Analysis Of Variance - ANOVA
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This document provides an overview of analysis of variance (ANOVA) techniques. It discusses Fisher's exact test and Lady Bristol's claim to distinguish teas through taste, noting the p-value was below 0.05, rejecting the null hypothesis that her ability was due to chance. It then outlines one-way ANOVA assumptions of normal distribution and independence between groups. Computing an F-score with sums of squares is explained, as is using ANOVA in R through functions like aov and manova. Reasons for using ANOVA are given, including exploring data, handling experimental error, and reducing Type 1 errors.
Analysis Of Variance - ANOVA
- 1. Presented by : Saumya Bhatnagar
- 2. Fisher’s exact test 1) Lady Bristol’s claim • H0 = her ability to distinguish the teas is mere chance • In Fisher’s approach, no Ha • 8 cups, p= 0.014 (=1/8C4 = 1/70) <0.05 (association is random = reject H0) 2) Lady tasting tea • two-alternative forced choice 3) Increase the #observations in each group • How these groups are varied? . . … . . . . … . . . . … . .
- 3. Table of Content 1) Fisher’s exact test 2) Anova 1) Anova & t-test 2) Anova Designs 3) One way Anova 4) Assumptions Anova 5) Computing F score & Anova table 3) Anova in R 4) Why Anova
- 4. Anova & t-test What is Anova? • Omnibus to t-tests or Generalized t-test • 𝑡 − 𝑠𝑐𝑜𝑟𝑒 = 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑔𝑟𝑜𝑢𝑝𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑤𝑖𝑡ℎ𝑖𝑛 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑝 ⇒ 𝑔𝑟𝑜𝑢𝑝𝑠 𝑎𝑟𝑒 𝑡 𝑡𝑖𝑚𝑒𝑠 𝑎𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 𝑎𝑠 𝑡ℎ𝑒𝑦 𝑎𝑟𝑒 𝑤𝑖𝑡ℎ𝑖𝑛 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟 • Analysis of data from two samples by both a t test and an ANOVA shows that the observed F values equals the observed t value squared F = t2, (degree of freedom=1)
- 5. T-test & type of errors 1) Null Hypothesis & errors 2) T-test 1) t-score 2) p-value(probability that results are by chance) 3) An Independent Samples t-test compares the means for two groups. 4) A Paired sample t-test compares means from the same group at different times 5) A One sample t-test tests the mean of a single group against a known mean Decision H0 = True H0 = False H0= reject Type 1 error, (false positive) Denoted as α Aka “significance level of a test” Correct inference (True Positive) H0 = Fail to reject Correct inference (True Negative) Type II error (False Negative) Denoted as β
- 6. One way Anova Two or more samples One factor or independent variable One dependent variable e.g. Tea taste is being tested at one level – what was poured first? Factorial Anova Two or more samples Multiple independent variable One dependent variable e.g. Tea taste is being tested at various levels - sugar, tea content etc. Repeated Measures Anova Same sample e.g. Tea being poured to the same set thrice a day and tested M-AN-O-VA Two or more samples Multiple independent variable Multiple dependent variables e.g. Tea being tested for color + taste at various levels AN-O-VA designs
- 7. One-way Anova • Do differences exist between two or more groups on one DV? • a priori vs. post-hoc (a posteriori) tests • Post hoc is the multiple comparison test of groups. it compares all possible pairs of means. othersthefromdifferentismeanstheofoneleastAt: : 321 a k o H H
- 8. Assumptions One-way Anova One way ANOVA is based on the following assumptions: 1) Normal distribution of the population 2) Two or more than two categorical independent groups in an independent variable. 3) Independence of samples 4) σ1 2 = σ2 2 = σ3 2 = ….
- 9. Computing F-score ANOVA is computed with the three sums of squares • Total – Total Sum of Squares, a measure of all variations in the dependent variable, SST • Treatment (Between) – Sum of Squares Treatments (Between), SSC • Error (Within) – Sum of Squares of Errors; yields the variations within treatments (or columns), SSE k i n j iij xxSSE 1 1 2 )( k i n j ij xxSST 1 1 2 )( k i ii k i n j i xxnxxSSC 1 2 1 1 2 )()( SST = SSC + SSE
- 10. Anova Table The table is as follows:
- 11. Anova in R One way Anova Import data set Check null values and mine data Plot box-plots Run aov Save in another variable Export the summary
- 12. Anova in R (where y = dependent variable, A=independent variable)One way Anova • fit <- aov(y ~ A, data) (For two way Anova)Factorial Anova • fit <- aov(y ~ A + B + A:B, data) fit <- aov(y ~ A*B, data) (Considering two dependent variables Y & Y’)MANOVA • fit <- manova(cbind(Y, Y’) ~ A*B, data)
- 13. Why Anova? 1) Gives an exploratory data analysis 1) Effect of many variables at once 2) Generates ‘F’ statistics : allows testing of a nested sequence of models 3) organization of an additive data decomposition, 4) sums of squares indicate the variance of each component of the decomposition (or, equivalently, each set of terms of a linear model). 2) Analysis of a variety of experimental designs. 3) Handles experimental error 1) Reduces chances of Type 1 error 2) The more statistical tests run, the greater likelihood that the researcher will obtain seemingly significant effects due to chance alone. (ANOVA determines whether the amount of variance between the groups is greater than the variance within the groups)
- 14. Thank you