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Chapter 3 Experiments with a Single Factor: The Analysis of Variance 1 3.1 An Example • Chapter 2: A signal-factor experiment with two levels of the factor • Consider signal-factor experiments with a levels of the factor, a 2 • Example: – The tensile strength of a new synthetic fiber. – The weight percent of cotton – Five levels: 15%, 20%, 25%, 30%, 35% – a = 5 and n = 5 2 • Does changing the cotton weight percent change the mean tensile strength? • Is there an optimum level for cotton content? 3 An Example (See pg. 61) • An engineer is interested in investigating the relationship between the RF power setting and the etch rate for this tool. The objective of an experiment like this is to model the relationship between etch rate and RF power, and to specify the power setting that will give a desired target etch rate. • The response variable is etch rate. 4 • She is interested in a particular gas (C2F6) and gap (0.80 cm), and wants to test four levels of RF power: 160W, 180W, 200W, and 220W. She decided to test five wafers at each level of RF power. • The experimenter chooses 4 levels of RF power 160W, 180W, 200W, and 220W • The experiment is replicated 5 times – runs made in random order 5 6 7 • Does changing the power change the mean etch rate? • Is there an optimum level for power? • We would like to have an objective way to answer these questions • The t-test really doesn’t apply here – more than two factor levels 8 3.2 The Analysis of Variance • a levels (treatments) of a factor and n replicates for each level. • yij: the jth observation taken under factor level or treatment i. 9 Models for the Data • Means model: i 1,2,..., a yij i ij , j 1,2,..., n – yij is the ijth observation, – i is the mean of the ith factor level – ij is a random error with mean zero • Let μ i = μ + τ i , is the overall mean and τ i is the ith treatment effect • Effects model: i 1,2,..., a yij i ij , j 1,2,..., n 10 • Linear statistical model • One-way or Signal-factor analysis of variance model • Completely randomized design: the experiments are performed in random order so that the environment in which the treatment are applied is as uniform as possible. • For hypothesis testing, the model errors are assumed to be normally and independently distributed random variables with mean zero and variance, σ 2, i.e. yij ~ N(μ + τi, σ 2) • Fixed effect model: a levels have been specifically chosen by the experimenter. 11 3.3 Analysis of the Fixed Effects Model • Interested in testing the equality of the a treatment means, and E(yij) = μ i = μ + τ i, i = 1,2, …, a H0: μ 1 = … = μ a v.s. H1: μ i ≠ μ j, for at least one pair (i, j) • Constraint: i a i i 0 i • H0: τ1 = … = τa =0 v.s. H1: τ i ≠ 0, for at least one i 12 • n a n Notations: yi yij , y yij j 1 i 1 j 1 yi y i / n, y y / N , N na 3.3.1 Decomposition of the Total Sum of Squares • • Total variability into its component parts. The total sum of squares (a measure of overall variability in the data) a n SST ( yij y.. )2 i 1 j 1 • Degree of freedom: an – 1 = N – 1 13 a n a n 2 ( y y ) [( y y ) ( y y )] ij .. i. .. ij i. 2 i 1 j 1 i 1 j 1 a a n n ( yi. y.. ) ( yij yi. ) 2 2 i 1 i 1 j 1 SST SSTreatments SS E • SSTreatment: sum of squares of the differences between the treatment averages (sum of squares due to treatments) and the grand average, and a – 1 degree of freedom • SSE: sum of squares of the differences of observations within treatments from the treatment average (sum of squares due to error), and a(n - 1) = N – a degrees of freedom. 14 SST SSTreatments SS E • A large value of SSTreatments reflects large differences in treatment means • A small value of SSTreatments likely indicates no differences in treatment means • dfTotal = dfTreatment + dfError 2 2 ( n 1 ) S ( n 1 ) S SS • 1 a E N a (n 1) (n 1) • No differences between a treatment means: variance cane be estimated by n ( yi y ) 2 SSTreatments a 1 i a 115 SSTreatments SS E • Mean squares: MS , MS E Treatments a 1 N a a n a 1 1 E ( MS E ) E ( y ij2 y i2 ) 2 N a i 1 j 1 n i 1 a E ( MS Treatments ) 2 n( i ) /( a 1) i 1 3.3.2 Statistical Analysis • Assumption: εij are normally and independently distributed with mean zero and variance σ 2 • Cochran’s Thm (p. 69) 16 • SST/σ2 ~ Chi-square (N – 1), SSE/σ2 ~ Chi-square (N – a), SSTreatments/σ2 ~ Chi-square (a – 1), and SSE/σ2 and SSTreatments/σ2 are independent (Theorem 3.1) • H0: τ1 = … = τa =0 v.s. H1: τi ≠ 0, for at least one i 17 • Reject H0 if F0 > F α , a-1, N-a • Rewrite the sum of squares: y2 SS T y ij N i 1 j 1 a n 1 a 2 y2 SS Treatments y i n i 1 N SS E SS T SS Treatments • See page 71 • Randomization test 18 ANOVA Table of Example 3-1 19 20 3.3.3 Estimation of the Model Parameters • Model: yij = μ + τ i + ε ij • Estimators: ˆ y ˆi yi y ˆ i yi • Confidence intervals: y i ~ N ( i , 2 / n) y i t / 2 , N a MS E MS E i y i t / 2, N a n n y i y j t / 2 , N a MS E MS E i j y i y j t / 2, N a n n 21 • Example 3.3 (page 74) • Simultaneous Confidence Intervals (Bonferroni method): Construct a set of r simultaneous confidence intervals on treatment means which is at least 100(1-): 100(1-/r) C.I.’s 3.3.4 Unbalanced Data • Let ni observations be taken under treatment i, i=1,2,…,a, N = i ni, 2 y SS T yij2 N i 1 j 1 a ni yi2 y2 N i 1 ni a SS Treatments 22 1. The test statistic is relatively insensitive to small departures from the assumption of equal variance for the a treatments if the sample sizes are equal. 2. The power of the test is maximized if the samples are of equal size. 23 3.4 Model Adequacy Checking • Assumptions: yij ~ N(μ + τ i, σ2) • The examination of residuals • Definition of residual: eij yij yˆ ij , yˆ ij ˆ ˆi y ( yi y ) yi • The residuals should be structureless. 24 3.4.1 The Normality Assumption • Plot a histogram of the residuals • Plot a normal probability plot of the residuals • See Table 3-6 25 • May be – the left tail of error is thinner than the tail part of standard normal • Outliers • The possible causes of outliers: calculations, data coding, copy error,…. • Sometimes outliers are more informative than the rest of the data. 26 • Detect outliers: Examine the standardized eij residuals, d ij MS E 3.4.2 Plot of Residuals in Time Sequence • Plotting the residuals in time order of data collection is helpful in detecting correlation between the residuals. • Independence assumption 27 28 • Nonconstant variance: the variance of the observations increases as the magnitude of the observation increase, i.e. yij 2 • If the factor levels having the larger variance also have small sample sizes, the actual type I error rate is larger than anticipated. • Variance-stabilizing transformation Poisson Square root transformation, Lognormal Logarithmic transformation, log y ij Binomial Arcsin transformation, arcsin y ij y ij 29 • Statistical Tests for Equality Variance: H 0 : 12 a2 v.s. H1 : above not true for at least one i2 – Bartlett’s test: 02 2.3026 q c a q ( N a) log S (ni 1) log S i2 2 P i 1 1 a 1 1 c 1 (ni 1) ( N a) 3(a 1) i 1 a S (ni 1) S i2 /( N a) 2 p i 1 2 2 – Reject null hypothesis if 0 ,a1 30 • Example 3.4: the test statistic is 02 0.43 and 02.05,3 7.81 • Bartlett’s test is sensitive to the normality assumption • The modified Levene test: – Use the absolute deviation of the observation in each treatment from the treatment median. d ij y ij ~ y i , i 1,2,, a, j 1,2,, ni – Mean deviations are equal => the variance of the observations in all treatments will be the same. – The test statistic for Levene’s test is the ANOVA F statistic for testing equality of means. 31 • Example 3.5: – Four methods of estimating flood flow frequency procedure (see Table 3.7) – ANOVA table (Table 3.8) – The plot of residuals v.s. fitted values (Figure 3.7) – Modified Levene’s test: F0 = 4.55 with P-value = 0.0137. Reject the null hypothesis of equal variances. 32 • • • • Let E(y) = and y Find y* = y that yields a constant variance. y* +-1 Variance-Stabilizing Transformations y and = 1 - Transformation y constant 0 1 No transformation y 1/2 ½ ½ Square root y 1 0 Log y 3/2 3/2 -1/2 Reciprocal square root y 2 2 -1 Reciprocal http://www.stat.ufl.edu/~winner/sta6207/transform.pdf 33 • How to find : log yi log log i • Use S i i and yi i • See Figure 3.8, Table 3.10 and Figure 3.9 34 3.5 Practical Interpretation of Results • Conduct the experiment => perform the statistical analysis => investigate the underlying assumptions => draw practical conclusion 3.5.1 A Regression Model • Qualitative factor: compare the difference between the levels of the factors. • Quantitative factor: develop an interpolation equation for the response variable. 35 The Regression Model 36 3.5.2 Comparisons Among Treatment Means • If that hypothesis is rejected, we don’t know which specific means are different • Determining which specific means differ following an ANOVA is called the multiple comparisons problem 3.5.3 Graphical Comparisons of Means 37 3.5.4 Contrast • A contrast: a linear combination of the parameters of the form a a i 1 i 1 ci i , ci 0 • H0: = 0 v.s. H1: 0 • Two methods for this testing. 38 • The first method: a a i 1 i 1 Let C ci y i Then Var (C ) n 2 ci2 a c y Under H 0 , i 1 i i a ~ N (0,1) n 2 ci2 i 1 a Hence the statistic, t 0 c y i 1 i i a nMS E ci2 ~ t N a i 1 39 • The second method: a F0 t 02 ( ci y i ) 2 i 1 a nMS E ci2 ~F1,N a i 1 a ci y i MS C SS C / 1 F0 , SS C i 1 a MS E MS E n ci2 i 1 40 • The C.I. for a contrast, a ci i i 1 σ2 Let C ci y i . Then Var(C) n i 1 a a Hence C.I. ci y i t / 2, N a i 1 MS E n a 2 c i i 1 a 2 c i i 1 • Unequal Sample Size ci y i ci y i i 1 3. SSC i a1 a 2 2 n c MS E ni ci ii a a a 1. ni ci 0 2. t 0 i 1 i 1 i 1 41 2 3.5.5 Orthogonal Contrast • Two contrasts with coefficients, {ci} and {di}, are orthogonal if ci di = 0 • For a treatments, the set of a – 1 orthogonal contrasts partition the sum of squares due to treatments into a – 1 independent single-degreeof-freedom components. Thus, tests performed on orthogonal contrasts are independent. • See Example 3.6 (Page 90) 42 43 3.5.6 Scheffe’s Method for Comparing All Contrasts • Scheffe (1953) proposed a method for comparing any and all possible contrasts between treatment means. Suppose u c1u 1 c au a , u 1,2, , m a C u ciu y i and S Cu MS E (ciu2 / ni ) i 1 i 1 The critical value : S ,u S Cu (a 1) F ,a 1, N a If C u S ,u , then reject H 0 : u 0 • See Page 91 and 92 44 3.5.7 Comparing Pairs of Treatment Means • Compare all pairs of a treatment means • Tukey’s Test: – The studentized range statistic: q y max y min MS E / n , y max and y min are the largest and smallest sample means out of a group of p sample means MS E The critical point is T q (a, f ) n or T q (a, f ) MS E (1 / ni 1 / n j ) – See Example 3.7 45 • Sometimes overall F test from ANOVA is significant, but the pairwise comparison of mean fails to reveal any significant differences. • The F test is simultaneously considering all possible contrasts involving the treatment means, not just pairwise comparisons. The Fisher Least Significant Difference (LSD) Method • For H0: i = j t0 y i y j MS E (1 / ni 1 / n j ) 46 • The least significant difference (LSD): LSD t / 2, N a 1 1 MS E n n j i • See Example 3.8 Duncan’s Multiple Range Test • The a treatment averages are arranged in ascending order, and the standard error of each average is determined as S yi MS E , nh nh a a 1 / n i 1 i 47 • Assume equal sample size, the significant ranges are RP r p, f S yi , p 2,3,, a • Total a(a-1)/2 pairs • Example 3.9 The Newman-Keuls Test • Similar as Duncan’s multiple range test • The critical values: K P q ( p, f ) S yi 48 3.5.8 Comparing Treatment Means with a Control • Assume one of the treatments is a control, and the analyst is interested in comparing each of the other a – 1 treatment means with the control. • Test H0: i = a v.s. H1: i a, i = 1,2,…, a – 1 • Dunnett (1964) • Compute yi ya , i 1,2,, a 1 • Reject H0 if y i y a 1 1 d (a 1, f ) MS E ni na • Example 3.9 49 3.7 Determining Sample Size • Determine the number of replicates to run 3.7.1 Operating Characteristic Curves (OC Curves) • OC curves: a plot of type II error probability of a statistical test, 1 PReject H 0 | H 0 is false 1 P( F0 F ,a 1, N a | H 0 is false) 50 • If H0 is false, then F0 = MSTreatment / MSE ~ noncentral F with degree of freedom a – 1 and N – a and noncentrality parameter • Chart V of the Appendix a • Determine 2 2 n i i 1 a 2 • Let i be the specified treatments. Then estimates a of i : i i , i / a i 1 2 • For , from prior experience, a previous experiment or a preliminary test or a judgment 51 estimate. • Example 3.11 • Difficulty: How to select a set of treatment means on which the sample size decision should be based. • Another approach: Select a sample size such that if the difference between any two treatment means exceeds a specified value the null hypothesis should be rejected. 2 nD 2 a 2 52 3.7.2 Specifying a Standard Deviation Increase • Let P be a percentage for increase in standard deviation of an observation. Then a 2 i /a i 1 / n 1 0.01P 2 1 n • For example (Page 110): If P = 20, then 1.2 2 1 n 0.66 n 53 3.7.3 Confidence Interval Estimation Method • Use Confidence interval. y i y j t / 2 , N a MS E MS E i j y i y j t / 2 , N a n n • For example: we want 95% C.I. on the difference in mean tensile strength for any two cotton weight percentages to be 5 psi and = 3. See Page 110. 54 3.8 A Real Application 55 56 57 58 3.10 The Regression Approach to the Analysis of Variance • Model: yij = μ + τ i + ij 2 • a n a n L ij2 y ij i i 1 j 1 i 1 j 1 L L 0, i 1,2, , a i y a n i 1 j 1 ˆ ˆi 0 & y ij ˆ ˆi 0, i 1,2,, a n ij j 1 59 • The normal equations Nˆ nˆ nˆ nˆ1 nˆ1 nˆ2 nˆa nˆ2 y y1 y 2 nˆ a nˆa • Apply the constraint ˆi 0 Then estimations are i 1 ˆ y ,ˆi yi y y a • Regression sum of squares (the reduction due to fitting the full model) 2 a a y i R( , ) ˆy ˆi yi i 1 i 1 n 60 • The error sum of squares: a n SS E yij2 R , i 1 j 1 • Find the sum of squares resulting from the treatment effects: R( | ) R( , ) R( ) R(Full Model) - R(Reduced Model) 2 y y /n N i 1 2 i 61 • The testing statistic for H0: 1 = … = a R( | ) /( a 1) F0 ~ Fa 1, N a a n 2 yij R( , ) /( N a) i 1 j 1 62