We are estimating the density of long distant migrant (LDM) birds in wetlands along Lake Michigan.

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1 Ch 17 Random ffecs and Mxed Models 17. Random ffecs Models We are esmang he densy of long dsan mgran (LDM) brds n welands along Lake Mchgan. μ + = LDM per hecaren h weland ~ N(0, ) The varably of expeced LDM s from weland o weland s. A each weland LDM s are couned a n ses. The se o se varably of measured LDM a ses whn a weland s h Y = j se n j h weland Y N o j = μ+ + j j ~ (, ) and j are ndependen Y ( ) = μ j Var( Y ) = Var( ) + Var( ) = + j j We vs random welands. In each weland we make observaons a n random ses. Source df SS Welands 1 Σn( y y ) = SST = SS for Treamens ( n 1) Toal n 1 There are reamens n Table 17. Σ( n 1) s = SS ΣΣ = SS ( yj y ) TSS = Toal MS s he pooled varance, so s expeced value s he common varance. (MS) = (pooled varance) = As shown on he nex page (MST) = + n

2 ( y y ) MST = n = ns = 1 1 ( MST ) = nvar( y) j y = + = + n Var( y) = + n MST j y ( ) = n + wrong on page 980 Source (MS) (T) Welands + n () To es MST Ho : F = MS F = 1 f f no weland effec > > 1 f 0 f weland effec The F-es s he same as 1-way fxed effecs ANOVA Source Trea (MS) = 1 + n = + nθt 1 θt s he book s noaon for some fxed effec. Ho : = 1 MST F = MS

3 For he weland example, wha s he varance of he overall LDM esmae? ( μ j) ΣΣ + + Var( y ) = Var n 1 = Var n j ( n) +ΣΣ = 1 n n = + = n n = + n S y To esmae hs S = + = S n ˆ μ MST n + = + n n n n MST n S ˆ μ The conrbuon of weland o weland varably,, o uncerany n our esmae of average LDM s s reduced by he number of welands vsed,. The conrbuon of se o se varably,, s reduced by he number of ses vsed, n.

4 Wha s he opmal number of ses per weland? I coss less me/money o vs anoher se n he same weland raher han ravel o anoher weland. Le C w = cos of addng a weland mappng, ravel ec. Le C S = cos of samplng anoher se n same weland Mnmze Var( y ) Subjec oal cos C w + nc S Calculus n C w = (No n ex) CS Cw Cs If addng anoher weland s expensve, C W spend more me a each weland number of samples a each weland. s w If here s grea varably from weland o weland, w go o more welands o reduce conrbuon o error; number of samples a each weland.

5 smang and Source ˆ ˆ (MS) + n T = MS MST MS = n Toal varance = Var( y) = + Source of Varaon smaor % ˆ T ˆ ( + ) ˆ and Toal ˆ ˆ ˆ + ˆ ( ) + can be used o fgure opmal allocaon for anoher expermen.

6 17.3, 17.4 More Random ffecs Two Facors No replcaon. For example a randomzed block desgn. A fxed A fxed A random Source df B fxed B Random B random A a-1 B b-1 (a-1)(b-1) Toal ab-1 + bθ A + aθ B + bθ A + a B + b + a B Σ Σβ j θa = θb = a 1 b 1 In any even o es for A effec, F = MSA MS The mulplers are only used for componens as above. To smplfy maers, le Q( ) = some quadrac (squared) funcon of ' s Q ( β) = funcon of β j ' s ype random varances when we wan o esmae he varance MS A fxed Source A df a-1 B random + Q( ) B b-1 Toal (a-1)(b-1) ab-1 + a B

7 Two facors wh replcaon (MS) (MS) A Fxed A Fxed df B Fxed B Random A a-1 Q( ) + n + Q( ) B b-1 AB Toal (a-1)(b-1) ab(n-1) abn-1 For all fxed effecs F = MS MS effec error + β + Q( β ) + Q( β, ) + an β + n β For A effecs: MSA s expeced o be bgger han MS only f here s an A effec, > 0. When B s a random effec, e.g. B = person and A = medcne. = 1 Ho : Σ F = MSA / MS AB (MSA) s he same as (MS AB) excep for he poenal A effec, Σ. If we used MSA/MS, he rao could be bg eher because of A effecs or AB effecs. Ths rao would no solae poenal A effecs. Ho Ho : β : β F = MSB / MS F = MS AB / MS Ineracons wh any random effecs are random.

8 17.6 Nesed ffecs Fxed A = Pescde Levels a levels Random B = Tanks b levels. b anks whn each pescde Source df MS A a-1 + n + Q( ) B(A) Toal a(b-1) ab(n-1) abn-1 + n β β Ho : Σ F = MSA / MS B(A) Pescde means are dfferen parly because of pescde effecs Q( ) and parly because dfferen anks were used β If we ook he rao of MSA o MS, he rao could be larger han 1 eher because of pescde effecs or ank o ank varably Usng he rao of MSA o MS B(A) solaes he poenal A effec Σ. The pescde es s he same as f we ook averages for each ank (expermenal un) and dd a 1- way ANOVA wh b replcaes for each pescde level gvng a(b-1) df for error. Ho : β 0 = F = MS B(A) / MS There s no neracon beween A and B(A), no neracon beween an effec and self.

9 Spl Plo Desgn A = 1 n whole plos A = n whole plos δ Whole Plo 1 na T=1 T=3 T= T=3 T=1 T= T= T=1 T=3 A = 1 A = A = 1 Source df MS A a Q( ) T -1 AT Whole Plo(A) Toal (a-1)(-1) a(n-1) a(-1)(n-1) an-1 δ + Q( τ ) + Q( τ ) + δ The error source s essenally T * Whole Plo(A) a(-1)(n-1) df We have more precson for esmang subplo reamens T=1, T=, T=3 are compared n a more conrolled envronmen he same whole plo. The denomnaor n F es for T s smaller (n expeced value) han denomnaor n F es for A. A ffec: Ho : Σ F = MSA / MS Whole Plo(A) The es for A effecs s he same as f we averaged each whole plo, and hen dd a 1-way ANOVA wh n replcaes for each A reamen gvng a(n-1) df for error. T effec: Ho : Στ F = MST / MS AT effec: Ho τ j : Σ F = MS AT / MS