5. Sample Size, Power & Thresholds

Transcription

1 5. Saple Size, Power & Tresolds review of saple size for t-test (known QTL) saple size, arker spacing & power Zeng talk at Plant & Anial Genoe XI, 2003 (statgen.ncsu.edu/zeng/qtlpower-presentation.pdf) QTL tresolds & genoe-wide error rates positive false detection rates selection bias c Broan, Curcill, Yandell, Zeng review saple size for t-test set up test under "null ypotesis" of no QTL two-sided test for difference size arbitrary--usually α =.05 deterine power or saple size under alternative power = cance to reject null if alternative true saple size (n/4 per MM,) follows square root idea µ MM µ d t = = σ 8/ n SE size/2 : pr( µ µ > c no QTL) = α / 2 power : pr( µ MM µ > c QTL) = 1 β MM 2 n = 2[( zα / 2 + zβ ) σ / d] c Broan, Curcill, Yandell, Zeng 2

2 roug guide: d = 2c = 4SE for power = 1 α/2 c Broan, Curcill, Yandell, Zeng 3 power increase wit saple size (nubers in figures are n/4) power c/σ c Broan, Curcill, Yandell, Zeng 4

3 5.3 QTL tresolds for IM IM scans across loci λ in genoe evidence for QTL in large LOD(λ) LOD(λ) = LR set genoe-wide LOD tresold protect against one or ore false positives rougly adjust size of eac individual test LOD distribution under null ypotesis at any particular location λ depends on design (1 df for BC, 2 df for F2) LOD( λ) ~ 2 or 2 = LR χ1 χ c Broan, Curcill, Yandell, Zeng 5 tresold: fro point to genoe (developed for BC) axiu LR witin arker interval χ 2 wit d.f. between 1 and 2 d.f. close to 1 for sall interval sparse arker ap across genoe assue M independent arker intervals use Bonferroni correction to level (α α/m) ax ax λ in interval λ in genoe 2 LR ( λ) ~ χ,1< ν < 2 LR( λ) ~? c Broan, Curcill, Yandell, Zeng 6 ν

4 LR, LOD and point-wise p-values p-value p-value LR LOD 1 d.f. 2 d.f < c Broan, Curcill, Yandell, Zeng 7 genoe-wide tresold: teory dense arker ap: arkers everywere LR test statistics are correlated correlation drops off quickly wit distance no correlation for unlinked arkers teory: Ornstein-Ulenbeck process C = nuber of croosoes, G = lengt of genoe in cm t = genoe-wide tresold value α t = corresponding point-wise significance level pr(ax LR( λ) > t) = α ( C + 2Gt) α λ in genoe t 2 wit pr( χ1 > t) = αt c Broan, Curcill, Yandell, Zeng 8

5 point-wise & genoe-wide distributions backcross one QTL one locus (solid) dense ap (das) pr(lod>t) = area under curve (fro Broan 2001 Lab Anial) c Broan, Curcill, Yandell, Zeng 9 p-value point-wise and genoe-wide p-values for backcross LOD tresold p-value 1e-07 1e LOD tresold c Broan, Curcill, Yandell, Zeng 10

6 point-wise and genoe-wide p-values for F2 intercross p-value p-value 1e-06 1e LOD tresold LOD tresold c Broan, Curcill, Yandell, Zeng 11 LOD wit QTL backcross genoe-wide 3=tresold das: no QTL solid: n=100 dot: n=200 (fro Broan 2001 Lab Anial) no QTL n=100 n=200 c Broan, Curcill, Yandell, Zeng 12

7 5.4 false detection rates and tresolds tresold: test QTL across genoe size = pr( LOD(λ) > tresold no QTL at λ ) guards against a single false detection very conservative on genoe-wide basis difficult to extend to ultiple QTL positive false discovery rate (Storey 2001) pfdr = pr( no QTL at λ LOD(λ) > tresold ) post-oc estiate of proportion of false positives easily extended to ultiple QTL in Bayesian context c Broan, Curcill, Yandell, Zeng 13 pfdr for interval apping pfdr = pr(no QTL)*size pr(no QTL)*size+pr(QTL)*power decide on tresold (e.g. genoe-wide) power = pr( LOD(λ) > tresold QTL at λ ) proportion of genoe wit LOD > tresold size = pr( LOD(λ) > tresold no QTL at λ ) point-wise p-value of tresold prior = pr( no QTL at λ ) proportion of genoe wit LOD < "sall value" divided by point-wise p-value for "sall value" c Broan, Curcill, Yandell, Zeng 14

8 IM for ypertense ice 8 6 lod LOD X genoe in cm c Broan, Curcill, Yandell, Zeng 15 istogras of LOD and p-value relative frequency 1.2 red lines at LOD=1.5,3 relative frequency red line at prior pr(no QTL) LOD p-value c Broan, Curcill, Yandell, Zeng 16

9 pfdr for yperactive ice pr(no QTL) BH pfdr(-) and size(.) LOD = Storey pfdr(-) p-value p-value c Broan, Curcill, Yandell, Zeng 17 false detection rates and tresolds ultiple coparisons: test QTL across genoe size = pr( LOD(λ) > tresold no QTL at λ ) tresold guards against a single false detection very conservative on genoe-wide basis difficult to extend to ultiple QTL positive false discovery rate (Storey 2001) pfdr = pr( no QTL at λ LOD(λ) > tresold ) Bayesian posterior HPD region based on tresold Λ ={λ LOD(λ) > tresold } {λ pr(λ Y,X, ) large } extends naturally to ultiple QTL c Broan, Curcill, Yandell, Zeng 18

10 SCD expression in B6/BTBR 4 3 lod LOD c2 c5 c9 genoe in cm c Broan, Curcill, Yandell, Zeng 19 istogras of LOD and p for SCD relative frequency relative frequency LOD p-value c Broan, Curcill, Yandell, Zeng 20

11 pfdr estiation curves for SCD pr(no QTL) BH pfdr(-) and size(.) Storey pfdr(-) p-value p-value c Broan, Curcill, Yandell, Zeng 21 pfdr and QTL posterior positive false detection rate pfdr = pr( no QTL at λ Y,X, λ in Λ ) pfdr = pr(h=0)*size pr(=0)*size+pr(>0)*power power = posterior = pr(qtl in Λ Y,X, >0 ) size = (lengt of Λ) / (lengt of genoe) extends to oter odel coparisons = 1 vs. = 2 or ore QTL pattern = c1,c2,c3 vs. pattern > 2*c1,c2,c3 c Broan, Curcill, Yandell, Zeng 22

12 trans-acting QTL for SCD1 doinance? c Broan, Curcill, Yandell, Zeng 23 SCD posterior for cr 2 density(x = data, bw = bw) istogra of posterior Density Frequency N = Bandwidt = 2 density c Broan, Curcill, Yandell, Zeng 24

13 SCD posterior for cr 5 SCD on c5 SCD on c5 Density Density N = Bandwidt = 2 Density c Broan, Curcill, Yandell, Zeng 25 SCD posterior for cr 9 SCD on c9 SCD on c9 Density Density N = Bandwidt = 2 Density c Broan, Curcill, Yandell, Zeng 26

14 pfdr for SCD1 analysis pr( H=0 p>size ) prior probability fraction of posterior found in tails BH pfdr(-) and size(.) Storey pfdr(-) relative size of HPD region pr( locus in HPD >0 ) c Broan, Curcill, Yandell, Zeng 27 1-D and 2-D arginals pr(qtl at λ Y,X, ) unlinked loci linked loci c Broan, Curcill, Yandell, Zeng 28

15 5.5 selection bias locus bias: locus picked via axiu LOD(λ) danger of gost QTL (Haley Knott 1992) CIM or Haley-Knott estiate can be biased effect bias: estiated effects θ often too large Utz, Melcinger Scön (2000); Broan (2001) consider oderate sized effect cance of issing QTL: undetected QTL not reported estiated effect is conditional on detection estiates of effects and genetic variance can be inflated confidence intervals for locus can be quite large c Broan, Curcill, Yandell, Zeng 29 selection bias wite=issed grey = observed above 8 (say) left das = true ean rigt das = obs. ean (fro Broan 2001 Lab Anial) c Broan, Curcill, Yandell, Zeng 30

16 independent validation separate study on sae parent lines Beavis (1994) siulation study Beavis et al. (1994 Crop Sci) field study Melcinger et al. (1998 Genetics) field study resapling evaluation Beavis (1994) siulation study Visscer et al. (1996) bootstrap Utz et al. (2000 Genetics) cross validation` c Broan, Curcill, Yandell, Zeng 31 any QTL of sall effect? repeated field studies yield different QTL environent effect? genetic differences: parents, independent cross? cance variation? siulation studies (Beavis 1994, 1998) 10 QTL of sae size only cance variation repeated trials yield different QTL again c Broan, Curcill, Yandell, Zeng 32

17 Beavis (1998) QTL Analyses estiated nuber & agnitudes nuber of QTL yy y y y y y genetic variation y y y yy y y y y y saple size saple size c Broan, Curcill, Yandell, Zeng 33 cross-validation idea saple fro genetic cross (e.g. F2 intercross) divide into several subsaples (e.g. k=5) 344 F2 69(S1),69(S2),69(S3),69(S4),68(S5) ES: estiate wit soe (4) subsaples predict QTL loci λ and effects θ using CIM predict eritability 2 : 2 = 2 TS: test wit oter (1) subsaples σ + re use loci fro ES to predict effects θ proportion of genotypic variance explained regress predicted effects on penotype adjusted R 2 proportion = adjusted R 2 / predicted 2 σ σ 2 g 2 ge e + σ 2 g c Broan, Curcill, Yandell, Zeng 34

18 Utz et al. (2000) design ES: estiation set u = 3 environents k 1 = 4 subsaples TS: test sets for CV CV/G: genotypic sapling sae environents 1 subsaple CV/E: environental sapling e = 1 environent sae subsaples CV/GE: unbiased new environent, subsaple c Broan, Curcill, Yandell, Zeng 35 Utz et al. (2000) resapling estiation ES( ) cross validation CV/E (box) CV/G (diaond) CV/GE (open tri) independent validation VS1 (up triangle) VS2 (down triangle) subsaples (k) to entire data set (DS) c Broan, Curcill, Yandell, Zeng 36

19 explained genotypic variance c Broan, Curcill, Yandell, Zeng 37 explained variance for plant eigt cross validation: 9 environents u environents for estiation, e = 9 u for test independent: VS1,VS2 = 4,5 environents c Broan, Curcill, Yandell, Zeng 38

20 1376 CV replicates estiated effects and LOD by locus c Broan, Curcill, Yandell, Zeng 39 siilar tests? a cautionary tale Goffinet Mangin (1998 Genetics) Haley-Knott regression test 1 vs 2 QTL on sae croosoe size = pr(detect first and second QTL only one QTL) power = pr(detect first and second QTL two QTL) siulations: power to detect ore tan 1 results: tests are not siilar Type I error depends on effects, loci 2 QTL odel better tan 1 QTL probles wit tis approac ay be artifact of using Haley-Knott regression size and power conditional on detection of first QTL c Broan, Curcill, Yandell, Zeng 40

21 tests for second QTL 1. first fixed QTL first QTL position known (estiated) 2. fixed first-qtl interval first QTL interval known 3. ultiple QTL CIM (or MQM) odel: cofactors 4. two QTL LR test wit unconstrained positions 5. likeliood sape correct for bias of QTL at oter arkers c Broan, Curcill, Yandell, Zeng 41 size depends on effect of QTL 1 (Type I error wit 5% critical value; Table 1, Goffinet & Mangin 1998) T1 T2 T3 percent percent percent position T position T position percent percent a=0.5 a=0.4 a=0.3 a= position position c Broan, Curcill, Yandell, Zeng 42

22 power depends on effect & distance to linked QTL (Table 3-4, Goffinet & Mangin 1998) T1 T2 T3 percent percent percent percent coupl repul coupl repul 80cM 40cM T4 coupl repul coupl repul 80cM 40cM percent coupl repul coupl repul 80cM 40cM T5 coupl repul coupl repul 80cM 40cM QTL effects coupl repul coupl repul 80cM 40cM a1 = 0.3, a2 = 0.3 a1 = 0.5, a2 = 0.5 a1 = 0.4, a2 = 0.4 a1 = 0.4, a2 = 0.4 a1 = 0.5, a2 = 0.5 a1 = 1, a2 = 1 a1 = 0.7, a2 = 0.7 a1 = 0.7, a2 = 0.7 c Broan, Curcill, Yandell, Zeng 43 proble: regression approxiation paradox goes away if we use likeliood recall probles wit Haley-Knott: wrong variance ixture over QTL genotypes ignored ML yields correct size under null of no QTL ig power under alternatives Kao (2000 Genetics) siulations wit 2 = 0.1,0.3,0.5 no epistasis REG/ML ratio wit epistasis c Broan, Curcill, Yandell, Zeng 44