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page 3 cheat sheet - To test Hg: ,8] = El [X and Y are...

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Unformatted text preview: To test Hg: ,8] = El [X and Y are independent} Term": Hg .3; 5'2 '3' “’9 115'? The 1: ETflfiE-Ti‘: In particular: confidence inter-:als for fl] are ofthe 1: _ ilk-ISM form _ MSE Point estimate : margin of error I :r a «SEi _ where SE = s has an? distribution with l and .II — 2 degrees of I, _ z 7 51 '7 -"“'-' ’-' ’- I'll _ 53'.- “If; freedom when HI]: ,8; = {I is true Recall that $3101.79; depends on a decomposition LlllEHl‘ l'E‘gTESSlDfl inference HSSLllflpllDflS Elf-sum Dfaquamd dermiflm 1. Linear relationship E, lJ-l- — = E — .57}: + E (F; ‘ 5"”. l'2 lCheck the scatterplot of Y versus 3: Total $5 = Model 55 + Error 55 lCheck the scatterplot of the residuals T.‘ersus X 53.1— = 53M ‘ 55E Similarly: degrees of freedom [DP] is partitioned: DPT = DFM - DFE. in this case {n—1}=rn—rn—2:I Finallj-fg for each source ofrarrance the ratio 2. lConstant variance lCheck the scatterplot of residuals 's'ersus predicted ralues of‘r’ 3- hidEPE'fldfl-‘T Teal-dim]? SS D? is called the mean square (MS). yielding lCheck how the data were collected (SR5?) MST {mean square total] 4. Normally distributed residuals HEM (meal-1' “ll-‘5“? modal and MSE {mean square error) lCheck normal probability plot of the residuals The test of H3: El = III rersus Ha: £1 # CI is a t—test Sum of I: bl = bl _ Source DF mums Mean square F 5 s, .- I IEmullS I ' 1 555 = EEG DFG E = MSG has a r distribution with H — 2 degrees of freedom when Error 1: -1 see . = we nrr MSE H0: ,9] = o is true 531' DPT - Note: for simple linear regression this is exactly the same as test ofthe os'erall model . i.e. Ilia-'3'} = Film-2) Confidence mien—315 for the mean Prediction intervals for a future responser - The model for the mean is - The model for a response 3" at value 1* is I . = — -I r _ - :I: #3 15h 1'5. Ir — fin + film +5 . _. ' as . _. '_. . . Est 3.11:1- point x the natural estrmate of ILLJ- 1s . At an}, Palm Is the uamfl] Predictor Of}. 15 n1. 2 inc, + Err * '1'“. = 5” + film. :5: - The confidence interval will he ofthe form _ The pmdmfiflfl interval will be CIme form F; i to 2.31—25EL'1 Or I]? i F9. ]__,,I._]S:E1- . CPI" ' Two related confidence 111TE’1‘1'fll concepts — giring _ We can test if Y and X are independent. hr; testing a confidence interval for a gir'en 's'fill'l-E ofa' ' _ _ _ Hg: p = Cl '-.'ersus H_,_: p ;' III using a t-te st - Confidence mten'als for the estnnated mean of}' . . r H — 2 atagisensalue ofa r: ' Prediction 111TE1‘L'fllS for a single future response “'1 — r"! ralue j; at a gir'en s'alue of): has a r-distrihution with H — 2 dfunder Hui p = D - The centers of these 1flTE‘fL‘fllS Will he the same. the margins of error will be different - the latter hating ' P- am he Dbmmea dire“? from the ANGIE-‘3" tame 5‘" a larger margin oferro-r ‘3," ' ‘. _—-~-" a o {I'll Is II - Most software gives confidence inters'als and 11-1 = 55M = -'-1 prediction inten'als — Data Desk does not SST I. I _|_ 1.- _ I I z . . . . . ' We will concentrate on interpretatron Set the log odds to he a linear eomhinatron ofthe ' Examine this ratio of's'ariance estimates as an F test predictor variable-3 51' _ MS between groups x F=§_ 1;] fi— 35+ ..X-.+_..— I 5,;- Is-IS within groups ELI-pl; ’5; ’al ' ’8' _ JEEP '3 Variance within groups ' Since we assumed equal is for allI groups. all sample SUs should be estimating the common 5 ' Thus we combine them in a pooled estimate (same as for the 2-samp1e pooled r—testj ' Pooled estimate of o2 [the variance within groups) is 1 . [ML—1:]sli—l'a3—l:]s§— +[iri3—lli'1l l-flJ—lill+l1?t3—l.:l+ . _ . ' The sub script W refers to this within groups estimate +[HI —l‘.I ' Also known as the “mean square {MS} within groups" Variance between groups If the null hypothesis. H0: .111 = u: = . . . = ILLI. is true then it is as if we are sampling Itimes from the same population. with mean !.L and standard deviation o Recall fiom the sampling distribution of sample means .5 so Hgf'e-N-‘JEI .o' . Iku. 5| . |_ u _| Thus under H5: :7: can be estimated by 1 Li“; [E —T}" +?t_._ [E —T]' + ._. +HI{JT'] —f_j. 1—1 The subscript E refers to this between groups estimate uIIJ I." .- _-'-"slso known as the “mean square EMS} between group s“ and “mean Equal-E mini {MEET- ' ICIInlsr a T.‘alid estimate of o: if I-L.: true. otherwise inflated Contrasts ' After the omnibus F test has shown orerall significance investigate other comparisons using contrasts ' A. contrast is a linear combination of Hf“: is: = E anti. where 2 :1I = D (is: is the Greek "psi":| - To test Hg: u] = In; =. . . = u: versus H_,_: not all Ill;- are equal we use the F statistic Corresponding contrast of sample means is c = E of} i For example: consider the bone density stud}r To compare group ‘1 s'eisus groups I + 3 use or = .HL — ’ui + #3 =[:1]ir-_ + (—1523,!!3 + (—1.5 211;; To compare group 2 versus group 3 use or = fl] — In; = (Dir; — {—lhg ' To test the hypothesis 1 3g _ MS between groups _ MSG :‘rIEi within groups MSE F: has an F distribution with I— 1 and N —1 degrees of freedom when H.) is true 1 For the contrast defined as {far :Eflflti}. where Eur. = CI Hg: or = 0 versus H_,_: or sf 1] _ _ we use a t—test ' The corresponding sample estimate is M‘s—an sE. a . ' The standard error of c is a- 7—5 wheres = Q P hiss has a t distii'oution with degrees of fleedom associated with s]: under H: [test can be 1—side-d or 3-sidedj 1-1 ssa ssa-ora Lisa-rise J_1 sss ssarors srsa-srsr (1-13m; ssan SEAR-DEAR sisan-‘s-ise s ssgore SE _ I F tests for main efi'ect r—‘i. main efi‘ect B and interaction 931E (note all are dirided by MSE} ' I'r’ISE is still our estimate ofo-T (ranance ofthe residuals) ...
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This note was uploaded on 04/10/2008 for the course STAT 578 taught by Professor Kirnan during the Spring '07 term at Harvard.

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page 3 cheat sheet - To test Hg: ,8] = El [X and Y are...

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