x yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.
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1 The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator and response varales and to predct the value of for a gven value of x. Recall that the equaton of a least-squares regresson lne has the form ŷ = a + x, where a s the -ntercept, s the slope, and we use -hat to remnd us that ths s onl an approxmate predcton. Also recall that the value of the correlaton coeffcent (r) tells us the drecton and strength of a lnear relatonshp etween the two quanttatve varales, and ts formula s gven 1 x x r =. n 1 sx s Addtonall, the value of r (called the coeffcent of determnaton ) tells us what fracton of the varaton n can e explaned the LSRL of on x. Fnall, rememer where the slope and ntercept come from n the equaton ŷ = a + x : s = r and a = x. sx In chapter 14, we want to perform nference (.e. calculate confdence ntervals and perform tests of sgnfcance) n ths settng Inference aout the Model As shown n the formulas aove, the slope and ntercept of the LSRL are statstcs ecause the were calculated from the sample data. If we dd repeated samples, we would proal end up wth dfferent data, and therefore dfferent values for a and (the ntercept and the slope). To do nference, we thnk of a and as estmates of certan unknown parameters, whch we name α and β, respectvel. For nference aout regresson, we have n oservatons on an explanator varale x and a response varale. Our goal s to stud or predct the ehavor of for gven values of x. As n other nference procedures, we encounter a numer of necessar condtons: For an fxed value of x, the values of var accordng to a normal dstruton and repeated values of are ndependent of each other. The mean response µ has a straght-lne relatonshp wth x so that µ = α + β x where the slope β and ntercept α are unknown parameters. The standard devaton of (call t σ ) s unknown, and s the same for all values of x, as shown n the fgure elow. Page 1 of 5
2 The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson We now have that a and are estmates of α and β, respectvel. Our estmate of σ s none other than s, the standard error aout the lne. Recall from chapter 1 that s, at least n the case of unvarate data, s gven the formula 1 s = ( x x) n 1 In the case of regresson, we now have two varales, and so the new formula for s s 1 s = ( ˆ ) n (ecause wth two varales there are n degrees of freedom nstead of n 1) where ˆ represents the resdual from the least-squares lne. 1 In summar, to each ( ) calculate s, fnd the predcted response for each x n our data set ( pluggng the x- value nto the equaton of the LSRL), then fnd the resdual ( sutractng the predcted response from the oserved data value), and fnall, use the formula to calculate s. In practce, ou wll almost certanl e gven the value of s, ether as part of a computer output or else as somethng ou could calculate easl usng technolog. Read and understand the Technolog Tp on page 786 for a lttle extra help. Confdence ntervals for regresson slope Usuall, the slope β s the most mportant parameter n a regresson prolem. Ths s ecause the ntercept α onl makes sense f the explanator varale can take on values at or ver near zero. A confdence nterval for the slope can tell us how accurate our estmate of β reall s. The confdence nterval stll has the form estmate ± t * SE estmate, and wth as our estmate, ths n turn ecomes ± t *SE. In ths case, the standard error of the least squares slope s s SE = and t* s the upper ( C) freedom. ( x x) 1 / crtcal value from the t dstruton wth n degrees of Agan, ou should rarel have to calculate the standard error hand t s tpcall gven n some form or another, as shown n the fgure elow. 1 Recall that a resdual s the dfference etween the oserved outcome and the predcted outcome, n that order. In other words, resdual = oserved predcted. Page of 5
3 The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson The 95% confdence nterval for β n the case aove would then e ( )( ) ± t * SE = ± = to.4873 We could also fnd the 95% confdence nterval for α usng and nstead. However, we wll rarel e nterested n estmatng α. Testng the hpothess of no lnear relatonshp The most common hpothess aout the slope β n regresson settngs s that the slope s zero,.e., H : 0 0 β =. If the slope s zero, the lne s horzontal, whch means that the value of does not change as x changes, so there reall s no true lnear relatonshp etween x and. Perhaps etter sad, ths means that straght-lne dependence on x s of no value for predctng. Alternatvel, we could sa that there s no correlaton etween x and n the populaton from whch we drew our data. Agan, the test statstc should look famlar t s smpl the estmated value of the slope dvded the standard error of the slope,.e., t =. The output from our computers SE or calculators usuall gves us these values n one wa or another: Ths means we can use the test for zero slope to test the hpothess of zero correlaton etween two quanttatve varales HOWEVER usng ths test onl makes sense f the oservatons are a random sample, whch s often not the case n regresson settngs ecause researchers mght fx (choose ahead of tme) the values of x the want to stud. Page 3 of 5
4 The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson As far as carrng out nference regardng regresson, use the followng outlne of steps: 1. Plot and nterpret. Alwas examne the data graphcall (almost alwas wth a scatterplot n the case of two related quanttatve varales). Look for and e ale to descre the form, strength, and drecton of an relatonshp that mght e vsle, and for outlers, nfluental ponts, and other devatons from the overall pattern.. Numercal summar. If (and onl f) there s graphcal evdence that a lnear relatonshp exsts, look at the values of r and r to determne how strong that lnear relatonshp mght e. 3. Mathematcal model. Fnd the desred LSRL (tpcall done usng technolog). 4. Inference. Perform an desred nference procedures, makng sure along the wa that all necessar condtons have een met. 14. Predctons and Condtons The most common reason for dong regresson s to predct the response to a partcular value of the explanator varale. To make a predcton usng the LSRL, we smpl plug n our x-value and see what we get for a -value. But how accurate s our predcton f t s ased on sample data? We can determne ths usng a confdence nterval. Frst, we must answer a ke queston. Do we want to predct the mean response for all memers of the populaton, or do we want to predct the sngle response for one partcular ndvdual n the populaton? The actual predcton would eld the same value, ut n terms of the confdence ntervals used to assess the accurac of the predctons, the margns of error are dfferent. To estmate the mean response µ = α + β x *, where µ s the mean of responses when x has the value x*, use a confdence nterval of the form 1 ( x * x) ˆ ± t * s + n ( x x). To estmate an ndvdual response for a gven value of x = x *, use a predcton nterval of the form 1 ( x * x) ˆ ± t * 1+ + n ( x x) (In oth cases, t* s the upper ( C). 1 / crtcal value of the t dstruton wth n degrees of freedom.) Note the slght dfference n the standard error for the two ntervals. The SE for the predcton nterval has an extra 1 under the square root, whch makes the margn of error larger and the nterval wder. Ths s ecause t s harder to predct one response than to predct a mean response. Page 4 of 5
5 The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Condtons revsted Before ou perform nference on regresson models, ou must check each of the followng condtons: Independent oservatons. Specfcall, repeated oservatons on the same ndvdual are not allowed. Trul lnear relatonshp. We wll proal never reall see ths exactl n practce. Be sure to draw a scatterplot AND a resdual plot to assess lneart and devatons from lneart. The standard devaton of the -values aout the true lne s the same everwhere. The est wa to assess ths, at least at our level of understandng, s usng the resdual plot. The scatter of the data ponts should reman the aout the same over the entre range of the data. The response vares normall aout the true regresson lne. Agan, the resduals show the varaton of the response aout the ftted lne, and f the responses are normall dstruted aout the lne, so should the resduals e. Make a stemplot, oxplot, or a hstogram of the resduals and look for strkng devatons from normalt. There are was to deal wth volatons of the aove condtons. Lke other t procedures, we can survve a relatvel mnor lack of normalt, especall f the sample sze s large enough. You should pa close attenton to nfluental oservatons (tpcall outlers n the x-drecton) as the can move the lne around qute a t, havng a drastc effect on our nference outcomes. Read and understand Examples 14.8 and Page 5 of 5
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