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j. Mean squares between columns,
MSC=SSC ̸ C-1
k. Mean squares between rows,
MSR=SSR ̸ r-1
l. Mean squares within samples,
MSE=SSE ̸ (C-1)(r-1)
m. Between columns F=MSC ̸ MSE
if Fcal < Ftab = accept H0
Degree of freedom for FC = [c-1,(c-1)*(r-1)]
n. Between rows F=MSR ̸ MSE
if Fcal < Ftab = accept H0
Degree of freedom for FR = [r-1,(c-1)*(r-1)]](https://image.slidesharecdn.com/shafeek-170716141354/85/ANOVA-TEST-by-shafeek-11-320.jpg)
ANOVA TEST by shafeek
- 1. ANALYSIS OF VARIANCE (ANOVA) 1 SUBMITTED BY shafeek.s
- 2. INTRODUCTION 2 The analysis of variance(ANOVA) is developed by R.A.Fisher in 1920. If the number of samples is more than two the Z-test and t- test cannot be used. The technique of variance analysis developed by fisher is very useful in such cases and with its help it is possible to study the significance of the difference of mean values of a large no.of samples at the same time. The techniques of variance analysis originated, in agricultural research where the effect of various types of soils on the output or the effect of different types of fertilizers on production had to be studied.
- 3. 3 The technique of the analysis of variance was extremely useful in all types of researches. The variance analysis studies the significance of the difference in means by analysing variance. The variances would differ only when the means are significantly different. The technique of the analysis of variance as developed by Fisher is capable of fruitful application in a variety of problems. H0: Variability w/i groups = variability b/t groups, this means that 1 = n Ha: Variability w/i groups does not = variability b/t groups, or, 1 n
- 4. F-STATISTICS 4 ANOVA measures two sources of variation in the data and compares their relative sizes. • variation BETWEEN groups: • for each data value look at the difference between its group mean and the overall mean. • variation WITHIN groups : • for each data value we look at the difference between that value and the mean of its group.
- 5. 5 The ANOVA F-statistic is a ratio of the Between Group Variaton divided by the Within Group Variation: F= A large F is evidence against H0, since it indicates that there is more difference between groups than within groups.
- 6. TECNIQUE OF ANALYSING VARIANCE 6 The technique of analysing the variance in case of a single variable and in case two variables is similar. In both cases a comparison is made between the variance of sample means with the residual variance. However, in case of a single variable, the total variance is divided in two parts only, viz.., variance between the samples and variance within the samples. The latter variance is the residual variance. In case of two variables the total variance is divided in three parts, viz. (i) Variance due to variable no.1 (ii) Variance due to variable no.2 (iii) Residual variance.
- 7. CLASSIFICATION OF ANOVA 7 The Analysis of variance is classified into two ways: a. One-way classification b. Two-way classification
- 8. TWO WAY CLASSIFICATION 8 1.In a one-way classification we take into account the effect of only one variable. 2.If there is a two way classification the effect of two variables can be studied. 3.The procedure' of analysis in a two-way classification is total both the columns and rows. 4.The effect of one factor is studied through the column wise figures and total's and of the other through the row wise figures and totals. 5.The variances are calculated for both the columns and rows and they are compared with the residual variance or error.
- 9. 9 a. We will start with the Null Hypothesis that is, the mean yield of the four fields is not different in the universe, or H0: µ1 = µ2 = µ3 = µ4 The alternate hypothesis will be H0: µ1 ≠ µ2 ≠ µ3 ≠ µ4 b. Compute T = Sum of all values. c. Total sum of samples (SST)= Sum of all the observations- T2 /N d. Sum of squares between samples(columns) SSC=B-D SSC=(∑x1 ) 2 ̸ n1 +(∑x2 ) 2 ̸ n2 + ∑x3 ) 2 ̸ n3 - T2 ̸N Where n1 = no. of elements in first column etc.
- 10. 10 e. Sum of the squares between rows SSR= ∑x1 ) 2 ̸ n1 +(∑x2 ) 2 ̸ n2 + ∑x3 ) 2 ̸ n3 - T2 ̸N n1= no. of elements in first row f. Sum of squares within samples, SSE=SST-SSC-SSR g. The no.of d.f for between samples ᶹ1 =C-1 h. The no.of d.f for between rows, ᶹ2 =r-1 i. The no.of d.f for within samples, ᶹ3 =(C-1)(r-1)
- 11. 11 j. Mean squares between columns, MSC=SSC ̸ C-1 k. Mean squares between rows, MSR=SSR ̸ r-1 l. Mean squares within samples, MSE=SSE ̸ (C-1)(r-1) m. Between columns F=MSC ̸ MSE if Fcal < Ftab = accept H0 Degree of freedom for FC = [c-1,(c-1)*(r-1)] n. Between rows F=MSR ̸ MSE if Fcal < Ftab = accept H0 Degree of freedom for FR = [r-1,(c-1)*(r-1)]
- 12. ANOVA TABLE FOR TWO-WAY 12 Source of variance d.f Sum of squares Mean sum of squares F-Ratio Between samples(column s) C-1 SSC=B-D MSC=SSC ̸ c-1 F=MSC ̸ MSE Between Replicants(rows) r-1 SSR=C-D MSR=SSR ̸ r- 1 Within samples(Residu al) c-1)(r-1) SSE=SST- SSC-SSR MSE=SSE ̸ (c- 1)(r-1) F=MSR ̸ MSE Total N-1 SST=A-D
- 13. APPLICATIONS OF ANOVA 13 Similar to t-test More versatile than t-test ANOVA is the synthesis of several ideas & it is used for multiple purposes. The statistical Analysis depends on the design and discussion of ANOVA therefore includes common statistical designs used in pharmaceutical research.
- 14. 14 This is particularly applicable to experiment otherwise difficult to implement such as is the case in Clinical trials. In the bioequelence studies the similarities between the samples will be analyzed with ANOVA only. Pharmacokinetic data also will be evaluated using ANOVA. Pharmacodynamics (what drugs does to the body) data also will be analyzed with ANOVA only. That means we can analyze our drug is showing significant pharmacological action (or) not.
- 15. 15 Compare heights of plants with and without galls. Compare birth weights of deer in different geographical regions. Compare responses of patients to real medication vs. placebo. Compare attention spans of undergraduate students in different programs at PC.
- 16. 16 General Applications: Pharmacy Biology Microbiology Agriculture Statistics Marketing Business research Finance Mechanical calculations
- 17. THANK YOU 17