Correlated data. Overview. Cross-over study. Repetition. Faculty of Health Sciences. Variance component models, II. More on variance component models

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1 Faculty of Health Sciences Overview Correlated data More on variance component models Variance component models, II Cross-over studies Non-normal data Comparing measurement devices Lene Theil Skovgaard Non-hierarchical models December 4, / 96 2 / 96 Repetition Cross-over study Patients with chronic headache are randomized into two groups: Severeal measurements on the same unit The traditional assumption of independence is violated Disregarding this may lead to erroneous conclusions is often quite simple to handle by introducing random effects sometimes more complicated Both groups receive LMMA and placebo, on two different days, with a suitable wash-out period in-between Group A was treated first with placebo (period 1), and then with LNMMA (period 2) Group B was treated first with LNMMA (period 1), and then with placebo (period 2) Pain was measured subjectively on a VAS-scale (small is good), at baseline and at 30, 60, 90 and 120 minutes after treatment. Ashina, Lassen, Bendtsen, Jensen og Olesen (1999), Lancet, pp / 96 4 / 96

2 Data We have repeated measurements over time (topic of next lecture) Here, we shall reduce the complexity by simply looking at difference between baseline and follow up (remember that for this to be a good idea, the correlation must be strong) Outcome: Difference between follow-up measurements and baseline, i.e. Y 30 + Y 60 + Y 120 3Y 0 5 / 96 6 / 96 Observations Average over patients 7 / 96 8 / 96

3 Ignoring periods: Paired T-test for treatment effect What about simple ANOVA? The TTEST Procedure Two-way anova in treat and period Difference: lnmma - placebo N Mean Std Dev Std Err Minimum Maximum Mean 95% CL Mean Std Dev 95% CL Std Dev DF t Value Pr > t LNMMA is more effective than placebo: 42.9 (14.6, 71.2) but we have ignored a possible effect of period... 9 / 96 Parameter Estimate Error t Value Pr > t Intercept B treat lnmma B treat placebo B... period B period B... Here, we have ignored the pairing / 96 Model for cross-over study Traditional approaches t = active, placebo, p = 1, 2(periods], i = 1, 2,... (individulas) Assuming no carry-over effect With no carry-over effect: Y tpi = α t + β p + A i + ε tpi where A i N (0, ω 2 ) ε tpi N (0, σ 2 ) With carry-over effect: Y tpi = α t + β p + γ tp + A i + ε tpi group Period 1 Period 2 period2-period1 A β 1 + α p + A i β 2 + α a + A i β 2 β 1 + α a α p B β 1 + α a + A j β 2 + α p + A j β 2 β 1 + α p α a group A-B α p α a + A α a α p + A 2(α a α p ) A i, A j and A refer to something with a subject effect Three possible comparisons of treatments, two of these ( ) include biological variations and are therefore less powerful 11 / / 96

4 Comparison in each period separately period=1 Variable: effect treat N Mean Std Dev Std Err Minimum Maximum lnmma placebo Diff (1-2) treat Method Mean 95% CL Mean Std Dev lnmma placebo Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F LNMMA is more effective than placebo: 8.2 (-37.6, 54.0) 13 / 96 period=2 Variable: effect treat N Mean Std Dev Std Err Minimum Maximum lnmma placebo Diff (1-2) treat Method Mean 95% CL Mean Std Dev lnmma placebo Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F LNMMA is more effective than placebo: 70.5 (22.5, 118.4) 14 / 96 Effect of treatment, adjusted for period Conclusion on treatment effect Using 1 2 period difference The TTEST Procedure Variable: half_period group N Mean Std Dev Std Err Minimum Maximum A B Diff (1-2) group Method Mean 95% CL Mean Std Dev A B Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F Method Effect Confidence Interval P-value Period (-37.59, 53.99) 0.71 Period (11.55, ) No adjustment (14.55, 71.19) T-test Period adjustment (-21.92, 50.25) No correlation Period adjustment (9.97, 68.70) With correlation 15 / / 96

5 Not much effect of period adjustment Assuming a carry-over effect (C) because the period effect (on average) is not large: 14.2 (11.8, 40.1) Variable: half_treat group N Mean Std Dev Std Err Minimum Maximum A B Diff (1-2) group Method Mean 95% CL Mean Std Dev A B Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F but the picture suggests...? 17 / 96 i.e. an interaction treat*period (γ p2 ) group LNMMA placebo sum A β 2 + α a + A i β 1 + α p + A i β 1 + β 2 + α p + α a + A i B β 1 + α a + A j β 2 + α p + C + A j β 1 + β 2 + α p + α a + γ p2 + A j group A-B β 1 β 2 + A β 2 β 1 + γ p2 + A γ p2 + A Test for carry-over effect: T-test for the sums 18 / 96 Test of carry-over effect, T-test Coded as a variance component model Variable: treat_sum group N Mean Std Dev Std Err Minimum Maximum A B Diff (1-2) group Method Mean 95% CL Mean Std Dev A B Diff (1-2) Pooled Diff (1-2) Satterthwaite Method Variances DF t Value Pr > t Pooled Equal Satterthwaite Unequal Equality of Variances Method Num DF Den DF F Value Pr > F Folded F Without carry-over effect: proc mixed data=a1 /*covtest*/; class patient group treat period; model effect=treat period / outpred=udp outpredm=udpm residual influence s cl; random intercept / subject=patient(group); Not significant, but... The carry-over effect is estimated to be an extra effect of placebo in period 2 of 62.3, with confidence interval ( 25.4, 149.9) 19 / 96 Include a carry-over effect as the interaction treat*period 20 / 96

6 Mixed, no carry-over effect Predictions in contrast to observations Cov Parm Subject Estimate Intercept patient(group) Residual Solution for Fixed Effects assuming no carry-over effect: Effect treat period Estimate Error DF t Value Pr > t Intercept treat lmmma treat placebo period period Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F treat period / / 96 Test of carry-over effect, Mixed Output, traditional parametrization proc mixed data=a1 /*covtest*/; class patient group treat period; model effect=treat period treat*period / s cl; random intercept / subject=patient(group); or data a1; set a1; active=(treat="lnmma"); period2=(period=2); carry_over=(treat="placebo")*(period=2); proc mixed data=a1; class patient group active period2 carry_over; model effect=active period2 carry_over / s cl; random intercept / subject=patient(group); 23 / 96 Cov Parm Subject Estimate Intercept patient(group) Residual Solution for Fixed Effects Effect treat period Estimate Error DF t Value Pr > t Intercept treat lmmma treat placebo period period treat*period lmmma treat*period lmmma treat*period placebo treat*period placebo Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F treat period treat*period / 96

7 Output, intuitive parametrization Output, intuitive parametrization II Class Level Information Class Levels Values patient group 2 A B active period carry_over Cov Parm Subject Estimate Intercept patient(group) Residual Solution for Fixed Effects carry_ Effect active period2 over Estimate Error DF t Value Intercept active active period period carry_over carry_over Solution for Fixed Effects carry_ Effect active period2 over Pr > t Alpha Lower Upper Intercept active active period period carry_over carry_over Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F active period carry_over / / 96 Predictions, without carry-over effect Paired T-test, revisited Tryptase level before and after operation Garvey et al. (2010a,b) 27 / / 96

8 Paired T-tests written as two-way anova The GLM Procedure Class Level Information The TTEST Procedure Difference: logbefore - logafter N Mean Std Dev Std Err Minimum Maximum Mean 95% CL Mean Std Dev 95% CL Std Dev DF t Value Pr > t <.0001 Tryptase levels decrease with a factor = 0.909, i.e. approximately 9% (CI: ) 29 / 96 Class Levels Values patient time 2 1:before 2:after Number of Observations Read 240 Number of Observations Used 240 Dependent Variable: logtryptase Source DF Type III SS Mean Square F Value Pr > F patient <.0001 time <.0001 Parameter Estimate Error t Value Pr > t Intercept B <.0001 patient B patient B patient B <.0001 patient B... time 1:before B <.0001 time 2:after B / 96 or mixed Missing values proc mixed data=b1; class patient time; model logtryptase=time / s cl; random patient; The Mixed Procedure Cov Parm Estimate patient Residual Solution for Fixed Effects Effect time Estimate Error DF t Value Pr > t Alpha Intercept < time 1:before < time 2:after Effect time Lower Upper Intercept time 1:before time 2:after.. 31 / 96 Now excluding a random number of observations: Analysis Variable : tryptase N N time Obs N Miss :before :after / 96

9 Paired T-test The TTEST Procedure Difference: logbefore - logafter N Mean Std Dev Std Err Minimum Maximum Mean 95% CL Mean Std Dev 95% CL Std Dev DF t Value Pr > t Only 79 patients with both observations before and after operation 33 / / 96 Unpaired T-test Random effects model in order to use all available observations Dependent Variable: logtryptase Sum of Source DF Squares Mean Square F Value Pr > F Model Error Corrected Total Class Level Information Class Levels Values time 2 1:before 2:after Number of Observations Read 240 Number of Observations Used 191 Parameter Estimate Error t Value Pr > t Intercept B <.0001 time 1:before B time 2:after B... to account for correlations/pairing and use all available observations: proc mixed data=b1; class patient time; model logtryptase=time / s cl; random patient; Parameter 95% Confidence Limits Intercept time 1:before time 2:after.. 35 / / 96

10 Output from mixed model Conclusion on tryptase effect The Mixed Procedure Dependent Variable logtryptase Number of Observations Number of Observations Read 240 Number of Observations Used 191 Number of Observations Not Used 49 Covariance Parameter Estimates Cov Parm Estimate patient Residual Solution for Fixed Effects Effect time Estimate Error DF t Value Pr > t Alpha Intercept < time 1:before time 2:after Effect time Lower Upper Intercept time 1:before time 2:after.. 37 / 96 Method Effect N Confidence Interval P-value Paired T-test (0.0112, ) Unpaired T-test ( , ) 0.17 Mixed model (0.0131, ) Back-transformed: Ratio before/after is estimated to = 1.087, or ratio after/before estimated to = 0.92, i.e. an 8% decrease, with confidence interval ( , ) = (0.87, 0.97), i.e. from 3-13% 38 / 96 When should we use what approach? Non-normal data Typical data from e.g. epidemiology are often not normally distributed (binary, ordinal, counts, survival...) Paired T-test: When the correlation is strong, and only few observations are missing Unpaired T-test: When the correlation is weak, and many observations are missing Random effects model: Always possible But note: The missingness has to be random! More on missing values later on... Generalised linear models in exponential families: Multiple regression models, on a scale that corresponds to the data: Normal (link=identity) Binomial (link=logit) Poisson (link=log) Mean value: µ Link funktion: g(µ) linear in covariates, i.e. g(µ i ) = β 0 + β 1 x i1 + + β k x ik + A i 39 / / 96

11 The Binomial distribution Poisson distribution N independent binary observations U i, all with P(U i = 1) = p (e.g. p = 0.51 for a baby boy) X = U U N = U i (number of ones, e.g. boys in a family) The distribution of X is called a Binomial distribution and is written X Bin(N, p) 41 / 96 P(X = x) = ( ) N p x (1 p) N x x Counts with no well-defined upper limit: the number of cancer cases in a specific community during a specific year the number of metastases following an experimentally induced cancer in laboratory rats Law of rare events: As the count parameter N in a Binomial distribution gets larger and the parameter p gets close to either 0 or 1, the Binomial probabilities are approximately P(u) = P(y = u) = mu exp( m), (1) u! where m = Np is the mean value. 42 / 96 Smoking among school children Possible covariates, at various levels Hierarchical (multilevel) design: 1498 children (i) 90 classes (c) 46 schools (s) Outcome: Individual smoking behaviour (0/1), y sci p sci ; the probability that child i in class c on school s is a smoker Mette Rasmussen Individual (i): sex, age, parental smoking behaviour (c2ab), parental smoking attitude (c1112a), parental labour market attachment (fsoc), best friend smoking (c2cr) Class (c): sex ratio, number of pupils, grade School (s): Type, school connectedness (con3ny) 43 / / 96

12 Multilevel model for binary outcomes Initial model y sci Bernoulli(p sci ) p sci = P(y sci = 1 η sci ) = exp(η sci) 1 + exp(η sci ) where η sci = β x sci + a s + b sc i-covariates + school + class a s = γ sz s + A s s-covariates + random school b sc = γ cz sc + B sc c-covariates + random class η sci = β x sci + γ 1z s + γ 2z sc + A s + B sc Two-level model: no covariates only random school proc glimmix data=a1; class school sclass; model dglryg(descending) = / dist=binary link=logit ddfm=satterth s; random school; A s N (0, ω 2 ) between school variation B sc N (0, τ 2 ) between classes (within school) variation 45 / / 96 Interesting part of output Interpretation of estimates Fit Statistics -2 Res Log Pseudo-Likelihood Generalized Chi-Square Gener. Chi-Square / DF 0.96 Cov Parm Estimate Error SCHOOL Solutions for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept < / 96 Fixed effects: Only intercept, i.e. overall level: Inverse logit-transformation: > exp(a)/(1+exp(a)) [1] Overall, approx. 18.6% of the pupils smoke Random effects (MOR) For two individuals from different schools, (and with identical covariates) we calculate median OR for a randomly chosen high risk individual compared to a randomly chosen low risk individual: 48 / 96

13 Median Odds Ratio, MOR Inclusion of variation between school classes Variation between schools: Variance component: ω 2 = ω = Typical difference D on logit scale: D N (0, 2ω 2 ) : ±2 2 ω 2 median numerical difference on logit scale: D 2 2ω 2 χ 2 1 2ω : 2 median(χ 2 1 ) median odds ratio (MOR): exp( 2ω 2 median(χ 2 1 )) and since median(χ 2 1 ) = , we get MOR = exp(0.954 ω) = 1.46 proc glimmix data=a1; class school sclass; model dglryg(descending) = / dist=binary link=logit ddfm =satterth s; random school sclass; Output: Fit Statistics -2 Res Log Pseudo-Likelihood Generalized Chi-Square Gener. Chi-Square / DF / / 96 Output, continued A possible third level... Cov Parm Estimate Error SCHOOL 0. sclass Solutions for Fixed Effects Effect Estimate Error DF t Value Pr > t Intercept <.0001 The variation between schools can be totally explained by the variation between school classes 51 / 96 Imagine an extra grouping: Gender group within class, i.e. a subgrouping in boys and girls Note: This is not the same as a gender effect it need not be a systematic difference the group definition is a substitute for cliques of which we know nothing Modify the Random-statement to: random school sclass ggroup; and remember ggroup in the Class-statement 52 / 96

14 Systematic sex effect The GLIMMIX Procedure Fit Statistics -2 Res Log Pseudo-Likelihood Generalized Chi-Square Gener. Chi-Square / DF 0.83 Cov Parm Estimate Error SCHOOL 0. sclass GGROUP Solutions for Fixed Effects proc glimmix data=a1; class school sclass ggroup sex c1112a c2ab c2cr; model dglryg(descending) = sex / dist=binary link=logit ddfm =satterth s; random school sclass ggroup; Effect Estimate Error DF t Value Pr > t Intercept < / / 96 Interpretation of results The GLIMMIX Procedure Fit Statistics -2 Res Log Pseudo-Likelihood Generalized Chi-Square Gener. Chi-Square / DF 0.84 Cov Parm Estimate Error SCHOOL 0. sclass GGROUP Solutions for Fixed Effects Effect sex Estimate Error DF t Value Pr > t Intercept <.0001 sex dreng sex pige Systematic effect of sex: OR=exp(0.4188) = 1.52 for girls vs. boys Random effects: MOR for two children of opposite sex in the same class: 1.52 exp( ) = 2.90 Random effects: MOR for two children of opposite sex in different classes (at same or different schools): 1.52 exp( ) = 3.10 How much does systematic sex effect explain of the random components? 55 / / 96

15 Variance component estimates Odds ratios (OR) and MOR model school school class gender group school alone school and school class school, class and gender group as above, with sex model school school class gender group sex school alone school and school class school, class and gender group as above, with sex 57 / / 96 Comparing measurement devices Illustration of all data Example: Peak expiratory flow rate, l/min: 17 subjects, 2 measurement devices Each measured twice subject Wright mini Wright id Y 1p1 Y 1p2 Y 2p1 Y 2p Average SD (Bland and Altman, 1986). 59 / / 96

16 Aim of investigation Variance component model Precision of each measuring device compare the two repetitions Agreement between the two devices compare individual measurements - or averages Practical advice for clinical use can we trust the devices, and use them interchangeably? Subject, p = 1,..., 17 Methods, m = 1, 2 Repetitions, j = 1, 2 Y pmj = β m + A p + C pm + ε pmj where A p N (0, ω 2 ), C pm N (0, τ 2 ), ε pmj N (0, σ 2 ) Note: Patients need not be random here..., why?? 61 / / 96 Correlation structure Correlation structure in the above model, for each subject if subjects are considered systematic: ω 2 + τ 2 + σ 2 ω 2 + τ 2 ω 2 ω 2 ω 2 + τ 2 ω 2 + τ 2 + σ 2 ω 2 ω 2 ω 2 ω 2 ω 2 + τ 2 + σ 2 ω 2 + τ 2 ω 2 ω 2 ω 2 + τ 2 ω 2 + τ 2 + σ 2 For each subject*method combination, i.e. for two repetitions: ( τ 2 + σ 2 τ 2 ) τ 2 τ 2 + σ 2 63 / / 96

17 SAS-programming Output proc mixed data=wright; class method id; model wr=method / ddfm=satterth s; random intercept method / subject=id; or proc mixed data=wright; class method id; model wr=id method / s; random id*method; Class Level Information Class Levels Values method 2 mini wright id Cov Parm Subject Estimate Intercept id method id Residual Fit Statistics -2 Res Log Likelihood AIC (smaller is better) Solution for Fixed Effects Effect method Estimate Error DF t Value Pr > t Intercept <.0001 method mini method wright / / 96 Estimates Precision of the methods Variance components: ω 2 = τ 2 = are assumed identical Difference between double measurements (identical repetitions): D pm = Y pmj1 Y pmj2 σ 2 = Systematic difference between measuring devices: = ε p1j1 ε p2j2 N (0, 2σ 2 ) ˆβ 1 ˆβ 2 = 6.03(8.05), P = 0.46 Limits-of-agreement: How can we use these?? ±2 2σ 2 = ± / / 96

18 Agreement between the two methods Agreement between averages Difference between single measurements by the two methods: D p = Y p1j1 Y p2j2 = β 1 β 2 + C p1 C p2 + ε p1j1 ε p2j2 N (β 1 β 2, 2τ 2 + 2σ 2 ) D p = X p1. X p2. = β 1 β 2 + C p1 C p2 + ε p1. ε p2. N (β 1 β 2, 2τ 2 + σ 2 ) Limits-of-agreement: ±2 2(τ 2 + σ 2 ) = ±75.31 (where we have ignored the nonsignificant systematic difference between the two, otherwise add 6.03) 69 / 96 Limits-of-agreement: ±2 2τ 2 + σ 2 = ±66.41 Only reasonable, if averages is the standard for clinical use! 70 / 96 Difference in precision?? Output, systematic subject effect New model (with systematic subject effects): Y pmj = µ + β m + α p + C pm + ε pmj C pm N (0, τ 2 ) ε pmj N (0, σ m 2 ) proc mixed data=wright; class method id; model wr=id method / ddfm=satterth s; random id*method; repeated / group=method type=simple subject=id*method; 71 / 96 Cov Parm Subject Group Estimate method*id Residual method*id method mini Residual method*id method wright Fit Statistics -2 Res Log Likelihood AIC (smaller is better) Solution for Fixed Effects Effect method id Estimate Error DF t Value Pr > t Intercept <.0001 id id id id method mini method wright Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F id <.0001 method / 96

19 Results Incorrect Bland-Altman approaches Precisions: Wright: σ 2 1 = mini Wright: σ 2 2 = Conclusion: Wright is better than mini Wright, but is it significantly better? F = σ2 2 σ 2 1 = = 1.69 F(17, 17) P = 0.14 No... Alternative test: 2 log Q = = 1.2 χ 2 (1) P = 0.27 Calculate agreement between averages: We have seen that these estimate 2τ 2 + σ 2 instead of 2τ 2 + 2σ 2 In general, with k repetitions: 2τ k σ2 Calculate all possible pairs D pj = Y p1j1 Y p2j2 = β 1 β 2 + C p1 C p2 + ε p1j1 ε p2j2 but these differences will be correlated due to the C s and can give erroneous results if the measurement devices react to subject characteristics 73 / / 96 Measurements taken in pairs Effect of the lens strength on visual acuity e.g. over time... Y pmt = β m + A p + C pm + E pt + ε pmt Precision becomes impossible, since we have no true replications Agreement: D pt = Y p1t Y p2t = β 1 β 2 + C p1 C p2 + +ε p1t ε p2t 7 individuals are looking at a screen, where a light flash appears. They are looking through 4 lenses, with powers 6/6, 6/18, 6/36 and 6/60, i.e. 4 magnifications: 1, 3, 6 and 10 with 2 eyes Outcome: Visual acuity, the time lag (milliseconds) between the stimulus and the electrical response at the back of the cortex but these differences will again be correlated due to the C s 75 / / 96

20 Data Factors to take into account Main effects: 7 individuals (person), 2 eyes for each individual (eye) 4 lens magnifications (power) Crowder & Hand (1990) Interactions? person*eye person*power eye*power 2-order interaction person*eye*power = Residual 77 / / 96 Model ingredients Outcome: Visual acuity Systematic: Mean value µ em eye α e, power β m eye*power γ em Random effects: patient A p patient*eye B pe, patient*power C pm Residual: patient*eye*power ε pem 79 / / 96

21 Model formulation Factor diagram where p = 1,..., 7, e = 1, 2, m = 1, 2, 3, 4 Y pem = µ em + A p + B pe + C pm + ε pem A p N (0, ω 2 ) B pe N (0, τe 2 ) C pm N (0, τm) 2 ε pem N (0, σ 2 ) [I ] = [Pa Ey Po] [Pa Ey] Ey Po [Pa Po] Ey [Pa] Po 0 81 / / 96 Not quite a multilevel model, but.. Level Unit Covariates 1 single measurements Ey*Po 2 interactions 2e [Pa*Ey] Ey 2m [Pa*Po] Po 3 individuals, [Pa] overall level proc mixed data=visual covtest; class patient eye power; model acuity=eye power eye*power / s; * random patient patient*eye patient*power; random intercept eye power / subject=patient; Z Cov Parm Subject Estimate Error Value Pr > Z Intercept patient eye patient power patient Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F eye power eye*power / / 96

22 Solution for Fixed Effects Predicted mean profiles Effect eye power Estimate Error DF t Value Pr > t Intercept <.0001 eye left eye right power power power power eye*power left eye*power left eye*power left eye*power left eye*power right eye*power right eye*power right eye*power right / / 96 Individual predictions Residual plot 87 / / 96

23 Omit the interaction eye*power Eye comparisons Z Cov Parm Subject Estimate Error Value Pr > Z Intercept patient eye patient power patient Residual Solution for Fixed Effects Effect eye power Estimate Error DF t Value Pr > t Intercept <.0001 eye left eye right power power power power Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F eye power / 96 Y pem = µ em + A p + B pe + C pm + ε pem where A p N (0, ω 2 ), B pe N (0, τe 2 ), C pm N (0, τm), 2 ε pem N (0, σ 2 ) Difference between eye averages: 90 / 96 Ȳ.e1. Ȳ.e 2. = µ stuff + B.e1 B.e2 + ε.e1. ε.e2. Consequence for eye comparisons Magnification comparisons Y pem = µ em + A p + B pe + C pm + ε pem Var(Ȳ.e 1. Ȳ.e 2.) = 2 7 τ 2 e σ2 τ 2 e is rather large (people have different eye preferences) We have to demand a larger difference in order to detect it where A p N (0, ω 2 ), B pe N (0, τe 2 ), C pm N (0, τm), 2 ε pem N (0, σ 2 ) Difference between magnification averages: Ȳ..m1 Ȳ..m 2 = µ stuff + C..m1 C..m2 + ε..m1 ε..m2 91 / / 96

24 Consequence for magnification comparisons If we ignore correlations Var(Ȳ..m 1 Ȳ..m 2 ) = 2 7 τ 2 m σ2 τ 2 m is not that large (people react more or less identically to the different magnifications) We can detect smaller differences i.e a model with no random effects Eye differences: but another σ 2 Magnification differences: Var(Ȳ.e 1. Ȳ.e 2.) = σ2 Var(Ȳ..m 1 Ȳ..m 2 ) = σ2 93 / / 96 Incorrect analysis, ignoring random effects Systematic vs. random effects Covariance Parameter Estimates Cov Parm Estimate Residual Solution for Fixed Effects Effect eye power Estimate Error DF t Value Pr > t Intercept <.0001 eye left eye right power power power power Type 3 Tests of Fixed Effects Num Den Effect DF DF Chi-Square F Value Pr > ChiSq Pr > F eye power Could the patients be treated as systematic here? Yes: Cov Parm Subject Estimate eye patient power patient Residual Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F patient eye power eye*power Can you think why? 95 / / 96