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Inference About 2 or More Normal Populations, Part 1

Inference About 2 or More Normal Populations, Part 1. BMTRY 726 2/4/14. Paired Samples. Common when we want to compare response to treatment before and after using the same subject. Helps control subject to subject variation. Univariate case :. Paired Samples. Multivariate case :

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Inference About 2 or More Normal Populations, Part 1

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  1. Inference About 2 or More Normal Populations, Part 1 BMTRY 726 2/4/14

  2. Paired Samples Common when we want to compare response to treatment before and after using the same subject. Helps control subject to subject variation. Univariate case:

  3. Paired Samples Multivariate case: • Notation

  4. Paired Samples Multivariate case: (2) Results 6.1: Assuming the differences D1, D2,…, Dn are a random sample from a Np(d,Sd), then This follows directly from the one sample Hotelling’sT2 test in chapter 5.

  5. Paired Samples Thus it is easy to see that the 100(1-a)% confidence region for d is Similarly the 100(1-a)% simultaneous confidence intervals for the individuals di’s are And the Bonferroni100(1-a)% CIs for the individuals di’s are

  6. Example Say we want to compare reliability of two faculty members grading dental student crown preparations

  7. Example Find T2

  8. Example What are 95% simultaneous CIs for d1 and d2 (F2,3(a)= 9.55)?

  9. Repeated Measures Consider a design with 1 response variable but multiple treatments for the same subject For example… -Measure outcome at different time points -Measure response under different treatments -Measure response for different raters

  10. Repeated Measures Cognitive performance in Parkinsons rats: -Three treatments to improve cognitive performance -Each rat receives each treatment for 1 week -Measure average cognitive performance at the end of each week of treatment Identification of seizure center onset in patients with epilepsy -70 intercranial electrodes in different regions of the brain -Record electrode activity in one second intervals across 10 seizures -Measure average connectivity for all electrode pairs

  11. Repeated Measures & Contrasts Test the hypothesis Write the null hypothesis in matrix form

  12. Repeated Measures & Contrasts Transform the vector of observed values Then Y1, Y2, …, Yn is a random sample form a q -1 dimensional normal with mean Cm and So our HotellingT2 test statistic is

  13. Repeated Measures & Contrasts Proof Then

  14. Repeated Measures & Contrasts So we start with Test: Reject if: Note, for

  15. Example Reaction times of n = 20 people have to visual stimuli driving in a simulator at 0, 30, and 60 minutes after consumption of 2 alcoholic beverages.

  16. Example Are the mean reaction times the same for all 3 stimuli?

  17. Example Are the mean reaction times the same for all 3 stimuli?

  18. More about Contrasts Note the choice of C is not unique. Any matrix C that is of full (row) rank will do and defines the null hypothesis in the same way In the example we could write These C matrices will give the same T2 value

  19. More about Contrasts • Proof

  20. Example 6.2 in the book -Four anesthetic treatments given to n =19 dogs: -T1 = High CO2, no Halothane -T2 = Low CO2, no Halothane -T3 = High CO2, with Halothane -T4 = Low CO2, with Halothane Milliseconds between heartbeats was measured for each treatment

  21. We want to know if average time between heartbeats the same for the four treatments? We could use something similar to our last example…

  22. Alternatively we could use our contrast matrix to estimate particular comparisons

  23. Repeated Measures: CIs For Cm, the 100(1-a)% confidence region is: Similarly the 100(1-a)% simultaneous confidence intervals for single contrasts are And the Bonferroni100(1-a)% CIs for the individuals contrasts are

  24. We can construct 95% simultaneous confidence intervals for these 3 contrasts… • Halothane versus no Halothane • High CO2 versus low CO2 • Interaction between Halothane and CO2

  25. However if a priori we decide we are only interested in these three effects, we can construct Bonferroni CIs • Halothane versus no Halothane • High CO2 versus low CO2 • Interaction between Halothane and CO2

  26. Conclusions • Use of halothane produces longer times between heartbeats. This is consistent across CO2 levels since the interaction was not significant • Lower levels of CO2 results in longer times between heartbeats whether or not halothane is used. -Note, we did not need T2 to come to this conclusion -The investigators in this study also did not randomized order in which the dogs received each treatment combination. Thus time or carry-over effects may be confounded with halothane or CO2 effects.

  27. What About ANOVA? ANOVA table Use of the F-test in this ANOVA is based on the assumptions (1) The 4 observations for each dog are independent (2) Observations taken in different dogs are independent (3) Homogeneous variance

  28. Repeated Measures Random Sample: Linear model: -random errors can have different variances and can also be correlated

  29. Mixed Model Analysis What happens if we have the following data We also assume {sj} are independent of {rij} Note: Observations taken on the same subject are correlated leading to… Fixed Trt Effect Random Sub Effect Random w/in Sub Effect

  30. Mixed Model Analysis For the jth subject

  31. Then Where Correlation between any 2 obs on the same unit -Note: the T2 test doesn’t assume this special covariance structure

  32. Mixed Model ANOVA Where Reject if

  33. When the mixed model covariance structure is correct

  34. When the mixed model F-test is applied when The type I error level often exceeds the specified a When is true Where the zj’s are the NID(0,1) random variables

  35. Otherwise Where

  36. Note that q will be close to when S has a large diversity in variance, or S has some unequal eigenvalues near zero, or both q= 1 if the mixed model assumptions are correct, or more generally when

  37. Since S is unknown, a conservative approach is to make q as small as possible, i.e. close to Then Reject H0: m1 = m2 = … = mp if Alternative approach: -Estimate q from the data

  38. Analysis of Repeated Measures Studies Mixed model ANOVA: -Familiar random subjects model -Simple, well known -Assumes Equal variances and equal correlations -Good power -Other covariance structures are available in PROC MIXED

  39. Analysis of Repeated Measures Studies Conservative degrees of freedom: -Familiar random subjects mixed model -Arbitrary covariance structure -Conservative -Actual type I error has less than nominal a -Loss of power -Could estimate correction to degrees of freedom

  40. Analysis of Repeated Measures Studies HotellingT2 (or MANOVA) -Arbitrary covariance structure -Exact type I error level -Less familiar -Power: -More power than conservative degree of freedom approach -Less power than random subjects mixed model

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