The calculation of real-time PCR ratios by means of Monte Carlo Simulation or high-order Taylor expansion

Transcription

1 The calculaton o real-tme PCR ratos by means o Monte Carlo Smulaton or hgh-order Taylor expanson Andrej-Nkola Spess Department o Andrology, Unversty Hosptal Hamburg-Eppendor

2 Do we need error propagaton or WHY THE FUSS...? I we have some nput varables 1,,..., each wth some nherent error ( 1, (,... (, then we nevtably must calculate (estmate) the error o Y = ( 1,,..., ). We COULD gnore ths, but we shouldn t, because t s good practce to descrbe the uncertanty o any acqured measurement.

3 Commonly used methods or estmatng the uncertanty o qpcr ratos Usng the rst-order Taylor expanson on the errors estmated rom threshold cycles and the ecency obtaned rom a dluton calbraton curve (Karlen et al., BMC Bonormatcs 007, 8:131) An approach smlar to 1) by usng Gaussan error propagaton as mplemented n the qbase quantcaton ramework (Hellemans et al., Genome Bology 007, 8:R19) Usng a permutaton regme by reallocatng the threshold cycles between sample and control groups as mplemented n the derent REST sotware versons (Pal et al., NAR 00, 30:E36) An extended Gaussan error propagaton approach that also takes the covarance structure between the estmated varables nto account (Nordgard et al, Anal Bochem 006, 356: )

4 The our derent approaches or estmatng uncertantes Gaussan error propagaton Permutaton approach Monte Carlo smulaton Hgh-order Taylor expanson assumes normalty dstrbuton ndependent dstrbuton ndependent dstrbuton ndependent parametrc stochastc stochastc parametrc computatonally (arly) smple computatonally (arly) demandng computatonally very demandng arthmetcally very demandng restrcted to small errors (1%) more tolerant to error range very tolerant to error range more tolerant to error range

5 e.c e.s The our approaches exempled... wth qpcr ratos #1: Gaussan error propagaton e.c e.s re.c re.s go.c go.s (go.c-go.s) (re.c-re.s) mean s.d Y 1.1% 1.3% 1.% 0.9% = 5.93 ± %

6 The our approaches exempled... wth qpcr ratos #: Permutaton approach (REST) e.c e.s re.c re.s go.c go.s e.c e.s (go.c-go.s) (re.c-re.s) Rato = 5.97 Condence 95% = [.983; 9.996] Beware: Only 80 possble combnatons, n > 80 repettve!

7 The our approaches exempled... wth qpcr ratos #3: Monte Carlo smulaton e.c e.s re.c re.s go.c go.s e.c e.s (go.c-go.s) (re.c-re.s) mean s.d Hstogram o x Hstogram o x Hstogram o x Hstogram o x 6.91 ± 4.? 5.89 ± 3.17? Frequency Hstogram o res$data.sm$eval Frequency x Frequency x Frequency x Frequency x res$data.sm $eval

8 The our approaches exempled... wth qpcr ratos #4: Hgh-order Taylor expanson e.c e.s re.c re.s go.c go.s e.c e.s (go.c-go.s) (re.c-re.s) mean s.d k 1 Y x x x k ( k T F C F ) = 5.93 ±.47! (5.93 ±.41)

9 Monte Carlo smulaton: Many Random samples that characterze my varable How does t work? I I have some varable ± C, I generate dataponts wth mean E and standard devaton C. I do ths or every varable n my ormula ( 1,,... ). Then I do my calculaton wth each o my generated dataponts,.e. 1 [ ] * [ ]. Ths gves a dataset o results. From ths dataset I calculate Y ± C. Advantage: I I plot a hstogram o all dataponts, I can see the dstrbuton o Y[ ]. Ths can tell me the mean and s.d. o Y s a realstc estmate.

10 Composte plot or Y random numbers wth mean = 5 and s.d. = mean = (315) s.d. = 95.5 (9.4%) skewness = Y Y

11 Composte plot or Y random numbers wth mean = 5 and s.d. = Y mean = (315) s.d. = 1744 (50%) skewness = Y

12 e.c e.s (go.c-go.s) (re.c-re.s) Monte Carlo smulaton: So what does that mean or rato calculaton? 0.1 % error e+00 e+04 4e+04 6e+04 8e+04 1e % error

13 e.c e.s (go.c-go.s) (re.c-re.s) Monte Carlo smulaton: So what does that mean or rato calculaton? % error % error 0e+00 e+04 4e+04 6e+04 8e+04 1e

14 Gaussan error propagaton aka Frst-order Taylor Expanson vs Hgh-order Taylor Expanson What`s the benet? It s possble that the Frst-order Taylor Expanson does not sucently estmate the propagated error (underestmaton). As we have seen, the assumpton o normalty holds only or VERY small errors that are n a range we seldomly encounter n qpcr analyss.

15 'propagate': an R uncton or general error propagaton General dentons: = Y Y = Gaussan Error/ Frst order Taylor + j jì j j Gaussan Error/ Frst order Taylor wth Covarance T Y = FxC xfx

16 k-order Taylor wth Covarance Y = + + j + j 1 j jì j j jì j ! j j j jì j k j jì j k k 1 = Y x x x k ( F C k F T )! DD <- uncton(expr, name, order = 1) { (order < 1) stop("'order' must be >= 1") (order == 1) D(expr,name) else DD(D(expr, name), name, order - 1) } C <- dag(sd^)

17 e.c e.s Hgher dervatves: maybe or your worst enemy... (go.c-go.s) (re.c-re.s) rst-order [[1]] [[1]][[1]] e.c^((go.c - go.s) - 1) * (go.c - go.s)/(e.r^(re.c - re.s)) [[1]][[]] -((e.c^(go.c - go.s)) * (e.r^((re.c - re.s) - 1) * (re.c - re.s))/(e.r^(re.c - re.s))^) [[1]][[3]] -((e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r))/(e.r^(re.c - re.s))^) [[1]][[4]] (e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r))/(e.r^(re.c - re.s))^ [[1]][[5]] e.c^(go.c - go.s) * log(e.c)/(e.r^(re.c - re.s)) [[1]][[6]] -(e.c^(go.c - go.s) * log(e.c)/(e.r^(re.c - re.s)))

18 (only) thrd-order partal dervatve o re.s! (e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r) * log(e.r) * log(e.r))/(e.r^(re.c - re.s))^ - (e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r) * log(e.r)) * ( * (e.r^(re.c - re.s) * log(e.r) * (e.r^(re.c - re.s))))/((e.r^(re.c - re.s))^)^ - (((e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r)) * ( * (e.r^(re.c - re.s) * log(e.r) * (e.r^(re.c - re.s) * log(e.r)) + e.r^(re.c - re.s) * log(e.r) * log(e.r) * (e.r^(re.c - re.s)))) + (e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r) * log(e.r)) * ( * (e.r^(re.c - re.s) * log(e.r) * (e.r^(re.c - re.s)))))/((e.r^(re.c - re.s))^)^ - (e.c^(go.c - go.s)) * (e.r^(re.c - re.s) * log(e.r)) * ( * (e.r^(re.c - re.s) * log(e.r) * (e.r^(re.c - re.s)))) * ( * ( * (e.r^(re.c - re.s) * log(e.r) * (e.r^(re.c - re.s))) * ((e.r^(re.c - re.s))^)))/(((e.r^(re.c - re.s))^)^)^)

19 What s more mportant or error propagaton: Reducng varance n ecency or ct s? % error on E % error on ct s 0.% error on E % error on ct s % error on E 0.% error on ct s

20 Fnally... How does t all compare? e.c e.s re.c re.s go.c go.s % error Gaussan permutaton (REST) Monte Carlo th-order Taylor

21 Please consder qpcr! ( Spess et al., BMC Bonormatcs (008) Rtz & Spess, Bonormatcs (008) * t 4- and asymmetrc 5-parameter sgmodal models * calculate ecency rom sgmodal, exponental and dluton curves * do model selecton * batch analyss o many runs * average multple reerence curves * calculate ratos and errors thereo (MC, propagaton) * many more...