Transcription

1

2 Group comparison test for independent samples The purpose of the Analysis of Variance (ANOVA) is to test for significant differences between means. Supposing that: samples come from normal populations with possibly different means but a common variance Two independent samples: z or t test on difference between means (or ANOVA) Three or more independent samples: ANalysis Of Variance (ANOVA)

3 ANOVA notation y ij = j-th observation of group i (independently on the role of rows and columns of data table) G = number of groups n = number of observation (equal) in each group In balanced design: each group contains the same number n of observations H 0 : µ = µ = = µ G H a : At least one inequality Under the null hypothesis all populations are supposed to have a common variance σ

4 Testing equality of each pair With 4 groups: 4 = 6 separate t tests would be required for testing the null hypothesis under consideration. Besides being tedious, 6 separate t tests on the same data would have an α level much higher than the α used in each t test: α = 0.05 in each test Total α = 0.05 x 6 = 0.3 (too high probability of I type error) One F test (α = 0.05): comparing the sample variance among groups with the sample variance within groups

5 Sources of variability Variance (or sum of squares) due to treatments: between groups Variance (or sum of squares) due to error: within groups The decomposition equation can be written as: SS T = SS TR + SS E where SS T is the total sum of squares n observations: SS T has n - d.f. k levels of factors: SS TR has k - d.f. k n j observations per group: SS E has ( n j ) = n k d.f j=

6 Conditional independence At least continuous variable X qualitative Categories of X (groups) mmmy continuous Conditional means of Y Conditional independence of Y on X: Conditional means of Y are invariant with respect to modalities of X X AREA Y INCOME ( ) Total NORTH 6 8 CENTRE 4 6 SOUTH Total 0 0 0

7 The partitioning of variance (i.e. of sums of squares) The total variance of Y is the sum of two components: Within variance = mean of groups variances Between variance = variance of groups means (with respect to the mean of Y) If: G = number of groups; µ i = mean of i-th group; n i = size of i-th group (i =,.,G); then: i.e.: σ = σ + µ µ G G y i n i ( i X ) n i n i = = n i W ITH IN B E T W E E N V A R IA N C E V A R IA N C E ( or ) σ = σ + σ σ + σ y WIT BET INT EX T

8 Why partitioning variance? Sales Constant mean and variance means variance σ BET = 0 variance mean σ WIT = σ The two groups have the same behaviour : A B brand Sales are the same for the two brands Sales Different means, constant variance means variance σ BET 0 variance mean σ WIT σ The two groups have different behaviours : Sales differ by brand A B brand

9 Example Sales (Y) Sector (X) Total Food 5 3 Drink Healt Care 6 Ice Packaging Total Food Food Food Drinks Sector Ice Packaging Health Care Ice Packaging. Sales X = 4 groups. Mean of Y (unconditional): n µ = y = ˆy n = Y i j j n i= n j= h ( ) = 50 = = 394,96

10 . Conditional means of Y X i j= ( ) h µ Y X = x = ŷ = = jnj n 348, 48 j= ( ) h µ Y X = x = ŷ = = jnj n 3 66,67 3 j= ( ) h µ Y X = x = ŷ = = 3 jn3j n 384,33 4 j= ( ) h µ Y X = x = ŷ = = 4 jn4j n 4 4

11 Remark: conditional means differ each other and with respect to the unconditional mean of Y, that there is some degree of conditional dependence between X and Y. Question: Is this dependence significant?

12 The F test H 0 : µ = µ = = µ G No effect of X on Y H a : At least one inequality Effect of X an Y Y = effect of X + error σ y = σ BET + σwit If means are equal, between groups variance is 0: σ = σ BET 0 = σ WIT The more means differ, the more: σ σ σ BET WIT 0

13 H 0 : µ = µ = = µ G No effect of X on Y H a : At least one inequality Effect of X an Y The decision is based on the sample ratio σ σ BET WIT. The lower the ratio, the more realistic the null hypothesis The higher the ratio, the less realistic the null hypothesis. Significance level of the decision: σ σ BET WIT G ~ F n G G ;n G

14 ANOVA output σ σ MS SS (k ) = = MS SS (n k) EXT EXT EXT INT INT INT Sum of squares DoF Mean of squares F (observed) Significance Variability Between groups (external) B Within groups (internal) W SS EXT SS INT k- n-k MS EXT = SS EXT /(k-) MS INT = SS INT /(n-k) F = MS EXT /MS INT P-value Total SS TOT n- MS TOT = SS TOT /(n-) = σ

15 Sales data results Variables: Sales (Y) by sector (X) Sector Ice Packaging Sales 0 Null hypothesis: mean sales are equal among sectors Food Food Food Health Care One-way ANOVA Ice Packaging Drinks Variabiliy SS Df MS F p-value Food Food 77 6 Between Within Total Health Care Ice Packaging Ice Packaging Food Health Care 493 Ice Packaging 85 F Decision: Ice Packaging ,36 0,807 Low value = low σ EXT σ σ EXT INT = means are close p-value is very high: We can accept the hypothesis of mean sales equal among sectors, as it s confirmed by observed sample.

16

17 k i= h j= 3. Index numerator ( Y X = x ) ( ) ( ) i Y i µ µ n = 348, , ,67 394, Index denominator ( ) ( ) + 384,33 394, ,96 4 = ,4 ( j Y ) j ( ) ( ) ŷ µ n = , , ( ) ( ) ( ) , , ,96 8 = Index k ( µ µ ) i n σ ,4 η = = = = Y X = x Y i EXTY i= Y X σ h Y ( yj µ Y ) n j j= 0, On average, sales don t depend on sector

18 Analysis of variance (inferential approach) A psychologist studying factors that influence the amount of time mice require to solve a new maze might be observing 4 groups of 3 mice each Hypothesis: learning has an effect on time required Previous experience at maze solving: Group : maze solved Group : mazes solved Group 3: 3 mazes solved Group 4: 4 mazes solved Learning would be indicated by a decrease in the time required to solve the maze

19 The data Time Time Time Time Group Group Group 3 Group 4

20 How to consider the differences The apparent differences in the graph could be due to sampling variability rather than learning Are differences in the sample averages significant? Hypothesis: no learning effect means µ = µ = µ 3 = µ 4 The 4 groups come from the same population The ANalysis Of VAriance (ANOVA) is a method for testing this hypothesis

21 Glossary Analysis of variance (ANOVA): statistical technique for deciding if G independent samples come from the same normal population. Experimental (or classification) factor: variable responsible for heterogeneity of means. Treatment: modality (categorical data) or level (ordinal data) of a factor. Random block: set of observations as homogeneous as possible. Each block includes as many observations as treatments; each observation is randomly assigned to one treatment. Sample observation: statistical unit that receives a treatment or a combination of treatments. Experimental design: set of rules for assigning sample observations to treatments, once factors are fixed

22 Hypotheses of ANOVA. Additivity: treatment effect is added to error effect, without interaction between error and treatment. Treatment effect is also independent from the intrinsic effect due to statistical units. Normality: Error is a Normal random variable, with null mean and constant variance among treatments. The G populations are normally distributed. 3. Homoschedasticity: error variance is constant among treatments and observations 4. Independence of observation and of samples: values in different samples are not in relation.

23 Results from mice experiment ANOVA SS d.f. MS F p-value Among , Within 8 8 Total 68 Decision This F statistic and p-value lead to rejection of H 0 The sample came from 4 populations among which there is at least one inequality Prior experience does affect the time required for the mice to solve a new maze

24 Remarks When G = ANOVA test is equal to the t-test for independent samples, since: F,m = t m In experimental science it is possible to select balanced (=same size) samples by a random experimental design; this is not always possible in social and economical science. Advantages of a balanced design Statistical test is less sensible towards small deviation from homoschedasticity. This is not true when samples have different sizes. Test power is maximized by equal size groups There are not serious consequences on results if group variances differ from population variance

25 ANOVA hypothesis violation Normality can be verified by residuals: y µ ˆ ij a i with an un histogram or a normal probability plot. If effects are not additive (e.g.: effects are multiplicative, or an interaction effect exists but are not included in the model), logarithmic transformation can be used. Observations independence assumption can be assured by randomly assigning statistical units to treatments, e.g. using random number.

26 For testing homoschedasticity we can use Hartley test, for equal size groups, or Bartlett test. For both, the null hypothesis is: H : σ = σ =... = σ = σ 0 k H: at least one variance different. If H 0 is rejected, we shouldn t proceed with ANOVA In some cases data can be transformed in order to fix variance. When causes are not identified experiment should be repeated

27 If the null hypothesis is rejected Conclusion: there is at least one inequality among the means of the treatment groups (or among the treatment effects) Further research Which pairs of treatments are different? (test the hypotheses H 0 : µ i = µ j for all i,j) Test some more complex hypotheses: how to compare one treatment effect with the average of some other treatments effects? (contrast analysis) Estimate of some parameters in the experiment? (Confidence intervals) Some examples Multiple-Comparison Procedures. Fisher s least significant difference. Duncan s new multiple-range test 3. Student Newman Keuls procedure 4. Tukey s honestly significant difference 5. Scheffè s method Mainly they differ for: test power type I error rate Equal sample sizes for the treatment groups

28 Example Precision resulting from operating hand-held chainsaws Experiment: measure of the kickback that occurs when a saw is used to cut a fiber board Response variable: angle (in degrees) to which the saw is deflected when it begins to cut the board 4 types of saws A B Σ j y ij Σ j (y ij ) (Σ j y ij ) C D Total H 0 : µ A = µ B = µ C = µ D = 0 H : at least one µ 0 Equal groups size n = 5

29 Results SS DoF MS F F 0.05;3;6 Among ,56 3,39 Within ,5 Total Decision: Conclusion: The null hypothesis is rejected. There is a significant difference among the average kickbacks of the four types of saws.

30 Pairwise Comparisons Procedures (post-hoc analysis). Fisher s least significant difference It is based on the t test. Cut off: y y > t ( ) k h α ;G n MS n WIT If the treatment groups are all of equal size n, then only samples where difference in means is greater than cut-off value can be tested for a significant difference by the test statistic: t y y y y = = s p s s p p + n n n Testing difference between means: H 0 : µ x = µ Y H : µ X µ Y ( X Y) ( µ x µ y ) s + n n x y ~ t ( nx + ny ) s p = pooled sample variance Equal groups size n s pooled = ( ) ( ) s n + s n X X Y Y n + n x y

31 Pairwise Comparison Procedures Chain saw example ya = = 33 yb = = yc = = 43 yd = = = 6 pairwise comparisons H : µ = µ H : µ = µ 0 A B 0 B C H : µ = µ H : µ = µ 0 A C 0 B D H : µ = µ H : µ = µ 0 A D 0 C D Least significant difference: WIT t = t ;G n 0,05;6 = 3,5 α ( ) MS 0,5 n 5 Means in increasing order: yd = 3 ya = 33 yc = 43 yb = 49 A C B Largest G D 3 8 Smallest G A C 43 6 Decision: the only pairs of means that are different are (µ A, µ C ) and (µ C, µ D )

32 Pairwise Comparison Procedures. Duncan s new multiple-range test Critical values depend on the span r of the two ranked averages being compared Cut-off: y y d ( ) i j α;r;g n MS n WIT d values come from a sampling distribution of the shortest differences between a set of means originating from the same population d is tabulated In the example: A C B D 3 8 A C 43 6 difference between 4 ranks difference between 3 ranks difference between (adjacent) ranks Means in increasing order: yd = 3 ya = 33 yc = 43 yb = 49 MS WIT d0,05;4;6 = 4,6 n MS WIT d0,05;3;6 = 4, MS WIT d0,05;;6 = 3,5 n n Slightly more conservative than Fisher s: it will sometimes find fewer significant differences About 95% agreement between them

33 3. Student Newman Keuls multiple-range test Pairwise Comparison Procedures Critical values depend on the span r of the two ranked averages being compared Cut-off: y y q ( ) i j α;r;g n MS n WIT q values come from a sampling distribution derived by Gosset and called the Studentized Range or Student s q, is similar to a t-distribution and corresponds to the sampling distribution of the largest differences between a set of means originating from the same population In the example: Means in increasing order: yd = 3 ya = 33 yc = 43 yb = 49 D A C A 33 C 43 0 B difference between 4 ranks difference between 3 ranks difference between (adjacent) ranks MS MS WIT q0,05;4;6 = 8, MS n WIT q0,05;3;6 = 6,4 WIT q0,05;;6 = 3,5 No significant difference, whereas the F test in the ANOVA indicated that a difference exists Still more conservative than Duncan s test n n

34 4. Tukey s Honestly Significant Difference (HSD test) Uses a single critical difference: Pairwise Comparison Procedures y y q ( ) i j α;g;g n MS n WIT that is only the largest critical difference in Student Newman Keuls s procedure In the example: A C D 3 A 33 0 C 43 B MS WIT q0,05;4;6 8, n = No significant difference, whereas the F test in the ANOVA indicated that a difference exists Still more conservative than Student-Newman-Keul s test

35 Pairwise Comparison Procedures 5. Scheffé s Method Can be used to compare means and also to make other types of contrasts, like: µ + µ H µ = 3 0 : that is, that treatment is the same as the average of treatments and 3. Cut-off: In the example: ( ) α ( ) y y G F i j ;G ;G n MS n WIT D A c A 33 C 43 0 B ( G ) Fα;G ;G ( n ) MS WIT = 0,5 = 3 F0,05;3;6 = 9, 8 5 n No significant difference, whereas the F test in the ANOVA indicated that a difference exists It the most conservative test

36 Pairwise Comparison Procedures Scheffé s approach is used more often for the other contrasts µ + µ H µ = B C 0 : A equivalent to: µ B µ C H 0 : µ A = 0 Cut-off: ( G ) Fα;G ;G ( n ) MS C n E The coefficient C is the sum of the squares of the coefficients in the linear combinations of the µ s: 3 C = + + = 3 MSE 3 0,5 G F = 3 F0,05;3;6 = 7,8 n 5 ( ) α;g ;G ( n ) yb yc to be compared with the sample statistic: ya = 33 = 3 Decision: the difference is not significant

37 Pairwise Comparison Procedures Which procedure should be used? It depends upon which type of error is more serious: Less conservative test: less probability II type error (more power) More conservative test: less probability I type error In the chain saw example, assume the prices are approximately the same. Then a Type I error is not serious; it would imply that we decide one model has less kickback than another when in fact the two models have the same amount of kickback. A Type II error would imply that a difference in kickback actually exists but we fail to detect it, a more serious error. Thus, in this experiment we want maximum power and we would probably use Fisher s least significant difference. The experimenter should decide before the experimentation which method will be used to compare the means.

38 Overall α level for all hypotheses m independent t tests each with α = 0.05 Probability that at least one will show significance by chance is: P(diff) = - ( -α) G = G G = P(diff) = α = 0.05 G = 6 P(diff) = = = 0.65

39 Bonferroni procedure Based on t tests We change the value of t a,g that will be used for statistical inference In the example: 4 = 6 possible comparisons H : µ = µ H : µ = µ 0 A B 0 B C H : µ = µ H : µ = µ 0 A C 0 B D H : µ = µ H : µ = µ 0 A D 0 C D α + α + + α6 α The critical t value for each two-sided test will be one with: and an α i = α/g = 0.05/x6 = 0,004 G(n - ) = 6 degrees of freedom Tables of the t distribution for such a value of α do not exist

40 The critical value used is the only thing that is different t 0.004;6 P values: we can use the P value for each of the 6 t tests and see if it is equal to or less than A C B D 3 t = P = t =.8856 P = t =.884 P = 0.0 A 33 t =.573 P = t =.54 P = C 43 t = P = None of the P values is equal to or smaller than None of the differences between model averages can be considered statistically significant

41 One-degree-of-freedom comparisons The multiple-comparison procedures are known as a posteriori tests, that is, they are run after the fact. Such tests will not be as powerful as those for planned orthogonal contrasts, and it seems reasonable that experiments which are well designed and which test specific hypotheses will have the greatest statistical power A priori approach Contrasts are planned before the experiment The experimenter believes prior to the investigation that certain factors may be related to differences in treatment groups. A significant F test is not a prerequisite for these one-degree-of-freedom tests

42 Contrasts analysis To determine which of the models are different with respect to kickback, a follow-up procedure will be needed. The experimenter believes prior to the investigation that certain factors may be related to differences in treatment groups. For example, he might want to know if the kickback from the home type (A and D) is the same as the kickback from the industrial type (B and C). In addition, he might also be interested in any differences in kickback within types Comparison H 0 3 Home vs. industrial Home model A vs. home model D Industrial model B vs. industrial model C µ + µ µ + µ B C = A D 0 µ µ = A D 0 µ µ = B C 0

43 Comparison H 0 3 Home vs. industrial Home model A vs. home model D Industrial model B vs. industrial model C µ + µ µ + µ B C = A D 0 µ µ = A D 0 µ µ = B C 0 Each of the null hypotheses can be expressed as a linear combination of the treatment means: 3 µ µ µ + µ µ + 0 µ + 0 µ µ A B C D A B C D 0 µ + µ µ + 0 µ A B C D

44 Orthogonal contrasts A set of linear combinations is called a set of orthogonal contrasts or orthogonal comparisons if it satisfies the following conditions A and B: A. The sum of the coefficients in each linear combination must be zero: : : 3: + = = = 0 Such a linear combination is called a contrast B. The sum of the products of the corresponding coefficients in any two contrasts must equal zero; this makes the contrasts orthogonal and : and 3: 3 and : = = 0 ( ) = 0 A set of contrasts is mutually orthogonal if every pair of contrasts is orthogonal

45 In general: Given any two linear combinations: L = a µ + a µ + + a µ G G M = b µ + b µ + + b µ G G they are orthogonal contrasts if: G i= a = 0 i G i= b = 0 i G i= ab = 0 i i And a set of contrasts is mutually orthogonal if every pair of contrasts is orthogonal An experiment involving G treatments can have several different sets of mutually orthogonal contrasts, but each set consists of at most G - orthogonal contrasts

46 Example Five toothpastes are being tested for their abrasiveness. The variable of interest is the time in minutes until mechanical brushing of a material similar to tooth enamel exhibits wear Absence or presence of certain additives Toothpaste Additive I II III IV V Whitener None Fluoride Fluoride with freshener Whitener with freshener Group totals and the basic ANOVA table are as follows for 4 observations per treatment group: Toothpaste: I II III IV V Ti = Σ j y ij 97,4 99,0,3 5,8 86,5 Source df SS MS F Among toothpastes 4 36,8 34, 39,8 Within toothpastes 5 3,0 0,86

47 Comparison Additive vs. no additive Whitener vs. fluoride Whitener vs. whitener with freshener Fluoride vs. fluoride with freshener H0 to Be Tested µ + µ 3 + µ 4 + µ 5 µ = 4 µ + µ 5 µ 3 + µ 4 = 0 µ µ = 5 0 µ µ = To test these comparisons within the ANOVA procedure, the among SS is partitioned into G - components which are each sums of squares for a onedegree-of-freedom F test This has an advantage over the multiple-comparison procedures of the previous section in that the partition divided into nonoverlapping parts can be used to determine the percentage of variability that is due to the different factors

48 The sum of squares for additive vs. no additive is found as follows: Null hypothesis H 0 : µ + µ 3 + µ 4 + µ 5-4µ = 0 Contrast L = µ + µ 3 + µ 4 + µ 5-4µ Coefficients a = a 3 = a 4 = a 5 = a = , 4,3 5, 8 86, SS = =,8 at i i Sum of squares ( ) i L n a i ( 4) i

49 Two-way ANOVA Effects of two factors Data are organized as follows: Each y ij is a Normal r.v. Y ij ~ N(µ ij ; σ ) Factor A A A... Aj... Ak B y y... y j... y k y. y. B y y... y j... y k y. y. Factor B... Bi... y i... y i y ij y ik... y i. y i Br y r y r... y rj... y rk y r. y r. y. y.... y.i... y.k y.. A,, Ak = levels of factor A B,, Br = levels of factor B y. y y... j..... y. k

50 The population mean y + y y i rk i= µ = = = r µ. k j= rk r k µ. j j =,, k i =,, r α = µ µ j. j β = µ µ i i. effect of level j of factor A effect of level i of factor B The additive model on heterogeneity of population means µ.j and µ i. µ ij = µ + α j + βi Yij = µ ij + ε ij = µ + α j + β i + ε ij Effects of factor A and block B i of factor B are supposed to be additive, i.e. there is any conjoint effect between α j e β i ε ij ~ N(0; σ ) and k α j = βi = j= i= r 0

51 SS partition ( Y Y ) ( Y Y ) ( Y Y ) ( Y Y Y Y ) ij.. =. j.. + i... + ij. j i. +.. k r k r k r k r ( Yij Y.. ) = ( Y.j Y.. ) + ( Y.i Y.. ) + ( Yij Y.j Yi. + Y.. ) j= i= j= i= j= i= j= i= SS * = SS * + SS * + SS * T K R E The amounts: ( Y Y ). j ( ).. is an estimator for ( Y Y ) i... ( Y Y Y Y ) ij. j i. +.. is an estimator for µ µ = α.j j ( ) µ i. µ = β measures the random effect i

52 The output of two-way ANOVA Source of variation Sum of squares DoF Means of squares Among columns Among rows Error (= within) Total SS * K k M S * = SS * / k SS * r M S * = SS * / r R SS * E ( k )( r ) SS * T rk K R M S * = SS * /( k )( r ) E K R E

53 The test ) Test on treatments of factor A H 0: µ i. = µ J. i,j =,...,k H : at least one α j 0 ) Test on treatments of factor B H 0: µ. i = µ.j i,j =,...,r H β i 0 : at least one Under H 0: µ i. = µ J. F = SS* K (k ) SS* E (k )(r ) = MS* MS* K E ~ χ (k ) (k ) χ (k )(r ) (k )(r ) ~ F (k );(k )(r ) Under H 0: µ. i = µ.j F = SS* R (r ) SS* E (k )(r ) = MS* MS* R E ~ χ (r ) (r ) χ (k )(r ) (k )(r ) ~ F (r );(k )(r )

54 Example IMS industrial vehicles manager wants to know which combination of diesel and carburetors performs better. He plans an experiment with 5 carburetors and 4 types of diesel. The same amount of each diesel is tested in each of the 5 carburetors. The performance are in the following table: Carburetors y. j y i ,4 Diesel , , ,4 y i y. j 6,5 0,75 5,75,5 7

55 Results C y.. rk = = 64 ( )(5) = 344, 8 4 SS = y C = ( ) 344, 8 = 9, T 4 5 ij i= j= SS K k y j =. (... ) C = r 4 j= 344, 8 = 08, SS R r yi ( ) =. C = 36 3 k 5 i= 344, 8 = 73, SS = SS ( SS + SS ) = 9, ( 08, + 73, ) = 9, 8 E T K R Source of variation Sum of squares DoF Means of squares F Among carburetors 08, 4 7,05 33, Among diesel types 73, 3 4,40 9,86 Error 9,8 0,87 Totale 9, 9

56 Decisions ) Test on treatments of factor A H 0:α j = 0 j =,..., k H : at last one α j 0 Being 33, > F 0.0; 4; = 5.4, we reject H0: αj = 0 at % level The 5 carburetors have different performances among diesel types ) Test on treatments of factor B H 0 :β i = 0 i =,...,r H β i 0 : at least one Being 9,86 > F 0.0; 3; = 5.95, we reject H 0 : β i = 0 at % level The 4 diesel types have different performances among carburetors We can choose the best combination on the data table

57 About the k levels of a factor Fixed-effects model (FEM, or Model I) The experimenter usually in the latter stages of experimentation narrows down the possible treatments to those in which he has a special interest. All levels of a treatment are included in the experiment. The inference is restricted to the treatments used in the experiment (Conclusion cannot be generalized to not observed treatments). Random-effects model (REM, or Model II) Treatments are a random sample of all possible treatments of interest. The model does not look for differences among the group means of the treatments being tested, but rather asks whether there is significant variability among all possible treatment groups (The investigator would be interested in the variability among all treatments). Results can be generalized to all treatments in the population even if they are not observed. If the experiment were to be repeated, treatments chosen at random would be used. In both models we assume that the experimental units are chosen at random from the population and assigned at random to the treatments.

58 In econometrics and statistics, a fixed effects model is a statistical model that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed models in which either all or some of the explanatory variables are treated as if they arise from random causes. Often the same structure of model, which is usually a linear regression model, can be treated as any of the three types depending on the analyst's viewpoint, although there may be a natural choice in any given situation.

59 Fixed-effects model H 0 : α = α = = α k = 0 H : at least one α 0 H 0 : σ A = 0 H :σ A > 0 Random-effects model µ: the mean of all possible experiments using the G designated treatments α i : for the i-th treatment group, the deviation from the mean due to the i-th treatment: Σ i a i = 0 µ: the population mean for all experiments involving all possible treatments of the type being considered α i : for the i-th treatment group, a random deviation from the population mean. The ai s are normal, with E(a i ) = 0 and V(a i ) = σ A ε ij : A random effect containing all uncontrolled sources of variability. The ε ij s are N (0, σ ), and they are independent of each other and of the a i s. ε ij : Same as for FEM