Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator

Transcription

1 South Dakota State Universit Open PRAIRIE: Open Public Research Access Institutional Repositor and Information Exchange Agricultural Experiment Station Technical Bulletins SDSU Agricultural Experiment Station 1978 Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator Paul Evenson Follow this and additional works at: Recommended Citation Evenson, Paul, "Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator" (1978). Agricultural Experiment Station Technical Bulletins This Article is brought to ou for free and open access b the SDSU Agricultural Experiment Station at Open PRAIRIE: Open Public Research Access Institutional Repositor and Information Exchange. It has been accepted for inclusion in Agricultural Experiment Station Technical Bulletins b an authorized administrator of Open PRAIRIE: Open Public Research Access Institutional Repositor and Information Exchange. For more information, please contact michael.biondo@sdstate.edu.

2 TB 50 Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator DDDDD DDDDD Agricultural Experiment Station South Dakota State Universit Brookings, South Dakota

3 3 Calculation of Multiple Regression with Three Independent Variables Using a Programable Pocket Calculator B: : Paul D. Evenson Assoc. Professor Plant Science and Statistics Multiple regression is to develop equations that describe relationships among several variables. This paper describes a multiple regression program for an equation with one dependent and three independent variables, which was written for a HewlettPackard 97 prograrnable "pocket" calculator. Once each variable is entered, the program calculates sums, sums of squares, sums of crossproducts and means of all variables, as well as the number of entries. It also computes the determinant of the matrix, elements of the inverted matrix, and regression coefficients. The intercept is calc1..llated after means of all variables are reentered, and a predicted value for the dependent variable can be determined for an set of independent variables. The multiple regression equation with three independent variables has the form Y =a+ b 1 X 1 + b 2 x 2 + b 3 where a is the intercept; b 1, b 2, and bj are regression coefficients; Y is the dependent variable; and x 1, x 2, and are independent variables. Calculation of Regression Coefficients The normal equations for this multiple regression are: xl x2 XJ LX~l + LXl x 2 b 2 + LXlXJbJ = LXl LX 1 x 2 bl + LX~ 2 + LX 2 xjbj = LX 2 LXlXJbl + LX 2 b 2 + LX}J = LXJY

4 4 where 2 Ex. 1 = EX. 2 l Ex. 1 _ ( EXi )( EX5 ) Ex. x. = EX. X. d 1 J 1 J n n = number of entries The following matrices are to solve this set of equations. 2 Exl 1 Ex 1 x 2 Ex 1 It Ex 1 A = Ex 1 x Ex 2 Ex x ' B = b2, and C = 2 Ex2 Ex 1 Ex 2 2 ExJ b3 E where A B = C To solve for B, multipl both sides of the equation b the inverse of A, A 1. A l A B = C Al Since A 1 A= I, the identit matrix, then I B = C A l or B = C A 1 all al2 al3 Ex 1 bl A 1. C = a21 a22 a23 Ex 2 = B = b2 a31 a32 a33 ExJ b3 IAI where a.. ' s are elements of A 1. 1J is the determinant of A and is in calculating A 1. Calculation of the Intercept where EY EX =~and x. =.:::_:i. n 1 n

5 use1 11~1i ue11uns Multiple Regression (Card 1) A & C Matrices IAI STEP INSTRUCTIONS 1. Enter card 1 2. Do 3 6 for each entr 3. Enter X 1 4. Enter X? 5. Enter X1 6. Enter Y and compute sums, sum of squares, and sum of crossproducts 7. Calculate statistics for A & C matrices ( continued) INPUT DATA/UNITS x1 KEYS 1 ll I L JI I Lt JI I CO i I I t 11 I l A_J CJ CJ CJ [LI CJ CJ CJ CJ CJ CJ CJ ~ I I CJ I I C] I I CJ I I CJ I I CJ I I C] I I CJ I I I 11 I I I ICJ IC] I ICJ L _JI I CJ I I OUTPUT DATA/UNITS x, EX1 0 EY 3 EXf 4 8 n 9 A X 1 X') B C x; D E I EX 1 X;, 0 fx 1 x1 1 EX1Y 2 EX2X A B C D E I

6 6 continued STEP INSTRUCTIONS INPUT DATA/UNITS KEYS OUTPUT DATA/UNITS Ex1 X? 0 I 11 I Ex1x1 1 I IC] EX1Y 2 EX?X1 3 CJ I I EX/ 4 EX1Y 5 ' Ex~ 6 Ex~ 7 C] [1 EX~ 8 E2 9 A ~ x, B X2 C c=jcj X1 D E 4 I C]c=J EX1 0 EX2 1 C] I I EX3 2 LY 3 x, 4 CJ I I x2 5 X1 6 CJ I I 7 l c=j I I 8 C] I I n 9 CJ I I, A I 1 1 I X1 B c=] I I x? C CJ I I X1 D E 8. Calculate determinant IAI CJ CJ I Ir. IC] IAI I IC] C] I I

7 User Instruetions a Multiple Regression Card ( 2) STEP INSTRUCTIONS Enter Card 2 Calculate elements of A 1 Calc~ate regression coefficients and store in Rn', R 1 ', and R? ', respectivel. Re enter x1 Re enter X;> Re enter X1 Re enter Calculate intercept,.. Calculate Y Enter X1 Enter X;, Enter X 1 INPUT DATA/UNITS KEYS I 11 I I A 11 I I 11 I ~~ c=j I I I 11 I ~c=j I 11 I c=] I I c=j I I ~c=j c=j c=]' x1 I r 11 a I x? IR/S 11 I xi []&] I I IB&] I I I f' 11 b I c=j I I X1 I t 11 I X2 I t 11 I Xi I f 11 C I I OUTPUT DATA/UNITS a11 0 a,? 1 a13 2 a21 3 a?? 4 a21 5 a,:n 6 a1? 7 a11 8 n 9 A X1 B x? C X1 D E IAI I T b '3 z b? b, X x1 x2 X1 a

8 8 Example Entr# x2 x Operation Output 1. Enter Card I *** G:) 0.94 Ent t 3.,~. 22 CD Ent t Ent t CD 8.23 GSBA CD 1.13 Ent t G:) 3.48 Ent t CD 1.28 Ent t (I) GSBA

9 (D 2.22 Ent t CD 3.35 Ent t CD 1.86 Ent t Q) 8.11 GSBA 61. CE> GSBB Del DC rx EY rxf DC~ rx n n A 2.22 B 3.35 C 1.86 D 8.11 E 0.00 I EX 1 X rx tx 1 Y rx EX 2 Y r Y A 2.22 B 3.35 C 1.86 D 8.11 E 0.00 I

10 10 Ex 1 x Ex Ex Ex x Ex E Exr Ex~ Exj E2 J A 2.22 B J.35 C 1.86 D 8.11 E 0.00 I EX EX JO 1 EXJ EY xl x XJ n A 2.22 B J.35 C 1.86 D 8.11 E 0.00 I 62. G) GSBC IAI ***

11 Enter Card CD GSBA 1 a of A a of A a of A a of A a of A a of A a of A a of A a of A n IAI 0.00 A 2.22 B 3.35 C 1.86 D 8.11 E I 65. G) b3 b2 bl GSBB 0.01 T 0.19 Z 0.47 Y 0.62 X f CDCD CD 72. J.00 Ci) G) xl 1.32 GSBa x R/S x R/S "l'.t 8.65 R/S,J GSBb a *** x Ent t X2 J.00 Ent t 1.50 GSBc ~ *** A Therefore, Y = x x When xl = 1.00, x2 = 3.00, and x3 = 1.50; Y = 8.72

12 12 Example Card 1 Card 2 94 ENTt 2.23 ENTt ENTt ENTt GSBA EHTt 8.68 GSBA B 8.23 GSBA 2.22 ENT j ENTt "'I 3.35 ENTt HU 1.86 ENTt ENTt 8.11 GSBA GSBA GSBB ENT ENTt B ?. 64 ENT A GSBA B ENTt 129.? C 8.00 A 2.20 EHTt D 2.22 B 08 ENTt E 3.35 C 8.92 GSBR I 1.86 D. 93 NH E 2.25 ENTt I 38 ENTt GSBA 0.80 A GSBB ~ ENTt 2.22 B j..., T ENT C 1 7? 1.45 ENTt D z S.34 GSBA 8.11 E NH 8.00 I X 2.62 ENTt ENTt GSBo GSBA A ~'.,r. R/ S ENTt B R/ S 2.9? ENTt C 8. 5 R/ S 3.68 ENTt D GSBtJ 9.15 bsba E ENT I ENTt ENTt 2.59 ENTt GSBC ENTt 9.40 GSBA 0.00 B Ut 1.58 GSBc 78 ENTt Ut ENTt 0.00 H ENH 2.22 B 9.01 GSBP C 1.18 ENTt 1.86 D 2.64 ENTt 8.11 E 3.16 ENTt 8.00 I 8.7? ~SBA ENTt 2.39 ENTt 23 ENTt 8.11 GSBA ENTt 2.76 ENTt?6 ENTt GSBA ENTt... ( ***

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