11.application of matrix algebra to multivariate data using standardize scores
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011 Application of Matrix Algebra to Multivariate Data Using Standardize Scores Aitusi Daniel Department of Statistics, P.M.B 13, Auchi polytectnic. Auchi. Phone: +2348032655601 E-mail: firstname.lastname@example.org Ehigie Timothy (Corresponding author) Department of Statistics, P.M.B 13, Auchi polytectnic. Auchi. Phone: +2348060357105 E-mail: email@example.com Ayobo Thiophillus. Department of Statistics, P.M.B 13, Auchi polytectnic. Auchi. Phone: +2348060357105AbstractThe aim of this work is to estimate the parameters in a regression equation plane y = α 0 + α 1 X 1 + α 2 X 2 + ... + α k X k + ei by formulating the correlation matrix R (R = X′sXs/m) andthe vector b* (b* = R-1r(y)) which is the vector of elements between the criterion and each predictor in turnwith elements. The parameters were estimated using b = b*(Sy/Sxi) (i = 1.2). This technique was applied todata extract and the Regression Plane estimated is ŷ = -2.263 + 1.550x1i – 0.239x2i using the standardizedscores.Key words: Plane, vector, criterion, correlation matrix, extract, standardized scores. 1. Introduction We often seek to measure the relationship (if any) between the dependent variable and sets ofvariables called the independent variables. The data sets collected for the purpose of measurement areusually collected in different units. The use of the original variables measured in the different units wereanalyzed by (Carrol & Green 1997) using the data on employees absenteeism, attitude towards the time andthe number of years employed by the firm. The estimated trend line obtained by them wasŶ = -2.263 + 1.550xi1 – 0.239xi2. The normal equation formulated by them was in terms of the originaldata.(Aitusi & Ehigie 2011) obtained the same trend line using mean-corrected score. (Koutsoyainis 1977) and(Carrol, and Green 1997) both suggested the method of standardized variable which was never applied. Inthis work, we seek to apply the standardized method to multivariate data, since it is more generalized andcan be applied to variables measured in different units.In multivariate data analysis where we simultaneously estimate the effect of variables on one another, casesof the variables being measured in different units does occur in measuring economic variables. For instancethe data set for one variable may be measured in rates while the others in millions, percentages, thousands,hundreds, monetary units etc.37 | P a g ewww.iiste.org
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011In statistics, a standard score indicates how many standard deviations an observation is above or below themean. It is a dimensionless quantity derived by subtracting the population mean from an individual rawscore and then dividing the difference by the population standard deviation. That is Xs = (Xi - X i)/SxiThis process of conversion is called standardizing or normalizing. The standard deviation is the unit ofmeasurement of the z-score. It allows comparison of observations from different normal distribution, whichis done frequently in research. Standard scores are also called z-values, z-scores, normal scores andstandardized variables. The use of “z” is because the normal distribution is also known as the “z-distribution”.The aim of this work is to estimate the parameters of the regression plane Ŷ =α0 + α1 X 1 + α 2 X 2 + ei ………………(1)i.e to obtain values for α 0, α 1 and α 2 using the standardized scores. ˆ ˆ ˆ 2. Research MethodologyThe multiple regression equation for two regressors is Ŷ = α 0 + α1 X 1 + α 2 X 2 + ei … (i)Where ei satisfies all the required assumptions and (i) is the linear equation for predicting the values of ythat minimizes the sum of square errors. m m ∑ e =∑ ( y − y ) i =1 2 ˆ i =1 2 = minimum …(ii)In this work, we shall estimate the parameters in (i) above as follows;The correlation matrix R is obtained thus,R = X′sXs/m … (iii)where Xs is the standardized variable and m is the number of observations.b* =R-1r(y) … (iv)where r(y) is the vector of the product-moment correlations between the criterion and each predictor inturn, with elements.R-1 is the inverse matrix of R Ys X si r ( y) = i = 1,2 ... (v ) Thus, we shall compute m Ys′X s1 Ys′X s 2 r1 = ... (vi ) r2 = ... (vii) m m r r r ( y ) = 1 and ... r (viii) b * = R −1 1 r ... (ix) 2 238 | P a g ewww.iiste.org
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011 s * s b1 = b1* y … (x) b2 = b2 y sx2 Hence, … (xi) s x1 * s generally, the parameters are obtained thus bi = bi y , i = 1,2,..., k ...( xii ) s xi Where sy and sxi are the standard derivations for variables y and xi’s respectively, and the vector b*measures the change in y per unit change in each of the predictors when all variables are expressed instandard units. The equation (xii) will only yield estimates for parameters b1…bk since we are employing thestandardized scores, hence, we shall obtain the value of b0(intercept of the equation) using b0 = y – b1 X 1 - b2 X 2 - … bk X k …(xiii) 3. Data Analysis and ResultsUsing the same data as (Carrol & Green 1997) we generate data and analyze as in the appendix. 4. ConclusionThe use of standardize scores in data analysis have been greatly emphasized. This is due to the fact that themethod converts values in their original form to a new form which is approximately normal. Similarly, theunits for which the data collected may be different, hence, the need to standardized the scores to unit-lessscores becomes imperative.Our result is the same as that of (Carrol & Green 1997) using original scores and that of (Aitusi & Ehigie2011) using mean-corrected score.ReferencesAitusi D.N and Ehigie T.O.,(2011) Application of Matrix Algebra To Multivariate Data Using Mean-Corrected Scores. Journal TEMAS Series Vol.2 No.2, kogi State University, Anyigba, Nigeria.Carroll, D. and Green, J., (1965) “Notes on Factor Analysis” Unpublished Paper, Bell Laboratories, Murray Hill, New Jersey.J. Douglas Carroll, and Paul E. Green.,(1997); Mathematical Tools For Applied Multivariate Analysis, Academic Press.Koutsoyiannis A..(1977); Theory Of Economics; An Introductory Exposition Of Econometric Methods, 2nd Edition, New York, palgrave Macmillian39 | P a g ewww.iiste.org
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011Appendix 5. Data Analysis and Results Y X1 X2 Ys Xs1 Xs2 ( X 1i − X 1 ) (X − X 2) =(yi- y ) = = 2i S X1 SX 2 1 1 1 -0.9663 -1.3938 -1.3133 0 2 1 -1.1503 -1.1283 -1.3133 1 2 2 -0.9663 -1.1283 -0.9783 4 3 2 -0.4141 -0.8628 -0.9783 3 5 4 -0.5982 -0.3319 -0.3082 2 5 6 -0.7822 -0.3319 0.3618 5 6 5 -0.2301 -0.0664 0.0268 6 7 4 -0.0460 0.1991 -0.3082 9 10 8 0.5061 0.9956 1.0319 13 11 7 1.2423 1.2611 0.6968 15 11 9 1.6104 1.2611 1.3669 16 12 10 1.7945 1.5266 1.7019 TOTAL 75 75 59 MEAN 6.25 6.25 4.92 STD DEV. 5.43 33.77 2.98Note: Xs1 = (X1i - X 1)/SX1, Xs2 = (X2i- X 2)/SX2The correlation matrix R = X′sXs/mwhere m = number or paired observation = 12 12.00 11.40 X′sXs = 11.407 12.00 12.00 11.407 1 0.9506 ′ R = ( X S X s ) / 12 = 12 = 0.9506 11.407 12.00 1 10.3747 − 9.8620 * -1R-1 = − 9.8620 , b = R r(y) 10.3747 r1 where r(y) is a vector ,thus ri = ( y1 Xsi)/m r s / r1 = ( y s Xs1)/m , r2 =( y s/ Xs2)/m 2 11.3974 11.3974 0.9498 ys/ Xs = 10.6826 ∴ r ( y) = 10.6826 / 12 = 0.8902 40 | P a g ewww.iiste.org
Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.1, No.1, 2011 10.3747 − 9.8620 0.9498 1.0743 ∴b*= = − 9.8620 10.3747 0.8902 − 0.1310 ∴b*1 = 1.0743 and b*2 = -0.1310Thus, the coefficients b1 & b2 are obtained as followsb = b* (Sy/Sxi) , Sy = 5.4333, Sx1 = 3.7666, Sx2 = 2.9849 Sy 5.4333 S 5.4333Hence, = = 1.4425 , y = = 1.8203 S x1 3.7666 S x 2 2.9849b1 = 1.0743 x 1.4425 = 1.5497 ~ 1.550 (3 decimal places)b2 = -0.1310 x 1.8203 = -0.2385 ~ -0.239 (3 decimal places)we shall obtain b0 using b0 = Y - b1 X 1 – b2 X 2∴ b0 = 6.25 – 1.5497(0.25) – (-0.2385 x 4.92) = -2.2630The estimated regression plane for the multiple regression model isŶ = -2.263 + 1.550x1 – 0.239x241 | P a g ewww.iiste.org
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