regression - Regression Distribution of Random Error r I ,1...

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Unformatted text preview: Regression Distribution of Random Error r I ,1 L 2) How is e distributed. for each single value ofx? y s: + E) Run cram de+€rwm3flb random Var-"aw A” 9"?“ H4 Lao ‘qufl‘r Wl‘q‘l’er X: 2 6’0 r €3.13! 721230 1‘ EIXITEZQ r15360+flfxz7+£zb Regression Assumptions Co’wfidrnd the mm»: m‘ ranolm cm 4mm 5 if qflumfi‘h‘on} are. vi‘thed; “"2 flgJe’ 05+:"Wi'99’ I*‘5 AM“ 3001/ 1. At any given value of x, the mean of the distribution of e = 0. 205,730 5”” £56” 2 L4 ,‘g‘a mean ‘1 é Cm: é (Ba Wm +6): 2‘ (50+5’5X1fi5“) ff”): 5096' 1L3; Y‘IJ'I‘O ((YJ: 80 +8,x 2. For all values of x, the variance of 6 (oz) is constant. an ,4 (pcqf‘u'ofl -_ 0- .“Ccrtfihm'r Icy)” OZCBQi-firffi) a... a. 6725),) : #2550 ffix) i (race!) (7' 1C)”: 0'3 Ce?) 4. e ’3 associated with any 2 observations are independent. E N0!” Values on X: “’9 i‘fi V‘Qféa 0M4" 9‘0“ H'e’ Va luv} 6'“ X2. In Summaryif-egficfs;w "WC/9’ ’“f’flqrs H‘s “ff/ms? at: )4“ Fa,“ fag flrqugr‘l.‘+y olffi‘i'h’bufi‘aw 0’ y = 1% T'ng + i gamefi ‘ 66m“; 50 1319,! but-wee Ila-an {5 ,t L and war: an (e .‘s 0" Ciark — BIT 2406 Page 14 Regression Variance of 6 (0‘2) The more variability in e, the greater the error of estimation for 9 . 5mm N M Vedanta 0-2 ha; éé-hko, 6741‘ m w'fr'on {or—r iar8€ varffince or? p {.5 no}— qfcureqf-e fie/‘46, and gm Population variance of 6(02) is estimated by $2: 2 SSE SSE S 2 degrees of freedom for error n ~ 2 where SSE = XUi * 9i )2 '7'» 522' -.- 76*75’1'0 .- 33,3951" 5'n-2/----'“*‘ (1‘ 5,; Standard Deviation of e(o-) is estimated by s: 5=Jg=% 5: V383675 1 G/‘i‘lo Clark - BIT 2406 Page 15 Regression Hypothesis Testing Testing the Slope B1 Hypothesis Test on [31 Ho: [51:0 :9 (5,), “51' {4567411 to [Moder/2 H3: [51 at 0 Choose a. ievel C (’0qu CELT,» 14’ Type I 3 rm» ) Test with statistic: .. [’0] .. kw: 51 S. S B] /\i SSxx Two —+crr‘l~eJ 4‘5"" Decision Rule: I will reject Ho if |tcalci > t% n _ 1 M U} 4mm. 7 inf/z, h-2. 0f“ Clark—BIT 2406 rr'caft, 4:; a} 4/7, r “.2, 0’74 "" +61: id f'fqi'fi 71-0“ n»; ‘4’qu Page 16 /V\~2 Regression Grade vs. Study Time Example Exam # Stud Time Grade on Exam x2 1 4 hr 41 164 16 1661 2 9 hr 63 567 61 3969 3 16 hr 87 1392 256 7569 4 23 hr 96 2254 529 9605 52 269 4377 662 22823 Recall SSE = 76.7306 and S = 6.1940 Two-tailedtestatoc=o.05::7 Ngecl’" “91 I" “add ’ 7mg" *’~ Ho: 61:9 +646 3 3/."0 ~: 7300?? :: 9; {’7'15’ C 7%? Ha: 61“) 55/337 l /Jz—06 +~+4ue ; 12.6% 2 “*1 2 “1°28; 7 “JOB .1. 69/5 7 fix/Z, h-2 (an c640”? fleJ'éC‘F Mr H0 1’ x13 " '. W/fé’d‘dy One-tailedtest(a=0.05)::;" flei'edh Ho‘ [f 4-0,. h: >fd 1 (7‘2 Ho: [31:0 f (41:16,?7‘45’ Ha:B1>0 . I‘dln-Z 3 +Orafi-Izg 9..on 1- am, 7 f—df n-2, Réfeoff our H0}(éfl£’ld(‘/'€ ‘9; 70 X “5 “NF”! l‘o ymdrc‘f“ Y Clark — BlT 2406 Page 17 Grade 1 a’q'IZB? N} Regression Confidence Interval for [31 Want to say: We are 911% confident that B, lies within the interval 100 (1-r1)% Confidence Interval: flit“ sB -; 3,009” 5 ‘ 4~ «H305 C-Hwo) - Y _. -- 3 W 7 Wm v } do, 47 : Q‘Jaa C “313(5) u .36“! ‘57: Confiden‘r Hurf [ If)? g 3, 5.. f ’P 4 (“'1 “If-thin fag twang”: Clark — BIT 2406 Page 18 Regression Coefficient of Determination Represents the proportion of vaniation explained by the model. _ .. ~ {Tcn / MOO/b mode] r2:m :1_SSE 5 2", IV‘df‘tq 6"!“ 7 SSW SSW mm Var/(‘qf-‘M ' 2 ssyy :Zu’i ‘VF : ZYEQ Y!) n W0 a .H 52: f’ ———-—-""76‘7 a [234); Z 0. 7695 3 76(55 /' 3. 6’5“; 7, of +"‘e Varl‘fiff‘d"! 6"" G? QXFKQD'IEGI 9y 0 th er Fad-0r; Clark — BIT 2406 Page 19 Regression Coefficient of Correlation The coefficient of oon'elation measures the strength of the linear relationship between x and y. n ’ ‘72 n" I“! {679% 5 my: 3st r z — (—1 3 rs 1) .rssxxssW s = o. ‘i' re! How Regression is Misused Extrapolation (“.0qu beyond fine (4,54 offer fampxp 0(ch Over-Reliance on Outliers and Extremes fa. chili-err Assume Cause-Effect Relationship diffmhg Ca “re ’2‘ Np" "‘e'ms r"; I“ h.“ H"? Nonsense Models flo j/gpeJ n4" fiehfioash¢ Clark — BIT 2406 Page 20 Regression Procedure to Develop a Model Step 1 Hypothesize deterministic nent of model y: 60 +fl! X 't E, 3th: da+ffmr.n;5;r‘5 PEN" + I3. ‘1 Line, 8mp2 Estimate parameters ,, ” am,“qu 59,3, one, 60 “J :9, 8mp3 Evaluate random error distribution Ervor {{qm Calcurqee 55‘; varl‘ah‘g or g (sm‘ver we éefle’; “"2 etc-"11h? Fe- Ffer/r‘c‘f ea'fihe‘fifl (“66:0 W5 € 8mp4 Determine model usefulness H {cream févfi' ~0'nt 1‘6"] (+/../ _ . V,_“ N‘% ,9 m4 l" 8wp5 Use model To We (Jaci- asf-‘Nfi‘fi o‘y Clark — BIT 2406 Page 21 Regression Regression Testing Review BIT 2406 A set of sample data has the fitted least squares line )7 = 49.1016 - 0.9590)(. n = 18, SSE = 218.1624, 2x: 346, 2x2 = 7614, By: 552. £y2 =18,032, ny = 9687 I want to test the slope of the least squares line to determine if there is a statistically significant negative relationship between x and y. 1. What is the null hypothesis? The alternative hypothesis? Halfifco Hq” 640 2. What is the decision rule in: = 0.025? ’(5 “F Ho f-F . t ML <- tan ,1 ;> +wrc < u tater/5‘, '0 41> fag/c <~— 2. 7.20 3. What is the value of the test statistic calculated from the sample data? {‘flw 6| ""‘ “a qO M“ " -~---—-** 1‘ '- 3105‘? 2. 5; gig, :iéqzé, 5synggz—g’fld': ‘ié'iH 4. Can we conclude that x is useful in predicting y with a negative relationship between the two variables at a = 0.025? kg 5. Whatisthe coefficient of determination? :3 -~.: /- fffi A I... &I?—I 62M 3. 03an '—‘ 5’03; Hg #- 55Yy Hwy T— £r2_§ffll V1 6. What is the coefficient of correlation? ~ 7": 7§KV wt: 2?,{9667 low-55w I catamarerer (HQ "fl 7 " .« Lula (“def-fur): "(/03ij :5 ‘— 0'5“??? 7. We are 80% confident that the true value of the slope lies within what interval? fl d i tar - 5mvar‘r56‘to . 76m t f) 5’12 F; f ) igfiflaiaffi‘ql I r: ‘erst- or 7W) t ‘0, 87521 F r.- f I... Clark — BIT 2406 ...
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This note was uploaded on 04/01/2008 for the course BIT 2406 taught by Professor Llclark during the Fall '07 term at Virginia Tech.

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regression - Regression Distribution of Random Error r I ,1...

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