Confidence Intervals for Linear Regression Slope

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1 Chapter 856 Cofidece Iterval for Liear Regreio Slope Itroductio Thi routie calculate the ample ize eceary to achieve a pecified ditace from the lope to the cofidece limit at a tated cofidece level for a cofidece iterval about the lope i imple liear regreio. Cautio: Thi procedure aume that the lope ad tadard deviatio/correlatio etimate of the future ample will be the ame a the lope ad tadard deviatio/correlatio etimate that are pecified. If the lope ad tadard deviatio/correlatio etimate are differet from thoe pecified whe ruig thi procedure, the iterval width may be arrower or wider tha pecified. Techical Detail For a igle lope i imple liear regreio aalyi, a two-ided, 00( α)% cofidece iterval i calculated by where b i the calculated lope ad b t ± α /, b b i the etimated tadard deviatio of b, or b = ( X i X ) where = ( ˆ Yi Y ) The value i ofte obtaied from regreio table a MSE. A oe-ided 00( α)% upper cofidece limit i calculated by b + t α, b 856- NCSS, LLC. All Right Reerved.

2 Cofidece Iterval for Liear Regreio Slope Similarly, the oe-ided 00( α)% lower cofidece limit i b t α, b Each cofidece iterval i calculated uig a etimate of the lope plu ad/or miu a quatity that repreet the ditace from the mea to the edge of the iterval. For two-ided cofidece iterval, thi ditace i ometime called the preciio, margi of error, or half-width. We will label thi ditace, D. The baic equatio for determiig ample ize whe D ha bee pecified i D = t α /, b Thi equatio ca be olved for ay of the ukow quatitie i term of the other. The value α / i replaced by α whe a oe-ided iterval i ued. I thi procedure, the lope ad the tadard deviatio of the X are etered a iput, where ad b ( X i X )( Yi Y ) = X = ( X i X ) ( X i X ) Oe of three differet additioal iput ca be ued to calculate.. Stadard deviatio of the Y : I thi cae i geerated uig. Correlatio: r = = Y = ( Yi Y ) ( b ) Y X ( X i X )( Yi Y ) ( X i X ) ( Yi Y ) I thi cae i geerated uig = b X r 856- NCSS, LLC. All Right Reerved.

3 Cofidece Iterval for Liear Regreio Slope 3. Directly uig : = ( ˆ Yi Y ) which i ofte obtaied from regreio table a the quare root of MSE. Cofidece Level The cofidece level, α, ha the followig iterpretatio. If thouad of ample of item are draw from a populatio uig imple radom amplig ad a cofidece iterval i calculated for each ample, the proportio of thoe iterval that will iclude the true populatio lope i α. Procedure Optio Thi ectio decribe the optio that are pecific to thi procedure. Thee are located o the Deig tab. For more iformatio about the optio of other tab, go to the Procedure Widow chapter. Deig Tab The Deig tab cotai mot of the parameter ad optio that you will be cocered with. Solve For Solve For Thi optio pecifie the parameter to be olved for from the other parameter. Oe-Sided or Two-Sided Iterval Iterval Type Specify whether the iterval to be ued will be a two-ided cofidece iterval, a iterval that ha oly a upper limit, or a iterval that ha oly a lower limit. Cofidece Cofidece Level The cofidece level, α, ha the followig iterpretatio. If thouad of ample of item are draw from a populatio uig imple radom amplig ad a cofidece iterval i calculated for each ample, the proportio of thoe iterval that will iclude the true populatio lope i α. Ofte, the value 0.95 or 0.99 are ued. You ca eter igle value or a rage of value uch a 0.90, 0.95 or 0.90 to 0.99 by NCSS, LLC. All Right Reerved.

4 Sample Size Cofidece Iterval for Liear Regreio Slope N (Sample Size) Eter oe or more value for the ample ize. Thi i the umber of idividual elected at radom from the populatio to be i the tudy. N mut be greater tha or equal to 3. You ca eter a igle value or a rage of value. Preciio Ditace from Slope to Limit() Thi i the ditace from the cofidece limit() to the ample lope. For two-ided iterval, it i alo kow a the preciio, half-width, or margi of error. You ca eter a igle value or a lit of value. The value() mut be greater tha zero. Regreio Slope B (Slope) Eter the magitude of the ample lope. I the liear equatio, Y=A+BX, B repreet the lope of the lie relatig the depedet variable Y to the idepedet variable X. Cautio: The ample ize etimate for thi procedure aume that the lope that i achieved whe the cofidece iterval i produced i the ame a the lope etered here. If the ample lope i differet from the oe pecified here, the width of the iterval may be arrower or wider tha pecified. You ca eter a igle value or a rage of value uch a,,3. Thi value mut be greater tha zero. Regreio Stadard Deviatio of X SX (Stadard Deviatio of X ) Thi i the tadard deviatio of the X value i the ample. Cautio: The ample ize etimate for thi procedure aume that the SX that i achieved whe the cofidece iterval i produced i the ame a the SX etered here. If the ample SX i differet from the oe pecified here, the width of the iterval may be arrower or wider tha pecified. It i aumed that the tadard deviatio formula ued i calculatig SX ue the divior N-. The value mut be greater tha zero. Regreio Reidual Variace Calculatio Reidual Variace Method Three method ca be ued to calculate the tadard deviatio of the reidual. Thi optio pecifie which method hould be ued NCSS, LLC. All Right Reerved.

5 Cofidece Iterval for Liear Regreio Slope SY Thi optio pecifie that the formula S = SQRT[SY^ (B*SX)^] hould be ued to calculate the tadard deviatio of the reidual. Notice that thi formula require oly SY, SX, ad B (the correlatio i ot ued). R Thi optio pecifie that the formula S=B(SX)SQRT[/R^ ] hould be ued to calculate the tadard deviatio of the reidual. Notice that thi formula require oly SX, B, ad R (the value of SY i ot ued). S Thi optio pecifie the value of S directly. The value of SY ad R are ot ued. SY (Stadard Deviatio of Y) Thi i the tadard deviatio of the Y value i the ample. Thi value i oly ued whe the Reidual Variace Method i et to 'SY (Std. Dev. of Y)'. Cautio: The ample ize etimate for thi procedure aume that the SY that i achieved whe the cofidece iterval i produced i the ame a the SY etered here. If the ample SY i differet from the oe pecified here, the width of the iterval may be arrower or wider tha pecified. It i aumed that the tadard deviatio formula ued i calculatig SY ue the divior N-. The value mut be greater tha zero. R (Correlatio) Thi i a etimate of the correlatio betwee Y ad X. Thi value i oly ued whe the Reidual Variace Method i et to 'R (Correlatio)'. Cautio: The ample ize etimate for thi procedure aume that the correlatio that i achieved whe the cofidece iterval i produced i the ame a the correlatio etered here. If the ample correlatio i differet from the oe pecified here, the width of the iterval may be arrower or wider tha pecified. Rage: 0 < R < Note that R caot be 0 or egative. S (Stadard Deviatio of Reidual) Thi i the tadard deviatio of the reidual from the regreio of Y o X. Thi value i oly ued whe the Reidual Variace Method i et to 'S (Std. Dev. of Reidual)'. Cautio: The ample ize etimate for thi procedure aume that the S that i achieved whe the cofidece iterval i produced i the ame a the S etered here. If the ample S i differet from the oe pecified here, the width of the iterval may be arrower or wider tha pecified. It i aumed that the tadard deviatio formula ued i calculatig S i qrt(mse), which ue the divior N-. The value (or value) mut be greater tha zero NCSS, LLC. All Right Reerved.

6 Cofidece Iterval for Liear Regreio Slope Example Calculatig Sample Size Suppoe a tudy i plaed i which the reearcher wihe to cotruct a two-ided 95% cofidece iterval for the lope uch that the ditace from the lope to the limit i o more tha uit. The cofidece level i et at 0.95, but 0.99 i icluded for comparative purpoe. The etimated lope i.7 ad the tadard deviatio of the X i.. The tadard deviatio of the reidual etimate, baed o the MSE from a imilar tudy, i Itead of examiig oly the iterval width of, a erie of width from 0.5 to.5 will alo be coidered. The goal i to determie the eceary ample ize. Setup Thi ectio preet the value of each of the parameter eeded to ru thi example. Firt, from the PASS Home widow, load the Cofidece Iterval for Liear Regreio Slope procedure widow by clickig o Regreio, ad the clickig o Cofidece Iterval for Liear Regreio Slope. You may the make the appropriate etrie a lited below, or ope Example by goig to the File meu ad chooig Ope Example Template. Optio Value Deig Tab Solve For... Sample Size Iterval Type... Two-Sided Cofidece Level Ditace from Slope to Limit to.5 by 0. B (Slope)....7 SX (Stadard Deviatio of X ).... Reidual Variace Method... S (Std. Dev. of Reidual) S (Stadard Deviatio of Reidual) Aotated Output Click the Calculate butto to perform the calculatio ad geerate the followig output. Numeric Reult Numeric Reult for Two-Sided Cofidece Iterval for the Slope i Simple Liear Regreio Target Actual Stadard Stadard Sample Ditace Ditace Deviatio Deviatio Cofidece Size from Slope from Slope Slope of X of Reidual Level (N) to Limit to Limit (B) (SX) (S) NCSS, LLC. All Right Reerved.

7 Cofidece Iterval for Liear Regreio Slope (Numeric Reult Cotiued) Referece Otle, B. ad Maloe, L.C Statitic i Reearch. Iowa State Uiverity Pre. Ame, Iowa. Report Defiitio Cofidece level i the proportio of cofidece iterval (cotructed with thi ame cofidece level, ample ize, etc.) that would cotai the true lope. N i the ize of the ample draw from the populatio. Ditace from Slope to Limit i the ditace from the cofidece limit() to the ample lope. For two-ided iterval, it i alo kow a the preciio, half-width, or margi of error. Target Ditace from Slope to Limit i the value of the ditace that i etered ito the procedure. Actual Ditace from Slope to Limit i the value of the ditace that i obtaied from the procedure. B i the ample lope. SX i the ample tadard deviatio of the X value. S i the tadard deviatio of the ample reidual. Summary Statemet A ample ize of 93 produce a two-ided 95% cofidece iterval with a ditace from the ample lope to the limit that i equal to whe the ample lope i.70, the tadard deviatio of the X' i.0, ad the tadard deviatio of the reidual i Thi report how the calculated ample ize for each of the ceario. Plot Sectio Thee plot how the ample ize veru the ditace from the ample lope to the limit for the two cofidece level NCSS, LLC. All Right Reerved.

8 Cofidece Iterval for Liear Regreio Slope Example Validatio uig Otle ad Maloe Otle ad Maloe (988) page 34 give a example of a calculatio for a cofidece iterval for the lope whe the cofidece level i 95%, the lope i 7.478, the tadard deviatio of the X i , the tadard deviatio of the reidual i , ad the ditace from the lope to the limit i The ample ize i 3. Setup Thi ectio preet the value of each of the parameter eeded to ru thi example. Firt, from the PASS Home widow, load the Cofidece Iterval for Liear Regreio Slope procedure widow by clickig o Regreio, ad the clickig o Cofidece Iterval for Liear Regreio Slope. You may the make the appropriate etrie a lited below, or ope Example by goig to the File meu ad chooig Ope Example Template. Optio Value Deig Tab Solve For... Sample Size Iterval Type... Two-Sided Cofidece Level Ditace from Slope to Limit B (Slope) SX (Stadard Deviatio of X ) Reidual Variace Method... S (Std. Dev. of Reidual) S (Stadard Deviatio of Reidual) Output Click the Calculate butto to perform the calculatio ad geerate the followig output. Numeric Reult Target Actual Stadard Stadard Sample Ditace Ditace Deviatio Deviatio Cofidece Size from Slope from Slope Slope of X of Reidual Level (N) to Limit to Limit (B) (SX) (S) PASS alo calculated the ample ize to be NCSS, LLC. All Right Reerved.

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