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III. Completely Randomized Design (CRD)

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  1. III. Completely Randomized Design (CRD) III.A Design of a CRD III.B Models and estimation for a CRD III.C Hypothesis testing using the ANOVA method III.D Diagnostic checking III.E Treatment differences Statistical Modelling Chapter III

  2. III.A Design of a CRD Definition III.1: An experiment is set up using a CRD when each treatment is applied a specified, possibly unequal, number of times, the particular units to receive a treatment being selected completely at random. Example III.1 Rat experiment • Experiment to investigate 3 rat diets with 6 rats: Diet A, B, C will have 3, 2, 1 rats, respectively. Statistical Modelling Chapter III

  3. Use R to obtain randomized layouts • How to do this is described in Appendix B, Randomized layouts and sample size computations in , for all the designs that will be covered in this course, and more besides. Statistical Modelling Chapter III

  4. R functions and output to produce randomized layout > # Obtaining randomized layout for a CRD > # > n <- 6 > CRDRat.unit <- list(Rat = n) > Diet <- factor(rep(c("A","B","C"), > times = c(3,2,1))) > CRDRat.lay <- fac.layout(unrandomized=CRDRat.unit, > randomized=Diet, seed=695) > CRDRat.lay • fac.layout from dae package produces the randomized layout. • unrandomized gives the single unrandomized factor indexing the units in the experiment. • randomized specifies the factor, Diets, that is to be randomized. • seed is used so that the same randomized layout for a particular experiment can be generated at a later date. (0–1023) Statistical Modelling Chapter III

  5. Randomized layout Units Permutation Rat Diet 1 1 4 1 A 2 2 1 2 C 3 3 5 3 B 4 4 3 4 A 5 5 6 5 A 6 6 2 6 B #remove Diet object in workspace to avoid using it by mistake remove(Diet) Statistical Modelling Chapter III

  6. III.B Models and estimation for a CRD • The analysis of CRD experiments uses: • least-squares or maximum likelihood estimation of the parameters of a linear model • hypothesis testing based on the ANOVA method or maximum likelihood ratio testing. • Use rat experiment to investigate linear models and the estimation of its parameters. Statistical Modelling Chapter III

  7. a) Maximal model • Definition III.2: The maximal expectation model is the most complicated model for the expectation that is to be considered in analysing an experiment. • We first consider the maximal model for the CRD. Statistical Modelling Chapter III

  8. Our model also involves assuming Example III.1 Rat experiment (continued) • Suppose, the experimenter measured the liver weight as a percentage of total body weight at the end of the experiment. • The results of the experiment are as follows: • The analysis based on a linear model, that is: • Trick is what are X and q going to be? Statistical Modelling Chapter III

  9. The fitted equation is: Perhaps? • Note numbering of Y's does not correspond to Rats; does not affect model but neater. • This model can then be fitted using simple linear regression techniques. Statistical Modelling Chapter III

  10. Using model to predict • However, does this make sense? • means that, for each unit increase in diet, % liver weight decreases by 0.120. • sensible only if the diets differences are based on equally spaced levels of some component; • For example, if the diets represent 2, 4 and 6 mg of copper added to each 100g of food • But, no good if diets unequally spaced (2, 4, and 10 mg Cu added) or diets differ qualitatively. Statistical Modelling Chapter III

  11. Regression on indicator variables • In this method the explanatory variables are called factors and the possible values they take levels. • Thus, we have a factor Diet with 3 levels: A, B, C. • Definition III.3: Indicator variables are formed from a factor: • create a variable for each level of the factor; • the values of a variable are either 1 or 0, 1 when the unit has the particular level for that variable and 0 otherwise. Statistical Modelling Chapter III

  12. Indicator-variable model • Hence E[Yi] =ak, var[Yi] =s2, cov[Yi, Yj] = 0, (i j) • Can be written as • Model suggests 3 different expected (or mean) values for the diets. Statistical Modelling Chapter III

  13. General form of X for CRD • For the general case of a set of t Treatments suppose Y is ordered so that all observations for: • 1st treatment occur in the first r1 rows, • 2nd treatment occur in the next r2 rows, • and so on with the last treatment occurring in the last rt rows. • i.e. order of systematic layout (prior to randomization) • Then XT given by the following partitioned matrix (1 only ever vector, but 0 can be matrix) Statistical Modelling Chapter III

  14. Still a linear model • In general, the model for the expected values is still of the general form E[Y] = Xq • and on assuming Y is • can use standard least squares or maximum likelihood estimation Statistical Modelling Chapter III

  15. Estimates of expectation parameters • Can be shown, by examining the OLS equation, that the estimates of the elements of a and y are the means of the treatments. • Example III.1 Rat experiment (continued) • OLS equation is • The estimates of a are: Statistical Modelling Chapter III

  16. Example III.1 Rat experiment (continued) • The estimates of the expected values, the fitted values, are given by: Statistical Modelling Chapter III

  17. Estimator of the expected values • In general, where is the n-vector consisting of the treatment means for each unit and • being least squares, this estimator can be written as a linear combination of Y. • that is, can be obtained as the product of an matrix and the n-vector Y. • let us write • M for mean because MT is the matrix that replaces each value of Y with the mean of the corresponding treatment. Statistical Modelling Chapter III

  18. Clear from the above expression that: • 1str1 elements of are the mean of the Yis for the 1st treatment, • next r2 elements are the mean of those for the 2nd treatment, • and so on. General form of mean operator • Can be shown that the general form of MT is • MT is a mean operator as it • computes the treatment means from the vector to which it is applied and • replaces each element of this vector with its treatment mean. Statistical Modelling Chapter III

  19. Estimator of the errors • The estimator of the random errors in the observed values of Y is, as before, the difference from the expected values. • That is, Statistical Modelling Chapter III

  20. Example III.1 Rat experimentAlternative expression for fitted values • We know that where Note not as estimates rather than estimators. Statistical Modelling Chapter III

  21. Residuals • Fitted values for orthogonal experiments are functions of means. • Residuals are differences between observations and fitted values Statistical Modelling Chapter III

  22. b) Alternative indicator-variable, expectation models • For the CRD, two expectation models are considered: • First model is minimal expectation model: population mean response is same for all observations, irrespective of diet. • Second model is the maximal expectation model. Statistical Modelling Chapter III

  23. Minimal expectation model • Definition III.4: The minimal expectation model is the simplest model for the expectation that is to be considered in analysing an experiment. • The minimal expectation model is the same as the intercept-only model given for the single sample in chapter I, Statistical inference. • Will be this for all analyses we consider. • Now the estimator of the expected values in the intercept-only model is where is the n-vector each of whose elements is the grand mean. • For Rat experiment Statistical Modelling Chapter III

  24. Marginal models • In regression case obtained marginal models by zeroing some of the parameters in the full model. • Here this is not the case. • Instead impose equality constraints. • Here simply set ak = m • That is, intercept only model is the special case where all ak s are equal. • Clear for getting E[Yi] = m from E[Yi] = ak. • What about yG from yT? • If replace each element of a with m, then a = 1tm. • So yT = XTa = XT1tm = XGm. • Now marginality expressed in the relationship between XT and XG as encapsulated in definition. Statistical Modelling Chapter III

  25. Marginality of models (in general) • Definition III.5: Let C(X) denote the column space of X. • For two models, y1 X1q1 and y2 X2q2, the first model is marginal to the second if C(X1)  C(X2) irrespective of the replication of the levels in the columns of the Xs, • That is if the columns of X1 can always be written as linear combinations of the columns of X2. • We write y1 y2. • Note marginality relationship is not symmetric — it is directional, like the less-than relation. • So while y1 y2, y2 is not marginal to y1 unless y1 y2. Statistical Modelling Chapter III

  26. Marginality of models for CRD • yG is marginal toyT or yG yT because C(XG)  C(XT) • in that an element from a row of XG is the sum of the elements in the corresponding row of XT • and this will occur irrespective of the replication of the levels in the columns of XG and XT. • So while yG yT, yT is not marginal to yG as C(XT) C(XG) so that yTyG. • In geometrical terms, C(XT) is a three-dimensional space and C(XG) is a line, the equiangular line, that is a subspace of C(XT). Statistical Modelling Chapter III

  27. III.C Hypothesis testing using the ANOVA method • Are there significant differences between the treatment means? • This is equivalent to deciding which of our two expectation models best describes the data. • We now perform the hypothesis test to do this for the example. Statistical Modelling Chapter III

  28. Analysis of the rat exampleExample III.1 Rat experiment (continued) Step 1: Set up hypotheses H0: aA = aB = aC = m (yG XGm) H1: not all population Diet means are equal (yD XDa) Set a = 0.05 Statistical Modelling Chapter III

  29. Example III.1 Rat experiment (continued) Step 2: Calculate test statistic • From table can see that (corrected) total variation amongst the 6 Rats is partitioned into 2 parts: • variance of difference between diet means and • the left-over (residual) rat variation. • Step 3: Decide between hypotheses • As probability of exceeding F of 3.60 with n1= 2 and n2= 3 is 0.1595 > a = 0.05, not much evidence of a diet difference. • Expectation model that appears to provide an adequate description of the data is yG XGm. Statistical Modelling Chapter III

  30. b) Sums of squares for the analysis of variance • From chapter I, Statistical inference, an SSq • is the SSq of the elements of a vector and • can write as the product of transpose of a column vector with original column vector. • Estimators of SSqs for the CRD ANOVA are SSqs of following vectors (cf ch.I): where Ds are n-vectors of deviations from Y and Te is the n-vector of Treatment effects. Definition III.6: An effect is a linear combination of means with a set of effects summing to zero. Statistical Modelling Chapter III

  31. SSqs as quadratic forms • Want to show estimators of all SSqs can be written as YQY. • Is product of 1n, nn and n1 vectors and matrix, so is 11 or a scalar. • Definition III.7: A quadratic form in a vector Y is a scalar function of Y of the form YAY where A is called the matrix of the quadratic form. Statistical Modelling Chapter III

  32. SSqs as quadratic forms (continued) • Firstly write • That is, each of the individual vectors on which the sums of squares are based can be written as an M matrix times Y. • These M matrices are mean operators that are symmetric and idempotent: M'=M and M2=M in all cases. Statistical Modelling Chapter III

  33. SSqs as quadratic forms (continued) • Then • Given Ms are symmetric and idempotent, it is relatively straightforward to show so are the three Q matrices. • It can also be shown that Statistical Modelling Chapter III

  34. SSqs as quadratic forms (continued) • Consequently obtain the following expressions for the SSqs: Statistical Modelling Chapter III

  35. SSqs as quadratic forms (continued) • Theorem III.1: For a completely randomized design, the sums of squares in the analysis of variance for Units, Treatments and Residual are given by the quadratic form: Proof: follows the argument given above. Statistical Modelling Chapter III

  36. In the notes show that • so that Residual SSq by difference • That is, Residual SSq = Units SSq - Treatments SSq. Statistical Modelling Chapter III

  37. ANOVA table construction • As in regression, Qs are orthogonal projection matrices. • QU orthogonally projects the data vector into the n-1 dimensional part of the n-dimensional data space that is orthogonal to equiangular line. • QT orthogonally projects data vector into the t-1 dimensional part of the t-dimensional Treatment space, that is orthogonal to equiangular line. (Here the Treatment space is the column space of XT.) • Finally, the matrix orthogonally projects the data vector into the n-t dimensional Residual subspace. • That is, Units space is divided into the two orthogonal subspaces, the Treatments and Residual subspaces. Statistical Modelling Chapter III

  38. Geometric interpretation • Of course, the SSqs are just the squared lengths of these vectors • Hence, according to Pythagoras’ theorem, the Treatments and Residual SSqs must sum to the Units SSq. Statistical Modelling Chapter III

  39. Example III.1 Rat experiment (continued) • Vectors for computing the SSqs are: • Total Rat deviations, Diet Effects and Residual Rats deviations are projections into Rats, Diets and Residual subspaces of dimension 5, 2 and 3, respectively. • Squared length of projection = SSq • Rats SSq is Y'QRY = 0.34 • Diets SSq is Y'QDY = 0.24 • Residual SSq is Exercise III.3 is similar example for you to try Statistical Modelling Chapter III

  40. c) Expected mean squares • Have an ANOVA in which we use F (= ratio of MSqs) to decide between models. • But why is this ratio appropriate? • One way of answering this question is to look at what the MSqs measure? • Use expected values of the MSqs, i.e. E[MSq]s, to do this. Statistical Modelling Chapter III

  41. Expected mean squares (cont’d) • Remember “expected value = population mean” • Need E[MSq]s • under the maximal model: • and the minimal model: • Similar to asking what is E[Yi]? • Know answer is E[Yi] = ak. • i.e. in population, under model, average value of Yi is ak. • So for Treatments, what is E[MSq]? • The E[MSq]s are the mean values of the MSqs in populations described by the model for which they are derived • i.e. an E[MSq] is the true mean value; • it depends on the model parameters. Statistical Modelling Chapter III

  42. E[MSq]s under the maximal model • So if we had the complete populations for all Treatments and computed the MSqs, the value of • the Residual MSq would equal s2 • the Treatment MSq would equal s2 + qT(y). • So that the population average value of both MSqs involves s2, the uncontrolled variation amongst units from the same treatment. • But what about q in Treatments E[MSq]. Statistical Modelling Chapter III

  43. The qT(y) function • Subscript T indicates the Q matrix on which function is based: • but no subscript on the y in qT(y), • because we will determine expressions for it under both the maximal (yT) and alternative models (yG). • That is, y in qT(y) will vary. • Numerator is same as the SSq except that it is a quadratic form in y instead of Y. • To see what this means want expressions in terms of individual parameters. • Will show that under the maximal model (yT) • and under the minimal model (yG) that Statistical Modelling Chapter III

  44. Example III.1 Rat experiment (continued) • The latter is just the mean of the elements of yT. • Actually, the quadratic form is the SSQ of the elements of vector When will the SSq be zero? Statistical Modelling Chapter III

  45. The qT(y) function • Now want to prove the following result: • As QT is symmetric and idempotent, • y'QTy= (QTy)'QTy • qT(y) is the SSq of QTy, divided by (t-1). • QTy=(MT – MG)y=MTy – MGy • MGy replaces each element of y with the grand mean of the elements of y • MTy replaces each element of y with the mean of the elements of y that received the same treatment as the element being replaced. Statistical Modelling Chapter III

  46. so that y'TQTyT is the SSq of the elements of The qT(y) function (continued) • Under the maximal model (yT) • Under the minimal model (yG=m1n) • MGyG=MTyG=yGso y'GQTyG= 0 and qT(y) = 0; • or ak = m so that • and so that qT(y) = 0. Statistical Modelling Chapter III

  47. Example III.1 Rat experiment (continued) Statistical Modelling Chapter III

  48. How qT(yT) depends on the as • qT(yT) is a quadratic form and is basically a sum of squares so that it must be nonnegative. • Indeed the magnitude of depends on the size of the differences between the population treatment means, the aks • if all the aks are similar they will be close to their mean, • whereas if they are widely scattered several will be some distance from their mean. Statistical Modelling Chapter III

  49. E[Msq]s in terms of parameters • Could compute population mean of MSq if knew aks and s2. • Treatment MSq will on average be greater than the Residual MSq • as it is influenced by both uncontrolled variation and the magnitude of treatment differences. • The quadratic form qT(y) will only be zero when all the as are equal, that is when the null hypothesis is true. • Then the E[MSq]s under the minimal model are equal so that the F value will be approximately one. • Not surprising if think about a particular experiment. Statistical Modelling Chapter III

  50. Example III.1 Rat experiment (continued) • So what can potentially contribute to the difference in the observed means of 3.1 and 2.7 for diets A and C? • Answer: • Obviously, the different diets; • not so obvious that differences arising from uncontrolled variation also contribute as 2 different groups of rats involved. • This is then reflected in E[MSq] in that it involves s2 and the "variance" of the 3 effects. Statistical Modelling Chapter III