Anova single factor
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This document discusses a single factor analysis of variance (ANOVA) model. It defines ANOVA and explains that it allows comparison of three or more population means. The key assumptions of ANOVA are normal distributions and equal variances across populations. The document provides an example of using a single factor ANOVA to compare final exam scores from students in different teaching formats. Total variability is partitioned into between and within treatment variability, and average variability is measured using mean square values.
![Remember that…
Standard deviation (s)
n
s = √[(Σ (xi – X)2)/(n-1)]
i = 1
In this case: Degrees of freedom (df)
df = Number of observations or groups
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Anova single factor
- 1. AANNOOVVAA OOnnee wwaayy SSiinnggllee FFaaccttoorr MMooddeellss KARAN DESAI-11BIE001 DHRUV PATEL-11BIE024 VISHAL DERASHRI -11BIE030 HARDIK MEHTA-11BIE037 MALAV BHATT-11BIE056
- 2. DEFINITION Analysis of variance (ANOVA) is a collection of statistical models used to analyze the differences between group means and their associated procedures (such as "variation" among and between groups), developed by R.A.Fisher .In the ANOVA setting, the observed variance in a particular variable is partitioned into components attributable to different sources of variation 22
- 3. -Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, geneticist, and eugenicist 33
- 4. Why ANOVA • Compare the mean of more than two population? • Compare populations each containing several subgroups or levels? 4
- 5. Problem with multiple T test • One problem with this approach is the increasing number of tests as the number of groups increases • The probability of making a Type I error increases as the number of tests increase. • If the probability of a Type I error for the analysis is set at 0.05 and 10 t-tests are done, the overall probability of a Type I error for the set of tests = 1 – (0.95)10 = 0.40* instead of 0.05 5
- 6. In its simplest form, ANOVA provides a statistical test of whether or not the means of several groups are equal, and therefore generalizes the t-test to more than two groups. As doing multiple two-sample t-tests would result in an increased chance of committing a statistical type-I error, ANOVAs are useful in comparing (testing) three or more means (groups or variables) for statistical significance. 66
- 7. • Another way to describe the multiple comparisons problem is to think about the meaning of an alpha level = 0.05 • Alpha of 0.05 implies that, by chance, there will be one Type I error in every 20 tests: 1/20 = 0.05. • This means that, by chance the null hypothesis will be incorrectly rejected once in every 20 tests • As the number of tests increases, the probability of finding a ‘significant’ result by chance increases. 7
- 8. Importance of ANOVA • The ANOVA is an important test because it enables us to see for example how effective two different types of treatment are and how durable they are. • Effectively a ANOVA can tell us how well a treatment work, how long it lasts and how budget friendly it will be an 8
- 9. CLASSIFICATION OF ANOVA MODEL 1. Fixed-effects models: The fixed-effects model of analysis of variance applies to situations in which the experimenter applies one or more treatments to the subjects of the experiment to see if the response variable values change. This allows the experimenter to estimate the ranges of response variable values that the treatment would generate in the population as a whole. 99
- 10. 2. Random-effects model: Random effects models are used when the treatments are not fixed. This occurs when the various factor levels are sampled from a larger population. Because the levels themselves are random variables , some assumptions and the method of contrasting the treatments (a multi-variable generalization of simple differences) differ from the fixed-effects model. 1100
- 11. 3.Mixed-effects models A mixed-effects model contains experimental factors of both fixed and random-effects types, with appropriately different interpretations and analysis for the two types. Example: Teaching experiments could be performed by a university department to find a good introductory textbook, with each text considered a treatment. The fixed-effects model would compare a list of candidate texts. The random-effects model would determine whether important differences exist among a list of randomly selected texts. The mixed-effects model would compare the (fixed) incumbent texts to randomly selected alternatives. 1111
- 12. ASSUMPTION Normal distribution Variances of dependent variable are equal in all populations Random samples; independent scores 1122
- 13. One way Single factor ANOVA 1133
- 14. ONE-WAY ANOVA One factor (manipulated variable) One response variable Two or more groups to compare 1144
- 15. USEFULLNESS Similar to t-test More versatile than t-test Compare one parameter (response variable) between two or more groups 1155
- 16. Remember that… Standard deviation (s) n s = √[(Σ (xi – X)2)/(n-1)] i = 1 In this case: Degrees of freedom (df) df = Number of observations or groups 1166
- 17. ANOVA ANOVA (ANalysis Of VAriance) is a natural extension used to compare the means more than 2 populations. Basic Question: Even if the true means of n populations were equal (i.e. m1 = m2 = m3 = m4) we cannot expect the sample means (`x1, `x2, `x3, `x4 ) to be equal. So when we get different values for the `x’s, How much is due to randomness? How much is due to the fact that we are sampling from different populations with possibly different mj’s.
- 18. ANOVA TERMINOLOGY Response Variable (y) WWhhaatt wwee aarree mmeeaassuurriinngg Experimental Units TThhee iinnddiivviidduuaall uunniitt tthhaatt wwee wwiillll mmeeaassuurree Factors IInnddeeppeennddeenntt vvaarriiaabblleess wwhhoossee vvaalluueess ccaann cchhaannggee ttoo aaffffeecctt tthhee oouuttccoommee ooff tthhee rreessppoonnssee vvaarriiaabbllee,, yy Levels of Factors VVaalluueess ooff tthhee ffaaccttoorrss Treatments TThhee ccoommbbiinnaattiioonn ooff tthhee lleevveellss ooff tthhee ffaaccttoorrss aapppplliieedd ttoo aann eexxppeerriimmeennttaall uunniitt
- 19. Example We want to know how combinations of different amounts of water (1 ac-ft, 3 ac-ft, 5 ac-ft) and different fertilizers (A, B, C) affect crop yields Response variable – ccrroopp yyiieelldd ((bbuusshheellss//aaccrree)) Experimental unit EEaacchh aaccrree tthhaatt rreecceeiivveess aa ttrreeaattmmeenntt Factors ((22)) WWaatteerr aanndd ffeerrttiilliizzeerr Levels ((33 ffoorr WWaatteerr;; 33 ffoorr FFeerrttiilliizzeerr)) WWaatteerr:: 11,, 33,, 55;; FFeerrttiilliizzeerr:: AA,, BB,, CC Treatments ((99 == 33xx33)) 11AA,, 33AA,, 55AA,, 11BB,, 33BB,, 55BB,, 11CC,, 33CC,, 55CC
- 20. Total Treatments Fertilizer A B C 1 AC-FT Treatment 1 Treatment 2 Treatment 3 Water 3 AC-FT Treatment 4 Treatment 5 Treatment 6 5 AC-FT Treatment 7 Treatment 8 Treatment 9
- 21. Single Factor ANOVA Basic Assumptions If we focus on only one factor (e.g. fertilizer type in the previous example), this is called single factor ANOVA. In this case, levels and treatments are the same thing since there are no combinations between factors. Assumptions for Single Factor ANOVA 1. The distribution of each population in the comparison has a normal distribution 2. The standard deviations of each population (although unknown) are assumed to be equal (i.e. s1 = s2 = s3 = s4) 3. Sampling is: Random Independent
- 22. Example The university would like to know if the delivery mode of the introductory statistics class affects the performance in the class as measured by the scores on the final exam. The class is given in four different formats: Lecture Text Reading Videotape Internet The final exam scores from random samples of students from each of the four teaching formats was recorded.
- 23. Samples
- 24. Summary There is a single factor under observation – teaching format There are k = 4 different treatments (or levels of teaching formats) The number of observations (experimental units) are n1 = 7, n2 = 8, n3 = 6, n4 = 5 total number of observations, n = 26 = = = = Treatment Means : x1 76, x2 65, x3 75, x4 74 = Grand mean (of all 26 observations) : x 72
- 25. Why aren’t all the`x’s the same? There is variability due to the different treatments -- Between TTrreeaattmmeenntt VVaarriiaabbiilliittyy ((TTrreeaattmmeenntt)) There is variability due to randomness within each treatment -- WWiitthhiinn TTrreeaattmmeenntt VVaarriiaabbiilliittyy ((EErrrroorr)) BBAASSIICC CCOONNCCEEPPTT If the average BBeettwweeeenn TTrreeaattmmeenntt VVaarriiaabbiilliittyy is “large” compared to the average WWiitthhiinn TTrreeaattmmeenntt VVaarriiaabbiilliittyy, we can reasonably conclude that there really are differences among the population means (i.e. at least one μj differs from the others).
- 26. Basic Questions Given this basic concept, the natural questions are: What is “variability” due to treatment and due to error and how are they measured? What is “average variability” due to treatment and due to error and how are they measured? What is “large”? How much larger than the observed average variability due to error does the observed average variability due to treatment have to be before we are convinced that there are differences in the true population means (the μ’s)?
- 27. How Is “Total” Variability Measured? Variability is defined as the Sum ooff SSqquuaarree DDeevviiaattiioonnss (from the grand mean). So, SSSSTT (Total Sum of Squares) Sum of Squared Deviations of all observations from the grand mean. (McClave uses SSTotal) SSSSTTrr (Between Treatment Sum of Squares) Sum of Square Deviations Due to Different Treatments. (McClave uses SST) SSSSEE (Within Treatment Sum of Squares) Sum of Square Deviations Due to Error SSSSTT == SSSSTTrr ++ SSSSEE
- 28. How is “Average” Variability Measured? “Average” Variability is measured in: MMeeaann SSqquuaarree VVaalluueess (MSTr and MSE) Found by dividing SSTr and SSE by their respective degrees of freedom AANNOOVVAA TTAABBLLEE # treatments -1 DFT - DFTR VVaarriiaabbiilliittyy SSSS DDFF MMeeaann SSqquuaarree ((MMSS)) Between Tr. (Treatment) SSTr k-1 SSTr/DFTR Within Tr. (Error) SSE n-k SSE/DFE TOTAL SST n-1 # observations -1
- 29. Formula for Calculating SST Calculating SST Just like the numerator of the variance assuming all (26) entries come from one population =åå - SST (x x) (82 72) 2 ... (81 72) 2 4394 2 ij = - + + - =
- 30. Formula for Calculating SSTr Calculating SSTr Between Treatment Variability Replace all entries within each treatment by its mean – now all the variability is between (not within) treatments 76 76 76 76 76 76 76 =å - SSTr n (x x) 2 j j 75 75 75 75 75 75 65 65 65 65 65 65 65 65 2 2 2 2 = - + - + - + - = 7(76 72) 8(65 72) 6(75 72) 5(74 72) 578 74 74 74 74 74
- 31. Formula for Calculating SSE Calculating SSE (Within Treatment Variability) The difference between the SST and SSTr --- = = SSE SST - SSTr = 4394 - 578 3816
- 32. Can we Conclude a Difference Among the 4 Teaching Formats? We conclude that at least one population mean differs from the others if the average between treatment variability is large compared to the average within treatment variability, that is if MSTr/MSE is “large”. The ratio of the two measures of variability for these normally distributed random variables has an FF ddiissttrriibbuuttiioonn and the FF--ssttaattiissttiicc ((==MMSSTTrr//MMSSEE)) is compared to a critical F-value from an F distribution with: Numerator degrees of freedom = DFTr Denominator degrees of freedom = DFE If the ratio of MSTr to MSE (the F-statistic) exceeds the critical F-value, we can conclude that aatt lleeaasstt oonnee ppooppuullaattiioonn mmeeaann ddiiffffeerrss ffrroomm tthhee ootthheerrss.
- 33. Can We Conclude Different Teaching Formats Affect Final Exam Scores? The F-test H0: m1 = m2 = m3 = m4 HA: At least one mj differs from the others Select α = .05. Reject H0 (Accept HA) if: F MSTr = > α,DFTr,DFE = .05,3,22 = F F 3.05 MSE
- 34. Hand Calculations for the F-test = = = 173.45 578 3816 22 MSTr SSTr MSE SSE DFE 192.67 3 DFTr = = = 1.11 F 192.67 = = 173.45 < 1.11 3.05 CCaannnnoott ccoonncclluuddee tthheerree iiss aa ddiiffffeerreennccee aammoonngg tthhee μμjj’’ss
- 35. Excel Approach
- 36. EXCEL OUTPUT pp--vvaalluuee == ..336655997755 >> ..0055 CCaannnnoott ccoonncclluuddee ddiiffffeerreenncceess
- 37. REVIEW ANOVA Situation and Terminology Response variable, Experimental Units, Factors, Levels, Treatments, Error Basic Concept If the “average variability” between treatments is “a lot” greater than the “average variability” due to error – conclude that at least one mean differs from the others. Single Factor Analysis By Hand By Excel