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23Models(1) - A collection of potentially useful models...

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A collection of potentially useful models I We’ve already seen two very common mixed models: I for subsampling I for designed experiments with multiple experimental units I Here are three more general classes of models I Random coefficient models, aka multi-level models I Models for repeated experiments I Models for repeated measures I For complicated problems, may need to combine ideas c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 1 / 26
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Random coefficient models I A regression where all coefficients vary between groups I Example: Strength of parachute lines. I Measure strength of a parachute line at 6 positions I Physical reasons to believe that strength varies linearly with position I Model by y i = β 0 + β 1 X i + i , where X is the position, y is the strength, and i indexes the measurement I What if have 6 lines, each with 6 observations? I Measurements nested in line I suggests: Y ij = β 0 + β 1 X ij + ν j + ij = ( β 0 + ν j ) + β 1 X ij + ij , where j indexes the line. I Intercept varies between lines, but slope does not c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 2 / 26
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1 2 3 4 5 6 900 1000 1100 1200 1300 Position Strength Parachute line strength c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 3 / 26
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I Random coefficient regression models allow slope to also vary Y ij = ( β 0 + α j 0 ) + ( β 1 + α j 1 ) X ij + ij u 0 = [ α 10 , α 11 , α 20 , α 21 , . . . , α 60 , α 61 ] u = 1 1 0 0 . . . 0 1 2 0 0 . . . 0 1 3 0 0 . . . 0 1 4 0 0 . . . 0 1 5 0 0 . . . 0 1 6 0 0 . . . 0 0 0 1 1 0 . . . 0 . . . . . . . . . . . . . . . 0 0 0 1 6 36 × 12 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 4 / 26
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I α j 0 α j 1 N 0 0 , G I G = σ 2 0 σ 01 σ 01 σ 2 1 I sometimes see model written as: Y ij = β 0 j + β 1 j X ij + ij , β j 0 β j 1 N β 0 β 1 , G I R usually assumed σ 2 e I . I Σ = ZGZ 0 + R is quite complicated, can write out but not enlightening. Features: I Var Y ij not constant, depends on X ij , even if R is σ 2 e I I Cov Y ij , Y i 0 j not constant, depends on X ij and X i 0 j I Cov Y ij , Y i 0 j 0 = 0 , since obs. on different lines assumed independent I so Σ is block diagonal, with non-zero blocks for obs. on same line c 2011 Dept. Statistics (Iowa State University) Stat 511 section 23 5 / 26
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I Customary to include a parameter for the covariance between intercept and slope random effects. I if omit, then model is not invariant to translation of X I i.e., fixed effect part of the regression is the same model even if shift X , e.g. X - 3 . I random effects part is the same only if include the covariance I some parameter values will change if X shifted, but structure stays the same. I Can extend model in at least two ways: 1. More parameters in regression model e.g. quadratic polynomial: Y ij = β 0 j + β 1 j X ij + β 2 j X 2 ij + ij , I Example: Allan Trapp’s MS. Longevity of stored seed, quadratic, 2833 seed lots of maize, each with 3 to 7 observations.
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