10%20Linear%20Model%202_14_08

10 Linear Model% - Statistical Models Introduction to Mixed Linear Models A statistical model describes a formal mathematical data generation

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1 1 Introduction to Mixed Linear Models 2/14/2008 2 Statistical Models ± A statistical model describes a formal mathematical data generation mechanism from which an observed set of data is assumed to have arisen. 3 Example 1: Two-Treatment CRD 4 Assign 8 Plants to Each Treatment Completely at Random 222 22 2 1 1 1 1 1 1 1 1
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2 5 Treatment effects about yield ± Suppose we have one measurement of yield (weight) for each plant and the question is whether the two treatments results in different yields. ± Data: y 11 , y 12 , y 13 , y 14 , y 15 , y 16 , y 17 , y 18 , and y 21 , y 22 , y 23 , y 24 , y 25 , y 26 , y 27 , y 28 ± Model: Y ij = μ + τ i + e ij 6 Model specification ± μ represents overall mean of yield. ± τ 1 and τ 2 represent the effects of treatments 1 and 2 on mean yield. ± There is treatment effect if ± e ij , i=1,2, j=1, …, 8, represent residual random effects that include any sources of variation unaccounted for by other terms. 7 Linear Model ± Our model is a linear model because the mean of the response variable may be written as a ± Suppose a model has parameters θ 1 , θ 2 , . .., θ m . ± A linear combination of the parameters is c 1 θ 1 +c 2 θ 2 + . .. +c m θ m where c 1 , c 2 , . .., c m are known constants . 8 Randomly Pair Plants Receiving Different Treatments 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1
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3 9 Randomly Assign Pairs to Slides Balancing the Two Dye Configurations 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 10 Observed Normalized Signal Intensities (NSI) for One Gene Y 111 Y 112 Y 113 Y 114 Y 221 Y 222 Y 223 Y 224 Y 125 Y 126 Y 127 Y 128 Y 215 Y 216 Y 217 Y 218 treatment dye slide 11 Modeling the Means of the Observed Normalized Signal Intensities (NSI) μ + τ 1 + δ 1 μ + τ 2 + δ 2 μ + τ 1 + δ 2 μ + τ 2 + δ 1 represents overall mean of NSI. represent the effects of treatments 1 and 2 on mean NSI. represents the effects of Cy3 and Cy5 dyes on mean NSI. A gene is differentially expressed if τ 1 ≠τ 2 . 12 Unknown Random Effects Underlying Observed NSI s 1 +e 111 s 2 +e 112 s 3 +e 113 s 4 +e 114 s 1 +e 221 s 2 +e 222 s 3 +e 223 s 4 +e 224 s 5 +e 125 s 6 +e 126 s 7 +e 127 s 8 +e 128 s 5 +e 215 s 6 +e 216 s 7 +e 217 s 8 +e 218 s 1 , s 2 , s 3 , s 4 , s 5 , s 6 , s 7 , and s 8 represent slide effects. e 111 ,...,e 218 represent residual random effects that include any sources of variation unaccounted for by other terms.
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4 13 We need to say more about the random effects for a complete model. ± We will almost always assume that random effects are independent and normally distributed with mean zero and a factor-specific variance . ± s 1 , s 2 , . .., s 8 ~ N(0, σ s ) and independent of e 111 , e 112 , e 113 , e 114 , e 221 , e 222 , e 223 , e 224 , e 125 , e 126 , e 127 , e 128 , e 215 , e 216 , e 217 , e 218 ~ N(0, σ e ). 2 iid 2 iid (or just to save time and space.) 14 What does s 1 , s 2 , . .., s 8 ~ N(0, σ s ) mean? 2 iid 15 Observed NSI are Means Plus Random Effects Y 111 = μ + τ 1 + δ 1 +s 1 +e 111 Y 112 = μ + τ 1 + δ 1 +s 2 +e 112 Y 113 = μ + τ 1 + δ 1 +s 3 +e 113 Y 114 = μ + τ 1 + δ 1 +s 4 +e 114 Y 221 = μ + τ 2 + δ 2 +s 1 +e 221 Y 222 = μ + τ 2 + δ 2 +s 2 +e 222 Y 223 = μ + τ 2 + δ 2 +s 3 +e 223 Y 224 = μ + τ 2 + δ 2 +s 4 +e 224 Y 125 = μ + τ 1 + δ 2 +s 5 +e 125 Y 126 = μ + τ 1 + δ 2 +s 6 +e 126 Y 127 = μ + τ 1 + δ 2 +s 7 +e 127 Y 128 = μ + τ 1 + δ 2 +s 8 +e 128 Y 215 = μ + τ 2 + δ 1 +s 5 +e 215 Y 216 = μ + τ 2 + δ 1 +s 6 +e 216 Y 217 = μ + τ 2 + δ 1 +s 7 +e 217 Y 218 = μ + τ 2 + δ
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This note was uploaded on 08/26/2009 for the course STAT 416 taught by Professor Peng,l during the Spring '08 term at Iowa State.

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10 Linear Model% - Statistical Models Introduction to Mixed Linear Models A statistical model describes a formal mathematical data generation

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