reasons: 1. There were more than two means as independent variable. means of the gel strength of three different stocking densities: 500 g/cm², 1000 g/cm² and 2000 g/cm²
  5. 5. The One-Way ANOVA is useful in this case for the following reasons: 2. There was only one independent variable involved stocking density of Gracilariopsis bailinae
  6. 6. Two-Way Analysis of Variance (Two-Way ANOVA) determine the main and simultaneous effects of two independent factors on one or more dependent variables
  7. 7. Example: A group studied the quality of mangoes produced at different physiological status (without flush, 25%-50% flush and 75%-90% flush) at different ages (105 DAFI, 110 DAFI and 120 DAFI). They determined the quality of mangoes in terms of weight of the mangoes produced. DAFI- Days After Flower Introduction
  8. 8. The Two-Way ANOVA is useful for the following reasons: 1. There were more than two means as independent variables. means of weights: three physiologic status for the first independent variable and three ages for the second independent variable (nine means)
  9. 9. The Two-Way ANOVA is useful for the following reasons: 2. There were two independent variable. physiologic status and ages, whose effect on the dependent variables of the study were simultaneously determined
  10. 10. Example: Taxonomic Classiffication and Determine of the Iodine Content of Seaweeds Collected along the Coast of Brgy. Sabang, Sibunag, Guimaras. Objectives: 1. To collect, classify and identify seaweeds found along the coast of Brgy. Sabang, Sibunag, Guimaras. 2. To measure the iodine connent of selected seaweeds collected along the coast of Brgy. Sabang, Sibunag, Guimaras. 3. To compare the iodine content of seaweeds collected along the coast of Brgy. Sabang, Sibunag, Guimaras.
  11. 11. Table 1. Iodine Content (percent) of Seaweeds Collected in the Coastal Area of Barangay Sabang, Sibunag, Guimaras. Gracilaria edulis 1.102 0.876 1.124 Gracilaria salicornia 0.737 0.859 1.056 Gracilaria acerosa 0.769 0.736 0.920 Gracilaria eucheumoides 1.149 0.790 0.721
  12. 12. ONE-WAY ANALYSIS OF VARIANCE 1.H0: There is no significant difference in the iodine content of seaweeds collected from the coastal areas of Barangay Sabang, Sibunag, Guimaras. H1: There is a significant difference in the iodine content of seaweeds collected from the coastal areas of Barangay Sabang, Sibunag, Guimaras. 2. Level of Significance: α=0.05
  13. 13. 3. Degrees of Freedom d.f for numerator = k-1 d.f for denominator = N-k k= numbers of groups N= total number of samples in all groups d.f for numerator = 4 – 1 = 3 d.f for denominator = 12 – 4 = 8
  14. 14. 4. Critical Region: F0.05= 4.07 4.07 If the computed F value ≥ 4.07, the null hypothesis is rejected.
  15. 15. 5. Test Statistics: One-way Analysis of Variance (ANOVA)
  16. 16. 6. Compute and fill-up the ANOVA Table Source of Sum of Degrees of Mean F Variation Squares Freedom Square Radio (Variance) Between 0.80250 3 0.026750 0.998283 Groups Within 0.214364 8 0.026796 Groups Total 0.294614 11
  17. 17. 7. Statistical Decision Since the computed F-value (0.998283) is less than the critical region (4.07), there is no sufficient evidence to reject the null hypothesis. 4.07 0.998283
  18. 18. 8. Conclusion There is no significant difference in the iodine content of seaweeds collected from the coastal areas of Barangay Sabang, Sibunag, Guimaras. Interpretation The iodine content is the same or equal for the four Gracilaria species.
  19. 19. Computing for F ... G. edulis G. salicornia G. Acerosa G. eucheumoides N1 = 3 N2 = 3 N3 = 3 N4 = 3 X1 X1² X2 X2² X3 X3² X4 X4² 1.102,, 1.214404 0.737 0.543169 0.769 0.591361 1.149 1.320201 0.876 0.767376 0.859 0.737881 0.736 0.541696 0.790 0.624100 1.124 1.263376 1.056 1.115136 0.920 0.846400 0.721 0.519841 ΣX1 = 3.102 ΣX1²= 3.245156 ΣX2= 2.652 ΣX2²= 2.396186 ΣX3= 2.425 ΣX3²= 1.979457 ΣX4= 2.660 ΣX4²= 2.464142 N= n1 + n2 +n3 + n4 = 12 ΣXTOT= 10.839 ΣX²TOT= 10.084941
  20. 20. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between SSBET d.f BET MSBET MSBET Groups Within SSW d.fW MSW Groups Total SST N-1 MDW
  21. 21. SST= ΣX²TOT (ΣXTOT)² N a. = 10.084941 – = 10.084941 – 9.790327 = 0.294614 (10.839)² 12 N = 12 ΣXTOT= 10.0839 ΣX²TOT= 10.084941
  22. 22. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between SSBET d.f BET MSBET MSBET Groups Within SSW d.fW MSW Groups Total 0.294614 N-1
  23. 23. SSBET= Σ (ΣX1)² N1[ ](ΣXTOT)² N (3.102)² 3 (2.652)² 3 (2.425)² 3 (10.839)² 12 = + + -(2.660)² 3 + = 3.207468 + 2.344368 + 1.960208 + 2.358533 – 9.790327 = 0.080250
  24. 24. G. edulis G. salicornia G. Acerosa G. eucheumoides N1 = 3 N2 = 3 N3 = 3 N4 = 3 X1 X1² X2 X2² X3 X3² X4 X4² 1.102,, 1.214404 0.737 0.543169 0.769 0.591361 1.149 1.320201 0.876 0.767376 0.859 0.737881 0.736 0.541696 0.790 0.624100 1.124 1.263376 1.056 1.115136 0.920 0.846400 0.721 0.519841 ΣX1 = 3.102 ΣX1²= 3.245156 ΣX2= 2.652 ΣX2²= 2.396186 ΣX3= 2.425 ΣX3²= 1.979457 ΣX4= 2.660 ΣX4²= 2.464142 N= 12 EXTOT= 10.839
  25. 25. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between 0.080250 d.f BET MSBET MSBET Groups Within SSW d.fW MSW Groups Total 0.294614 N-1 MSW
  26. 26. c. SSW= Σ SS1 = SS1 + SS2 + ... + SSk SS1= ΣX1² - (ΣX1)² N1
  27. 27. SS1= 3.245156 - = 0.037688 SS2= 2.396186 - = 0.051818 SS3= 1.979457 - = 0.019249 SS4= 2.464142 - = 0.105609 (3.102)² 3 (2.652)² 3 (2.425)² 3 (2.660)² 3 SSW = SS1 + SS2 + SS3 + SS4 = 0.037688 + 0.051818 + 0.019249 + 0.105609 = 0.214364
  28. 28. G. edulis G. salicornia G. Acerosa G. eucheumoides N1 = 3 N2 = 3 N3 = 3 N4 = 3 X1 X1² X2 X2² X3 X3² X4 X4² 1.102,, 1.214404 0.737 0.543169 0.769 0.591361 1.149 1.320201 0.876 0.767376 0.859 0.737881 0.736 0.541696 0.790 0.624100 1.124 1.263376 1.056 1.115136 0.920 0.846400 0.721 0.519841 ΣX1 = 3.102 ΣX1²= 3.245156 ΣX2= 2.652 ΣX2²= 2.396186 ΣX3= 2.425 ΣX3²= 1.979457 ΣX4= 2.660 ΣX4²= 2.464142
  29. 29. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between 0.080250 d.f BET MSBET MSBET Groups Within 0.214364 d.fW MSW Groups Total 0.294614 N-1 MSW
  30. 30. d. d.f.BET, d.f.W, and d.f.TOT d.f.BET= k-1 d.f.W= N-k d.f.TOT= d.f.BET + d.f.W= N - 1 = 4 – 1 = 3 = 12- 4 = 8 = 12 – 1 = 11
  31. 31. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between 0.080250 3 MSBET MSBET Groups Within 0.214364 8 MSW Groups Total 0.294614 11 MSW
  32. 32. e. Mean of Squares (MSBET and MSW) SSBET d.f.BET SSw d.f.W MSBET= MSW= = 0.080250 3 = 0.026750 = 0.214364 8 = 0.026796
  33. 33. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between 0.080250 3 0.26750 MSBET Groups Within 0.214364 8 0.026796 Groups Total 0.294614 11 MSW
  34. 34. f. F-value F = MSBET MSW = 0.026750 0.026796 = 0.998283
  35. 35. The ANOVA Table Source of Sum of Degrees of Mean F Ratio Variation Squares Freedom Square (Variance) Between 0.080250 3 0.26750 0.998283 Groups Within 0.214364 8 0.026796 Groups Total 0.294614 11 6. Compute and fill-up the ANOVA Table
  36. 36. References: Amparado, K.M., Pestaño, R., Suelo, J.K.. 2006. Taxonomic Classification and Determination of the Iodine Content of Seaweeds Collected along the Coast of Brgy. Sabang, Sibunag, Guimaras. Unpublished Research Paper. Philippines Science High School Western Visayas. Brase, C.H., Brase, C.P,. 1995. Understandable Statistics Concepts and Methods, D.c., Health and Company. Lexington, Massachussettes. Cadomigara, M., 2002. Fundamentals of Research, Methods and Models Mindset Publishing, Inc. Iloilo City Milton J.S., McTeer, p.m and Corbet, J.J., 1997. Introduction to Statistics McGraw-Hill Companies, Inc. United States of America

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