Topic- 11 The Analysis of Variance

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1 Topc- 11 The Analyss of Varance

2 Expermental Desgn The samplng plan or expermental desgn determnes the way that a sample s selected. In an observatonal study, the expermenter observes data that already exst. The samplng plan s a plan for collectng ths data. In a desgned experment, the expermenter mposes one or more expermental condtons on the expermental unts and records the response.

3 Expermental Desgn Must desgn an experment that wll test your hypothess. Ths experment wll allow you to change some condtons or varables to test your hypothess.

4 Varables Varables are thngs that change. The ndependent varable s the varable that s purposely changed. It s the manpulated varable. The dependent varable changes n response to the ndependent varable. It s the respondng varable.

5 The Basc Prncples of Expermental Desgn Two aspects to any expermental problem: 1. The desgn of the experment. Statstcal analyss of the data Three basc prncples of expermental desgn 1. Replcaton. Randomzaton 3. Blockng To reduce error

6 Defntons An expermental unt s the object on whch a measurement or measurements) s taken. A factor s an ndependent varable whose values are controlled and vared by the expermenter. A level s the ntensty settng of a factor. A treatment s a specfc combnaton of factor levels. The response s the varable beng measured by the expermenter.

7 Example A group of people s randomly dvded nto an expermental and a control group. The control group s gven an apttude test after havng eaten a full breakfast. The expermental group s gven the same test wthout havng eaten any breakfast. Expermental unt : person Factor Response : Score on test Levels Treatments : Breakfast or no breakfast meal Breakfast or no breakfast

8 Example The expermenter n the prevous example also records the person s gender. Descrbe the factors, levels and treatments. Expermental unt person Response score Factor #1 meal Factor # gender Levels breakfast or no Levels male or female breakfast Treatments: male and breakfast, female and breakfast, male and no breakfast, female and no breakfast

9 The Analyss of Varance (ANOVA) All measurements exhbt varablty. The total varaton n the response measurements s broken nto portons that can be attrbuted to varous factors. These portons are used to judge the effect of the varous factors on the expermental response.

10 The Analyss of Varance If an experment has been properly desgned, Factor 1 Total varaton Factor Factor Random varaton We compare the varaton due to any one factor to the typcal random varaton n the experment. The varaton between the sample means s larger than the typcal varaton wthn the samples. The varaton between the sample means s about the same as the typcal varaton wthn the samples.

11 Assumptons 1. The observatons wthn each populaton are normally dstrbuted wth a common varance s.. Assumptons regardng the samplng procedures are specfed for each desgn. Analyss of varance procedures are farly robust when sample szes are equal and when the data are farly moundshaped.

12 Two Desgns 1. Completely Randomzed Desgn (CRD) an extenson of the two ndependent sample t-test.. Completely Randomzed Block Desgn (CRBD) an extenson of the pared dfference test.

13 1. The Completely Randomzed Desgn (CRD) A one-way classfcaton n whch one factor s set at k dfferent levels. The k levels correspond to k dfferent normal populatons, whch are the treatments. Are the k populaton means the same, or s at least one mean dfferent from the others?

14 Randomzaton n CRD Factor : A A has 4 levels (a1, a,a3, and a4) There are 4 replcatons each level (balance) So, there fll be 4 x 4 16 EU a3 a1 a3 a4 a a4 a a1 a4 a3 a4 a a1 a a3 a1

15 Assumptons 1. Randomness & Independence of Errors Independent Random Samples are Drawn for each condton. Normalty Populatons (for each condton) are Normally Dstrbuted 3. Homogenety of Varance Populatons (for each condton) have Equal Varances

16 About CRD Random samples of sze n 1, n,,n k are drawn from k populatons wth means m 1, m,, m k and wth common varance s. Let x j be the j-th measurement (replcaton) and the -th sample. The total varaton n the experment s measured by the total sum of squares: Total SS å ( x - j x )

17 Example Is the attenton duraton of chldren affected by whether or not they had a good breakfast? Twelve chldren were randomly dvded nto three groups and assgned to a dfferent meal plan. The response was attenton duraton n mnutes durng the mornng readng tme. No Breakfast Lght Breakfast Full Breakfast 8 (x11) 14 (x1) 10 (x31) 7 (x1) 16 (x) 1 (x3) 9 (x13) 1 (x3) 16 (x33) k 3 treatments. Are the average attenton spans dfferent? 13 (x14) 17 (x4) 15 (x34)

18 The ANOVA Table of the CRD Source df SS MS F Treatments k -1 SST SST/(k-1) MST/MSE Error n -k SSE SSE/(n-k) Total n -1 Total SS SS(Sum of Squares) SST k å 1 TotalSS n T - ( å x ) j å x - j n ( å x ) j n MS(Mean Squares) MST SST/(k-1) MSE SSE/(n-k) SSE Total SS - SST

19 Computng Formulas Note: CM Common Mean CM CF CF Correcton Factor

20 The ANOVA Table of the CRD The Total SS s dvded nto two parts: ü SST (sum of squares for treatments): measures the varaton among the k sample means. ü SSE (sum of squares for error): measures the varaton wthn the k samples. Total SS SST SSE Thesesums of squares behave lke the numerator of a sample varance. When dvded by the approprate degrees of freedom, each provdes a mean square, an estmate of varaton n the experment. Degrees of freedom are addtve, just lke the sums of squares. Total df Trt df Error df

21 The Breakfast Problem No Breakfast Lght Breakfast Full Breakfast T 1 37 T 59 T 3 53 G CM 1 Total SS SST SSE Total SS-SST - CM CM CM

22 The Breakfast Problem 149 CM 1 Total SS SST SSE CM CM CM Total SS-SST 58.5 Source df SS MS F Treatments Error Total

23 Testng the Treatment Means H0 : m 1 m m3... mk versus : at least one mean s dfferent H 1 Remember that s s the common varance for all k populatons. The quantty MSE SSE/(n - k) s a pooled estmate of s, a weghted average of all k sample varances, whether or not H 0 s true.

24 If H 0 s true, then the varaton n the sample means, measured by MST [SST/ (k - 1)], also provdes an unbased estmate of s. However, f H 0 s false and the populaton means are dfferent, then MST whch measures the varance n the sample means s unusually large. The test statstc F MST/ MSE tends to be larger that usual.

25 The F Test Hence, you can reject H 0 for large values of F, usng a rghttaled statstcal test. When H 0 s true, ths test statstc has an F dstrbuton wth df 1 (k - 1) and df (n - k) degrees of freedom and rghttaled crtcal values of the F dstrbuton can be used. H0 : m 1 m m3... mk MST Test Statstc : F MSE Reject H 0 f F > F wth k -1 and n-k df a.

26 The Breakfast Problem Source df SS MS F Treatments Error Total H H 0 1 : m 1 :at least MST F MSE Rejecton m m one 3 versus mean s regon : F > F dfferent We reject H0 and conclude that there s a dfference n average of attenton spans

27 Confdence Intervals. based on error s MSE and where 1 1 ) ( : Dfference : mean, A sngle / / df t s n n s t x x n s t x j j j ø ö ç ç è æ ± - - ± a a m m m. based on error s MSE and where 1 1 ) ( : Dfference : mean, A sngle / / df t s n n s t x x n s t x j j j ø ö ç ç è æ ± - - ± a a m m m If a dfference exsts between the treatment means, we can explore t wth confdence ntervals.

28 Tukey s Method for Pared Comparsons Desgned to test all pars of populaton means smultaneously, wth an overall error rate of a. Based on the studentzed range, the dfference between the largest and smallest of the k sample means. Assume that the sample szes are equal and calculate a ruler that measures the dstance requred between any par of means to declare a sgnfcant dfference.

29 dfferent. are declared they, than by more dffer means of par any If 11. from Table value ), ( sze sample common error MSE means treatment of number where ), ( : Calculate w w a a - k n k q n df df s k n s df k q dfferent. are declared they, than by more dffer means of par any If 11. from Table value ), ( sze sample common error MSE means treatment of number where ), ( : Calculate w w a a - k n k q n df df s k n s df k q Tukey s Method

30 The Breakfast Problem Use Tukey s method to determne whch of the three populaton means dffer from the others. No Breakfast Lght Breakfast Full Breakfast T 1 37 T 59 T 3 53 Means 37/ / / s 6.47 w q. 05(3,9)

31 The Breakfast Problem Lst the sample means from smallest to largest. x w x 3 x Snce the dfference between 9.5 and 13.5 s less than w 5.0, there s no sgnfcant dfference. There s a dfference between populaton means 1 and however There s no dfference between 13.5 and We can declare a sgnfcant dfference n average attenton spans between no breakfast and lght breakfast, but not between the other pars.

32 . THE COMPLETELY RANDOMIZED BLOCK DESIGN (CRBD) A drect extenson of the pared dfference or matched pars desgn. A two-way classfcaton n whch k treatment means are compared. The desgn uses blocks of k expermental unts that are relatvely smlar or homogeneous, wth one unt wthn each block randomly assgned to each treatment.

33 RANDOMIZATION IN CRBD Ò Factor : A Ò A has 4 levels (a1, a,a3, and a4) Ò Block : 4 blocks (n CRD, ths s lke replcaton) Ò So, there fll be 4 x 4 16 EU a4 a1 a4 a a a3 a a4 a1 a4 a1 a3 a3 a a3 a1 Block1 Block Block3 Block4

34 ABOUT THE CRBD Let x j be the response for the -th treatment appled to the j-th block. 1,, k j 1,,, b The total varaton n the experment s measured by the total sum of squares: Total SS å ( x - j x )

35 THE ANOVA TABLE OF THE CRBD Source df SS MS F Treatments k -1 SST SST/(k-1) MST/MSE Blocks b -1 SSB SSB/(b-1) MSB/MSE Error (b-1)(k-1) SSE SSE/(b-1)(k-1) Total n -1 Total SS SS(Sum of Squares) SST SSB k å 1 b T TotalSS å - B ( å xj ) n å x - SSE Total SS - k j - j ( å x j ) n ( å x ) j n SST -SSB MS(Mean Squares) MST SST/(k-1) MSB SSB/(b-1) MSE SSE/(n-k)

36 THE ANOVA TABLE OF THE CRBD The Total SS s dvded nto 3 parts: SST (sum of squares for treatments): measures the varaton among the k treatment means SSB (sum of squares for blocks): measures the varaton among the b block means SSE (sum of squares for error): measures the random varaton or expermental error n such a way that: Total SS SST SSB SSE df Total df T df B df E

37 COMPUTING FORMULAS G CM where G n Total SS å x - CM j å åt SST - CM where T total for treatm ent b å B j SSB - CM where Bj total for block j k SSE Total SS-SST -SSB x j

38 THE SEEDLING PROBLEM 16 CM 1 Total SS 187 ( ) SST SSB SSE Locatons Sol Prep T A B C B j

39 THE SEEDLING PROBLEM-ANOVA Total df bk 1 n -1 Mean Squares Treatment df k 1 MST SST/(k-1) Block df b 1 MSB SSB/(b-1) MSE SSE/(k-1)(b-1) Error df bk (k 1) (b-1) (k-1)(b-1) Source df SS MS F Treatments k -1 SST SST/(k-1) MST/MSE Blocks b -1 SSB SSB/(b-1) MSB/MSE Error (b-1)(k-1) SSE SSE/(b-1)(k-1) Total n -1 Total SS

40 THE SEEDLING PROBLEM-ANOVA 16 CM Total SS SST SSB SSE Source df SS MS F Treatments Blocks Error Total

41 TESTING THE TREATMENT AND BLOCK MEANS For ether treatment or block means, we can test: H0 : m1 m m3... versus : at least one mean s dfferent H 1 To test the H0 that trea tment (or block) means are equal MST MSB Test Statstc :F (or F ) MSE MSE Reject H 0 f F > F wth k -1(or b -1) and ( b -1)( k -1) df a.

42 TESTING THE TREATMENT AND BLOCK MEANS Remember that s s the common varance for all bk treatment/block combnatons. MSE s the best estmate of s, whether or not H 0 s true. If H 0 s false and the populaton means are dfferent, then MST or MSB whchever you are testng wll unusually large. The test statstc F MST/ MSE(or F MSB/ MSE) tends to be larger that usual. We use a rght-taled F test wth the approprate degrees of freedom.

43 THE SEEDLING PROBLEM-ANOVA H H Source df SS MS F Sol Prep (Trts) Locaton (Blocks) Error Total To test for a dfference 0 a : m m m 1 :at least MST F MSE Rejecton We reject H dfference one mean s dfferent regon 0 of 3 versus : F > and conclude seedlng F due to sol growth preparato n : that ther e s a due to sol Although not of prmary mportance, notce that the blocks (locatons) were also sgnfcantly dfferent (F 10.88) preparato n.

44 CONFIDENCE INTERVALS If a dfference exsts between the treatment means or block means, we can explore t wth confdence ntervals or usng Tukey s method. Dfference n treatment means : ( T -T j ) ± t a / s æ ç è b ö ø Dfference n block means : ( B - B j ) ± t a / s æ ç è k ö ø where T T / b and B B / k are the necessary treatment or block means. s MSE and t s based on error df.

45 TUKEY S METHOD For comparng treatment means : w q ( k, df ) a s b For comparng block means : w q ( b, df ) a s k where : s MSE df error df q a ( k, df ) value from Table 11. If any par of means dffer by more they are declared dfferent. than w,

46 THE SEEDLING PROBLEM TUKEY S METHOD Use Tukey s method to determne whch of the three sol preparatons dffer from the others. A (no prep) B (fertlzaton) C (burnng) T 1 50 T 64 T 3 48 Means 50/ / /4 1 w q.05(3,6) s

47 THE SEEDLING PROBLEM TUKEY S METHOD Lst the sample means from smallest to largest. T C 1 T A 1.5 T B 16.0 w.98 Snce the dfference between 1 and 1.5 s less than w.98, there s no sgnfcant dfference. There s a dfference between populaton means C and B however. There s a sgnfcant dfference between A and B. A sgnfcant dfference n average growth only occurs when the sol has been fertlzed.

48 CAUTIONS ABOUT BLOCKING ü ü ü A randomzed block desgn should not be used when both treatments and blocks correspond to expermental factors of nterest to the researcher Remember that blockng may not always be benefcal. Remember that you cannot construct confdence ntervals for ndvdual treatment means unless t s reasonable to assume that the b blocks have been randomly selected from a populaton of blocks.

49 Normal Probablty Plot ü If the normalty assumpton s vald, the plot should resemble a straght lne, slopng upward to the rght. ü If not, you wll often see the pattern fal n the tals of the graph. 99 Normal Probablty Plot of the Resduals (response s Growth) Percent Resdual 1 3

50 Resduals versus Fts ü If the equal varance assumpton s vald, the plot should appear as a random scatter around the zero center lne. ü If not, you wll see a pattern n the resduals Resduals Versus the Ftted Values (response s Growth) 0.5 Resdual Ftted Value

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