4.4. Further Analysis within ANOVA

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1 4.4. Further Analysis within ANOVA 1) Estimation of the effects Fixed effects model: α i = µ i µ is estimated by a i = ( x i x) if H 0 : µ 1 = µ 2 = = µ k is rejected. Random effects model: If H 0 : σa 2 =0 is rejected, then we estimate the variability σa 2 among the population means by ki=1 n 2 i s 2 MSB MSW A = with n 0 = N 2, n 0 N(k 1) where N = n 1 + n n k, with n 0 = n in the special case of equal sampel sizes n i = n in all groups. 135

2 2) Planned Comparisons: Contrasts The hypothetical data below shows errors in a test made by subjects under the influence of two drugs in 4 groups of 8 subjects each. Group A1 is a control group, not given any drug. The other groups are experimental groups, group A2 gets drug A, group A3 gets drug B, and group A4 gets both drugs. 136

3 Suppose we are really interested in answering specific questions such as: 1. On the average do drugs have any effect on learning at all? 2. Do subjects make more errors if given both drugs than if given only one? 3. Do the two drugs differ in the number of errors they produce? All these questions can be formulated as null hypotheses of the form H 0 : λ 1 µ 1 + λ 2 µ λ k µ k = 0, where λ 1 + λ λ k = 0. For example, the first question asks whether the mean of group A1, µ 1, differes from the average of the means for the groups A2, A3, and A4, (µ 2 + µ 3 + µ 4 )/3. That is, we wish to test the null hypothesis H 0 (1) : µ µ µ µ 4 =

4 A contrast is a combination of population means of the form ψ = λ i µ i, where λ i = 0. The corresponding sample contrast is In our example: L = λ i x i. L 1 = X X X X 4. Now, since each individual observation X ij is distributed as N(µ i, σ 2 ) and all observations are independent of each other, each sample mean X is distributed as N(µ i, σ 2 /n i ), such that ( L N λi µ i, σ 2 λ 2 ) i, n i which implies that under the null hypothesis L σ λ 2 i n i N(0, 1). 138

5 Now recalling that the square of a standard normally distributed random variable is always χ 2 -distributed with df = 1, we obtain denoting Q = L 2 /( λ 2 i /n i): Q/σ 2 χ 2 (1). Furthermore, using SSW/σ 2 χ 2 (N k), we obtain that: Q/MSW F (1, N k), the square root of which has a t-distribution, that is: t = L t(n k), s(l) where s(l) = s λ 2 i which simplies to n i and s = MSW, λ 2 s(l) = s i for equal sample sizes n. n s(l) is called the standard error of the contrast. 139

6 This suggests to test the null hypothesis H 0 : λ 1 µ 1 + λ 2 µ λ k µ k = 0 with a conventional t-test of the form t = L t(n k). s(l) Example: (continued.) L = X X X X 4 = ( ) 3 = 4.167, MSW λ 2 s(l) = i (1 + 3/9) = n 8 = 1, 109 such that t = = The associated p-value in a two-sided test is TDIST(3.76;28;2)=0.08%, and in a onesided test against H 1 : µ 1 < (µ 2 + µ 3 + µ 4 )/3 0.04%, which we regard as clear statistical evidence that the drugs considered increase error rates. 140

7 Alternatively we may test the null hypothesis H 0 : λ 1 µ 1 + λ 2 µ λ k µ k = 0 with an F -test of the form F = Q MSW = t2 F (1, N k), where Q = which simplifies to L2 λ 2 i /n i, Q = nl2 λ 2 i in the case of identical sample sizes n i = n. Example: (continued.) Q = nl2 λ 2 i = /3 = 104.2, F = Q MSW = = The associated p-value may be found in Excel as FDIST(14.1;1;28)=0.08%, the same as before. 141

8 The remaining questions may be tackled in an analogous way: 1. Do subjects make more errors if given both drugs than if given only one? H 0 (2) : µ (µ 2 + µ 3 ) = 0 2. Do the two drugs differ in the number of errors they produce? H 0 (3) : µ 2 µ 3 = 0 The computational steps in conducting the test are summarized in the tables below: A1 A2 A3 A4 λ 2 i X i λ i (1) 1-1/3-1/3-1/3 4/3 λ i (2) 0-1/2-1/2 1 3/2 λ i (3) L s(l) t Q F p H 0 (1) H 0 (2) H 0 (3) Note that the F -test Q/MSW F (1, N k) results in the same p-value as the two-sided t-test L/s(L) t(n k), and that t 2 = F. 142

9 Orthogonal hypotheses The 3 hypotheses in the preceding section were special in the sense that the truth of each null hypothesis is unrelated to the truth of any others. If H 0 (1) is false, we know that drugs have some effect upon errors, although H 0 (1) says nothing about whether the effect of taking both drugs simultaneously is different from the average of the effects of the drugs taken seperately. Similarly, if H 0 (2) is false, we know that the effect of taking both drugs is different from taking only one, but we don t know which of the two drugs taken individually is more effective. A set of contrasts such that the truth of any one of them is unrelated to the truth of any other is called orthogonal. For equal sample sizes in each group the orthogonality of two contrasts L 1 = λ 1i x i and L 2 = λ 2i x i may be assessed by checking the orthogonality condition λ1i λ 2i = 0. ( λ1i λ 2i n i = 0 for n i const. 143 )

10 In an ANOVA with k groups, no more than k 1 orthogonal contrasts can be tested. Such a set of k 1 orthogonal contrasts is however not unique. In practice the researcher will select a set of orthogonal contrasts such that those contrasts of particular interest in the research question are included. Once such a set is found, it exploits all available information that can be extracted for answering the maximum amount of k 1 independent questions, that can be asked in form of contrasts. Once such a set is found, it has the property that the sum of squares for the individual kontrasts Q = L 2 /( λ 2 i /n i) sum up to the sum of squares for the model: Q 1 + Q Q k 1 = SSB. In our example: = (the difference to SSB= in the ANOVA table is only due to rounding error.) 144

11 To illustrate the importance of orthogonal hypothesis, assume that instead of testing the orthogonal hypotheses H 0 (1) H 0 (3) we would have tested the original hypothesis H 0 (1) : µ µ µ µ 4 = 0 together with H 0 (4) : µ 4 µ 1 = 0, that is that error rates are the same when taking both drugs or taking no drugs, upon the data below: 145

12 The two hypotheses are not orthogonal: A1 A2 A3 A4 X i λ i (1) 1-1/3-1/3-1/3 λ i (4) We may therefore get conflicting results from both hypothesis tests: L Q F p H 0 (1) H 0 (4) In this example, we safely accept H 0 (1) that taking drugs has no impact on error rates, while we strongly reject H 0 (4) that taking both drugs has no impact. 146

13 Constructing Orthogonal Tests If all sample sizes are equal (n i = n=const.), orthogonal tests may be constructed based on the principle that the test on the differences among a given set of means is orthogonal to any test involving their average. Consider our original set of hypotheses: H 0 (1) : µ (µ 2 + µ 3 + µ 4 ) = 0 H 0 (2) : µ (µ 2 + µ 3 ) = 0 H 0 (3) : µ 2 µ 3 = 0 H 0 (1) involves the average of µ 2, µ 3, and µ 4, while H 0 (2) and H 0 (3) involve differences among them. Similarly, H 0 (2) involves the average of µ 2 and µ 3, while H 0 (3) involves the difference between them. 147

14 A Word of Caution Contrasts are a special form of planned comparisons, that is, the hypotheses with the contrasts to be tested must be set up before taking a look at the data, otherwise the associated p-values will not be valid. The situation is similar to whether applying a onesided or a two-sided test. Assume you want to test whether the means in two samples are the same using the conventional significance level of α = 5%. You apply a two-sided test and get a p-value of 8%, so you may not reject. Let s say as a step in your calculations you figured out that x 1 > x 2. You may be tempted then to replace your original two-sided test against H 1 : µ 1 µ 2 by a one-sided test against H 1 : µ 1 > µ 2, which, technically, would allow you to divide your p- value by 2 and get a significant result at 4%. But that p-value of 4% is fraud, because the reason that it is only one half of the two sided p-value is exactly that without looking at the data, you also had a 50% chance that the sample means would have come out as x 1 < x

15 3) Multiple Comparisons As seen above, contrasts must be formulated in advance of the analysis in order for the p-values to be valid. Multiple-comparisons procedures, on the other hand, are designed for testing effects suggested by the data after the ANOVA F -test led to a rejection of the hypothesis that all sample means are equal. All multiple comparisons methods we will discuss consist of building t-ratios of the form t ij = x i x j SE ij, or, equivalently, setting up confidence intervals of the form for all ( k) k(k 1) = 2 2 within the k groups. [( x i x j ) ± t SE ij ] pairs that can be built The methods differ in the calculation of the standard errors SE ij and the distributions and critical levels used in the determination of t. 149

16 Fisher s LSD = least-significant difference One obvious choice is to extend the 2 independent sample t-test to k independent samples with test statistics t ij = s x i x j 1ni + 1 nj, where s = MSW, and to declare µ i and µ j different whenever t ij > t α/2 (N k), or, equivalently, whenever 0 / ( x i x j ) ± t α/2 (N k) s n i n j This procedure fixes the probability of a false rejection for each single pair of means being compared as α.. This is a problem if the numbers of means being compared is large. For example, if we use LSD with a significance of α=5% to compare k=20 means, then there are k(k 1) 2 = 190 pairs of means and we expect 5% 190 = 9.5 false rejections! 150

17 Bonferroni Method The Bonferroni method uses the same test statistic as Fisher s LSD, but replaces the significance level α for each single pair with α = α/ ( ) k 2 or equivalently the original p-value with p = ( ) k 2 p as a conservative estimate of the probability that any false rejection among all k(k 1) 2 comparisons will occur. An obvious disadvantage is that the test becomes weak when k is large. Example: Comparison of A1 and A2 (original data): t ij = LSD: s x i x j 1ni + 1 nj = = p LSD = TDIST(2.668; 28; 2) = 1.25%. Bonferroni: p B = p LSD = % = 7.5%. 151

18 Tukey s HSD = honestly significant difference Tukey s method for multiple comparisons delivers the most precise estimate of the probability that any false rejection among all paired comparisons will occur. In its original form it is only applicable in the case of equal sample sizes in each group (n i = n), but corrections for unequal sample sizes have been suggested and are implemented in SAS EG. Tukey s honestly significant difference is: HSD = q α (k, N k) s n, where q α denotes the α-critical value from the so called Studentized range distribution tabulated e.g. in table 6 of Azcel, and s is estimated by MSW, as usual. Two group means µ i and µ j are declared different at significance level α if or, equivalently, if 0 / [ ( x i x j ) ± HSD ], x i x j s/ n > q α(k, N k). 152

19 Example: Comparison of A1 and A2 (continued). LSD and Bonferroni gave conflicting results whether the difference between µ 1 and µ 2 is significant at 5% or not (recall p LSD = 1.25% and p B = 7.5%), because LSD underestimates the probability of any false rejection among all paired comparisons, whereas Bonferroni overestimates it. Applying Tukey s procedure yields x i x j s/ n = /8 = Looking up in a table of critical values for the Studentized range distribution reveals that q 0.05 (k, N k) = q 0.05 (4; 28) = 3.86, which is larger than the statistic calculated above. The difference between µ 1 and µ 2 is therefore not significant at 5% level, if the test was first suggested by the data. Note: A planned comparison before looking at the data in form of a contrast would have found a significant difference with a p-value of 1.25%(=p LSD ). 153

20 Simultaneous Confidence Intervals The idea with multiple comparisons in posthoc tests is usually to determine, which groups may be combined into larger groups, that are homogeneous in the sense that their group means are not significantly different. For that purpose, one constructs confidence intervals of the form [( x i x j ) ± smallest significant difference], where the smallest significant differences are: LSD : tα(n k) s n i n j Bonferroni: t α2 /( k 2 )(N k) s n i n j s HSD : q α (k, N k) n Combinations of groups are considered homogeneous at confidence level (1 α), if none of their paired comparisons lies outside the corresponding confidence intervals; that is, pairs of means, the confidence intervals of which include the value 0 will not be declared significantly different, and vice versa. 154

21 Homogeneous Subgroups in SAS EG If your only reason for doing multiple comparisons is to identify homogeneous groups in the sense that their group means are not significantly different, then the easiest way to do so is in the Means/Comparison tab of the One-Way ANOVA procedure. This will not provide you with p-values for the differences in means though, and it will not allow you to define your own contrasts. Below is some output for Tukey s HSD and Fisher s LSD: One-Way Analysis of Variance Results Tukey's Studentized Range (HSD) Test for errors NOTE: This test controls the Type I experimentwise error rate, but it generally has a higher Type II error rate than REGWQ. Alpha 0.05 Error Degrees of Freedom 28 Error Mean Square Critical Value of Studentized Range Minimum Significant Difference Means with the same letter are not significantly different. Tukey Grouping Mean N drug A A B A B B B B One-Way Analysis of Variance Results t Tests (LSD) for errors NOTE: This test controls the Type I comparisonwise error rate, not the experimentwise error rate. Alpha 0.05 Error Degrees of Freedom 28 Error Mean Square Critical Value of t Least Significant Difference Means with the same letter are not significantly different. t Grouping Mean N drug A B B C B C C

22 Multiple Comparisons in SAS EG Post hoc tests in SAS EG may be found in Analyze/ANOVA/Linear Models. For the overall ANOVA F -test (the same as under Analyze/ANOVA/One-Way Anova) select the dependent variable (e.g. errors) and classification variable (e.g. drug) under Task Roles. You will also have to reselect your classification variable as Main Effect under Model. To perform the multiple comparisons push Add under Effects to estimate within Post Hoc Tests/Least Squares. This will open four drop-down menues under Options for means tests. Select Yes under Class effects to use, All pairwise differences for Show p- values for differences, and True for Show confidence limits. The option No adjustment under Adjustment method for comparison stands for LSD. Homogeneous subgroups in our example are (A1,A3) and (A2,A3) under LSD, and (A1,A2, A3), (A2,A4) under Tukey and Bonferroni. 156

23 Analyze => ANOVA => Linear Models

24

25 drug Linear Models Least Squares Means errors LSMEAN LSMEAN Number Least Squares Means for effect drug Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: errors i/j < < drug errors LSMEAN 95% Confidence Limits i Least Squares Means for Effect drug j Difference Between Means 95% Confidence Limits for LSMean(i)- LSMean(j)

26 Linear Models Least Squares Means Adjustment for Multiple Comparisons: Bonferroni drug errors LSMEAN LSMEAN Number Least Squares Means for effect drug Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: errors i/j drug errors LSMEAN 95% Confidence Limits i Least Squares Means for Effect drug j Difference Between Means Simultaneous 95% Confidence Limits for LSMean(i)-LSMean(j)

27 Linear Models Least Squares Means Adjustment for Multiple Comparisons: Tukey drug errors LSMEAN LSMEAN Number Least Squares Means for effect drug Pr > t for H0: LSMean(i)=LSMean(j) Dependent Variable: errors i/j drug errors LSMEAN 95% Confidence Limits i Least Squares Means for Effect drug j Difference Between Means Simultaneous 95% Confidence Limits for LSMean(i)-LSMean(j)

28 Contrasts in SAS EG Calculating contrasts in SAS EG requires a (tiny) little bit of programming. Proceed as if you were going to perform some multiple comparisons. Then, after having made the same choices as described on the preceding pages, press the Preview code buttom in the lower left corner of whichever tab you are in. Next push the Insert Code buttom in the upper left corner of the Code preview for Task window. This will bring up the program code generated by SAS EG in such a way that it indicates spaces to insert the users own code. Double-click on the space immediately above the line beginning with the command LSMEANS <independent variables name>. This will bring up a window where you can enter your own code. The syntax to be entered is: Contrast <table text> <indep. variable> λ 1, λ 2,..., λ k ; The coefficients λ 1, λ 2,..., λ k must be entered as real numbers seperated by spaces (no commas); fractions (such as 1/2) are not allowed. 162

29 Analyze => ANOVA => Linear Models... Preview Code => Insert Code OK => OK => Run

30 Linear Models Dependent Variable: errors errors Source DF Sum of Squares Mean Square F Value Pr > F Model Error Corrected Total R-Square Coeff Var Root MSE errors Mean Source DF Type III SS Mean Square F Value Pr > F drug Contrast DF Contrast SS Mean Square F Value Pr > F H1: control vs other H2: 2 drugs vs 1 drug H3: drug A vs drug B Parameter Estimate Standard Error t Value Pr > t Intercept B <.0001 drug B <.0001 drug B drug B drug B... The X'X matrix has been found to be singular, and a generalized inverse was used to solve the normal equations. Terms whose estimates are followed by the letter 'B' are not uniquely estimable.

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