Gene Expression Data Analysis (II) statistical issues in spotted arrays

Transcription

1 STATC4 Sprg 005 Lecture Data ad fgures are from Wg Wog s computatoal bology course at Harvard Gee Expresso Data Aalyss (II) statstcal ssues spotted arrays Below shows part of a result fle from mage aalyss Note that. Gree laser beam at 53m. Red laser beam at 635 m 3. F635 Mea the mea value of red testes of frot pxels 4. B635 Mea the mea value of red testes of backgroud pxels 5. Spot red testy s usually defed as F635 Mea - B635 Mea, a backgroud corrected red testy

2 Normalzato The purpose of ormalzato s to adjust mcroarray data for effects whch arse from varato the techology (e.g. dye effects, slde effects) rather tha from bologcal dffereces betwee the two samples. Why ormalzato s eeded? Commoly used otatos:. Cy5 ad Cy3 deote the backgroud-corrected red ad gree testes for each spot. M = log ( Cy5) log ( Cy3) measures the dfferece betwee the red ad gree testes of the spot.. 3. A = (log ( Cy5) + log ( Cy3)) / measures the overall brghtess of the spot. O ths scale, M=0 represets equal expresso; M= represets a -fold chage betwee the two samples. Calbrato expermets: the same samples are appled o both dyes (so we should expect M=0) ad Purple ad orage represet two replcate sldes (so we should expect o dfferece betwee purple ad orage pots). However the above fgure s dfferet from what we have expected. The devato of M from zero mples the dye effects, ad the dfferece betwee the orage ad purple pots mples the slde effects. The calbrato expermets show that ormalzato s ecessary cdna array aalyss. Geeral dea behd ormalzato: Dye effect : Cy5 s usually more bleached tha Cy3. Slde effect: The ormalzato factor s slde depedet. Usually eed to assume that most gees are ot dfferetally expressed or up- ad dow-regulated gees roughly cacel out the expresso effect. Curret popular methods of ormalzato (overvew): House-keepg gees: Select a set of o-dfferetally expressed gees accordg to expereces. The use these gees to ormalze. Costat ormalzato factor : Use mea or meda of each dye to ormalze. ANOVA model (Churchll s group. Paper readg s suggested ot requred ) Average-testy-depedet ormalzato: Robust olear regresso (Lowess) appled o whole geome. (Speed s group) Select varat gees computatoally (rak-varat method). The apply Lowess. (Wog s group)

3 Normalzato method : P-wse ormalzato usg all the gees (Speed Group, paper readg s suggested, ot requred: It requres the assumpto that up- ad dow-regulated gees wth smlar average testes (deoted A) are roughly cacelled out or otherwse most gees rema uchaged. Normalzato method : Rak-varat method (Tseg et al. 00; readg s suggested ot requred Idea: If a partcular gee s up- or dow- regulated, the ts Cy5 rak amog whole geome wll sgfcatly dfferet from Cy3 rak. Iteratve selecto helps to select a more coserved varat set whe umber of gees s large. Blue pots are varat gees selected by rak-varat method. Red curves are estmated by Lowess ad extrapolato.

4 Dowstream aalyss (Approaches to assess expresso level) Wth replcate sldes: Tradtoal t-test Permutato t-test (Please read! ) ANOVA model (Suggested. ) Appedx I: Z-test, t-test, ad cofdece terval A test statstcs s used to measure the dfferece betwee the data ad what s expected o the ull hypothess. The choce of test statstcs depeds o the model ad the hypothess beg cosdered. Oe Sample Z-test Statstcs The oe sample z-test statstcs s: observed expected z =. z says stadard errors away a observed values s from stadard error ts expected value, where the expected value s calculated usg the ull hypothess. z-test s ofte used as a hypothess testg for the mea of oe sample wth kow varace or large sample sze. The observed sgfcace level s the chace of gettg a test statstc as extreme as, or more extreme tha, the observed oe. The chace s computed o the bass that the ull hypothess s rght. The smaller the chace s, the stroger the evdece agast the ull. As z statstcs s approxmately stadard ormal dstrbuted, we ca use the ormal dstrbuto to compute the sgfcace level of the z value we obtaed. For example: We toss a co 0,000 tmes ad we observe 5,67 heads. We wat to kow whether the chace of heads equal to 50%? Or whether the co s a far co? Null hypothess: the co we tossed s a far co. The observed chace of heads equal to 5,67 0, 000 = The expected chace of heads for a far co s 0.5. Letx = for a head at the th toss, x = 0 otherwse. Let head chace estmate ( x ) s: 0,000 x = x /0, 000. The the stadard error for our 0,000 ( x 0.567) (0, 000 ) s SE = = = , 000 So we have z = = As z s approxmately N(0,), P(z>3.34)= So we reject the ull hypothess There are too may heads to expla as chace varato. Note that the SE ca also be calculated aother way as we kow the varace of observg a head uder the ull hypothess:

5 σ = Var( X ) = E( X E( X )) = E( X 0.5) = 0.5( 0.5) + 0.5(0 0.5) = 0.5. Thus σ SE = = So sample varace z = = = Ths tells that whe sample s large, the σ / 0, s = ( x x) ( ) s a good approxmato to populato varace σ. That s why z-test s ofte used as a hypothess testg for the mea of oe sample wth kow varace or large sample sze. The t-test Wth small samples, the z-test has to be modfed. t-test s ofte used as a hypothess testg for the mea of oe sample wth small sample sze ad wthout kow varace. T-test was veted by W. S. Gossett (Eglad, ). Gossett worked as a executve the Gues Brewery, where he wet after takg hs degree at Oxford. He publshed uder the pe ame Studet because hs employers dd ot wat the competto to realze how useful the results could be. Studet's t-test ( observed expected z = ) deals wth the problems assocated wth ferece based o small stadard error samples: the calculated mea ( X ) ad stadard error may by chace devate from the real mea ad stadard devato (σ ), or other words, the sample varace s = ( x x) ( ) s ot a good estmate for whe the sample sze s small. I ths stuato, observed expected z = s ot approxmately ormally stadard error dstrbuted. Istead, t s t-dstrbuted by W. S. Gossett. Normal vs Studet σ desty Normal Curve studet Curve x I above fgure, the black le s Studet s curve for 4 degrees of freedom. The red le s a ormal curve for comparso. The oly parameter the t-dstrbuto s the degrees of freedom. I the preset cotext, degrees of freedom = sample sze - =. As goes to fty, the t-dstrbuto coverges to the stadard ormal dstrbuto.

6 From above, eve we have the same formula of Statstcs for z-test ad t-test, the sgfcace level wll be determed by dfferet dstrbuto curves uder dfferet crcumstaces. For a large sample, a ormal curve s used to determe the sgfcace level. For a small sample, a studet curve s used to determe the sgfcace level. Cofdece Itervals A cofdece terval s a rage of values that has a hgh probablty of cotag the parameter beg estmated. The 95% cofdece terval s costructed such a way that 95% of such tervals wll cota the parameter.

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