# CTL.SC0x Supply Chain Analytics

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1 CTL.SC0x Supply Chai Aalytics Key Cocepts Documet V1.1 This documet cotais the Key Cocepts documets for week 6, lessos 1 ad 2 withi the SC0x course. These are meat to complemet, ot replace, the lesso videos ad slides. They are iteded to be refereces for you to use goig forward ad are based o the assumptio that you have leared the cocepts ad completed the practice problems. The first draft was created by Dr. Alexis Batema i the Fall of This is a draft of the material, so please post ay suggestios, correctios, or recommedatios to the Discussio Forum uder the topic thread Key Cocept Documets Improvemets. Thaks, Chris Caplice, Eva Poce ad the SC0x Teachig Commuity Fall 2016 v1 v1.1 Fall 2016 This work is licesed uder a Creative Commos Attributio NoCommercial ShareAlike 4.0 Iteratioal Licese.

5 Calculatig Cofidece Itervals Whe the >30 We ca assume: X ~N(μ,σ/ ) The level c of a cofidece iterval gives the probability that the iterval produced icludes the true value of the parameter. Where z is the correspodig z score correspodig to the area aroud the mea: z=1.65 for =.90, z=1.96 for =.95, z=2.81 for =.995 x zs, x zs For spreadsheets use: z = NORM.S.INV((1+β)/2) Whe 30 The we eed to use the t distributio, which is bell shaped ad symmetric aroud 0. Mea = 0, but Std Dev = (k/k 2) Where k is the degrees of freedom ad, geerally, k= 1 The value of c is a fuctio of β ad k Where c is the correspodig t statistic correspodig to the area aroud the mea. x cs, x cs CTL.SC0x Supply Chai Aalytics 5

7 Spreadsheet Fuctios: Fuctio Microsoft Excel Google Sheets LibreOffice >Calc Returs p value for Chi Square Test =CHISQ.TEST(observed_values, expected_values) =CHITEST(observed_values, expected_values) =CHISQ.TEST(observed_values; expected_values) Ordiary Least Squares Liear Regressio Regressio is a statistical method that allows users to summarize ad study relatioships betwee a depedet (Y) variable ad oe or more idepedet (X) variables. The depedet variable Y is a fuctio of the idepedet variables X. It is importat to keep i mid that variables have differet scales (omial/ordial/ratio). For liear regressio, the depedet variable is always a ratio. The idepedet variables ca be combiatios of the differet umber types. Liear Regressio Model The data (, y i ) are the observed pairs from which we try to estimate the Β coefficiets to fid the best fit. The error term, ε, is the uaccouted or uexplaied portio. Liear Model: y i 0 1 Y i 0 1 i for i 1, 2,... Residuals Because a liear regressio model is ot always appropriate for the data, you should assess the appropriateess of the model by defiig residuals. The differece betwee the observed value of the depedet variable ad predicted value is called the residual. ŷ i b 0 b 1 for i 1,2,... e i y i ŷ i y i b 0 b 1 for i 1,2,... Ordiary Least Squares (OLS) Regressio Ordiary least squares is a method for estimatig the ukow parameters i a liear regressio model. It fids the optimal value of the coefficiets (b 0 ad b 1 ) that miimize the sum of the squares of the errors: 2 e i y i ŷ i y i b 0 b 1 i1 i1 2 i1 2 y b 1 x b 1 i1 ( x)( y i y) i1 ( x) 2 CTL.SC0x Supply Chai Aalytics 7

8 Multiple Variables These relatioships traslate also to multiple variables. Y i 0 1 x 1i... k x ki i for i 1,2,... E(Y x 1,x 2,..., x k ) 0 1 x 1 2 x 2... k x k StdDev(Y x 1,x 2,..., x k ) y i ŷ i y i b 0 b 1 x 1i... b k x ki 2 e i1 i i1 2 i1 2 Validatig a Model All statistical software packages will provide statistics for evaluatio (ames ad format will vary by package). But the model output typically icludes: model statistics (regressio statistics or summary of fit), aalysis of variace (ANOVA), ad parameter statistics (coefficiet statistics). Overall Fit Overall fit = how much variatio i the depedet variable (y), ca we explai? Total variatio of CPL fid the dispersio aroud the mea. Total Sum of Squares Make estimate for of Y for each x. Error or Residual Sum of Squares TSS (y i y) 2 e i y i ŷ i 2 RSS e i y i ŷ i 2 Model explais % of total variatio of the depedet variables. Coefficiet of Determiatio or Goodess of Fit (R 2 ) R 2 1 RSS TSS 1 y ŷ i i 2 y i y 2 Adjusted R 2 corrects for additioal variables adjr 2 1 RSS 1 TSS k 1 i1 CTL.SC0x Supply Chai Aalytics 8

9 Idividual Coefficiets Each Idepedet variable (ad b 0 ) will have: A estimate of coefficiet (b 1 ), A stadard error (s bi ) o s e = Stadard error of the model s e y i ŷ i 2 N 2 o s x = Stadard deviatio of the idepedet variables = umber of observatios The t statistic o k = umber of idepedet variables o b i = estimate or coefficiet of idepedet variable Correspodig p value Testig the Slope o We wat to see if there is a liear relatioship, i.e. we wat to see if the slope (b 1 ) is somethig other tha zero. So: H 0 : b 1 = 0 ad H 1 b 1 0 o Cofidece itervals estimate a iterval for the slope parameter. Multi Colliearity, Autocorrelatio ad Heterscedasticity Multi Colliearity is whe two or more variables i a multiple regressio model are highly correlated. The model might have a high R 2 but the explaatory variables might fail the t test. It ca also result i strage results for correlated variables. Autocorrelatio is a characteristics of data i which the correlatio betwee the values of the same variables is based o related objects. It is typically a time series issue. Heterscedasticity is whe the variability of a variable is uequal across the rage of values of a secod variable that predicts it. Some tell tale sigs iclude: observatios are supposed to have the same variace. Examie scatter plots ad look for fa shaped distributios. Refereces Hillier ad Lieberma (2012) Itroductio to Operatios Research, McGraw Hill. Bertsimas ad Freud (2003) Data, Models, ad Decisios: The Fudametals of Maagemet Sciece, Dyamic Ideas. CTL.SC0x Supply Chai Aalytics 9

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