ĐĎॹá>ţ˙ lnţ˙˙˙k˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙˙ěĽÁ` řż‰2bjbjËsËs 7ZŠŠ‰*˙˙˙˙˙˙¤€€€€€€€”ü:ü:ü:ü:D@;T”DCş ;ś;ś;ś;ś;ś;ś;ś;ĂBĹBĹBĹBĹBĹBĹB$ţDhfG´éB€=ś;ś;==éB€€ś;ś;ţBSBSBSB=ň€ś;€ś;ĂBSB=ĂBSBSB€€SBś;”; 0´äĺ™Çü:s@úSBĂBC0DCSBHmAÜHSBSBH€gB\ś;v,<TSB€<DÄ<˝ś;ś;ś;éBéBIB ś;ś;ś;DC====”””ä x&„”””x&”””€€€€€€˙˙˙˙ T Tests, ANOVA, and Regression Analysis  Here is a one-sample t test of the null hypothesis that mu = 0: DATA ONESAMPLE; INPUT Y @@; CARDS; 1 2 3 4 5 6 7 8 9 10 PROC MEANS T PRT; RUN; ------------------------------------------------------------------------------------------------ The SAS System The MEANS Procedure Analysis Variable : Y t Value Pr > |t| ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ 5.74 0.0003 ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ------------------------------------------------------------------------------------------------ Now an ANOVA on the same data but with no grouping variable: PROC ANOVA; MODEL Y = ; run; ------------------------------------------------------------------------------------------------ The SAS System The ANOVA Procedure Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 1 302.5000000 302.5000000 33.00 0.0003 Error 9 82.5000000 9.1666667 Uncorrected Total 10 385.0000000 R-Square Coeff Var Root MSE Y Mean 0.000000 55.04819 3.027650 5.500000 Source DF Anova SS Mean Square F Value Pr > F Intercept 1 302.5000000 302.5000000 33.00 0.0003 ------------------------------------------------------------------------------------------------ Notice that the ANOVA F is simply the square of the one-sample t, and the one-tailed p from the ANOVA is identical to the two-tailed p from the t. Now an Regression analysis with Model Y = intercept + error. PROC REG; MODEL Y = ; run; ------------------------------------------------------------------------------------------------ The REG Procedure Model: MODEL1 Dependent Variable: Y Sum of Mean Source DF Squares Square F Value Pr > F Model 0 0 . . . Error 9 82.50000 9.16667 Corrected Total 9 82.50000 Root MSE 3.02765 R-Square 0.0000 Dependent Mean 5.50000 Adj R-Sq 0.0000 Coeff Var 55.04819 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 5.50000 0.95743 5.74 0.0003 ------------------------------------------------------------------------------------------------ Notice that the ANOVA is replicated. Now consider a two independent groups t test with pooled variances, null is mu1-mu2 = 0: DATA TWOSAMPLE; INPUT X Y @@; CARDS; 1 1 1 2 1 3 1 4 1 5 2 6 2 7 2 8 2 9 2 10 PROC TTEST; CLASS X; VAR Y; RUN; ------------------------------------------------------------------------------------------------ The SAS System T-Tests Variable Method Variances DF t Value Pr > |t| Y Pooled Equal 8 -5.00 0.0011 ------------------------------------------------------------------------------------------------ Now an ANOVA on the same data: PROC ANOVA; CLASS X; MODEL Y = X; RUN; ------------------------------------------------------------------------------------------------ The ANOVA Procedure Dependent Variable: Y Sum of Source DF Squares Mean Square F Value Pr > F Model 1 62.50000000 62.50000000 25.00 0.0011 Error 8 20.00000000 2.50000000 Corrected Total 9 82.50000000 R-Square Coeff Var Root MSE Y Mean 0.757576 28.74798 1.581139 5.500000 Source DF Anova SS Mean Square F Value Pr > F X 1 62.50000000 62.50000000 25.00 0.0011 ------------------------------------------------------------------------------------------------ Notice that the ANOVA F is simply the square of the independent samples t and the one-tailed ANOVA p identical to the two-tailed p from t. And finally replication of the ANOVA with a regression analysis: PROC REG; MODEL Y = X; run; ------------------------------------------------------------------------------------------------ The SAS System The REG Procedure Model: MODEL1 Dependent Variable: Y Number of Observations Read 10 Number of Observations Used 10 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model 1 62.50000 62.50000 25.00 0.0011 Error 8 20.00000 2.50000 Corrected Total 9 82.50000 Root MSE 1.58114 R-Square 0.7576 Dependent Mean 5.50000 Adj R-Sq 0.7273 Coeff Var 28.74798 Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > |t| Intercept 1 -2.00000 1.58114 -1.26 0.2415 X 1 5.00000 1.00000 5.00 0.0011  OK, but what if we have more than two groups? Show me that the ANOVA is a regression analysis in that case. Here is the SAS program, with data: data Lotus; input Dose N; Do I=1 to N; Input Illness @@; output; end; cards; 0 20 101 101 101 104 104 105 110 111 111 113 114 79 89 91 94 95 96 99 99 99 10 20 100 65 65 67 68 80 81 82 85 87 87 88 88 91 92 94 95 94 96 96 20 20 64 75 75 76 77 79 79 80 80 81 81 81 82 83 83 85 87 88 90 96 30 20 100 105 108 80 82 85 87 87 87 89 90 90 92 92 92 95 95 97 98 99 40 20 101 102 102 105 108 109 112 119 119 123 82 89 92 94 94 95 95 97 98 99 *****************************************************************************; proc GLM data=Lotus; class Dose; model Illness = Dose / ss1; title 'Here we have a traditional one-way independent samples ANOVA'; run; *****************************************************************************; data Polynomial; set Lotus; Quadratic=Dose*Dose; Cubic=Dose**3; Quartic=Dose**4; proc GLM data=Polynomial; model Illness = Dose Quadratic Cubic Quartic / ss1; title 'Here we have a polynomial regression analysis.'; run; ***************************************************************************** Here is the output: Here we have a traditional one-way independent samples ANOVA 2 The GLM Procedure Dependent Variable: Illness Sum of Source DF Squares Mean Square F Value Pr > F Model 4 6791.54000 1697.88500 20.78 <.0001 Error 95 7762.70000 81.71263 Corrected Total 99 14554.24000 R-Square Coeff Var Root MSE Illness Mean 0.466637 9.799983 9.039504 92.24000 Source DF Type I SS Mean Square F Value Pr > F Dose 4 6791.540000 1697.885000 20.78 <.0001 ------------------------------------------------------------------------------------------------ Here we have a polynomial regression analysis. 3 The GLM Procedure Number of observations 100 ------------------------------------------------------------------------------------------------ Here we have a polynomial regression analysis. 4 The GLM Procedure Dependent Variable: Illness Sum of Source DF Squares Mean Square F Value Pr > F Model 4 6791.54000 1697.88500 20.78 <.0001 Error 95 7762.70000 81.71263 Corrected Total 99 14554.24000 Note that the polynomial regression produced exactly the same F, p, SS, MS, as the traditional ANOVA. R-Square Coeff Var Root MSE Illness Mean 0.466637 9.799983 9.039504 92.24000 Source DF Type I SS Mean Square F Value Pr > F Dose 1 174.845000 174.845000 2.14 0.1468 Quadratic 1 6100.889286 6100.889286 74.66 <.0001 Cubic 1 389.205000 389.205000 4.76 0.0315 Quartic 1 126.600714 126.600714 1.55 0.2163 ------------------------------------------------------------------------------------------------  HYPERLINK "http://core.ecu.edu/psyc/wuenschk/StatsLessons.htm" Return to Wuensch’s Stats Lessons Page November, 2006 ()*+@Aijko{€ˆęŘęÇ˝łŚ˜ŚŽŚrY@Y@Y0h¸őB*CJOJQJ^JfHph˙qĘ ˙˙˙˙0h¸őB*CJOJQJ^JfHphqĘ ˙˙˙˙6h¸ő5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙h¸őOJQJ^JhÁHŮh¸ő6OJQJ^Jh¸őh¸őOJQJ^Jh }:OJQJ^JhătOJQJ^J!jhăthătOJQJU^J#hD4.B* CJOJQJ^JaJph€)h }:h¸őB* CJOJQJ^JaJph€(*k‡Ľź   € ş ť ö ÷ 1 k Ľ ß @ A  œ ý đääÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÚÓÓÚÚ¤gdĹ'X ¤5$9DgdĹ'X ¤¤[$\$gdĹ'X$¤¤[$\$a$gdĹ'X‰2ţĽŠŞŻ°ąˇşź“ ˜ ž ¤ @ A B  ƒ „ ‰ ‹  — š œ ‹ ç˲˲™˛Ë˛‹t‹]‹YUY˲˲™˛Ë˛D hÁHŮh¸őCJOJQJ^JaJh }:h¸ő,hăth¸ő5B* CJOJQJ^JaJph€€,hăth¸ő5B*CJOJQJ^JaJph˙h¸őCJOJQJ^JaJ0h¸őB*CJOJQJ^JfHph˙qĘ ˙˙˙˙0h¸őB*CJOJQJ^JfHphqĘ ˙˙˙˙6h¸ő5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙0h¸őB*CJOJQJ^JfHŕphqĘ ˙˙˙Ŕý ţ _ ` š › ą ˛ ç A B œ  â ă deŻ°ą  fgČ\őőőőőőőőőőőőőőőőőőőőőőőőőőőő ¤5$9DgdĹ'X‹ ‘ • › Čßŕ NOYZ]›Ÿ ŁĽŞą´śéŘÁŘš­š­š­š­š­šĽ‰p‰pWp‰pF hʢh }:CJOJQJ^JaJ0h }:B*CJOJQJ^JfHph˙qĘ ˙˙˙˙0h }:B*CJOJQJ^JfHphqĘ ˙˙˙˙6h }:5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙h }:^JaJh¸őh¸ő6^JaJh¸ő^JaJ,hăth¸ő5B* CJOJQJ^JaJph€€ hÁHŮh¸őCJOJQJ^JaJ,hăth¸ő5B*CJOJQJ^JaJph˙\]›śR‰ÄĹ`aľř,-.yÄřůú45qÄĹőőőőőőőőőőőőőőőőőőőőőőőőőőőő ¤5$9DgdĹ'X zČÉäůű˙  Q]éŘÁŘš­šĽš‰pWpWp>p0h }:B*CJOJQJ^JfHŕphqĘ ˙˙˙Ŕ0h }:B*CJOJQJ^JfHph˙qĘ ˙˙˙˙0h }:B*CJOJQJ^JfHphqĘ ˙˙˙˙6h }:5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙hś|Ľ^JaJh }:h }:6^JaJh }:^JaJ,hăth }:5B* CJOJQJ^JaJph€€ hʢh }:CJOJQJ^JaJ,hăth }:5B*CJOJQJ^JaJph˙Ĺz Ąű 8QrÔ  @A”•čIk’óô./EFőőőőőőőőőőőőőőőőőőőőőőőőőőőő ¤5$9DgdĹ'X]bfimprsÔ×ŰáçIikopuw|€…çÎçβՐĄyĄbĄZR˛Î˛Î9Î90h }:B*CJOJQJ^JfHph˙qĘ ˙˙˙˙h¸ő^JaJh }:^JaJ,hăth }:5B* CJOJQJ^JaJph€€,hăth }:5B*CJOJQJ^JaJph˙ hʢhʢCJOJQJ^JaJ hʢh }:CJOJQJ^JaJ6h }:5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙0h }:B*CJOJQJ^JfHphqĘ ˙˙˙˙0h }:B*CJOJQJ^JfHph˙qĘ ˙˙˙˙…‘’óô %)/\stĽŚŔÁŢßĺć+/0çË粥“‚k‚T‚L@L@L@L@L@LËçh }:h }:6^JaJh }:^JaJ,hăth }:5B* CJOJQJ^JaJph€€,hăth }:5B*CJOJQJ^JaJph˙ hʢh }:CJOJQJ^JaJh }:CJOJQJ^JaJ hʢhʢCJOJQJ^JaJ0h¸őB*CJOJQJ^JfHphqĘ ˙˙˙˙6h }:5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙0h }:B*CJOJQJ^JfHphqĘ ˙˙˙˙F{ŐÖ01vwŹ­ŽřůCDEŸ úű\čé+G¨Šáâőőőőőőőőőőőőőőőőőőőőőőőőőőőő ¤5$9DgdĹ'X035:BEG8 = A G ý"ţ"˙"#”#ĺ%!&'ä˲ËäËĄŠĄsĄeWH@:0:hĹ'XB*PJph˙ hĹ'XPJhĹ'X^JaJjœhĹ'XhĹ'XU^JaJhĹ'XCJOJQJ^JaJh }:CJOJQJ^JaJ,hăth }:5B* CJOJQJ^JaJph€€,hăth }:5B*CJOJQJ^JaJph˙ hʢh }:CJOJQJ^JaJ0h }:B*CJOJQJ^JfHph˙qĘ ˙˙˙˙0h }:B*CJOJQJ^JfHphqĘ ˙˙˙˙6h }:5B* CJOJQJ\^JfHph€qĘ ˙˙˙˙âRŽŇST—ďđH ‹ ż Ŕ Á !W!‹!Œ!!Ç!Č!"W"X"őőőőőőőőőőőőőőőőőőőőőőőőőőőő ¤5$9DgdĹ'XX"Ť"ţ"˙"#o#”#Ą#Ű#â#ç#0$6$u${$š$ż$%%O%ž%Ŕ%Ţ%)&x&y&Ę&'őőőőőőîîîîîîîîîîîîîîîîîîîîî¤gdĹ'X ¤5$9DgdĹ'X'L'š'Ë'(L)Q)â,-\.a.é.'/(/*/+/-///1/3/O/Q/22O2P2Q2w2x2y2ˆ2‰2őďßČßČßČßČßȮȮȮȮȠߔŒz”m”eŒZhĹ'Xhăt^JaJhĹ'X^JaJhăthăt0J^JaJ#j8hăthătU^JaJhăt^JaJjhătU^JaJhĹ'XCJOJPJQJaJ3hĹ'X56B*CJOJPJQJ\]^JaJph˙-hĹ'X5B*CJOJPJQJ\^JaJph˙hĹ'XCJOJPJQJ^JaJ hĹ'XPJhĹ'XB*PJph˙'U'Ł'¸'š'((T(U(q(r(§())\)])˘)Ł)Ř)Ů)Ú)%*&*q*r*s*Í*Î*řîîîééééééééééééééééééééééégdĹ'X ¤5$9DgdĹ'X¤gdĹ'XÎ*(+)+Š+‹+ě+í+&,',f,g,Č,É,*-+-d-e--‚-ˇ-..l.m.˛.ł.č.é.O/P/úúúúúúúúúúúúúúúúúúúúúúúúúúúúúgdĹ'XP/Q/œ//č/é/ę/D0E0Ÿ0ů0S1­12y2z2‰2úúúúúúúúúúúúúđđđ ¤5$9DgdĹ'XgdĹ'X501F:pžjĚ°Đ/ °ŕ=!° "° # $ %°°Đ°Đ ĐœDdčô$<đP đ 3 đŹ¨™ż˙3"ń”•ż((đ€œDdčô$<đP đ 3 đŹ¨™ż˙3"ń”•ż((đ€ďDĐÉęyůşÎŒ‚ŞKŠ ŕÉęyůşÎŒ‚ŞKŠ ~http://core.ecu.edu/psyc/wuenschk/StatsLessons.htmyXô;HŻ,‚]ą'cĽŤ†œX@ń˙X Normal¤x5$7$8$9DH$CJOJQJ_HmH sH tH DA@ň˙ĄD Default Paragraph FontVió˙łV  Table Normal :V ö4Ö4Ö laö (kô˙Á(No List 6ţOň6 SAS¤CJOJPJQJ@Z@@ Plain TextCJOJQJ^Jd$@d Envelope Address!„@ „ü˙„ô˙„đ&€+Dź/„´^„@ ^JaJF%@"F Envelope Return¤CJ^J^^@2^ ¸ő Normal (Web) ¤d¤d5$7$8$9DH$[$\$ OJQJaJ6U@˘A6 ăt Hyperlink >*B*ph˙‰*Z˙˙˙˙(*k‡Ľź€şťö÷1kĽß@Aœýţ_`š›ą˛çABœâădeŻ°ą  fgČ\]›ś   R ‰ Ä Ĺ  ` a ľ ř , - 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