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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: ETﬂfiETi‘: In particular: conﬁdence inter:als for ﬂ] are ofthe
1: _ ilkISM 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 Elfsum Dfaquamd dermiﬂm 1. Linear relationship E, lJl — = 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 Finalljfg for each source ofrarrance the ratio 2. lConstant variance lCheck the scatterplot of residuals 's'ersus
predicted ralues of‘r’ 3 hidEPE'ﬂdﬂ‘T Tealdim]? 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 (meal1' “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'} = Film2) 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'lE ofa' ' _ _ _ Hg: p = Cl '.'ersus H_,_: p ;' III using a tte 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 rdistrihution 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 oferror ‘3," ' ‘. _—~"
a o {I'll Is II
 Most software gives confidence inters'als and 111 = 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 variable3 51' _ MS between groups x F=§_ 1;] ﬁ— 35+ ..X.+_..— I
5,; IsIS within groups ELIpl; ’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 2samp1e pooled r—testj ' Pooled estimate of o2 [the variance within groups) is
1 . [ML—1:]sli—l'a3—l:]s§— +[iri3—lli'1l
lﬂJ—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'eN‘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 EqualE mini {MEET ' ICIInlsr a T.‘alid estimate of o: if IL.: 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—sided or 3sidedj 11 ssa ssaora Lisarise J_1 sss ssarors srsasrsr (113m; ssan SEARDEAR sisan‘sise
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 ofoT (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.
 Spring '07
 Kirnan

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