ANOVA Analysis of Varianace Biostatistics
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The document explains Analysis of Variance (ANOVA), a statistical method developed by Sir Ronald A. Fisher in the 1920s for comparing means across three or more independent groups. It outlines two types of ANOVA: one-way and two-way, detailing the steps involved in conducting the analysis, including hypothesis formulation, calculation of variances, mean sums of squares, and the F-statistic. An example is provided to illustrate the application of ANOVA, demonstrating how to determine if significant differences exist among sample means.
ANOVA Analysis of Varianace Biostatistics
- 1. Analysis Of Variance (ANOVA) Presented By: Sir Abuzar Tabassum
- 2. 1 What is ANOVA? ANOVA 2 Two Types of ANOVA Types 3 Steps Involved in ANOVA Steps
- 3. A statistical method in which the total variation in a set of observations is divided into distinct components 01 What is ANOVA? Sir Ronald A. Fisher (1920’s) Developed By
- 4. One Way ANOVA Two Way ANOVA 02 Types of ANOVA Determine the significant differences between the means of three or more independent groups A) One Way ANOVA Use only one factor. i.e., only 1 independent variable (factor) One Way ANOVA
- 5. Two Way ANOVA 02 Types of ANOVA Comparisons between the means of 3 or more than 3 independent groups of data B) Two Way ANOVA Two independent variables are considered (Two factors) Two Way ANOVA
- 6. Analysis Of Variance (ANOVA) Steps Involved in ANOVA Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA
- 7. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA State null & alternative hypotheses H0 = μ1 = μ2 = μ3…. = μi i.e., H0 : all sample means are equal HA = Not all of the means are equal i.e., At least one sample has different mean
- 9. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA Two types k = No. of Samples K-1 n = Total No. of observations n-k
- 10. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA 01 02 03 Calculated value of F Table value of F If cal. F > table value of F, then reject Ho
- 11. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA Calculate test Statistics Total variation = variation among treatments + variation within treatments
- 12. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Inference after ANOVA Calculate variance between the samples Calculate variance within the samples Calculation of F statistic
- 13. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Calculating variance between the samples
- 14. 01 STEP Calculate the mean of each sample 02 STEP Calculate the Grand average or overall mean of all samples means 03 STEP Take the difference between means of various samples & grand average 04 STEP Square these differences & obtain total which will give sum of squares between samples (SSbetween) 05 STEP Divide the total obtained in step 4 by the degrees of freedom (K-1) to calculate the mean sum of square between samples (Msbetween) 06 STEP 𝐌𝐒𝐛𝐞𝐭𝐰𝐞𝐞𝐧 = 𝑺𝑺𝒃𝒆𝒕𝒘𝒆𝒆𝒏 / − 𝒌 𝟏
- 15. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Calculating variance within the samples
- 16. 01 STEP Calculate mean value of each sample 02 STEP Take the difference of the various items in a sample from the mean values of the respective samples 03 STEP Square these differences & obtain total which gives the sum of square within the samples (SSE) or (SSerror) 04 STEP Divide the total obtained in 3rd step by the degrees of freedom (n-K) to calculate the mean sum of squares within samples (MSE) or (MSerror) 05 STEP 𝐌𝐒𝐄 = / − 𝑺𝑺𝑬 𝒏 𝒌
- 17. Hypothesis State Alpha Degree of Freedom Decision Rule Test Statistics Calculation of F statistics
- 18. 01 STEP 𝐅( ) = 𝐜𝐚𝐥 𝐌𝐒𝐛𝐞𝐭𝐰𝐞𝐞𝐧 /𝐌𝐒𝐞𝐫𝐫𝐨𝐫 02 STEP Compare the F(cal) value with F(tab) value 03 STEP F(tab) value is obtained by looking for it in F distribution tables against degrees of freedom 04 OPTION If the calculated value of F > table value H0 is rejected
- 19. Example • Three samples obtained from normal populations with equal variances. Test the hypothesis that sample means are equal. X1 X2 X3 8 7 12 10 5 9 7 10 13 14 9 12 11 9 14
- 20. Solution 1. Null hypothesis • No significant difference in the means of 3 samples. 2. State Alpha • The level of significance, i.e., 0.05 3. Calculate degrees of Freedom k-1 = 2 & n-k = 12 (Because K = 3 & n= 15) 4. State decision rule • Table value of F at alpha = 0.05 or 5% level of significance for d.f 2 & 12 is 3.89. • If the calculated value of F > 3.89, H0 will be rejected
- 22. 5. Calculate test statistic • This step is further divided into three steps A. Calculate variance between samples B. Calculate variance within the samples C. Calculate F-statistic
- 23. • Grand average or overall mean = 𝟏𝟎+𝟖+𝟏𝟐 / 𝟑 = 10 X1 X2 X3 8 7 12 10 5 9 7 10 13 14 9 12 11 9 14 Total = 50 Mean1 = 10 Total = 40 Mean2 = 8 Total = 60 Mean3 = 12
- 24. A) Calculate variance between samples • The formula for calculating the variance between samples is • j denotes a group mean • is the overall mean • nj is the sample size per group For our example this result is: SSbetween = n1 (M1 – Grand avg)2 + n2 (M2– Grand avg)2 + n3(M3– Grand avg)2 SSbetween = 5(10 - 10)2 + 5(8 - 10)2 + 5(12 - 10)2 = 40 D.fbetween = k-1 = 3-1 = 2
- 25. Mean sum of squares between the samples (MSbetween) MSbetween = 𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛 / − 1 𝑘 MSbetween = 40 / 2 = 20
- 26. B) Calculate variance within the samples • The formula for calculating Variance WITHIN the samples is Here • Xi Denotes an individual observation • j denotes a group mean (M1= 10, M2 = 8, M3 = 12) For our example, this'll be SSwithin = SSerror = (7-8)2 + (5-8)2 + (10-8)2 + (9-8)2 + (9-8)2 + (12-12)2 + (9-12)2 + (13-12)2 + (12-12)2 + (14-12)2 SSwithin = (8-10)2 + (10-10)2 + (7-10)2 + (14-10)2 + (11-10)2 + (-2)2 +(0)2 +(-3)2 +(4)2 (1)2 + ………… (2)2 = 60
- 27. Calculation of Mean Sum of Squares within samples (MSE) 𝐌𝐒within =MSerror = / − 𝑺𝑺𝑬 𝒏 𝒌 𝐌𝐒within = / 𝟔𝟎 𝟏𝟐 = 5
- 28. C) Calculate F-statistic 𝐅𝐜𝐚𝐥 = 𝐌𝐒𝐛𝐞𝐭𝐰𝐞𝐞𝐧 / 𝐌𝐒𝐞𝐫𝐫𝐨𝐫 • 𝐅𝐜𝐚𝐥 = / 𝟐𝟎 𝟓 • 𝐅𝐜𝐚𝐥 = 4 • The Table value of F at 5% level of significance for d.f 2 & 12 is 3.89. • The F𝐜𝐚𝐥 > Ftable value. • H0 is rejected • It means that there is significant difference in sample means.
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