Experiments with a Single Factor: The Analysis of Variance (ANOVA) Dr. Mohammad Abuhaiba 1
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1 Experiments with Single Fctor: The Anlsis of Vrince (ANOVA) Dr. Mohmmd Abuhib 1
2 Wht If There Are More Thn Two Fctor Levels? The t-test does not directl ppl There re lots of prcticl situtions where there re either more thn two levels of interest, or there re severl fctors of simultneous interest Single fctor experiments with multiple levels The nlsis of vrince (ANOVA) is the pproprite nlsis engine for these tpes of experiments Dr. Mohmmd Abuhib 2
3 An Exmple Tensile strength of New Snthetic Fiber Test specimens t five levels of cotton weight percent: 15, 20, 25, 30, nd 35%. Test 5 specimens t ech level. Single fctor experiment with = 5 levels nd n = 5 replictes. 25 runs in rndom order Dr. Mohmmd Abuhib 3
4 Dt from Tensile Experiment In clss box nd sctter plots Cotton weight % Observtions Totl Averge Dr. Mohmmd Abuhib 4
5 Grphicl Representtion We strongl suspect tht: 1. Cotton content ffects tensile strength 2. Around 30% cotton results in mx strength The t-test is not the best solution for this problem becuse it would led to considerble distortion in tpe I error. Exmple Dr. Mohmmd Abuhib 5
6 The Anlsis of Vrince In generl, there will be levels of the fctor, or tretments, nd n replictes of the experiment, run in rndom order completel rndomized design (CRD) N = n totl runs We consider the fixed effects cse Objective is to test hpotheses bout the equlit of the tretment mens Dr. Mohmmd Abuhib 6
7 Models for the Dt There re severl ws to write model for the dt: Let ij i ij, then i ij i ij i is clled the effects model is clled the mens model Dr. Mohmmd Abuhib 7
8 The Anlsis of Vrince Effects Model The nme nlsis of vrince stems from prtitioning of the totl vribilit in the response vrible into components tht re consistent with model for the experiment The bsic single-fctor ANOVA model is i 1,2,..., ij i ij, j 1,2,..., n ij n overll men, 2 experimentl error, NID (0, ) i i th tretment effect, Dr. Mohmmd Abuhib 8
9 The Anlsis of Vrince Effects Model For hpothesis testing, the model errors re ssumed to be normll nd independentl distributed rndom vribles with men zero nd vrince 2. The vrince 2 is ssumed to be constnt for ll levels 2 N, ij i Dr. Mohmmd Abuhib 9
10 THE ANALYSIS OF VARIANCE Fixed Effects Model Bsic Definitions: i i = Totl of observtions under ith tretment = = Averge of observtions under ith tretment = = Grnd totl of ll observtions = n i 1 j1 ij n j=1 ij n i = Grnd verge of ll observtions = N Dr. Mohmmd Abuhib 10
11 THE ANALYSIS OF VARIANCE Fixed Effects Model Hpothesises: H H o 1 : : for t lest one pir (i,j) i i i 1 i i,, i 0 i 1 Another w to write the bove hpothesis is in terms of tretment effects i : H H o 1 j : : 0 for t lest one i i Dr. Mohmmd Abuhib 11
12 THE ANALYSIS OF VARIANCE Fixed Effects Model Totl Sum of Squres Totl vribilit is mesured b totl sum of n squres: 2 SS ( ) T i1 j1 The bsic ANOVA prtitioning is: n n 2 2 T ( ij..) [( i...) ( ij i.)] i 1 j 1 i 1 j 1 SS n 2 2 (...) i ( ij i.) Tretments E i 1 i 1 j 1 n SS SS ij.. Dr. Mohmmd Abuhib 12
13 THE ANALYSIS OF VARIANCE Fixed Effects Model Totl Sum of Squres SS Tretments : sum of squres of differences between tretment verges nd grnd verge. It is mesure of differences between tretment mens. It hs (-1) DOF SS E : sum of squres of differences of observtions within tretments from the tretment verge. It is due to rndom error. It hs (N-) DOF. Dr. Mohmmd Abuhib 13
14 The Anlsis of Vrince SS SS SS T Tretments E A lrge vlue of SS Tretments reflects lrge differences in tretment mens A smll vlue of SS Tretments likel indictes no differences in tretment mens Forml sttisticl hpotheses re: H H : : At lest one men is different Dr. Mohmmd Abuhib 14
15 The Anlsis of Vrince While sums of squres cnnot be directl compred to test the hpothesis of equl mens, men squres cn be compred. A men squre is sum of squres divided b its degrees of freedom: MS DOF DOF DOF Totl Tretments Error n 1 1 ( n 1) Tretments SSTretments SSE, MS E 1 ( n 1) If the tretment mens re equl, the tretment nd error men squres will be (theoreticll) equl. If tretment mens differ, the tretment men squre will be lrger thn the error men squre. Dr. Mohmmd Abuhib 15
16 The Anlsis of Vrince E(MS E ) = 2 E MS Tretments n 2 i 1 2 i 1 If there re no differences in tretment mens ( i =0), then E(MS tretments ) = 2 If tretment mens do differ, the expected vlue of tretment men squres is greter thn 2 Dr. Mohmmd Abuhib 16
17 The Anlsis of Vrince Test Sttistic: F o SSTretments /( 1) MS SS /( N ) MS E Tretments E SS SS T Tretments 1 n n i 1 j1 i 1 2 ij 2 i SS SS SS N N 2 2 E T Tretments Dr. Mohmmd Abuhib 17
18 The Anlsis of Vrince The reference distribution for F 0 is the F -1, (n-1) distribution Reject the null hpothesis (equl tretment mens) if F F 0, 1, ( n 1) Dr. Mohmmd Abuhib 18
19 ANOVA Tble Exmple 3-1: The tensile test experiment Cotton weight % Sum of squres DOF Men Squre F o P-vlue Cotton weight % Fo = <0.01 Error Totl Observtions Totl Averge Dr. Mohmmd Abuhib 19
20 The Reference Distribution Dr. Mohmmd Abuhib 20
21 Coding the Observtions Exmple 3.2 Subtrcting constnt from the originl dt does not chnge the sum of squres. Multipl the observtions in the originl dt does not chnge the F rtio Dr. Mohmmd Abuhib 21
22 Estimtion of the Model Prmeters A 100(1-)% confidence intervl on the ith tretment men, i : A 100(1-)% confidence intervl on the difference in n tretment mens: Exmple 3-3 MS t t n E i / 2, N i i / 2, N MS n 2MS E 2MS i j t/ 2, N i j i j t/ 2, N n n E E Dr. Mohmmd Abuhib 22
23 Unblnced Dt The number of observtions tken within ech tretment m be different Let ni observtions be tken under tretment I, i=1,2,,: SS SS T Tretments N i1 j1 i 1 i 1 ni n n i 2 ij 2 2 i i N N 2 Dr. Mohmmd Abuhib 23
24 Model Adequc Checking in ANOVA The use of prtitioning to test for no differences in tretment mens requires the stisfction of the following ssumptions: Observtions re dequetel described b the fixed effects model Normlit of errors Constnt unknown vrince Independence distributrion with zero men It is unwise to rel on ANOVA until the vlidit of these ssumptions hs been checked. Dr. Mohmmd Abuhib 24
25 Model Adequc Checking in ANOVA Exmintion of residuls e ˆ ij ij ij ij If the model is dequte, residuls should be structureless: the should contin no obvious pttern Residul plots re ver useful i. Dr. Mohmmd Abuhib 25
26 Model Adequc Checking Norml Probbilit Plot (NPP) of residuls If the error distribution is norml, the plot will resemble stright line. Emphsis should be on the centrl vlues of the plot rther on the extremes Tble 3-6 Error distribution slightl skewed with longer right til The plot tend to bend down slightl on the left impling tht the left til of the error distribution is thinner thn would be nticipted in norml distributiom. The negtive residuls re not s lrge s expected. Dr. Mohmmd Abuhib 26
27 Model Adequc Checking Norml Probbilit Plot (NPP) of Residuls An error distribution tht hs considerbl thicker or thinner tils thn the norml is of more concern thn skewd distribution F test (ANOVA) is robust to normlit ssumption Devition from normlit cuse the true significnce level nd the power to differslightl from the dvertised ones, with the power being lower. Dr. Mohmmd Abuhib 27
28 Model Adequc Checking NPP of residuls - Outliers Outlier: residul tht is ver much lrger thn n of the others. Cuses: Mistke in clcultions, dt coding, or tpo If this is not the cuse, the experimentl circumestnces surrounding this run must be studied If the outling response is prticulr desirble vlue, the outlier m be more informtive thn the rest of the dt Not to reject n outling observtion unless we hve solid ground for doing so. We m end up with two nlses: one with outlier nd one without Dr. Mohmmd Abuhib 28
29 Model Adequc Checking NPP of residuls - Outliers Stndrdized residuls: If ij re N(0, 2 ), the Stndrdized residuls should be nerl norml with zero men nd unit vrince 68% of Stndrdized residuls slould fll within the limits ± 95% of them within ±2 All of them within ±3 A residul bigger thn 3 or 4 is potentil outlier Exmple d ij e ij MS E Dr. Mohmmd Abuhib 29
30 Model Adequc Checking Plot of Residuls in Time Sequence Non constnt vrince: Sometimes the skill of the experimenter m chnge s the experiment progresses, or the process being studied m become more ertic. This will often result in chnge in the error vrince over time, this condition often leds to plot of resduls tht exbits more spred t one end thn t the other. In our exmple ther is no evidence of n violtion of independence or constnt vrince ssumption Dr. Mohmmd Abuhib 30
31 Model Adequc Checking Plot of Residuls vs Fitted Vlues If the model is dequte nd the ssumptions re stisfied, the residuls should be unrelted to n other vrible Sometimes, Vrince of observtions increses s the mgnitude of the observtion increses When Nonconstnt vrince cse occurs, ppl vrince stblizing trnsformtion Dr. Mohmmd Abuhib 31
32 Model Adequc Checking Plot of Residuls vs Fitted Vlues- Trnsformtion Experimenter knows theoriticl distribution of observtions: Distribution Trnsformtion Poisson lognorml Binomil Squre root Logrithmic Arcsin or 1 * * ij ij ij ij * ij log sin * 1 ij ij ij When there is no obvious trnsformtion, experimentl empiricll seeks trnsformtion tht equlizes vrince regrdless of vlue of men Trnsformtion brings error distribution to norml Dr. Mohmmd Abuhib 32
33 Model Adequc Checking Sttisticl Test for Equlit of VARIANCE-BARTLETT S Test H o : H 1 : bove not true for t lest one Test sttistic: q x q N S n S , log10 p i 1log10 i c i 1 ni 1 S i 1 c 1 ni 1 N, S p 3( 1) i 1 N Q i lrge when smple vrinces differ gretl, nd is equl to zero when ll vrinces re equl 2 2 Reject H o when x1 x, 1 When normlit ssumption is not vlid, Brtlett s test not used Exmple 3-4 Dr. Mohmmod Abuhib 33 2 i
34 Model Adequc Checking Sttisticl Test for Equlit of Vrince Modified Levene test It is robust to deprtures from normlit Absolute devition of observtions ij in ech tretment from i 1,2,..., tretment medin: d ij ij i j 1,2,..., n If men devitions re equl, vrinces in ll tretments will be the sme Test sttistic is the usul ANOVA F sttistic pplied to d ij Exmple 3-5 Dr. Mohmmod Abuhib 34
35 Model Adequc Checking Empiricl Selection of Trnsformtion Suppose tht the stndrd devition of dt is proportionl to power of the men Find trnsformtion tht ields constnt vrince. * Suppose tht the trnsformtion is power of originl dt: l 1 This ields * If we set l 1, the vrince of trnsformed dt is constnt Tble 3-9: Vrince stbilizing trnsformtions Appl trnsformtion to Exmple 3-5 In prctice: tr severl lterntives nd observe the effect of ech trnsformtion on the plot of residuls vs predicted response. l Dr. Mohmmod Abuhib 35
36 The Regression Model Fctors of n experiment: 1. Qunttive: one whose levels cn be ssocited with points on numericl scle 2. Qulttive: levels cn not rrnged in order of mgnitude Model: 2 1. Qudrtic: o 1x 2x 2. Cubic: 2 3 o 1x 2x 3x The constnt prmeters re estimted b minimizing the sum of squres of errors Dr. Mohmmd Abuhib 36
37 The Regression Model Dr. Mohmmd Abuhib 37
38 Post-ANOVA Comprison of Mens The nlsis of vrince tests the hpothesis of equl tretment mens Assume tht residul nlsis is stisfctor If tht hpothesis is rejected, we don t know which specific mens re different Determining which specific mens differ following n ANOVA is clled the multiple comprisons problem There re mn ws to do this Comprisons between tretment mens re mde in terms of: 1. Tretment totls or 2. Tretment verges Dr. Mohmmd Abuhib 38
39 Grphicl Comprison of Mens Known Stndrd devition of tretment verge = / n If ll fctor level mens re identicl, the observed smple mens i. would behve s if the were set of observtions drwn t rndom from norml distribution with men nd stndrd devition.. / n Visulize norml distribution cpble of being slid long n xis below which tretment mens re plotted. If tretment mens re ll equl, there should be some position for this distribution tht mkes it obvious tht the tretment mens were drwn from the sme distribution. If this is not the cse, tretment men vlues tht pper not to hve been drwn from this distribution re ssocited with fctor levels tht produce different men responses. Dr. Mohmmd Abuhib 39
40 Grphicl Comprison of Mens unknown Replce with scle fctor MS E MS / n Sketch of t-distribution in Figure 3-11: Multipl bsciss t b scle fctor E Plot this ginst ordinte of t t tht point The distribution cn hve n rbitrr origin from ANOVA nd use t-distribution with insted of the norml MS / n 8.06/ E Dr. Mohmmd Abuhib 40
41 Grphicl Comprison of Mens Dr. Mohmmd Abuhib 41
42 Contrsts A contrst is liner combintion of prmeters of the form: c, c 0 Hpothesises: Hpothesises testing b: 1. t-test 2. F test i i i i1 i1 H : c 0, H : c 0 o i i 1 i i i1 i1 Dr. Mohmmd Abuhib 42
43 Contrsts- Hpothesises testing t-test Contrst is in terms of tretment totls: If H o is true, then the rtio Test sttistic H o is rejected if t o 2 2 i i., ( ) i i1 i1 C c V C n c i 1 nms c i E i. i 1 t t o / 2, N c 2 i i 1 n c i i. 2 2 ci i 1 hs N(0,1) Dr. Mohmmd Abuhib 43
44 Contrsts- Hpothesises Testing F-test Squre of t rndom vrible with n DOF is n F rndom vrible with 1.0 numertor nd n denomintor DOF Test sttistic 2 2 c i i. c i i. 2 i 1 MSC SSC /1, i1 o o C 2 MS E MS E 2 nms E ci nci i1 i1 F t SS H o is rejected if F F o,1, N Dr. Mohmmd Abuhib 44
45 Contrsts- Hpothesises Testing Confidence Intervl Contrst in terms of tretment verges: 2 2 c i i, C c i i., V ( C ) ci i 1 i 1 n i 1 The 100(1-) confidence intervl MS E 2 E 2 c i i. t/ 2, N ci c i i c i i. t/ 2, N ci i 1 n i 1 i 1 n i 1 MS If the confidence intervl includes zero, we would be unble to reject H o Dr. Mohmmd Abuhib 45
46 Contrsts- Hpothesises Testing Orthogonl Contrsts Two contrts with coefficients c i nd d i re orthogonl if i 1 c d Exmple 3-6 i i 0 Dr. Mohmmd Abuhib 46
47 Contrsts- Hpothesises Testing Sheffe s Method for Compring All Contrsts A method for compring n nd ll possible contrsts between tretment mens. Tpe I error is t most for n of the possible comprisons Set of m constrsts in the tretment mens c c... c, u 1,2,..., m Stndrd error Exmple: P95 u 1u 1 2u 2 u C c c... c u 1u 1. 2u 2. u. 2 C / u E iu i i 1 S MS c n Dr. Mohmmd Abuhib 47
48 Contrsts- Hpothesises Testing Compring pirs of Tretment Mens Tuke s Following ANOVA in which we hve rejected the null hpothesis of equl tretment mens Ho : i j, H1 : i j The overll significnce level is exctl when smple sizes re equl nd t most when smple sizes re unequl Confidence level is 100(1- )% when smple sizes re equl nd t lest 100(1- )% when smple sizes re unequl Two mens re different if MS E i. j. T q, f 100(1- )% confidence intervls: n MS E MS i. j. q, f i j i. j. q, f n n Exmple 3-7 Dr. Mohmmd Abuhib 48 E
49 Contrsts- Hpothesises testing Compring pirs of Tretment Mens Fisher lest Significnt Difference (LSD) Mens Two mens re significntl different if 1 1 i. j. LSD t / 2, N MS E ni nj Exmple 3-8 Dr. Mohmmd Abuhib 49
50 Contrsts- Hpothesises Testing Compring pirs of Tretment Mens Duncn s Multiple Rnge Test Order tretment verges in scending order Stndrd error of ech verge: S Tble VII: r (p,f), p=2,3,, R p =r (p,f) Si. MS E i. h nh, n i 1 1/ n i Exmple 3-9 Dr. Mohmmd Abuhib 50
51 Contrsts- Hpothesises Testing Which Pirwise Comprison Method to Use? No cler cut nswer LSD method is ver effective test for detecting true differences in mens if it is pplied onl fter the F test in ANOVA is sginificnt t 5%. Good performnce in detecting true differences with Duncn s multiple rnge. Dr. Mohmmd Abuhib 51
52 Choice of Smple Size 1 2 = P(tpe I error) = P(reject H o while H o is true) = P(tpe II error) = P(fil to reject H o while H o is flse) Power of the test = 1 = P(reject H o while H o is flse) Choice of smple size is relted to Suppose tht the mens re not equl, d = 1 2. Becuse H o is not true, we re concerned bout wrongl filing to reject H o. The probbilt of tpe II error depends on the true difference in mens d. Operting Chrcteristic Curve (OCC): grph of vs d for prticulr smple size Dr. Mohmmd Abuhib 52
53 Choice of Smple Size n 1 = n 2 nd 1 2 The error is function of the smple size For given vlue of d, the error decreses s the smple size increses. A specified diffenece in mens is esier to detect for lrger smple sizes thn for smll ones. Figure 2-12 d ; d 1 2 n * n n*: smple size used to construct the curve The greter the difference in mens, the smller the probbilit of tpe II error As the smple size gets lrger, the probbilt of tpe II error gets smller Dr. Mohmmd Abuhib 53
54 Choice of Smple Size n 1 = n 2 nd 1 2 Exmple: Portlnd cement Mortr A difference in men strength b 0.5kgf/cm2 cb be detected with high probbilit d Preior experience: s = 0.25Kgf/cm 2, d = 1.0 If we wish to reject the null hpothesis 95% of the time when 1 2 = 0.5, then = 0.05 Figure 2-12: n* = 16, n = 9 Dr. Mohmmd Abuhib 54
55 Smple Size Determintion FAQ in designed experiments Answer depends on lots of things; including: 1. Tpe of experiment is being contemplted 2. how it will be conducted 3. resources, 4. desired sensitivit Sensitivit refers to the difference in mens tht the experimenter wishes to detect Generll, incresing the number of replictions increses the sensitivit or it mkes it esier to detect smll differences in mens Dr. Mohmmd Abuhib 55
56 Smple Size Determintion Fixed Effects Cse Equl Smple Sizes Cn choose the smple size to detect specific difference in mens nd chieve desired vlues of tpe I nd tpe II errors Tpe I error reject H 0 when it is true () Tpe II error fil to reject H 0 when it is flse () Power = 1 We consider the probbilit of tpe II error o o P Fo F, 1, N o = 1-P{reject H H is flse} 1 { H is flse} If H o is flse, F o is distributed s noncentrl F rndom vrible with -1 nd N- DOF nd noncentrlit prmeter d. If d=0, the noncentrl F distribution becomes centrl Dr. Mohmmd Abuhib 56
57 Smple Size Determintion Fixed Effects Cse Equl Smple Sizes Operting chrcteristic curves plot ginst 2 prmeter F where ni 2 i1 F 2 2 F is relted to the noncentrlit prmeter d. Determintion of f: 1. Given 1, 2,, i i, (1/ ) i 2. Estimte 2 i 1 bsed on: Prior experience Previous experiment or preliminr test Judgement If uncertin bout vlue of 2, smple sizes cn be determined for rnge of likel vlues of 2 Dr. Mohmmd Abuhib 57
58 Smple Size Determintion Fixed Effects Cse---Use of OC Curves The OC curves for the fixed effects model re in the Appendix, Tble V, pg. 613 Exmple 3-11 A ver common w to use these chrts is to define difference in two mens D of interest, then the 2 minimum vlue of F is 2 2 nd F 2 2 Tpicll work in term of the rtio of D / nd tr vlues of n until the desired power is chieved Minitb will perform power nd smple size clcultions see pge 103 Exmple Dr. Mohmmd Abuhib 58
59 Smple Size Determintion Fixed Effects Cse - Specif Stndrd Devition Increse If the tretment mens do not differ, the stndrd devition of n observtion chosen t rndom is. If the tretment mens re different, the stndrd devition of n observtion chosen t rndom is given b: 2 2 / i 1 If we choose %P for the increse in stndrd devition of n observtion beond which we wish to reject the hpothesis tht ll tretment mens re equl: Exmple: P110 i i i i1 P i1 / / P 1, F / n 100 Dr. Mohmmd Abuhib 59 n
60 Power nd Smple Size Clcultions from Minitb Dr. Mohmmd Abuhib 60
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