MAT3378 ANOVA Summary
|
|
- Bartholomew Arnold
- 6 years ago
- Views:
Transcription
1 MAT3378 ANOVA Summary April 18, 2016 Before you do the analysis: How many factors? (one-factor/one-way ANOVA, two-factor ANOVA etc.) Fixed or Random or Mixed effects? Crossed-factors; nested factors or partially nested factors? The same or different sample sizes for each treatment? Contents 1 One Factor Anova Fixed effects Random effects Two-Factor ANOVA with crossed-factors Fixed effects - equal sample sizes for each treatment Random effects - equal sample sizes for each treatment Mixed effects - equal sample sizes for each treatment (Chapter 25.2) 15 3 Two-Factor ANOVA with nested design Fixed effects Random effects Mixed effects Complicated designs: Three-Factor ANOVA with partially nested design and mixed effects Fixed-effects only Mixed-effects
2 1 One Factor Anova 1.1 Fixed effects Textbook: Chapter 16; p. 681 Model: Estimates: µ is a constant, Y ij = µ i + ε ij = µ + α i + ε ij, µ i are factor level means (fixed, deterministic), α i are factor level effects, α i = µ i µ, ε ij are indepndent N(0, σ 2 ), i = 1,..., r, r is the number of factor levels, j = 1,..., n i, n i is the sample size for the level i, N = n n r. Y i = Y = ni, i = 1,..., r, sample mean for level i. n i ni, i = 1,..., r, overall mean. N r Sums of Squares: Decomposition Y ij Y = Y ij Y i + Y i Y }{{ } Total deviation Deviation of each observation from the factor level mean Deviation of the factor level mean from the overall mean leads to SSTO = SSE + SSTR, 2
3 SSTO = SSTR = SSE = r n i ( 2 Yij Y ) r ( 2 n i Y i Y ) n i r ( 2 Yij Y i ) Degrees of freedom: SSTO has n 1 degrees of freedom, since there are n observations and one constrain: r ni (Y ij Y ) = 0; SSTR has r 1 degrees of freedom, since there are r levels and one constrain: r n i(y i Y ) = 0; The number of degrees of freedom is n r since we must have N 1 = (N r) + (r 1). Mean Squares: Sums of squares and the number of degrees of freedom lead to: Expected Values: MSTR = SSTR r 1, MSE = SSE N r E[MSE] = σ 2, (1) r E[MSTR] = σ 2 + n i(µ i µ ) r 1 µ is a weighted mean: µ = r n iµ i /N. Test: H 0 : µ 1 = = µ r. If all factor level means are the same, then E[MSTR] = σ 2. Hence, the test statistics for H 0 can be constructed by comparing MSTR with MSE. ANOVA table: 3
4 source df SS MS F Treatment r 1 SSTR MSTR F Error (Residuals) N r SSE MSE Total N 1 SSTO F = MSTR MSE has F distribution with (r 1, N r) degrees of freedom. Estimation: Goal: estimate µ i ; Estimator: Y i ; σ 2 [Y i ] = σ 2 /N; By 1, σ 2 can be estimated by MSE which has N r degrees of freedom; The confidence interval for µ i is R implementation: aov(y~x) Y i ± t(1 α/2, N r) MSE 1 n i. 4
5 1.2 Random effects Textbook: Chapter 25.1; p Model: Y ij = µ i + ε ij, µ i are independent N(µ, σµ) 2, ε ij are independent N(0, σ 2 ), µ i and ε ij are independent, i = 1,..., r, r is the number of factor levels, j = 1,..., n i, n is the sample size for the level i, N = nr. Estimates: Y i = Y = ni, i = 1,..., r, sample mean for level i. n i ni, i = 1,..., r, overall mean. N r Sums of Squares: Decomposition Y ij Y = Y ij Y i + Y i Y }{{ } Total deviation Deviation of each observation from the factor level mean Deviation of the factor level mean from the overall mean leads to SSTO = SSE + SSTR, 5
6 SSTO = SSTR = SSE = r n i ( 2 Yij Y ) r ( 2 n i Y i Y ) n i r ( 2 Yij Y i ) Degrees of freedom: SSTO has N 1 degrees of freedom, since there are n observations and one constrain: r ni (Y ij Y ) = 0; SSTR has r 1 degrees of freedom, since there are r levels and one constrain: r n i(y i Y ) = 0; The number of degrees of freedom is N r since we must have N 1 = (N r) + (r 1). Mean Squares: Sums of squares and the number of degrees of freedom lead to: Expected Values: MSTR = SSTR r 1, MSE = SSE N r E[MSE] = σ 2, (2) E[MSTR] = σ 2 + nσ 2 µ. (3) Note the difference in E[MSTR] as compared to the fixed effects ANOVA. Test: ANOVA table: H 0 : σ µ = 0. 6
7 source df SS MS F Treatment r 1 SSTR MSTR F Error (Residuals) N r SSE MSE Total N 1 SSTO F = MSTR MSE has F distribution with (r 1, N r) degrees of freedom. Estimation I: Goal: estimate µ ; Estimator: Y ; σ 2 [Y ] = nσ2 µ +σ2 rn ; By 3, the right hand-side can be estimated by MSTR nr of freedom; which has r 1 degrees The confidence interval for µ is Y ± t(1 α/2, r 1) 1 MSTR rn. Note the difference in as compared to the fixed effects ANOVA. Estimation II: Goal: estimate R implementation: aov(y~x) σ 2 µ σ 2 µ +σ2, σ 2, σ 2 µ; 7
8 2 Two-Factor ANOVA with crossed-factors 2.1 Fixed effects - equal sample sizes for each treatment Textbook: Chapter 19; p. 812 Model: µ is a constant, Y ijk = µ + α i + β j + (αβ) ij + ε ijk, α i are main effects for Factor A, β i are main effects for Factor B, a α i = 0, b β j = 0, (αβ) ij are interactions between A, B, a (αβ) ij = 0, j = 1,..., b b (αβ) ij = 0, i = 1,..., a ε ijk are independent N(0, σ 2 ), i = 1,..., a, a is the number of factor levels for A, j = 1,..., b, b is the number of factor levels for B, k = 1,..., n, n is the sample size for the level treatment (i, j), N = nab, the total sample size. Estimates: n k=1 Y ij = Y ijk, i = 1,..., a, j = 1,..., b, sample mean for treatment (i, j), n b n k=1 Y i = Y ijk, i = 1,..., a, sample mean for level i of factor A, bn a n k=1 Y j = Y ijk, j = 1,..., b, sample mean for level j of factor B. an Y = a b n k=1 N, i = 1,..., r, overall mean. 8
9 Sums of Squares: Decomposition Y ijk Y = Y ijk Y ij + Y ij Y Total deviation Deviation of each observation from the treatment mean Deviation of the treatment mean from the overall mean leads to SSTO = SSE + SSTR, a b n ( 2 SSTO = Yijk Y ) k=1 a b ( 2 SSTR = n Y ij Y ) a b n ( 2 SSE = Yijk Y ij ). k=1 Further decomposition Y ij Y Deviation of the treatment mean from the overall mean leads to = Y i Y }{{ } + Y j Y Factor A main effect Factor B main effect SSTR = SSA + SSB + SSAB, + Y ij Y i Y j + Y Interaction a ( 2 SSA = nb Yi Y ) SSB = na b ( 2 Y j Y ) a b ( 2 SSAB = n Yij Y i Y j + Y ). Degrees of freedom: 9
10 SSTO has N 1 degrees of freedom, since there are N observations and one constrain; SSTR has ab 1 degrees of freedom, since there are ab treatments and one constrain: SSE: The number of degrees of freedom is N ab since we must have N 1 = (N ab) + (ab 1). SSA has a 1 degrees of freedom since there are a levels and 1 constrain; SSB has b 1 degrees of freedom since there are b levels and 1 constrain; SSAB has (a 1)(b 1) degrees of freedom since we must have ab 1 = (a 1) + (b 1) + (ab 1), Mean Squares: Sums of squares and the number of degrees of freedom lead to: MSTR = SSTR ab 1, MSE = SSE N ab, MSA = SSA a 1, MSB = SSB b 1, MSAB = SSAB (a 1)(b 1), Expected Values: E[MSE] = σ 2, (4) a E[MSA] = σ 2 + nb (µ i µ ) 2 a = σ 2 + nb α2 i, (5) a 1 a 1 b E[MSB] = σ 2 + na (µ j µ ) 2 b = σ 2 + na β2 j, (6) b 1 b 1 E[MSAB] = σ 2 + n a b (αβ) ij (a 1)(b 1), (7) 10
11 Tests: H 0 : α 1 = = α a = 0 (equivalently) H 0 : µ 1 = = µ a. H 0 : β 1 = = β b = 0 (equivalently) H 0 : µ 1 = = µ b, H 0 : (αβ) ij = 0. Equations (4)-(7) suggest evaluation of MSA/MSE; MSB/MSE; MSAB/MSE. ANOVA table: source df SS MS F Factor A a 1 SSA MSA F A Factor B b 1 SSB MSB F B Interactions AB (a 1)(b 1) SSAB MSAB F AB Error (Residuals) N ab SSE MSE Total N 1 SSTO F A = MSA MSE, F B = MSB MSE, F AB = MSAB MSE, have F distribution with (a 1, N ab), (b 1, N ab) and ((a 1)(b 1), N ab) degrees of freedom, respectively. Estimation: Goal: estimate L = a c iµ i ; Estimator: ˆL = a c iy i ; σ 2 [ˆL] = σ2 bn a c2 i ; We estimate σ 2 by MSE which has N ab degrees of freedom; The confidence interval for L is ˆL ± t(1 α/2, N ab) 1 MSE bn. R implementation: 11
12 aov(y~xa*xb) aov(y~xa+xb+xa*xb) aov(y~xa+xb+xa:xb) Note: aov(y~xa+xb) produces the output for the model aov(y~xa:xb) produces the output for the model Y ijk = µ + α i + β j + ε ijk, Y ijk = µ + (αβ) ij + ε ijk, 12
13 2.2 Random effects - equal sample sizes for each treatment Textbook: Chapter 25.2; p Model: µ is a constant, Y ijk = µ + α i + β j + (αβ) ij + ε ijk, α i, β j, (αβ) ij, are independent zero-mean normal random variables with variances σ 2 α, σ 2 β, σ2 αβ, ε ij are independent N(0, σ 2 ), α i, β j, (αβ) ij and ε ijk are independent, i = 1,..., a, j = 1,..., b, k = 1,..., n, N = nab, the total sample size. Expected Values: Instead of (4)-(7), we have Tests: E[MSE] = σ 2, (8) E[MSA] = σ 2 + nbσ 2 α + nσ 2 αβ, (9) E[MSB] = σ 2 + naσ 2 β + nσ2 αβ, (10) E[MSAB] = σ 2 + nσ 2 αβ, (11) H 0 : σ α = 0, H 0 : σ β = 0, H 0 : σ αβ = 0. Equations (8)-(11) suggest evaluation of MSA/MSAB; MSB/MSAB; MSAB/MSE. ANOVA table: source df SS MS F Factor A a 1 SSA MSA F A Factor B b 1 SSB MSB F B Interactions AB (a 1)(b 1) SSAB MSAB F AB Error (Residuals) N ab SSE MSE Total N 1 SSTO 13
14 F A = MSA MSAB, F B = MSB MSAB, F AB = MSAB MSE, have F distribution with (a 1, (a 1)(b 1)), (b 1, (a 1)(b 1)) and ((a 1)(b 1), N ab) degrees of freedom, respectively. R implementation: Use the fixed-effects commands: aov(y~xa*xb) aov(y~xa+xb+xa*xb) aov(y~xa+xb+xa:xb) Ignore F statistics and p-values and compute them on your own. 14
15 2.3 Mixed effects - equal sample sizes for each treatment (Chapter 25.2) Textbook: Chapter 25.2; p Model: µ is a constant, α i are fixed Factor A main effects, Y ijk = µ + α i + β j + (αβ) ij + ε ijk, a α i = 0, β j, (αβ) ij, are independent zero-mean normal random variables with variances σ 2 β, (a 1)σ2 αβ /a, ε ijk are independent N(0, σ 2 ), β j, (αβ) ij and ε ijk are independent, i = 1,..., a, j = 1,..., b, k = 1,..., n, N = nab, the total sample size. Expected Values: Instead of (8)-(11), we have Tests: E[MSE] = σ 2, (12) a α2 i E[MSA] = σ 2 + nb + nσαβ 2 a 1, (13) E[MSB] = σ 2 + naσβ 2, (14) E[MSAB] = σ 2 + nσ 2 αβ, (15) H 0 : α 1 = = α a = 0, H 0 : σ β = 0, H 0 : σ αβ = 0. Equations (8)-(11) suggest evaluation of MSA/MSAB; MSB/MSE; MSAB/MSE. ANOVA table: 15
16 source df SS MS F Factor A a 1 SSA MSA F A Factor B b 1 SSB MSB F B Interactions AB (a 1)(b 1) SSAB MSAB F AB Error (Residuals) N ab SSE MSE Total N 1 SSTO F A = MSA MSAB, F B = MSB MSE, F AB = MSAB MSE, have F distribution with (a 1, (a 1)(b 1)), (b 1, N ab) and ((a 1)(b 1), N ab) degrees of freedom, respectively. R implementation: Use the fixed-effects commands: aov(y~xa*xb) aov(y~xa+xb+xa*xb) aov(y~xa+xb+xa:xb) Ignore F statistics and p-values and compute them on your own. 16
17 3 Two-Factor ANOVA with nested design 3.1 Fixed effects Textbook: Chapter 26.1; p For random and mixed effects see Table 26.5 on page Model: µ is a constant, α i are fixed Factor A main effects, β j(i) are within levels effects, ε ijk are independent N(0, σ 2 ), i = 1,..., a, Y ijk = µ + α i + β j(i) + ε ijk, a α i = 0, b β j(i) = 0, j = 1,..., b, b is the number of levels of Factor B within each level of Factor A; k = 1,..., n, N = nab, the total sample size. Sums of Squares: Decomposition Y ijk Y = Y ijk Y ij + Y ij Y Total deviation Deviation of each observation from the treatment mean Deviation of the treatment mean from the overall mean leads to SSTO = SSE + SSTR, 17
18 a b n ( 2 SSTO = Yijk Y ) k=1 a b ( 2 SSTR = n Y ij Y ) a b n ( 2 SSE = Yijk Y ij ). k=1 Further decomposition Y ij Y Deviation of the treatment mean from the overall mean leads to SSTR = SSA + SSB(A), = Y i Y }{{ } + Y ij Y i Factor A main effect Factor B effects within A a ( 2 SSA = nb Yi Y ) a SSB(A) = n b ( 2 Y ij Y i ). Degrees of freedom: SSTO has N 1 degrees of freedom, since there are N observations and one constrain; SSTR has ab 1 degrees of freedom, since there are ab treatments and one constrain: SSE: The number of degrees of freedom is N ab since we must have N 1 = (N ab) + (ab 1). SSA has a 1 degrees of freedom since there are a levels and 1 constrain; SSB(A) has a(b 1) degrees of freedom since for each level of A we have b levels less 1 constrain; 18
19 Mean Squares: Sums of squares and the number of degrees of freedom lead to: MSTR = SSTR ab 1, MSE = SSE N ab, MSA = SSA a 1, MSB = SSB b 1, MSB(A) = SSB(A) a(b 1). Expected Values: E[MSE] = σ 2, (16) E[MSA] = σ 2 + nb a α2 i a 1, (17) Tests: E[MSB(A)] = σ 2 + n a b β2 j(i) a(b 1) H 0 : α 1 = = α a = 0, H 0 : β j(i) = 0,, (18) Equations (16)-(18) suggest evaluation of MSA/MSE; MSB(A)/MSE. ANOVA table: source df SS MS F Factor A a 1 SSA MSA Factor B within A a(b 1) SSB MSB FA FB(A) Error (Residuals) N ab SSE MSE Total N 1 SSTO F A = MSA MSE, F B(A) = MSB(A) MSE, 19
20 have F distribution with (a 1, N ab), (a(b 1), N ab) degrees of freedom, respectively. R implementation: aov(y~xa+xa/xb) For random and mixed effects ignore the F and p-value part and calculate manually. 3.2 Random effects Expected Values: E[MSE] = σ 2, (19) E[MSA] = σ 2 + nbσα 2 + nσβ 2, (20) E[MSB(A)] = σ 2 + nσβ 2, (21) 3.3 Mixed effects Model: Y ijk = µ + α i + β j(i) + ε ijk, µ is a constant, α i are fixed Factor A main effects, β j(i) are N(0, σβ 2 ), ε ijk are independent N(0, σ 2 ), ε ijk, β j(i) are independent, a α i = 0, j = 1,..., b, b is the number of levels of Factor B within each level of Factor A; k = 1,..., n, N = nab, the total sample size. Expected Values: 20
21 A - fixed; B - random E[MSE] = σ 2, (22) a α2 i E[MSA] = σ 2 + nb + nσβ 2 a 1, (23) E[MSB(A)] = σ 2 + nσβ 2, (24) 4 Complicated designs: Three-Factor ANOVA with partially nested design and mixed effects Textbook: Chapter 26.9; p Fixed-effects only Model: Set-up: All effects are fixed, A and B interact, C nested in A. Y ijkm = µ + α i + β j + (αβ) ij + γ k(i) + (βγ) jk(i) + ε ijkm, 21
22 µ is a constant, α i are fixed Factor A main effects, β j are fixed Factor B main effects, (αβ) ij are interactions between A, B, a (αβ) ij = 0, j = 1,..., b b (αβ) ij = 0, i = 1,..., a γ k(i) are C within A effects, a α i = 0, b β j = 0, c γ k(i) = 0, (βγ) jk(i) are B and C interactions within A, ε ijkm are independent N(0, σ 2 ), i = 1,..., a, j = 1,..., b, k = 1,..., c, m = 1,..., n, N = nabc, the total sample size. c (βγ) jk(i) = 0, Tests: H 0 : α 1 = = α a = 0. ANOVA table: 22
23 source df SS MS F Factor A a 1 SSA MSA FA Factor B b 1 SSB MSB FB Interactions A and B (a 1)(b 1) SSAB MSAB FAB C nested in A a(c-1) SSC(A) MSC(A) FC(A) Interactions B and C nested in A a(b 1)(c 1) SSBC(A) MSBC(A) FBC(A) Error (Residuals) N ab SSE MSE Total N 1 SSTO F A = MSA MSE, F B = MSB MSE, F AB = MSAB MSE, F C(A) = MSC(A) MSE, F BC(A) = MSBC(A), MSE have F distribution with (a 1, N ab), (b 1, N ab), ((a 1)(b 1), N ab), (a(c 1), N ab), (a(b 1)(c 1), N ab) degrees of freedom, respectively. 4.2 Mixed-effects Model: Set-up: A and B are fixed, C is random; A and B interact, C nested in A. Y ijkm = µ + α i + β j + (αβ) ij + γ k(i) + (βγ) jk(i) + ε ijkm, 23
24 µ is a constant, α i are fixed Factor A main effects, β j are fixed Factor B main effects, (αβ) ij are interactions between A, B, a (αβ) ij = 0, j = 1,..., b b (αβ) ij = 0, i = 1,..., a a α i = 0, b β j = 0, γ k(i) are random C within A effects, N(0, σγ) 2, (βγ) jk(i) are random B and C interactions within A, N(0, σβγ 2 ), ε ijkm are independent N(0, σ 2 ), all random variables are considered to be normal and independent, i = 1,..., a, j = 1,..., b, k = 1,..., c, m = 1,..., n, N = nabc, the total sample size. Expected Values: E[MSE] = σ 2, (25) E[MSA] = σ 2 + nbc a α2 i a 1 + bnσ 2 γ, (26) b β2 j E[MSB] = σ 2 + nac + nσβγ 2 b 1, (27) E[MSC(A)] = σ 2 + bnσγ 2, (28) a b E[MSAB] = σ 2 + cn (αβ) ij + nbσβγ 2 (a 1)(b 1), (29) E[MSBC(A)] = σ 2 + nσ 2 βγ, (30) 24
25 Tests: H 0 : α 1 = = α a = 0, ANOVA table: source df SS MS F Factor A a 1 SSA MSA FA Factor B b 1 SSB MSB FB Interactions A and B (a 1)(b 1) SSAB MSAB FAB C nested in A a(c-1) SSC(A) MSC(A) Interactions B and C nested in A a(b 1)(c 1) SSBC(A) MSBC(A) Error (Residuals) N ab SSE MSE Total N 1 SSTO F A = MSA MSC(A), F B = MSB MSBC(A), F AB = MSAB MSBC(A), have F distribution with (a 1, a(c 1)), (b 1, a(b 1)(c 1)), ((a 1)(b 1), a(b 1)(c 1)) degrees of freedom, respectively. R implementation: aov(y~xa+xb+xa:xb+xa/xc+xa/(xb:xc)) For random and mixed effects ignore the F and p-value part and calculate manually. 25
Theorem A: Expectations of Sums of Squares Under the two-way ANOVA model, E(X i X) 2 = (µ i µ) 2 + n 1 n σ2
identity Y ijk Ȳ = (Y ijk Ȳij ) + (Ȳi Ȳ ) + (Ȳ j Ȳ ) + (Ȳij Ȳi Ȳ j + Ȳ ) Theorem A: Expectations of Sums of Squares Under the two-way ANOVA model, (1) E(MSE) = E(SSE/[IJ(K 1)]) = (2) E(MSA) = E(SSA/(I
More informationFactorial ANOVA. Psychology 3256
Factorial ANOVA Psychology 3256 Made up data.. Say you have collected data on the effects of Retention Interval 5 min 1 hr 24 hr on memory So, you do the ANOVA and conclude that RI affects memory % corr
More informationUnit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs
Unit 7: Random Effects, Subsampling, Nested and Crossed Factor Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Understand how to interpret a random effect Know the different
More informationTwo or more categorical predictors. 2.1 Two fixed effects
Two or more categorical predictors Here we extend the ANOVA methods to handle multiple categorical predictors. The statistician has to watch carefully to see whether the effects being considered are properly
More informationAnalysis of Variance and Design of Experiments-I
Analysis of Variance and Design of Experiments-I MODULE VIII LECTURE - 35 ANALYSIS OF VARIANCE IN RANDOM-EFFECTS MODEL AND MIXED-EFFECTS MODEL Dr. Shalabh Department of Mathematics and Statistics Indian
More informationTWO OR MORE RANDOM EFFECTS. The two-way complete model for two random effects:
TWO OR MORE RANDOM EFFECTS Example: The factors that influence the breaking strength of a synthetic fiber are being studied. Four production machines and three operators are randomly selected. A two-way
More informationStatistics 210 Part 3 Statistical Methods Hal S. Stern Department of Statistics University of California, Irvine
Thus far: Statistics 210 Part 3 Statistical Methods Hal S. Stern Department of Statistics University of California, Irvine sternh@uci.edu design of experiments two sample methods one factor ANOVA pairing/blocking
More informationOutline Topic 21 - Two Factor ANOVA
Outline Topic 21 - Two Factor ANOVA Data Model Parameter Estimates - Fall 2013 Equal Sample Size One replicate per cell Unequal Sample size Topic 21 2 Overview Now have two factors (A and B) Suppose each
More informationSTAT Final Practice Problems
STAT 48 -- Final Practice Problems.Out of 5 women who had uterine cancer, 0 claimed to have used estrogens. Out of 30 women without uterine cancer 5 claimed to have used estrogens. Exposure Outcome (Cancer)
More informationSTAT 705 Chapter 19: Two-way ANOVA
STAT 705 Chapter 19: Two-way ANOVA Adapted from Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 41 Two-way ANOVA This material is covered in Sections
More informationSTAT 705 Chapter 19: Two-way ANOVA
STAT 705 Chapter 19: Two-way ANOVA Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 38 Two-way ANOVA Material covered in Sections 19.2 19.4, but a bit
More informationStatistics 512: Applied Linear Models. Topic 9
Topic Overview Statistics 51: Applied Linear Models Topic 9 This topic will cover Random vs. Fixed Effects Using E(MS) to obtain appropriate tests in a Random or Mixed Effects Model. Chapter 5: One-way
More informationiron retention (log) high Fe2+ medium Fe2+ high Fe3+ medium Fe3+ low Fe2+ low Fe3+ 2 Two-way ANOVA
iron retention (log) 0 1 2 3 high Fe2+ high Fe3+ low Fe2+ low Fe3+ medium Fe2+ medium Fe3+ 2 Two-way ANOVA In the one-way design there is only one factor. What if there are several factors? Often, we are
More informationChapter 11 - Lecture 1 Single Factor ANOVA
Chapter 11 - Lecture 1 Single Factor ANOVA April 7th, 2010 Means Variance Sum of Squares Review In Chapter 9 we have seen how to make hypothesis testing for one population mean. In Chapter 10 we have seen
More informationTopic 32: Two-Way Mixed Effects Model
Topic 3: Two-Way Mixed Effects Model Outline Two-way mixed models Three-way mixed models Data for two-way design Y is the response variable Factor A with levels i = 1 to a Factor B with levels j = 1 to
More informationTwo-Way Analysis of Variance - no interaction
1 Two-Way Analysis of Variance - no interaction Example: Tests were conducted to assess the effects of two factors, engine type, and propellant type, on propellant burn rate in fired missiles. Three engine
More informationUnbalanced Designs Mechanics. Estimate of σ 2 becomes weighted average of treatment combination sample variances.
Unbalanced Designs Mechanics Estimate of σ 2 becomes weighted average of treatment combination sample variances. Types of SS Difference depends on what hypotheses are tested and how differing sample sizes
More informationTwo-Factor Full Factorial Design with Replications
Two-Factor Full Factorial Design with Replications Dr. John Mellor-Crummey Department of Computer Science Rice University johnmc@cs.rice.edu COMP 58 Lecture 17 March 005 Goals for Today Understand Two-factor
More information3. Design Experiments and Variance Analysis
3. Design Experiments and Variance Analysis Isabel M. Rodrigues 1 / 46 3.1. Completely randomized experiment. Experimentation allows an investigator to find out what happens to the output variables when
More informationIf we have many sets of populations, we may compare the means of populations in each set with one experiment.
Statistical Methods in Business Lecture 3. Factorial Design: If we have many sets of populations we may compare the means of populations in each set with one experiment. Assume we have two factors with
More informationNested Designs & Random Effects
Nested Designs & Random Effects Timothy Hanson Department of Statistics, University of South Carolina Stat 506: Introduction to Design of Experiments 1 / 17 Bottling plant production A production engineer
More informationSection 7.3 Nested Design for Models with Fixed Effects, Mixed Effects and Random Effects
Section 7.3 Nested Design for Models with Fixed Effects, Mixed Effects and Random Effects 1 TABLE 25.5 Mean Square MSA MSB Expected Mean Squares for Balanced Two-Factor ANOVA Models. Fixed AN OVA Model
More informationHomework 3 - Solution
STAT 526 - Spring 2011 Homework 3 - Solution Olga Vitek Each part of the problems 5 points 1. KNNL 25.17 (Note: you can choose either the restricted or the unrestricted version of the model. Please state
More informationTwo-factor studies. STAT 525 Chapter 19 and 20. Professor Olga Vitek
Two-factor studies STAT 525 Chapter 19 and 20 Professor Olga Vitek December 2, 2010 19 Overview Now have two factors (A and B) Suppose each factor has two levels Could analyze as one factor with 4 levels
More informationAllow the investigation of the effects of a number of variables on some response
Lecture 12 Topic 9: Factorial treatment structures (Part I) Factorial experiments Allow the investigation of the effects of a number of variables on some response in a highly efficient manner, and in a
More informationFractional Factorial Designs
k-p Fractional Factorial Designs Fractional Factorial Designs If we have 7 factors, a 7 factorial design will require 8 experiments How much information can we obtain from fewer experiments, e.g. 7-4 =
More informationStatistics 512: Applied Linear Models. Topic 7
Topic Overview This topic will cover Statistics 512: Applied Linear Models Topic 7 Two-way Analysis of Variance (ANOVA) Interactions Chapter 19: Two-way ANOVA The response variable Y is continuous. There
More informationFactorial and Unbalanced Analysis of Variance
Factorial and Unbalanced Analysis of Variance Nathaniel E. Helwig Assistant Professor of Psychology and Statistics University of Minnesota (Twin Cities) Updated 04-Jan-2017 Nathaniel E. Helwig (U of Minnesota)
More informationSTAT 705 Chapter 16: One-way ANOVA
STAT 705 Chapter 16: One-way ANOVA Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis II 1 / 21 What is ANOVA? Analysis of variance (ANOVA) models are regression
More informationChapter 8: Hypothesis Testing Lecture 9: Likelihood ratio tests
Chapter 8: Hypothesis Testing Lecture 9: Likelihood ratio tests Throughout this chapter we consider a sample X taken from a population indexed by θ Θ R k. Instead of estimating the unknown parameter, we
More informationTopic 9: Factorial treatment structures. Introduction. Terminology. Example of a 2x2 factorial
Topic 9: Factorial treatment structures Introduction A common objective in research is to investigate the effect of each of a number of variables, or factors, on some response variable. In earlier times,
More informationG. Nested Designs. 1 Introduction. 2 Two-Way Nested Designs (Balanced Cases) 1.1 Definition (Nested Factors) 1.2 Notation. 1.3 Example. 2.
G. Nested Designs 1 Introduction. 1.1 Definition (Nested Factors) When each level of one factor B is associated with one and only one level of another factor A, we say that B is nested within factor A.
More informationBIOSTATISTICAL METHODS
BIOSTATISTICAL METHODS FOR TRANSLATIONAL & CLINICAL RESEARCH Cross-over Designs #: DESIGNING CLINICAL RESEARCH The subtraction of measurements from the same subject will mostly cancel or minimize effects
More informationKey Features: More than one type of experimental unit and more than one randomization.
1 SPLIT PLOT DESIGNS Key Features: More than one type of experimental unit and more than one randomization. Typical Use: When one factor is difficult to change. Example (and terminology): An agricultural
More informationDefinitions of terms and examples. Experimental Design. Sampling versus experiments. For each experimental unit, measures of the variables of
Experimental Design Sampling versus experiments similar to sampling and inventor design in that information about forest variables is gathered and analzed experiments presuppose intervention through appling
More informationCS 147: Computer Systems Performance Analysis
CS 147: Computer Systems Performance Analysis CS 147: Computer Systems Performance Analysis 1 / 34 Overview Overview Overview Adding Replications Adding Replications 2 / 34 Two-Factor Design Without Replications
More informationNested 2-Way ANOVA as Linear Models - Unbalanced Example
Linear Models Nested -Way ANOVA ORIGIN As with other linear models, unbalanced data require use of the regression approach, in this case by contrast coding of independent variables using a scheme not described
More informationCHAPTER 4 Analysis of Variance. One-way ANOVA Two-way ANOVA i) Two way ANOVA without replication ii) Two way ANOVA with replication
CHAPTER 4 Analysis of Variance One-way ANOVA Two-way ANOVA i) Two way ANOVA without replication ii) Two way ANOVA with replication 1 Introduction In this chapter, expand the idea of hypothesis tests. We
More informationLecture 10: Factorial Designs with Random Factors
Lecture 10: Factorial Designs with Random Factors Montgomery, Section 13.2 and 13.3 1 Lecture 10 Page 1 Factorial Experiments with Random Effects Lecture 9 has focused on fixed effects Always use MSE in
More informationFactorial designs. Experiments
Chapter 5: Factorial designs Petter Mostad mostad@chalmers.se Experiments Actively making changes and observing the result, to find causal relationships. Many types of experimental plans Measuring response
More informationStatistics For Economics & Business
Statistics For Economics & Business Analysis of Variance In this chapter, you learn: Learning Objectives The basic concepts of experimental design How to use one-way analysis of variance to test for differences
More informationCh. 5 Two-way ANOVA: Fixed effect model Equal sample sizes
Ch. 5 Two-way ANOVA: Fixed effect model Equal sample sizes 1 Assumptions and models There are two factors, factors A and B, that are of interest. Factor A is studied at a levels, and factor B at b levels;
More informationChapter 15: Analysis of Variance
Chapter 5: Analysis of Variance 5. Introduction In this chapter, we introduced the analysis of variance technique, which deals with problems whose objective is to compare two or more populations of quantitative
More information3. Factorial Experiments (Ch.5. Factorial Experiments)
3. Factorial Experiments (Ch.5. Factorial Experiments) Hae-Jin Choi School of Mechanical Engineering, Chung-Ang University DOE and Optimization 1 Introduction to Factorials Most experiments for process
More informationANOVA Randomized Block Design
Biostatistics 301 ANOVA Randomized Block Design 1 ORIGIN 1 Data Structure: Let index i,j indicate the ith column (treatment class) and jth row (block). For each i,j combination, there are n replicates.
More informationUnit 8: 2 k Factorial Designs, Single or Unequal Replications in Factorial Designs, and Incomplete Block Designs
Unit 8: 2 k Factorial Designs, Single or Unequal Replications in Factorial Designs, and Incomplete Block Designs STA 643: Advanced Experimental Design Derek S. Young 1 Learning Objectives Revisit your
More informationNesting and Mixed Effects: Part I. Lukas Meier, Seminar für Statistik
Nesting and Mixed Effects: Part I Lukas Meier, Seminar für Statistik Where do we stand? So far: Fixed effects Random effects Both in the factorial context Now: Nested factor structure Mixed models: a combination
More informationRandom and Mixed Effects Models - Part III
Random and Mixed Effects Models - Part III Statistics 149 Spring 2006 Copyright 2006 by Mark E. Irwin Quasi-F Tests When we get to more than two categorical factors, some times there are not nice F tests
More informationResearch Methods II MICHAEL BERNSTEIN CS 376
Research Methods II MICHAEL BERNSTEIN CS 376 Goal Understand and use statistical techniques common to HCI research 2 Last time How to plan an evaluation What is a statistical test? Chi-square t-test Paired
More informationStat 511 HW#10 Spring 2003 (corrected)
Stat 511 HW#10 Spring 003 (corrected) 1. Below is a small set of fake unbalanced -way factorial (in factors A and B) data from (unbalanced) blocks. Level of A Level of B Block Response 1 1 1 9.5 1 1 11.3
More informationAnalysis of Variance
Analysis of Variance Blood coagulation time T avg A 62 60 63 59 61 B 63 67 71 64 65 66 66 C 68 66 71 67 68 68 68 D 56 62 60 61 63 64 63 59 61 64 Blood coagulation time A B C D Combined 56 57 58 59 60 61
More informationGraduate Lectures and Problems in Quality Control and Engineering Statistics: Theory and Methods
Graduate Lectures Problems in Quality Control Engineering Statistics: Theory Methods To Accompany Statistical Quality Assurance Methods for Engineers by Vardeman Jobe Stephen B. Vardeman V2.0: January
More information16.3 One-Way ANOVA: The Procedure
16.3 One-Way ANOVA: The Procedure Tom Lewis Fall Term 2009 Tom Lewis () 16.3 One-Way ANOVA: The Procedure Fall Term 2009 1 / 10 Outline 1 The background 2 Computing formulas 3 The ANOVA Identity 4 Tom
More informationTwo Factor Full Factorial Design with Replications
Two Factor Full Factorial Design with Replications Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu These slides are available on-line at: http://www.cse.wustl.edu/~jain/cse567-08/
More informationChapter 11 - Lecture 1 Single Factor ANOVA
April 5, 2013 Chapter 9 : hypothesis testing for one population mean. Chapter 10: hypothesis testing for two population means. What comes next? Chapter 9 : hypothesis testing for one population mean. Chapter
More information2017 Financial Mathematics Orientation - Statistics
2017 Financial Mathematics Orientation - Statistics Written by Long Wang Edited by Joshua Agterberg August 21, 2018 Contents 1 Preliminaries 5 1.1 Samples and Population............................. 5
More informationLecture 22 Mixed Effects Models III Nested designs
Lecture 22 Mixed Effects Models III Nested designs 94 Introduction: Crossed Designs The two-factor designs considered so far involve every level of the first factor occurring with every level of the second
More informationChapter 6 Randomized Block Design Two Factor ANOVA Interaction in ANOVA
Chapter 6 Randomized Block Design Two Factor ANOVA Interaction in ANOVA Two factor (two way) ANOVA Two factor ANOVA is used when: Y is a quantitative response variable There are two categorical explanatory
More informationn i n T Note: You can use the fact that t(.975; 10) = 2.228, t(.95; 10) = 1.813, t(.975; 12) = 2.179, t(.95; 12) =
MAT 3378 3X Midterm Examination (Solutions) 1. An experiment with a completely randomized design was run to determine whether four specific firing temperatures affect the density of a certain type of brick.
More informationMultiple Comparisons. The Interaction Effects of more than two factors in an analysis of variance experiment. Submitted by: Anna Pashley
Multiple Comparisons The Interaction Effects of more than two factors in an analysis of variance experiment. Submitted by: Anna Pashley One way Analysis of Variance (ANOVA) Testing the hypothesis that
More informationAN IMPROVEMENT TO THE ALIGNED RANK STATISTIC
Journal of Applied Statistical Science ISSN 1067-5817 Volume 14, Number 3/4, pp. 225-235 2005 Nova Science Publishers, Inc. AN IMPROVEMENT TO THE ALIGNED RANK STATISTIC FOR TWO-FACTOR ANALYSIS OF VARIANCE
More informationMuch of the material we will be covering for a while has to do with designing an experimental study that concerns some phenomenon of interest.
Experimental Design: Much of the material we will be covering for a while has to do with designing an experimental study that concerns some phenomenon of interest We wish to use our subjects in the best
More informationFormal Statement of Simple Linear Regression Model
Formal Statement of Simple Linear Regression Model Y i = β 0 + β 1 X i + ɛ i Y i value of the response variable in the i th trial β 0 and β 1 are parameters X i is a known constant, the value of the predictor
More informationEcon 3790: Business and Economic Statistics. Instructor: Yogesh Uppal
Econ 3790: Business and Economic Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu Chapter 13, Part A: Analysis of Variance and Experimental Design Introduction to Analysis of Variance Analysis
More informationWITHIN-PARTICIPANT EXPERIMENTAL DESIGNS
1 WITHIN-PARTICIPANT EXPERIMENTAL DESIGNS I. Single-factor designs: the model is: yij i j ij ij where: yij score for person j under treatment level i (i = 1,..., I; j = 1,..., n) overall mean βi treatment
More informationTopic 17 - Single Factor Analysis of Variance. Outline. One-way ANOVA. The Data / Notation. One way ANOVA Cell means model Factor effects model
Topic 17 - Single Factor Analysis of Variance - Fall 2013 One way ANOVA Cell means model Factor effects model Outline Topic 17 2 One-way ANOVA Response variable Y is continuous Explanatory variable is
More informationSTAT 506: Randomized complete block designs
STAT 506: Randomized complete block designs Timothy Hanson Department of Statistics, University of South Carolina STAT 506: Introduction to Experimental Design 1 / 10 Randomized complete block designs
More informationOne-Way Analysis of Variance. With regression, we related two quantitative, typically continuous variables.
One-Way Analysis of Variance With regression, we related two quantitative, typically continuous variables. Often we wish to relate a quantitative response variable with a qualitative (or simply discrete)
More informationThe Dependence of Repair Times of Safety-significant Devices on Repair Class in the Loviisa Nuclear Power Plant
HELSINKI UNIVERSITY OF TECHNOLOGY Systems Analysis Laboratory Mat-2.4108 Individual Research Projects in Applied Mathematics The Dependence of Repair Times of Safety-significant Devices on Repair Class
More informationChapter 20 : Two factor studies one case per treatment Chapter 21: Randomized complete block designs
Chapter 20 : Two factor studies one case per treatment Chapter 21: Randomized complete block designs Adapted from Timothy Hanson Department of Statistics, University of South Carolina Stat 705: Data Analysis
More informationOrthogonal contrasts for a 2x2 factorial design Example p130
Week 9: Orthogonal comparisons for a 2x2 factorial design. The general two-factor factorial arrangement. Interaction and additivity. ANOVA summary table, tests, CIs. Planned/post-hoc comparisons for the
More informationSummary of Chapter 7 (Sections ) and Chapter 8 (Section 8.1)
Summary of Chapter 7 (Sections 7.2-7.5) and Chapter 8 (Section 8.1) Chapter 7. Tests of Statistical Hypotheses 7.2. Tests about One Mean (1) Test about One Mean Case 1: σ is known. Assume that X N(µ, σ
More informationChap The McGraw-Hill Companies, Inc. All rights reserved.
11 pter11 Chap Analysis of Variance Overview of ANOVA Multiple Comparisons Tests for Homogeneity of Variances Two-Factor ANOVA Without Replication General Linear Model Experimental Design: An Overview
More informationFactorial Designs. Prof. Daniel A. Menasce Dept. of fcomputer Science George Mason University. studied simultaneously.
Desig of Expeimets: Factoial Desigs Pof. Daiel A. Measce Dept. of fcompute Sciece Geoge Maso Uivesity Basic Cocepts Factoial desig: moe tha oe facto is studied simultaeously. k umbe of factos umbe of levels
More information1 Introduction to One-way ANOVA
Review Source: Chapter 10 - Analysis of Variance (ANOVA). Example Data Source: Example problem 10.1 (dataset: exp10-1.mtw) Link to Data: http://www.auburn.edu/~carpedm/courses/stat3610/textbookdata/minitab/
More informationV. Experiments With Two Crossed Treatment Factors
V. Experiments With Two Crossed Treatment Factors A.The Experimental Design Completely Randomized Design (CRD) Let A be a factor with levels i = 1,,a B be a factor with levels j = 1,,b Y ijt = the response
More informationSTAT 401A - Statistical Methods for Research Workers
STAT 401A - Statistical Methods for Research Workers One-way ANOVA Jarad Niemi (Dr. J) Iowa State University last updated: October 10, 2014 Jarad Niemi (Iowa State) One-way ANOVA October 10, 2014 1 / 39
More information4.1. Introduction: Comparing Means
4. Analysis of Variance (ANOVA) 4.1. Introduction: Comparing Means Consider the problem of testing H 0 : µ 1 = µ 2 against H 1 : µ 1 µ 2 in two independent samples of two different populations of possibly
More informationPower & Sample Size Calculation
Chapter 7 Power & Sample Size Calculation Yibi Huang Chapter 7 Section 10.3 Power & Sample Size Calculation for CRDs Power & Sample Size for Factorial Designs Chapter 7-1 Power & Sample Size Calculation
More informationChapter 10: Analysis of variance (ANOVA)
Chapter 10: Analysis of variance (ANOVA) ANOVA (Analysis of variance) is a collection of techniques for dealing with more general experiments than the previous one-sample or two-sample tests. We first
More informationStatistical Analysis of Unreplicated Factorial Designs Using Contrasts
Georgia Southern University Digital Commons@Georgia Southern Electronic Theses & Dissertations Jack N. Averitt College of Graduate Studies COGS) Summer 204 Statistical Analysis of Unreplicated Factorial
More informationTwo-Way Factorial Designs
81-86 Two-Way Factorial Designs Yibi Huang 81-86 Two-Way Factorial Designs Chapter 8A - 1 Problem 81 Sprouting Barley (p166 in Oehlert) Brewer s malt is produced from germinating barley, so brewers like
More informationNotes for Week 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1
Notes for Wee 13 Analysis of Variance (ANOVA) continued WEEK 13 page 1 Exam 3 is on Friday May 1. A part of one of the exam problems is on Predictiontervals : When randomly sampling from a normal population
More informationdf=degrees of freedom = n - 1
One sample t-test test of the mean Assumptions: Independent, random samples Approximately normal distribution (from intro class: σ is unknown, need to calculate and use s (sample standard deviation)) Hypotheses:
More information1 The Randomized Block Design
1 The Randomized Block Design When introducing ANOVA, we mentioned that this model will allow us to include more than one categorical factor(explanatory) or confounding variables in the model. In a first
More informationMultiple Sample Numerical Data
Multiple Sample Numerical Data Analysis of Variance, Kruskal-Wallis test, Friedman test University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html 1 /
More informationGrant MacEwan University STAT 252 Dr. Karen Buro Formula Sheet
Grat MacEwa Uiversity STAT 5 Dr. Kare Buro Formula Sheet Descriptive Statistics Sample Mea: x = x i i= Sample Variace: s = i= (x i x) = Σ i=x i (Σ i= x i) Sample Stadard Deviatio: s = Sample Variace =
More informationUnit 8: A Mixed Two-Factor Design
Minitab Notes for STAT 6305 Dept. of Statistics CSU East Bay Unit 8: A Mixed Two-Factor Design 8.1. The Data We use data quoted in Brownlee: Statistical Theory and Methodology in Science and Engineering,
More informationChapter 11: Factorial Designs
Chapter : Factorial Designs. Two factor factorial designs ( levels factors ) This situation is similar to the randomized block design from the previous chapter. However, in addition to the effects within
More informationModel II (or random effects) one-way ANOVA:
Model II (or random effects) one-way ANOVA: As noted earlier, if we have a random effects model, the treatments are chosen from a larger population of treatments; we wish to generalize to this larger population.
More informationLec 5: Factorial Experiment
November 21, 2011 Example Study of the battery life vs the factors temperatures and types of material. A: Types of material, 3 levels. B: Temperatures, 3 levels. Example Study of the battery life vs the
More informationIn a one-way ANOVA, the total sums of squares among observations is partitioned into two components: Sums of squares represent:
Activity #10: AxS ANOVA (Repeated subjects design) Resources: optimism.sav So far in MATH 300 and 301, we have studied the following hypothesis testing procedures: 1) Binomial test, sign-test, Fisher s
More informationEX1. One way ANOVA: miles versus Plug. a) What are the hypotheses to be tested? b) What are df 1 and df 2? Verify by hand. , y 3
EX. Chapter 8 Examples In an experiment to investigate the performance of four different brands of spark plugs intended for the use on a motorcycle, plugs of each brand were tested and the number of miles
More informationChapter 4: Randomized Blocks and Latin Squares
Chapter 4: Randomized Blocks and Latin Squares 1 Design of Engineering Experiments The Blocking Principle Blocking and nuisance factors The randomized complete block design or the RCBD Extension of the
More information2 k, 2 k r and 2 k-p Factorial Designs
2 k, 2 k r and 2 k-p Factorial Designs 1 Types of Experimental Designs! Full Factorial Design: " Uses all possible combinations of all levels of all factors. n=3*2*2=12 Too costly! 2 Types of Experimental
More informationSTAT22200 Spring 2014 Chapter 8A
STAT22200 Spring 2014 Chapter 8A Yibi Huang May 13, 2014 81-86 Two-Way Factorial Designs Chapter 8A - 1 Problem 81 Sprouting Barley (p166 in Oehlert) Brewer s malt is produced from germinating barley,
More informationLektion 6. Measurement system! Measurement systems analysis _3 Chapter 7. Statistical process control requires measurement of good quality!
Lektion 6 007-1-06_3 Chapter 7 Measurement systems analysis Measurement system! Statistical process control requires measurement of good quality! Wrong conclusion about the process due to measurement error!
More informationDESAIN EKSPERIMEN Analysis of Variances (ANOVA) Semester Genap 2017/2018 Jurusan Teknik Industri Universitas Brawijaya
DESAIN EKSPERIMEN Analysis of Variances (ANOVA) Semester Jurusan Teknik Industri Universitas Brawijaya Outline Introduction The Analysis of Variance Models for the Data Post-ANOVA Comparison of Means Sample
More informationStatistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data
Statistics for Managers Using Microsoft Excel Chapter 10 ANOVA and Other C-Sample Tests With Numerical Data 1999 Prentice-Hall, Inc. Chap. 10-1 Chapter Topics The Completely Randomized Model: One-Factor
More informationDesign & Analysis of Experiments 7E 2009 Montgomery
Chapter 5 1 Introduction to Factorial Design Study the effects of 2 or more factors All possible combinations of factor levels are investigated For example, if there are a levels of factor A and b levels
More information